I[ [ 2o113-FR1 FINAL REPORT For Period l August 1964 to 31 March 1965 I- L Sr_.UDY OF AN ATTITUDE REFERENCE SYSTEM _i Jet Propu._sion Laboratory Contract No. 950915 [ __r__00_ _L GPO PRICE $ I OTS PRICE{S) $ 1 Microfiche (MF_[ 1| I 1965011367 https://ntrs.nasa.gov/search.jsp?R=19650011367 2020-03-16T23:53:39+00:00Z 20113-FRI 31 March 1965 i, ; FINAL REPORT • For Period 1 August 1964 to 31 March 1965 .i :! "! STUDY OF AN ATTITUDE REFERENCE SYSTEM UTILIZING AN ELECTRICALLY SUSPENq:)ED GYRO 1 ! Jet Propulsion Laboratory Contract No. 950915 -, Thisworkwaspe_onnedfox"theJet pmpulsio,xLsbo,mn_, : California Imtimte of Technology, sponsored by the National Aeronau_ _ _ _ !_ J! J.(_._Olson---- Sec_n Head R. C. Amos I Program Administrator t Honeywell Inc. • Aeronautical Division Minneapolis, Minnesota 1965011367-002 . - il- l CONTENTS "_ Page SECTION I GENERAL 1 SECTION II OBJECTIVES AND CONCLUSIONS 3 Objectives 3 ]" Conclusiofis 7 L SECTION III SDMEG-ARS PARAI_._ETER STUDY 9 r", Introduction 9 (,/ Independent Parameters 9 Gyro Configurations 10 _" System Outputs 11/ _,: System Accuracy 11Parameter Combinations 12 , Summary of Results 13 _: Harr_ware Configurations 13 I, Inde _endent Pa_'ameters for Hardware 14 Configurat._ ons _ _ Accuracy Versus Dependent Parameters 15 if _,. By- Product Results 17 Selectionof theARS Configuration 18 !_ # Introduction 18 i. £ Coordinate Frames 19 _ Euler Angle Approach I 20 ,_ Euler Angle Approach II 25 [ Gyro Spin Axi_ Commands, Q1, Q2 32Vehicle _teering in Gyro Fram_ 7 _: ARS System Equations 45b Discussion of System Equations 45_ Cosine Computer 45 Attitude Error and Body Rate Computer 51 " _ Computations for the Twelve Systems of 53 _:i the Parameter Study _ _. Performance Analysis 59 _= Introduction 59 !_ _ Summary 59 _. Discussion of Error Sources 62 _: Mathematical Considerations 88 _ t ARS Performance in Attitude Mode 92 _ ARS Performance in the Rate Mode 100 _ Derivations of Error Equations 104 ! . 1965011367-003 "i { .t • } ... • . } - 111- • _ ARS Computer and I[O Logic 115 }!} Introduction 115 Gyro Interface Logic 124 Gyro Startup 132 ' The ARS Computer 135 I Telemetry-Computer Interface 157 Vehicle Body Rate From the End &xis Rotor 160 ;: Pattern .i Introduction 160 Effect of Attitude on Derive Rate Signal 160 Pickoff Field of View 164i _ Effect of Noise on Derive Rate Signal 165 Conclusions 168 . Gyro Caging and Torqaing Capability 169 ' Caging 16 9 Torquing 172 Conclusions 173 !, Size. Weight, Power and Reliability 174 .:! Size, Weight, Power and Reliability of Gyros 174 Computer Size, Weight, Power and 177 ,! Reliabilityi Power Supply Size, Weight, Power and 182 "' Reliability Estimate Summary 183 _,I Filters for Body Rate Output 186| Introduction 186 Digital Pseudo Rate 187 ,}• j Derivation Of Body Rates 188Implementing Sampled Data Filters 202 Checkout of SDMEG-ARS 209 ,,_ Introduction 209 !_ Checkout and Monitoring of ARS Computer 210 " Checkout of Gyro-Computer Interface Logic 211 Summary 212 . SECTION IV SDMEG ARS FOR A CAPSULE LANDER 213 Introduction and Summary 213 i'i_ System Configuration 217 '( _ End-Axis Patterns 218 E quatoria_ Patt erns 226 "l ! [ System Description 236• Gyro Modifications Electronics Des cription 237 }i; Performance Analysis 251Introduction and Summary 251 Error Representations 256 Error Sources 266 I} System Physical Characteristics 274Size and Weight 274 Power Requirements 275 'i} Reliability 277 1965011367-004 | - iv - _ SECTION V SDMEG MODIF:CATIONS 279 Readout improvement 279 Shaping Electronics 281Adjustment for InitialAlignment 282Modifications for Singte Gyro Application 282 Simultaneous Piekoff Readout 283 _ [ SECTION VI RECOMMENDATIONS 287 Introduction 287 _ Summary 289 , Background 293 _ Simulation and Noise Study 294 =" [ Reliability Study 300Supporting Programs 7 r Preliminary Work Statement for Follow-on Effort 309 _ _ Task I -,Simulation and Noise Study 310Task II - Reliability Study 312 A PPENDLX A DERIVATION OF EQUATIONS FOR CALCULATION {_i OF TItE THIRD SPIN AX_z_ DIRECTION COSINE APPENDIX B DERIVATION OF EQUA_I iONS FOR ATTITUDE "' I" ERROR AND BODY RATE APPENDLX C COORDINATE TRANSFORMATION BY "METHOD • } [, OF LEAST WORK" 5 t all I. i ] 9650] ] 367-005 - V - ' ILLUSTRATIONS 'i Figure Page 2- i Pro,gram Schedule 5 .1 ._ 3-1 System Accuracy Versus System Sizing 16 ! _, 3- 2 Platform Mechanization 21 i' 3-3 SDMEG-ARS Mechanization 21i _t I 3-4 Graphic Coordinate Transformation 25 3-5 Geometry Used In Control of Vehicle Using QI and Q2 a3 _ Commands 3-6 The Vehicle X-Axis Steering in Gyro Frame 38 i) 3-7 Cosine Computer Flow Chart 46 3-8 Attitude Error and Body Rate Computer Flow Chart 47 q I; 3-9 Pulse Shape From a Pickoff Preamplifier 74 , ' 3-I0 Modified Cosine Pattern 79 !f, 3-II Locus of Pickoff Under Arbitrary Vehicle Motion 80 il 3-12 Counter Timing 80 ¥. 3-13 -Pickoff Locus Under Vehicle Motion 83 i' [, 3-14 One Counter Timing 811 t } 3-15 SDMEG-ARS Error Model 90 3-16 Attitude and Rate Modes of ARS Performance 93 3-18 Cosine Computer Flow Ch_rt 117 ft . 3-19 AttitUde Error and Body Rate Computer Flow Chart lt9 ; _ 3-20 Single Channel of Six-Counter System of G.yro-Comp,lter 130 J Interface Logic i'il :"_, 20 t l 3-FR1 1965011367-006 [_ J • _ - Vl - Figure Page 3-21 One-Counter System of Gyro-Computer Interface Logic 13_ I_ '!-22 Flow Diagram For Gyro Startup Program 134p : 3-.23 Computer _lGck Diagram 139 ¢. . 3-24 Te]em,ztry Data by Local Exerciser 158 3-25 Telemetry Data by Remote Interrogation 158 C |: 3-26 Conventional End-Axis Pattern and Pickoff Geometry 161 -'_ j_ 3-27 Pseudo-Rate Computational Flow Diagram 189 _ 3-28 Typical Phase-Plane Portrait of Digital Pseudo-Rate 190 (_ Response for a Pure Inertia Plant l: 3-29 Counter Binary Noise n(t) 191 -30 A ,ocor. la*io i 3-31 Normalized Spectral Density of n(t) .191 I ;. ! I 3-32 Represeutative Input Signals 194 3-33 Block Diagram Established to Derive the Desired 197 Filter M(Z) b • 3-34 Rate Averaging Filter Block Diagram 203i f" i 3-35 Effect of Finite Memory Averaging Rate Filter on 203 Quantized Noise Filter 3-36 S - z Plane Map 208 S t 3-37 Z - w Plane Map 208 4-i Two-Axis ESG-ARS Block Diagram 215 _ {]2 4-2 Functional ESG-ARS Block Diagram 219 t,_ 4-3 Conventional End-Axis Pattern and Pickoff Geometry 220 I i_ 4-4 Output for Conventional End-Axis Readout 222 Pickoff 4-5 End-Axis Pickoff Readout Electronics 223 I i 4-6 Modified i_nd-Axis Pattern Geometry 225 ;F _ i! 2o113-FRI 1965011367-007 , -- vii- FiF ure Page 4-7 "Constant Slope:' Equatorial Pattern Geometry 226 4-8 Conceptual Readout Mechanization 228 4-9 Error Contribution - Great Circle Approximation of 230 "Constant Slope" 4-10 Combined Great Circle - Constant Slope Pattern 231 4-11 Symmetric Great Circle - Constant Slope Pattern 233 4-12 Platform Components for Readout Electronics 239 4-13 Analog Attitude Computing Module Functional Block 240 Diagram 4-14 Digital Control Signal Waveforms 241 4-15 _gital Control Logic Functional Block Diagram 243 ," 4-16 Pitch and Yaw Rate Compu/er 245 4-17 Roll Rate Calculation 247 4-18 Control Signals for Roll Rate Caiculation 2,19 4-19 SDMEG-ARS Error Diagram 252 4-20 SDMEG-ARS Error Functiors 259 4-21 Power Spectral Density Function 262 6-1 Over-all Program Plan 288 6-2 Block Diagram of Attitude Control System Simulation 298 6-3 Flow Diagram of the Data Processing to Obtain the 299 Attitude Error Signal 6-4 Pre] iminary Schedule 309 20113-FRI 1965011367-008 - Vlll - I- TABLES ; ! Table Page 2-1 Def:mition of Study Tasks 4 i, 3-I System Sizing Estimates 14 3-2 Hardware for Original 12 ARS 15i 3-3 Summary of Major Arithmetic Operations if Guidance 24 Provides Three Euler Angles _:_ 3-4 Summary of Major Arithmetic Operations if Guidance 31 ; Provides Three Euler Angles _; 3-5 Summary of Major Arithmetic Operations if the 36 Guidance Provides Three Euler Angles _L 3-6 Summary of Major Arithmetic Operations if Guidance 42 Provides Three Euler Angles ? " |_ 3-7 ARS Performance in Attitude Mode 61 3-6 ARS Performance in Rate Mode 61 ]- 3-9 ARS Error Sources 64 3- 10 Instruction Count 152 _ 3-11 Storage Requirements 154 i_. , 3-12 Computer Speed and Instructio,_ I_equirerneJ_ts 156 3-13 Final Computer Configurations 156 :' 3-14 Sumr of Size, arm 175ary Gyro Weight Reliability 3-15 Sizing Estimates for Gyro Torquing Hardware 175 3-16 Power Inputs Required for SDMEG pe_" Gyro !'/,. ? _. f- 3-.17 Component Count for the Three Systems i ' 3-18 Component Count for Mechanizatien with Discrete 179 _' 3-19 Power Requirements for Computer Configurations 180 _ _ I! 20113-FRIt 1965011367-009 - iX - Table Page 3-20 Size, Weight, and Power Comparisons 180 3-21 Reliability Comparisons 182 3-22 Sur.ltnary of Sizing Fstimates 184 3-23 Sizing Estimate for Total Systems 184 .3-24 Summary of Size, Weight, Power and Reliability for ARS 185 4-1 Specified Requirements for a Capsule Lander ARS 214 4-2 Summary Characteristics for Selected ARS 217 4-3 Attitude Output Errors 253 4-4 SDMEG-ARS Performance Characteristics 255 _ 4-5 Pitch and Yaw Rate Errors 264 4-6 SDMEG ARS Size and Weight Characteristics 275 : 4-7 SDMEG ARS Power Summary 276 4-8 Relia:_ility Predictions for SDMEG Subassemblies 278 : 20113-FRI 1965011367-010 - I " ? ; I. SECTION I _, ) GENERAL This is the final report submitted in accordance with the Statement of Work at Jet Propulsion Laboratory Contr-act No. 950915. It covers work performed ; I!i. during the period i August 1964 through 31 March 1965. L _ _ _ - ¢ i 1965011367-011 _" 3 '. SE CTION 'LI ' OBJECTIVES AND CON CLUS!ONS t OBJECTIVESI ,, The goal of this study program is to establis[J on a factual basis the feasi- :! bility uf using miniature electl o-static gyros (MEG) in a spacecraft attitude i eference system. In order to accomplish this goal the program was divided • into t_o tasks. Task I was a study to establish the relationships between major system parameters, and Task II was the prelimina:y design of ai strapped down MEG attitude reference sy_:.ern for a _pecific application - a landing capsule. Table 2-1 shows the breal_do%n of the £wo major tasks into subtasks for the purpose of carrying out the program. Figure 2-1 shows the schedule used in performing the study. .{ Since a parameter study such as selected for Task I can have almost any :. desired magnitude, it was necessary to constrain the study to contain it I , within the available time and funds. Al_o constraints were necessary to ensure that the study results would be within the category of JPL's interest. Following are the constraints or ground rules used during the course of the } Task I parameter study: • The Attitude Refere :ce System (AIRS) will be all attitude • The system will be instrumented with strapped down MEG's i l • The attitude reference system will output to the at+Atude control system only • • The attitude reference system will provide outputs for three axis attitude control of the vehicle II (I 20113.-FRI I 1965011367-012 -4- Table 2-I. Definitionof Study Tasks i il, Talk I Definition . | i , ,.,. IA Select the parameters ariel ",.herange and manner of variation for a paran,etrlr study of the SDMEG-ARS. ) IB Discuss factors affecting system design. , IC System definition {I) Derive block diagrams and system equations to cover the range of parameters selected in Task IA (2) Produce preliminary design of lo_|c for gyro computer Interfdce (3) Produce preliminary designs of computers for systems defined m (1). Design to the level of block diagrams and size, weight, power and reliability estimates. (4) Produce preliminary design of system other than computer for systems in (I). Design to level of system detail block diagram and size, welght, { power and reliability estimates.I I (5) Conduct a performance analysis to estimate the accuracy of each o' " ithe systems derived in (I). : (6) Tabulate size, weight, power, rellabzlity, and accuracy for' all systems in,restigated. . ID Discuss system additions and/or alterations required to utilize a SDMEG I i cagingcapability, IE Investigate and discuss the advantages of SDMEG modifications other than _ } . caging and torquing. ._ Task II "- .... i IIA Define functional requirements of an ARS for a capsule lander. List or tabulate ARS requirements. lib _electa system configurationtomeet requlrerr-ntsdefinelinIIA, Writ,. a description ofthe system and its operation. IIC Make detail block diagrams of the _ystem selected in liB. lid Determine characteristics of selected system (1) Estimate size, weight, and power. Tabulate and d, ecues basis for est, mates, ii(2,_ Estimate system reliability. Tabulate estimates for subsystem_ and discus_ basis of estimates. (3) E_timate system accuracy. Discuss basis for estimates and present r,_' resultt_. ] (4) Make a summary tabulation of liD(1), (2), and (3). _j lIE D,_,scuss any modificationsthatmay be necessary inthe SDMEG fortn(, !.I t _ system selectedinTas,_II. i! 1965011367-013 -6- e The attitude reference system "Hill accept commands to change attitude • Optical sensors will provide the information for initial alignment uf the ARS • Initial vehicle tumbling will be arrested by a means separate from the ARS • The gyros will not be operating during launch but will be started and put in operation in space following launch. The application selected for the preliminary design to be performed in Task II of the program is an attitude reference system for a .l_nding capsule. The mission for the landing capsu]e assumes it is separated from a parent space vehicle m transit to, or wh_le orbiting a target planet. After separation, the capsule is two-axis attitude stabilized in pitch and yaw, and roll-rate stabilized during retrofire and free-fall entry into the planet's atmosphere,if any. Once into the planet's atmosphere the aei_odynamic forces w,ll overpower the reaction jet system and the capsule will reorient as dictated by physical shape, and fall to the planet's surface. The attitude reference system's function is to provide two-ax_ attitude hold and three-axis rate stabilization signals to the capsule autopilot or stabilization ai,d control system. Events during the mission are assume¢_ to be as follows: • The orbiter attitude will be oriented to give the iand:_,g _,_tpsule the des ire d attitude • The landing Capsule gyro will be started and omented to the capsule body axes - • The landing capsule will be separated Irom the orbiter and the ARS _ will maintain the attitude established by the orbiter :, • The ARS will hold the attitude established before separation while i_ the retr0 rockets are fired to decelerate the capsule l I i 20113-FRI #- 1965011367-015 - 7- ° • The ARS will maintain the same attitude as the capsule falls to the surface. r CON CLUSIONS In addmon to a better general understanding and appreciation of t,he require- ments that must be satisfied by the instrumentation of a spacecraft strapped down ESG attitude reference system, the study produced five rather specific conclusions. 1) The parameter study produced results showing that the ARS configurattons selected for the study can be built for a system pointing accuracy in the range 0.04 to 0.07 degree within a volume l 0.5 cubic feet, a weight of 45 pounds a maximum power of and consumption of 140 watts. 1 2) The parameter study results showed that the system size, weight, power and reliability was increased by less than 10 percent when theARS was changed from .an output of attitude error o_y to an output of attitude error and body rate.I 3) The investigation of possible uses for a caging and torquing i.[ capabihty in the gyro showed that caging capability beyond that available in the current gyro design would not simplify or reduce il the _lze of the _ystem configuration considered in the parameter study. The investigation covered possible use of caging and il torquing to simplify imtial conditions _nd the possible use of torquing to assist in obtaining a less noisy vehicle body rate signal. i 4) T}_e results of the preliminary design performed in Ta_k !I show that, except for the rate stabilization signals, the performanceI J 20113-FR1 1965011367-016 -8- requirements of the attitude reference system for the landing •: capsule can be satisfied by a system instrumented with a single electro-static gyro. 5) Results from both the perforn_ance analyses performed for the parameter study of Task I and the preliminary design of Task II revealed that the quality and usefulness of the ARS rate outputs were limited by the level of noise on the gyro readout. The investigations conducted in the course of performing Tasks I and II also produced some "by product" type results. 1) In order to provide output information at a rate of 100 iterations per second it will be necessary to supply each gyro pickoff with it's own set of shaping electronics rather than attempt to time share two sets of electronics between three pickoffs. 21 In the course of the parameter study it was necessary to establish the characteristics of a digital computer which would be suitable : for the selected ARS concept. It was concluded that the computer wLll be a general purpose, serial/parallel machine with a 12-to- 16-bit word length. The arithmetic would be done with 4 bits per bite at a bit rate of 1.5 megacycles per second. The memory would |i have a capacity of 512 to 1024 words. The smaller memory will be used in the ARS sys,',em that provides only attitude error output. L!I 20113-FRI - i I 1965011367-017 - 9 - • ; SECTION III i! SDMEG-ARS PARAMETER STUDY _J INTRODUCTION The objective of a parameter study for an SDMEG-ARS or any other system is to establish in some way the relationships between the various system parameters or characteristics. Since even a modest system may have many parameters, attenti_ must be focused on the most prominent ones to keep the study within manageable proportions. In the parameter study of theSDM_:G-ARS three independent parameters and four dependent paramete_'s were selected for investigation. INDEPENDENT PARAMETERS _rhe independent parameters were the configuration, the system accur-gyro acy and the form of the system output. The dependent parameters were _ system size, weight, power consumption and reliability. Since the selected independent parameters are not continuous throughout their range and since it was impractical Within_the scope of the study to Lnvestigate a large number of parameter va]ues, the study considered the following discrete values of the independent parameters. A. Gyro Configuration i l 'A(1) Gyro developed under Contract No. 950_07/ A(2) Gyro with Caging Capability }! • A(3) Gyro with Torquing Capability B. System Outputsf) (( B(1) Three Axes Attitude Error and Body Rate il B(2) Three Axes AttitudeError .......... -..,--,----, _ Nil --- -- ...... i 1965011367-018 :, 10- C. System Accuracy i C(1) High _ ccuracy System _' C(2) Low Accuracy Sy._tem GYHO CO." TIGURATIONS The gyro developed under JPL Contract No, 950607 was not designed to ! provide caging or torquing as such. However, the gyro was designed with i the customary damping trails which cause the gyro to spin in a stable man- i: ner about its preferred axis. In addition to this function the damping coils provide a limited caging capability as a by product. In reducing "wobble" :. in the rotor motion the damping coils tend toalignthe gyro spinvector alongthe '; magnetic axis of the damping coils. _. i"or the purposes of this parameter study a gyro with a caging capability , has as part of its aesign the ability to precise].y align the gyro spin axis with a fixed line in the gyro case. With the gyro having this capability it will be possible to retur._ the gyro spin axis to a known orientation re!ative to the gyrc case. It is anticipated that caging capability will be mechanize_l with coils similar to the damping coils, However, th_ coils will be preciqely made and mounted to the gyro case ,so that the coil magnetic axis can be properly aligned with the desired caging orientation. _ A torquing capability in the strapped down miniature electrostatic gyro is defined as the ability to orient the gyro spin vector in any selected direction relative to the gyro case, It is anticipated that this capability will be instrumented with three sets of coils similar to the damping coils. O?he coils would be mounted to the gyro with their magnetic axes forming an orthogonal triad. Thus, the current in the coils could be adjusted to I produce a resulting magnetic force vector that could have any orientation ! relative to the gyro case. Since the torques on the gyro rotor (caused by the magnetic force vector)arasuch as to align the spin vector along the magnetic vector, the desired torquing action results. $0113-FR1 1 1965011367-019 ! 11 1 I It was felt that these three levels of gyro complexity would provide usefulinformation for making future gyro development decisions. Also, it would provide generally useful trade-off information between gyro caging and I torquing capability and system complexity. ! SYSTEM OUTPUTS I System output was selected as a parameter to be varied since not all space- craft applications are likely co need both attitude error signal and a signalproportional body rate. It was therefore desirable to determine how omitting i the body rate output signal would affect system size and power. Also, this information could be combined with information on the size, weight, etc., to obtain body rate from another source, such as a rate gyro, and determined which source of rate signal best suits the considered application. SYSTEM ACC URACY } ' System output accuracy is Jmportan* in selecting an attitude reference i l system for a spacecraft application. The required accuracy may rangefrom two or three seconds of arc, as re0uired for the AOSO program, to two or three degrees, as required for the Mercury spacecraft. Therefore, the information resulting from a study of the trade-off between il system accuracy and size, weight, and reliability, will be u_eful in power future decisions regal'ding the use of the _trapped down e]ectrostatic gyro attitude reference° l! Twolevel_ofaccuracy, labeled"high" and"low", wereselectedforinvestigation in the parameter study. It was assumed the "high" level system would include mathematical compensation for misalignments in ;i i! 20 ,3FR1 1965011367-020 -12 - the gyro pickoffs and mountings aboard the space_raft, _s well as any non- orthogonality of the spin vectors. The system with the "low" level accuracy would neglect these corrections. Also, it was assumed the "high" accuracy system would have an accuracy in the computer or data processor equivalent to 0.01 degrees error in the attitude c,,_put. And it would be capable of processing data fast enough as to provide the attitude information 10 times per second and the body rake information 100 times per second. The "low'" accuracy system was assumed to have a compating accuracy equivalent to 0.1 degrees error in attitude output and iteration rates on the outputs of one per second for attitude and 10 per second for body rate. PA RAME2:ER COMBhNATIONS If all combinat'ons of the independent parameters are used to define attitude reference systems, there are 12 systems which can b_:. studied. With the symbols previously used to indicate the variations of the independent para- meters the systems a .'e: ARS No. I = A(1) B(_) C(I) ARSNo. 2 = A(1) B(1) C(2) , ARS No. 3 = A(1) B(2) _(') ARS No. 4 ._ A(1) B(2) C(2) AI_S No. 5 -- A(2) B(1) C(i) ARS No. 6 = A(2) ]B(1) C(2) ARS No. 7 = A(2) B{2) C(1) ARS No. 8 = A(2) B(2_ C(2) ARSNo. 9--A(3) B(1) C(1) ARS No. i0 = A(3) B(1) C(2) ARS No. 11 = A(3) B(2) C(I) ARS No. 12 - A(3) B(2) C(2) " 20II3-FRI I 1965011367-021 '! -13- Although there are 12 combinations of the independent parameters defining _2 attitude reference systems, this does not necessarily mean that 12 distinct sets of hardware will be required to satisfy the 12 system require- ments. SUMMARY OF RESULTS I - 'il The parameter study's objective was to establish the relationships between 'I I_ the independent and dependent parameters. These parameters were definedJ and discussed in the preceding introduction. Counting all combinations of ] the independent parameters there were 12 attitude reference systems to be !_ considered in the study. Itwas determined thatthree distinctsets of system hardware wou,]dsatisfythe requirement.sof all12 sys_,ems. 1 H_ RDWA R E CONFIGURAT IONS Although itwas originallythought that&-jrocaging and torquingwould significantlysimplifysystem mechanization, the investigationdid not /0 support such a conclusion. Itwas therefore concluded thatthe systems would allbe instrumented with the same type gyros, without caginZ or torquing. Therefore.,for hardware and instrumentationthe original!2 systems were reduced to four. The study and prelimin_,.rydesign of the ' data processor required by the strapped down ESG attitude reference [ system revealed that the requirements of the remaining four systems could be satisfied with th,-ee different designs of the data processor. As II a result, the requirements of the original 12 combinations of independent parameters can be satisfied by three distinct sets of hardware. These !1 sets of hardware were labeled systems A, B and C. Table 3-] shows thevalues of the dependent parameters of size, weigbt_ power and reliability for each of the three syste:ns of hardware. _} 20113-FRI 1965011367-022 -i4- Table 3-1. System Sizing Estimates Size Weight I Input Power [I B_ailure Rate, = (in. 3) {pounds) (watts) ( _] i000 hours_ l i Min. Standby Ma-.:imum ! System A 898 46.7 I 7.2 62 i37 i0.4202 System B 765 43.5 I 7.2 43 ll8 8. 9102System C 833 45.0 7.2 56 .21 9. 4720 INDEPENDENT PARAMETERS FOR HARDWARE CONFIGURATIONS The independent parameters associated with each of the systems of Table 3-1 . are: System A is a high accuracy system that provides both attitude error and -body rate outputs; System B is a low accuracy system that provides both attif.ude ' error and body rate outputs; and System C is a high accuracy system that pro- ?. rides attitude error output only. The parameter combination of low accuracy and attitude error output did not result in system hardware distinctly different : ': from system B. Table 3-1 indicates three levels of power input to the systems. The maximum power assumes everything operating with the gyros in the high "g" mode. The _ standby power assumes the computer "on" but not operating and the gyro in "_ the low "g" mode and pickoffs turned off. Minimum power assumes everything "off" except the gyro low "g" suspenQion. i Table 3-2 indicates the relationship between the originally defined 12 attitude references systems and the three sets of system hardware listed in Table 3-1. 20113-FRI ,i _: . i ] 9650] ] 367-023 " 15- ,! _J Table 3-2. Hardware for Original 12 ARSL System Hardware ARS No. 1 A i ARS No. 2 B _ ARS No. 3 C i ARS No. 4 B il ARS No. 5 A !! ARS 6 B ARS No. 7 C ' ARS No. 8 B •i ARS No. 9 A ,} : ARS No. 10 B ARS No. Ii C : ARS No. 12 B i __, J ACCURACY VERSUS DEPENDENT PARAMETERS ! Figure 3-1 is a plot showing the relationship between the independent parameter accuracy and the dependent parameters of system size, weight, _ : power and reliability. The information presented in th_s plot is limited since, due to practical considerations, the parameter study was restricted I i -to tws values of system accuracy. The performance analysis conducted -during the parameter study showed these two accuracies to be0.036 and .! } 0.066 degrees, for the high and low accuracy systems_ respectively.1 These are one sigma errors in the accuracy with which a given axis of t 1 the space vehicle can be pointed in a commanded direction. I! I ' 201i3-FR! ., " ''" 1 An A. ,, , . . , - .... _. :%::_ .... 1965011367-024 -16 - _" I ...... '../'1 t',_ ,-4 •,¢ _" " II t I -° I z t,a4 .,,-a .,.4 "1" :._ .._ ," % 111 U (SONflOd}.LHOI3MIN3LSAS _ '_ ,4 c; _, r._ {l-i "n:3)3ZIS W3.I.SAS ! , w i I l ,- J ¢_ !* 0 ,0 _1" ¢_ O0 ,_" 0 e.,I r.41 ,'4 ._ (SJ.L¥1_J,ndNl_3MOd IN_IJ.SASXVW i I I' I i ! ! I ,'4 e..I (SIJHO00Z/J.N3Oll3d) 31YIJ 3_AllY_I IN31SAS 20113- FR1 1965011367-025 ; -17 -- BY-PRODUCT R ESULTS In addition to the parameter study'smain objecLive of establishing the relatio,_- ship between the independent and dependent parameters the study had several other by-product results. One of these was the discovery that the noise level on the gyro readout was the limiting factor in the usefulness of the body rate signal derived from the gyro readout. The perforrnance analysis indicated the error in the body rate signal to have a one sigma value of 1.4 deg/sec. The readout noise contributed 70 times as much to this error as the RSS of all other sources of errors. i It was also discovered that in order to read the gyro 100 times a second each 1 gyro pickoff must be equipped with its own set of shaping electronics, rather than to time share t_o sets of electronics between three pickoff_ as originaliy planned. Also, since it was necessary to carry out a preliminary design of the data processor to establish its size, weight, power and reliability, the other characteristic determined in the process is useful by-products. For examp__e, it was concluded that the most accurate data processor would be instrumented with a 16-bit word, and would be a serial parallel machine operating with four bits per bite at a bit rate of 1.5 megacycles. _) 20113-FR1 1965011367-026 - 18- SELE(YrION OF THE ARS CONFIGURATION INTR ODU (._FION It has been assumed the SDMEG-ARS is capable of accepting commands in a digital form from the Central Control and Sequencer (CC&S). or m,- equivalent subsystem and provide torque comrJ-.and outputs to a Stabilization and Control Subsystem (SC_) which will affect the required attitude change of the vehicle. Conceivably, the CC&S can output several types of command signals which all uniquely define a given attitude change of the vehicle. The ARS design will ch&uge in complexity depending on the sc!ected type of input from the CC&S. Therefore. the purpose of this discussion is to systematically consider reasonable inputs fro_ the CC&S and determine the corresponding necessary design changes in the AI_S. The type of a_ib.zdecommand s'tgnalswhich are considered in thisdi3cussion are: • Three-Euler angles • Three sines and three cosines of Euler angles • Six direction cosines of the gyro spin axes in body axes • A complete 3 by 3 matrix which can be used to transform vectors .from the vehicle reference frame to the frame oriented to the desired attitude of the vehicle. The following discussion is arranged from the point of view of the AR$. design with the following approaches: • Euler angle approach I i • Euler angle apl_roach I.! • Gyro 3pin axis commands . : • Vehicle steeringin-thegyro frame The use of the four types of CC_ commands with each of the. above: four 'ARS design approaches will be poir_ted out at apprepri-ate places in the_iscussion. !, E t ..................... ...... ......... _, ......... .Jl:l L_. U ........... 1965011367-027 19- COOR DINATE FRAMES Four different frames of reference are used in the discussion of the ARS design approaches. These frames will carry the subscript: o, f, g, b. o-frame - is the basic reference frame and, in this discussion will also be used as a navigational frame. It is assumed that this frame is established by optical sensors on board the } vehicle independently of the gyro readout. For instance, thL o-frame could be the sun-canopus h'ame. "l ) f-frame - defines the vehicle's desired attitude. The orientation of the f-frame is defined by the guidance subsystem relative to the o-frame. By the assumphons of o-frame the def- nition of this frame is independent of the gyro readout. Let the relative attitudes of the f and o-frame be defined by the transformation To2 f (transformation from o to fL. i frame). The transformation To2 f consists of nine direction co_mes; however, it requires only three Euler angle:, to define the nine direction cosines. J g-frame .- is the inertial reference frame whose two axes are along ! the two gyro spin axes; the third is along a vector which I is perpendicular to the two gyro spin axes. J The ,,rientation of this frame is defined by the transforma- 'T tlon _ _b2g) relative to the vehicle body axes. The elements) i) of the Tb2g are determined in the ARS computer from tile gyro readout information. b.-frame - zs the vehicle body frame who_e axes are the roll, ptgeh /i and yaw axes. I ) The transformations between the four frames contain the necessary information "_I for the vehicle attitude control. This information is summarized below. fi 2011a-FRl r_,_" _, ,_I__ _ i it,,.. ....... • iI l iiilllmli m 1965011367-028 - 20 - (command) (alignment) (;eadout) T02f %02g Tb2g T02 f - Defined by a guidance subsystem T02g - Defined by the initial alignment of the ARS Tb2g - Defined by the gyro readout EULER ANGLE APPROACH I Concept The Euler Angle Approach I is identical in concept to the control system mechanization using a four-gimbal platform for attitude reference. The , similarity in mechanizations is depicted in Figures 3-2 and 3-3. The basic difference between the two mechanizations in Figures 3-2 and .Figure 3-3 is that the four-gimbal platform in Figure 3-2 is replaced by the two SDMEG's and the digital computer in Figure 3-3. In this discussion it is not the intent to arrive at the trade-offs of *.he two mechanizations. The four-gimbal mechanization is introduced to aid the understanding of the SDMEG-ARS mechanization using the Euler Angle : Approach I ,! Conceivably, the mechanization in Figure 3-3 can be a hybrid system (analog)P, and digital)where the digitalcomputer outputs are the three Euler angles. :,, The comparison of commanded Euler Angles to measured Euler angles and • _ the transformation to body axes can conceivably be an electromechanical _ 'l 20113-FR1"5. 1965011367-029 i - 21 - COMMANDED EULER ANGLES EULERTO BOOY 5CS • _ FOUR GIMBALPLATFORM TRANSFORMATION Te2b I ] 'l Figure 3-2. Platform Mechani.zation i COMMANDED EULER ANGLES I COSINE 80])7 EULER TRANSFORMATION TO SCS _,OMP TRANSFORMATION ANGLES Te2b Figure 3-3. SDMEG-ARS Mechanization I 20113-FR1 1965011367-030 - 22 - analog system. Howe:,er, in the present discussion all meehanizetions wUl be assumed to be of digital type to arrive at the intended trade-offs among the different approaches of mechanizing the SDMEG-ARS system. Computationa! Requirements To mechanize the ARS concept shown in Figure 3-3 the following computations must be performed by the digital computer. P (1) Compute six direction cosines, (2) Perform a vector cross-product; hence, complete the definition of the body-to-gyro transformation (Tb2g). This transforma_Aon also defines the T02g ARS alignment matrix when the vehicle is held to *he basic i reference attitude by opticai sensors. (3) Invert matrix T02g and obtain Tg2o. This is a comparatively trivial computation if the gyro spin axes are orthogona]. (4) Multiply two 3 by 3 matrixes and obtain body-ta-reference transformation Tb2 o = Tg2o Tb2g ? f (5) The elements of the Tb2 ° transformation a_Je functions of the:desired i Euler Angles. And. in particular, three elements (a21, a31, a32) of the transformation will be Sin 0 _ = -a31 J i Sin _ cos 0 = a32 _ Sin_cos 0 = a21 1965011367-031 :J , -23 - t ' Use the "method of least work" Jn Appendix C to verify the above relation- shin. Therefore, compute the Euler angles as follows: I (5.1) 0 = -sin-I ", a31 .; (5.2) cos O = cos (-sin -1 a31) i , }I (5.3)_ ._n_%2/oos0) (5.4) VJ = sin-1 (a21/cos e) Ji (6) Compute Euler Transformation Te2 b To compute this transformation it is required to determine (6.I) sin @ f_ It'" (6.2) sin _ cos 0 (*6.3) cos ecos ¢ (6.4) sin ¢ ! ('6.5) cos @ _, (7) Compute the difference (A¢, _8, A_ ) between the commanded and i computed Euler angles. '!_t (8) Transform A¢, A0, A_ to body axes to be used by the SCS as the vehicle rate commands il d _ ?o, _-.%..ooI! L%d "* o ] The _xc t _)yc and ¢0zc are the rate commands to tile autopilet. *Quantities to be computed in Step (6); the other quantities were already computed in Step (5). 20113-FRI i J9650JJ367-032 - 24 - All of the computation steps, except for Step (3), must be done at a. compu- tation rate which is consistent with the autopilot bandwidths reo_uirements. Step (4) does not have to be completely carried out since cnly three elements (a21, a31, a32) are required for the Euler angle computatlon. , Table 3-3 does not include operations in the cosine computer (Step 1 above). Table 3-3. Summary of Major Arithmetic Operations if Guidance Provides Three E'uler Angles High Speed Computations Low Speed Computations / , { (Co, reputations During Maneuver) (Calculations Prior to a Maneuver) ti-- Multiplications = 25 (None Required) 7 Divisions -- 2 ARC sine = 3 Cos -- 2 ( Interface with Guidance Subsystem Guidance Subsystem serds three number_ (Euler angles) to the SDMEG-ARS for the re-orientation maneuver, ARS initial alignment procedure will require that the guidance subsystem: • Command vehicle hold attitude using optical sensors with vehicle body axes along the body reference attitude (body axes along o-frame). I • Instruct the A RS that it is in the alignment mode. 20113-FRI 1965011367-033 - 25 - 11 '1 EULER ANGLE APPROACH II ] Concept { In this concept, the Euler angle error signals are computed without tile explicit computation of the Euler angles and comparing to the commanded i i angles. Examination of the computation of the torque command signalsF '_ (attitude error in body axis) shows that the F,uler angle transformation (Te2 b) is not required.I /.J The concept is explained using the pictorial representation _f the coordinate i rotations in Figure 3-4. e _ A_'* Ae AO _0\!/_ i/ i "\i/_, / , - _, . '0 , -/__/ i -/k_l" REFERENCE FRAME OR DESIRED BODY i I NAV FRAME BODY AXES ATTITUDEATTITUDE " i. %. , i& ...... . ,t Tolb Tb2f * - • . T02 f - COMMAND MATRIX _. Figure 3-4. Graphic Coordinate Transformation 'II fl , 20113-FRI . _:__._o _ I " - : iiimr....... m. A__ I 1965011367-034 "4 - 26 - The method used in Figure 3-4 to describe the coordinate rotation is explained in Appendix C. However, at this point it Will suffice to say that this figure is _,nalogous co resolver mechanization of the transformations, namely 3, ¢ ! xc -- -_ x i = x° cos0 + Yo sin0 ; Yo _ Yl = -Xo sine' + Yo cos 0 Z O Z 1 = Z O Following a signal through the reso!ver in a diagonal direction gives a sine function. A dot can inchoate which function requires the negative sign. A 5 signal going straight acros._ is multiplied by the cosine. In Figure 3-4 the transformation T02 f is the guidance subsystem command " matrix and is always uniquely defined by a consistent set-- of command signals. _, The transformation T02 b is computed in the ARS com_uter using gyro read- out and ARS alignment data, The body axes _xb, Yb' Zb ) attitude at any time during the maneuver.is known both in the o and f frames and the autopilot has to torque the vehicle sach as to drive the Euler angle errors A_J, A0 and A_ to z_ro or drive the trans- formation Tb2 f to a unit matrix. The concept of the Euler angle Approach II then involves the computation of the transformation Tb2 f and computation of the steering com_lands which will drive the Tb2 f to unit matrix. ,¢ k, :i I - i 201_13-FRI ] 9650] ] 367-035 - 27 - i Computaticnal Requirements The mechaniza',ion of the ARS using the Euler Angle Approach II requires the fo!lowmg computations in the ARS d _ltal computer. (1) Compute six dJrection cosines. (2) Perform a vector cross product, hence, define the body-to-gyro trans- formation (Tb2g). This transformation also defines the To2g ARS align- ment m_trix when the vehicle is held to the basic reference attitude by optical sensors. i t (3) Invert matrix To2g and obtain Tg2o. This is a comparatively trivial computation if the gyro spinaxes are orthogonal. The orthogonality of the spin axes can be assured if the gyros have caging or torquing cap- ability. This computation can be done prior to a re-orientation maneuver-. (4) Multiply two 3 by 3 matrixes and obtain body-to-reference transformation, Tb2 o = Tg2o Tb2g (5) Establish the command matrix To2 f. Depending on *.he form of the guidance command signals, *.he ARS com-- i i purer wiil be required to perform different amounts of computations. However, these computations do not have to be done simultaneously with il the computations which enter into real time vehicle control. In other words, the To2 f can be precompuZed by the ARS GP computer prior to :ii a maneuver when the computer is riot used for the vehicle control purposes. 0 If _bc, Oc' Cbc are the commanded Euler angles, then (-S0c) I{ {c_ c roe) (s¢c roe) _} -s_c cO +c_ c sec) (s_c sec S¢c+. c_ c c¢c) (re c S¢c)| To2 f := L( s_e s¢ c+c$ c c_ c) (s_ c se e c¢ c - c_ c S¢c)*(c0 c C¢c) jli ......... --........................... _...',. , .,._ 2:_::_ _-:-v:-. , ..... i i I , . L_ L_,.,, 1965011367-036 - 28 - Use the "Method of Least Wo:-k '' in Appendix C to verify the above trans- formation. Note that in order to evaluate the To2 f matrix the computer must have in its memory the sine subroutine which is the new addition to the functions the computer must perform in its primary mode of operation {vehicle con- trol mode). In order to eliminate the storage of sine subroutine the guidance system command signals must have one of the following forms: sin_b c, cos _bc, sin e c, cos 9c, sir _bc. cos _bc sin_b c, sin 0c, s.m _bc and sign of cos @e sign of cos e c sign of cos C Guidance provides the nine elements of-the To2 f transformation. (6) Compute Tb2 f transformation. Tb2 f = To2 f Tb2o The matrix Tb2 f is a 3 by 3 matrix and consists of nine elements, namely ell el2 el31 Tb2f = e21 e22 e23 L¢31 e32 e33_ : 20113-FR1 ] 9650] ] 367-037 - 29- From Figure 3-4 the Tb2 f matrix can be written as follows r (cA_ cA*) (s_ cA0) (-s_0) I Tb2f =](-cA_b_S_+SAd_s_O c_b)(CA* c_b+sAd_ sA8 SZ_)(sAsh CAR) | I(c_¢s_oc_$+s_¢sa_) (c_¢shes_- sA_c..4,)(cn¢cno) L_ The ¢'s are the computed elements of the Tb2 f transformation and comparing term by term with Tb2 f transformation exy _ssed in terms of the Euler angle errors it can be seen that \ ¢12 = sin_cos A8 _- A_ / Approximation fori ':i el$ = -sin A8 _ -A9 I small errors }I ¢23 = sin Ad_ cos _8 _" A_b 1 ('l) Determine vehicle steering law which will drive ¢12" el3 and ¢23 terms to zero and thus align the vehicle to the desired attitude. It is shown below that no further calculations are required to derive vehicle rate command signals in order to null the ¢12 ° ¢13 and ¢23 terms. And, in fact, Wxc = -¢23 -- Xaxis attitude error in body axis ; wyc +el3 --, Yaxis attitude error in body axis : _ Wzc = -¢12 -, Zaxis attitude error in body axis ill PROOF il ell el2 e13 _o! ^ I -- ¢21 e22 e23 YbI] L_f. _ 3 3 3 _Zb I! 20113-FR1 ) i 1965011367-038 - 30- Where Xf _(f 7.f Are unit vectors along f frame Xb Yb Zb Are unit vectors along b frame It is desired to align Xf with Y_b" Yf with Yb mid 7.f with 7"b" To accomplish this maneuver two axes must be aligned and the third axis represents the ,'e;dundant information. For this derivation assume that the Xts and Y's are aligned. " To align XSs i_ is sufficient to rotate the vehicle about the vector XfX._L° until : _he magnitude of Xfx:_ b assumes the zero value. Slmilari[y for Y's rntate the vehicle about the Yf×YbVector until its magnitude is zero. , Smc_. the relative attitude of f and the b frame is defined by the Tb2 f trans- ;. rmat_,)n the vector cross products can be evaluated as follows: ! _(f × _(b = (¢11 Xb + ¢12 _(b + ¢13 Zb ) × Xb : -_12 Zb _(b Therefore WZc -¢12 _ _Yc = +el3 Similarly Yf XYb = (¢21 Xb + e22 _/b + ¢23 7"b ) x _(b '_ _ ¢21 _'b - ¢23 "_b r I 20113-FRI ! ,]L 1965011367-039 t-31 - Therefo re a_Xe = -c23 Table 3-4 does not include operations in the cosine computer (Step 1 above). Table 3-4. Summary of Major Arithmetic Operations if Gaidance Provides Three Euler Angles . High Speed Computations Low Speed Computationb : ! (CalculationsPrior to a Maneuver) Multiplications = 45 Multiplications = 16 Sin _ 3 i Cos = 3 Interface With Guidance Subsystem The guidance subsystem attitude commands can have any one of the following forms" (1) Three Euler angles. (2) Three sine and three cosine functions of the Euler angies. ]1 (3) Three sine functions of the Euler angles and three signs of the corr_s- x X ponding cosine functions. ''_ril (4) Nine numbers which define the To2 f transformation. ] _] The ARS imtiat alignment procedure will require that the guidance subsystem: i-i• (1) Command vehicle hold usi.,g optical s_-_sors with the vehicle body axes aligned along the body reference attitude (body axes along o frame). "i[ (2) Instruct the ARS that it is in the alignment ,node. ! 20113-FRI I 1965011367-040 - 32 - YRosPIN coMiNg. QI'Q2 Concept For this concept the vehicle attitude error is computed by co,mparing the current _'r_ readout with the commanded readout and then _teering the vehicle ° '. to null the difference. The geometry is depicted in Figure 3-5. • _- In Figure 3-5 the gyro spin vectors SI and S 2 are known in the body axes by the virtue of the gyro readout and some data processing. _he vectors Q1 and Q2 are the desired attitudes of _1 and _2 in axes.body ! These attitudes are defined by processing the command signai-s (e. g. Euler angles} from the guidance subsystem together with the initiai ARb alignment data. The vector difference (_-S) between Q and S is $ormed for each gyro and i is used in computation of a unique vector 0 such tb + if the veh::cIe is rotated .. ' -- -- "-Y+"approach null values iabout 0 the readout errors represented by A$1 ar ..... 2 -t simultaneously. Another property oi the 0 vet-t, s that it becomes a null ' vector (zero magnitude) when both the ._S 1 and _,_ ar_ zero. In fact _'is the i +_ desired attitude error in body axes .o be used as an input :_, '- .,CS. i It should be obvious from the abo re discussion that during a given maneuver the and are constants and tbey are precomputed before the maneuver. i- Consequently, these constants ca-a be precomputed by the Guidance Subsystem _ or by the ARS G. P. Computer. The ARS computer, before a maneuver may be in the "S*.audby" Mode, therefore+ could be available for the computation : of Q! and Q2 constants. ! 20113- F_ I -I -! +1 .......... i+ :±- ._2 _ _+,,_,_, 1965011367-041 ; 33- _(b'Yb' tb UNIT VECTORS ALONG T-HEBODYAXES ' _;1' _;2 UNIT VECTORS ALONG ,::YROSPIN AXES QI' 42 COMMANDEDATTITUDES OF $1 AND $2'RESPECTIVELY _i i $11' S12'S13 S1 DIRECTION COSINES ^ DIRECTION COSINES $21'S22'S23 S2 (NOT SHOWNIN THE FIGURE._ '!1 qll'¢12'q13 _11 DIRECTIONCOSINES " q'_l' q22' q23 Q2 DIRECTION COSINES " (NOT SHOWN IN THE FIGURE) t'l i ATTITL;DE ERROR IN BODY AXES !l ;J! Figure 3-5. Geometry Use_i in Control of Vehicle Using Q1 and Q2 Commands 1 -1 -! • 20113-FR1 ,._- ..... ,_..... i iL _ _ L i i _.u_:-:::::-._ ........." " " I 1965011367-042 !- 34 - Computational Requirements The computational requirements for this ARS concept are summarized below: t (i) Compute six directioncosines. l (2) Compute the command matrix To2 f. Depending on the form of the guidance command signals, the ARS computer will be required to perform different amounts of computations. However, these computations do not have to be done simultaneously with the computations which enter into real time vehicle contrcl. In other words, the To2 f can be precomputed by the ARS G.P. computer prior to a maneuver when the computer is not used for the vehicle control purposes. @ 8c, _bc are the commanded Euler angles then __ C ' (c@o c) (s¢,c | Cec) (-Sec) 1 "- + sO s_c) (s_cse s¢,.+ c¢c) (cOcs_ c)To2 f (-s_b c COc c_ c c c - c_ c (s_ cs¢ c+c_ cso cc_ c) (s_ cso cc_bc-c_ cs¢ c) (cOcc_ c) Use the "Method of Least Work" in Appendix C to verify the above transformation. Note that in order to evaluate the To2 f matrix the computer must have in it.s memory the sine subroutine which is the new addition to the functions the computer must perform in its primary mode of operation (vehicle control mode). ' In order to eliminate the storage of sine subroutine the guidance system coro- t i mand signals must have one of the following forms: ! sin@ c, cos@ c, sin ec,'COS e c, sin_b e, cos _bc sin@ , sin e , sin _c andc c ':_ 20113-FRI ,j' _;ILI ........... --_- L5 III I I ,.i_.-' , - ,:_, +_- ; "- ...... " " ' ' -_. I '_ ...... I'.,'_,' " ' i ' , " ' '" , ' ' _'_ , ' ' ' , ' ' ' " ' • " " "f_,_,_+_'._"._7 4_'lI't" "" '",, ' ' =....... Aii]ll _ L___TL 1965011367-043 - 35 - . sign of cos _c sign of cos e C i i sign of cos _c i]_._ Guidance provides the nine elements of the To2 f transformation. A l (3) Compute the command vectors Q1 and Q2 " These computations do not have to be done simultaneously with the computations which enter into the real time vehicle control. In other words, the six direction cosines which describe the Q1 and Q2 command vectors can be precomputed by the ARS G. P. computer prior to a maneuver when the computer is not used for the vehicle control purposes. The computation involves multiplication of a 3 by 3 matrix (To2 f) by a 2 by 3 matrix % , Tht, six direction cosines (Sij o) are available from the ARS computer • i memory and are the initial ARS alignment data. Before storage of these constants in the computer memory they are computed by th_ Cosine Com- _i] puter (Step 1 above) during the ARS alignment mode. i Note thatifthe Q1 and Q2 command signalsare establishedby the guidanceI subsystem then the guidance "ARS interface must accomodate the tr_ns- il misslon of six numbers (Sij O) from ARS to the guidance and six numbers(qi3)fromguidance to the AI_S. Consequently, the ARS design trade-off involvesthe storage of a subroutine in %he AI_S computer to solve for Q1 and Q2 versus the additional requirement on the guidance ARS interface to transmit ARS alignment data to the guidance subsystem and to compute the command vectors Q1 and Q2 in the guidance computer. I] 20113-FR1 1965011367-044 r- 36 - (4) Steer the vehicle according to the law - _ xs1 -s_(q2 _ xs2) See Attitude Error and Body Rate Computer in Progress Report 20113-PR2. Table 3-5. Summary of Major Arithmetic Operations if the Guidance Provides Three Euler Angles High Speed Computations Low Speed Computations 1V[ultiplications = 18 Multiplications = 34 Sin = 3 * Cos = 3 :,. , Table 3-5 does not include operations in the cosine computer (Step 1 above). ' Interface with Guidance Subsystem ,,, ', ; In the discussion of the "Computational Requirements" section, two approaches were pointed out: (I) The command vectors (_1 and (_2 are computed by the2 guidancesubsystem,and(2)the_1andU2vectorsar_.com,utedbyth,ARS computer. The interface requirements will differ for the two approaches. Interface Requirements for Approach 1 -- _ (1) Six numbers which describe ARS initial ahgnr_ent are transmitted i to the guidance subsystem.(2} Six numbers which describe Q_I and 0_ vectors are transmitted 1_', from guidance to the ARS computer. " 20113-FR1 t _ .... .J£_l // I / _11 I II I I I, IIIII II --q I - / _ ' I / I II lUam,_ • _ m _nnn_n '_ nn , ' .. '_ . ' y2' '.:' ' _ ,' _,'* 9650_ _367-045 37- , , (3) The ARS initialalignment procedure willrequire ,ha*the _Jidance subsystem: : (3. I). Commands vehicle hold using optical sensors with the vehicle body axes aligned to the o frame. 'i (3.2) Ins,_ructsthe ARS thatitis in the alignment mode. i InterfaceRequirements for Appr_oach 2 -- J (1) The Guidance system wil! transmit t_ the ARS: (1.1) Three Euier angles, ori.i 11.2) Three sine and three cosine functions of the Euler angles, or (1.3) Three sine functions of the Euler ,ngles and three signs of } the corresponding cosine functions, or (1.4) Nine numbers which define the To2 f transformation. ]! Ir (2) The ARS initial alignment procedure will reqlfire that the guidance _] subsystem: (2.1) Commana_-_ehicle hold using optical sensors with the vehicle ]_ body axes aligned to the o frame. II (2.2) Instrucmtl _-=ARS that it is in the alignment mode. [_ V_HICLE STEERING IN GYRO FRAME In this concept, the inertial gyro frame is used as a primary reference frame for the computation of the vehicle steering commands. Since the computed Ii steeringsignalsareintheinertialgyroframe0 theymustbe resolvedalong the vehicle body axes. I1 ¢ -_='T ....... : : _i_,.,,L!IL..,._,:.. _JlJ._ in i J In --ill - I I _' ........... Ill " ' .._._,._.': 1965011367-046 o;- 38- "- Con cept :, The concept is depicted in Figure 3-6 which shows the vehicle X-axis steering. A similar diagram can bc drawn for Y-axis (or Z-axis) steering to control the vehicle about the X-axis. 2 ,,.-Iliy,I i SI = X i ib=%"i. O. - AVECTOItAGOUTWHICHTHEVEHICLEMUSTBEIOTATEDTOALIGNXbWI111X! ,_ _99g_i - c'_o VRAU( Fi_re 3-6, The Vehicle X-Axis Steering in Gyro Frame !,'or (he X-axis steering, the actual and desired X-axis attitudes _re computed c 4b in th(, gyro frame. The vector cross-product of X b and Xf is formed, which, iftransformed to the body frame, can be used as input to the Y and Z controllers of the vehicle. % A Sire _,']y,the cross-product of Yb and Yf, after suitable transformatlon, can he u 2d as an input to the X-ayis controller. ) ii _-1 20t 13-FRI 1965011367-047 : - 39 - Computational Requirements The MRS mechanization using the vehicle steering in the gyro frame r_ quires the following computation steps by the digital computer. (!) Compute six direction cosines (Cosine Computer in Progress Report 20113-PR2). J i (2) Perform vector cross-product, hence, definethe body-to-gyro trans-A formation (Tb2g). This transformation also definesthe To2g ARS alignment matrix when the vehlcleis held to the body referenceJ attitudeby opticalsensors. .[ (3) Invertmatrix Tb2g and obtain Tg2b. This is a comparatively trivial : computation if the gyro spin axes are orthogonal. The orthogonality i of the spin axes can be assured if the gyros have caging or torquing capability. (4) Establish the command matrix To2 f, ! Depending on the form of the guidance command signals, the ARS computer will be required to perform different amounts o_ computations. However, • [ these computations do not have to be done simultaneously with the computa- tions which enter into real time vehicle control. In other words, the To2 f "j canbeprecomputedbytheARSG. P. computerpriortoamaneuverwhen the computer is not used for the vehicle control purposes.} If _c' 0c' qbc are the commanded Euler angles then J - toe) i : ('_c COc+ C¢cSOcS4'c)(S¢'cSecS_'c+ C_cC4'c)(c°csd,_) _, To2 f = ! + C ' •_ (s% s,c % Secc%.__% sec C'c- % S'c) (°% %,. I / J ,. 20113-FRI I J I 19650] 1367-048 - 40- lit;,, thf. "M:_lhod of l.east Work" in Appendix C to verify the above transforma- I i (Jn, N, t( that in order i( evaluate the To2 f matriy the computer must have in its m:-m.,'y the sine subroutine which is the new addition to the functions the ,, .mputer must perform Ln its primary mc_do of operations (vehicle cr,ntr¢! iTl,'de). .- In order to eiimLuate the storage of sine subroutine the guidance system command signals must have one of the fol_,owing forms: sin lf_:, cos _c' sin 0c, cos 0c, sin 4)c, cos 41 sin 0c, sin 0c, sin 4>c and } sigll _f cos _c sign of cos 0 c sign of cos 4 c 2 Guidance provides the rdne elements of the To2 f transformation. : (5) Muitit,ly the 3 by 3 To2 f matrix by the 3 by 3 matrix T g2o"t i" Tg2f = To2 f Tg2o " (6) list: inf,.,rmation from Step 15) and Step (3) and from two vector cross= i p r,,dlic ts. ! = oyz f 'jl ;1 :i - 20113-FRI ( 1965011367-049 i!, - 41 f (7) Calculatp the rate commands to be used as inputs t,,SCS. g2b _ Oyz • g t _xc : Tg2b _x " _ ;! , _ - g.. i.} Note that the computations wKich are described in Steps (4) and (5) can be pre-computed before a re-orientation maneuver either in the ARS or i : guidance computer. The computations in these steps transform the ! guidance command sigrm, ls from the reference frame (o-frame) to the inertially fixed gyro frame (g-frame). The ARS gllidance interface requirements will change depending on whelher ! I the ARS or guidance computer transforms the guidance commands to the' gyro frame. i). -;1 I ] 9650] ] 367-050 - 42 - : Table 3-6. Summary of Major Arithmetic Operations if Guidance Provides Three Euler Angles ,'_ ,, , { ] Low Speed Computations |hgh Speed Computations t(Calculations Pric. to a Maneuver) ! _' i_ , ..... Muliiphciations -- 27 MuItiplica_.ions = 43 Sin = 3 Cos = 3 Ini(.rf;,¢.'.. wilh (;uich, nve Subsyslem Ii, Ih(, di._.us.,don of lhe "Compulational Requirements" section, two approaches wt, re i),.,ir_lt, d oul: (1) lhe guidance specifies altitude (.ommand in gyro frame, (2) lh(..¢1(S c',),'npdler transfot'ms guidance ('ommands from the reference to rh,. gvro frame. The interface requiremenls will differ for the two approaches. I inter far(: RequiremenLs for Approach 1 -- /1) dix numbers wrath describe ARS initial alignment are trans- i = mitred t_ the guidance subsystem. • 7 (2) Six n,Jmbers v_dch describe the desired attitude of the vehicle [_ ww,-, b(,dy axes in the gb'ro frame (g-frame) are transmitted ) f_,,m guidance _o the ARS computer. _ (3'.' "l'_.,' ARS initial alignment procedure will require that the guidance _" i s_Jbsyst(.m: r (3_ 1) Commands vehicle hold using optical sensors with the _ ,.,vhJcir- body axis aligned to the O-frame. i',: _3, 2) lns',ucts the ARS computer that it is in the alignment [nt.,tle_ } 20.113- i,'_ 1i 9650 367-05 43- J hiterface Requirements for Approach 2 -- The guidance subsystem attitude , commands can have any one of the following forms: _ (1) Three Euler angles ! (2) Three sine and three cosine functions of the Eulcr angles l (3) Three sine functions of the Euler angles and three signs of the , corresponding cosine functions j (4) Nine numbers which define the To2 f transformation The ARS initial ahp.nment procedure will require that the guidance subsystem: i (1) Commands vehicle ho?d using optical sensors with the vehicle body axes aligned along the body reference attitude (body axes along o frame) !l "(2) Instructs the ARS that it is in _he alignment mode J CON CLUSIONS -i The computer operations which are shown in Tables 3-2 through 3-6 have i een selected to show the tradeoffs among the four ARS design approaches. • In all tables the assumed attitude command signals from the guidance sub- "1 system were assumed to be Euler angles. The airborne computer operations i.| required to calculate spin vector direction cosines are not included in the tables since these computations are required by all four design approaches1 _[ of the ARS. The computations are divided into high and low speed categories. t '1 I I) 20113-FRI J _,q 4 _=!::, _ I ..... I[ I ............... ,J II ,_ / i 1965011367-052 44- The high-speed computations are those which are done on-board the vehicle and whose rate is c,_msistentwiln the cgntrol system stabilityrcquirements. The low-speed computations do not enter into the real time control of the vehicle, and can be done either ,'n-board the vehicle prior to a maneuver or in a guidance computer which may be located on the ground. If the low.-speed computatioas are done on-board the vehicle, in general only airborne computer memory will increase in order to store the necessary pro- gram. However, if these computations are performed in a ground based com- puter, the telemetry link will be required to accommodate six ARS alignment} numbers from vehicle to ground and six or nine numbers from ground to the _- vehicle. Assume that the low-speed computations are done in the airborne computer.Then by inspection of Tables 3-3 through 3-6 it can be seen that Table 3-5 is {- the best compromise. Table 3-3 is definitely out because of the relatively _ complicated high-speed section of the computer. Table 3-5, ifc'_mpaycd with Tables 3-4 and 3-6, has not ot_!y less high-speed multiplications, but alsoj* i_ requires less high-plus-slow multiplications. $ Therefore, the ARS design approach represented by Table 3-5 has ' 'oee._ selected as the approach for the SDMEG-ARS study. } 2 2 | 20113-FRI i ] 9650] ] 367-053 i - 45- "' ARS SYSTEM EQUATI()NS ) This subsection presents the equations and logic' statements that must b_' mechanized in the computer section of the 12 attitud_' rcf_,rcnc_, sysh,ms. "' This work is a prerequisite for determining siz( _, weight, power consuml)- tion and other parameters for the various systems. DISCUSSION OF SYSTEM EQUATIONS I-or the purposes of this discussion, the data processor hns been arbitrarily divided into two cetegories: the Cosine Computer and the Attitude Error and Body Hate Computer. Flow charts for the two computer sections are shc)wn in Figures 3-7 and 3-8. The flow charts represent a maximum design of the '.[ data processing by including all detail necessary for the highest ac.c:ur'acy ADMEG attitude refe-ence system. Therefore, the data procesaor or corn- / puters for all the systems can be discussed as variations of th(,se compute, vs.) Figure 3-7 indicates calculations that would be performed by the cosine ccm_-i! puter for one gyro. In a complete system, the computer se,-_ion performing this function could be duplicated for each gyro or shar_,d by both gyros. "l'h(, _"!_ choice is dictated by system accuracy requirements and the delay time the. attitude control system can tolerate. In this study it is assumed il is ad('ctu_l(' "}i to time share the computer between the gyros. portional to the direction cosines of tht; gyro spin vectors in th(, vehicle b()cly axes. The eosine compute- receives its inputs from the interface logic thattr! 'i.i couples the gyros and cornpater. The interface logic provides digital signals proportional to the time interval between the time at which the :otor rcfcrc,m,c l line passes under a pickoff and the time at which the rotor cosine pattern passes under the same pickoff. This information is supplied lot two pic.kcfffs on each gyro (referred to here as ' r, artial counts"). Also, the interface logic. provides digital inputs proportional to the time of one revolution of each gy:'c) .j rotor. These inputs will be referred to as "total counts". _' 20113-Ht 1 I 1965011367-054 - 46 - !!,il _ " ] L._ "' E.... _...... -L ° _ -_i_ -- - _ _u i 0 _ Z i i I|! 1_! .;..,l, ,: t, - i _.. 20113-FRI -] _-- ,,.77-:, :::::::-_,-:......."-"._ ....... _---_,-............. -.,.-,--.... _ ......... '-.-..... _,r.LJ.<.l_-_....-:.. J../ [.. I _ 1965011367-055 I + : - 47 - . i - ll _ |+ l " ia _- .no a_ -; - - - _ l I " , _ + o, ,,_ i1 +. . . . . '_ h_ - - % 3" '1 3 I - Lh . - _ . _ - . 3Yl :L --X z2 'l_ 'X r_ s1 2 - - _ = _ 1 " Yl z: x 13' n 'x 11 s1 3Y I"_ '_J + }I ' ° " '+ "+ +' '+ 'I , o = ,, , , + , o ,- -, ,° o + = o .: + , , I + ' + i , + + , , + + .,, +I.- _- _ J , . , + + s, , + + ,< + + ,+ = iJ I I W iT HR ES PE CT TO I i.. | sl l I TI M E /_ _ I + + . - _ + 7_' _ + r - - + _ I + + < _ ,s ,, .@ 1I ! _ _ _ L, . , , • . . . . . + , i I + [! _ i + ,,, , , = + ,° ,,. . ' : + ' i . . . . i + - , , o . c+ + ,,. ,, 'I u ' I _ + u , ,, , - , , , I-- Z. L/ I_ T, o .I , co ., o ,= , , , :, = , , , - , , . , = , ,_ + ,j ' ° , :', , " , ,' . L+ _i . _ _ do: :+ ,, o _ I + i + o + , o ,' - ,- - - . . . . . . . " ;l + [ + " I I ! r _ . ®i - m 'J , , + , _ I I + i_ i i . o o , , , o o o c. _ I, [;"F m ,, ,_ ,+ II ,+ c, , . ,, i; I • ° '" ' _ i + _ , , , , + x, , , , ,e li - _ - I"' ,_ _ ', . , '" , , ' . _ I i! CO M M A N D I. I- = " _ I '2 '" '2 _ . . . . L . . . . . 2 I _ :2 '" I_ ' Y O :_ + + < _ _ J ^ ., ,' _ , , " , 2 " J_ + J G ', , . o ,P ,. ,x , s I N O N- O RT H O O O NA LI TY CO M PE NS AT IO N - - _ Fi gu re 3- 8. A tt itu de Er ro r a n d Bo dy R a te C om pu te _ Fl ow C ha rt _ + + l I 20 11 3- FR 1 i II I I II I II/ . . . . . . . . . . . . . . T _ _l -_ 1965011367-058 49 - The pr. acipal fm ctions of the cosine ,;omputer are: • To com_ute the spin a.xi._direc-ioncosines relativeto ,heP'co _.ctivepickoffsfrom which data are obtained. • ,r_,'omputeT.bo __pinaxiJ oirectioncosine relative the in_.ctivepickoffaxis • 'recompute the spin axis airectioncosLnes re]ativet,)tne vehiclebody axe_ ' The computation of the spin axis directioncos_nes from the outputsof the interfacelogic is definedby the follo_ng equations and indicatedin _-'igure3-7 inblocks (i)through (7). a I = 4P 1 {_)- 2 (1} ¢ _2 = 4 P2 -2 (2) : In Equations (1) and (2), a 1 aud o 2 are the directior: cosines of the gyro spi_ vector relative _o the axes of the number t and 2 pickoffs, respectively. P1 and P2 are _he partial counts from pickoffs 1 and 2 and T is the :otal _:ount for the gyro. ? ? i_ The d;rection cosine of the gyro spin axis relative to the ax:.s of the third _., !_ pickoff is computed from the following equation: _Ii WC 2 a 3 = -D2a 1 - B2_ 2 ... 2ai A2a 2 (3) Ecaation (3) is derived inAppendix A, In Figare 3-7, blocks (f;) through (I ')are a graphical representationof the sotutionof Equation (3)for the third ._ di: ection cosine, _3" 20113-FRI J I I 1965011367-059 - 5O - If the available partial counts a;e for two pi<.koffs o .her than pickoffz 1 and 2, fl,e equations for calculating di.rectior cosines are cf the sam'. form. "[he avaflabie partial counts may be used in Equat:.o:_ (1) and (2) and corre- sp .r _-ng direction cosines will be computed. '_ if the partial count from. pickoff number I Ls not _vatlable and if the dirccuon cosine of the spin axis to the number 1 pic._off a_m must be computed from dir_-ction cosines a 2 and a 3, Equation (3) becom,, s ': "_3_1 = " Daa2 - B3a3 + " a3 2 - IE3a2 "" (4) and in a similar manner, if a 2 must be co'.npu_c : from a I and a 3 _a2 = o D1.a3-_lal ±'_1 .-_12- [El, 3 -A_.ai}2 (5) I _, The coefficients for Equations (3), 14) and i5) will l>.= s_ored and the com_na,_d Ithat seiects the pickoffs to provide the partial coants to Equations (1) and {2) i will also select the proper coefficients from sto,'age and the proper program as represented by Equations (3), (4) or (5). I In Figure 3-7, blocks 20 through _1 show the caicvlation._ neceseary to trans-- I form the spin ax._s direction cosine._ from the r(:ference frame represent_d by the axes of the three pickoffs to the vehJ.cle reference frame. ! Since the gyros are strapped down in the vehicle, _here is a fixed relationship !I ' between the pickoff axes and the vehicle reference frame. This relationship can be expressed mathematically by a 3 x 3 m_trix whose elements can be i i determined Zrom measurements made durin Z ':he installation of the _fros on tj : the vehicle. The transformation of the direction ,aosine_ to body ec_ordinates is the product of this alignment matrix and thaw, ctor whose components are !i _, the direction cos_es relative t,_ the pickoff axes. • I; ,) I 1965011367-060 -51- Tnat is: _sI All ,._A13 s21 = A21 A2Z A2: • a 2 I z s:.i %i %2 %3 "3J 1 where: A S = (_II' s2' s3) " spin vector in body axes IAI3j = alignment matmx .f ATTITUDE ERROR AND BODY RATE COMPUTER ] The attitude error and body rate computer's function is to generate digital signals at the computer's output which are suitably relat_:d :o the vehicle i attitude and body rate so that they may drive the vehicle _._abi_ization ant control system to produce stable vehicle attitude. The basis fo::' the attitude error signals genera.ted by t'ae computer is the cross products of the commanded and observed spin vec,.ors in body co- ordinates. The basis for the body rate signals generated in the computer is the difference between successive observations of the gyro spin vectors. The compu_,alion of the attitude error sigl_,tl from the outputs of the cosine •i computer section and the attitude commands supplied to the ARS is defined J by the following vector equation: f 2011 3-FI:_ 1 -I 1965011367-061 - 52 - Whe "_ : (_ = attitudeerror v.._ctor Q1 & Q2 = co;,.}manded spin _e_ors $! & $2 = obser_,edspin vectors The solution of this equation i s irdicated grap:Really in Figure 3-8 when switches 1 and 2 are in the position malked A. The derivation of *.he attitude error equation is pre_ented late: in tlua sectior,_ ", The computation of tLe body rate signal from the observed positions of -- the. gyro spin vectors in body r_.ference axes is defined by the fo;i_wing ve ctor equation: ISl s2/" where: o_ "_' = body ang_alarvelocityvector ! _1 & $2 -" derivatives of _ & '_2 in body axes Figure 3-7 is a graphic displayof the _olutionof the above equation when ( switches 1 and 2 are in position,_, • The derivation of the body rate equation is presented later in this section. ,, 1 I 20113--F[_ ! "'1 1965011367-062 - 53 - It i_ obvious from the presence of switches 1 and 2 inFigure 3-8 tha: a large portion of the computation necessa_, to determ3ne the attitude error' signaland the body rate.signalis ider4icaL Whether the equipment per- forming this computation willactuIHv¢be shared in time betwe_.:nthe atti- tude error and body rate calculations(as imp;led by the switches)e _ll depend on the system a_curacv and performance n_cessary to the appllcatlom it is anticipated that blocks 36 and 38 "n Figure 3-8 will approximate the, spin vector derivatives by finite differences between observations c,f the spin vectors in bo_ axes. The interval between observations will depend on tne quality requL-ed in the body rate Information and may vary between applications. Blocks 41 through 45 indicate the computations necessary- as a reeult o_" the gyro spD, vectors being non-orthogona!. These computa*_ons wi!] _? omitted for application that do not req,:ire high accuracy, or if the gyros have the facihty for making the spin vectors orthogonaL J J COMPUTATIONS FOIl THE TWELVE SYSTEMS O_ THE PAHAMETFR STUDY ) i Twelve attitude reference systems were defined for the paraineter study of the SDMEG attitude reference s:¢stem. The systems were defined on the basis of three variations of gyro complexity, two variations of system output arm two var,.'ations of system accuracy. In order to discuss the computers co_nposed for these independent parameter _ariations, the following _ymbols are used: A. Gyro Complexity A(1) Gyro definedunder Contract No. 950607 A(2) Gyro with caging 2apa.bility ', A(3) Gyro with torquing capability ,# 201t3-FR] o0 J / [ i 1965011367-063 - 54 - B. System (utputs B(1) Th,:ee-axe_ attitude error arid bod_, r_.e B{2 _ ThJ,ee-axes attitude error C. Sys';.en', Accuracy C(!) High system accuracy 0_ _ a ,_,,_= ratc of !00/sec(C()mputer error less than 0. '° ^*_ on body ra+.e informatiox_., and 1O.,sec on att:,,tud,,_ er_'or in _or marion) C(:_) Lc,w sysLem accuracy (.i •(.omputer error less than ') I ° . a data rate of 10/see for body rate and 1/_,ec on attitude error) The 12 att'itude _reference systems may be indi,,_'azed _,.s folk_ws: ARS Ne. i = A(1) B(1) C(1) ARS No. 2 = A(1) B(1) C(2) ARS No. 3 - A(1) B(2) C(1) i ARS No. 4 = A(1) B(2) C(2) -_ ARS No. 5 = A(2) B(1) C(1) ARS No. 6 = A(2) B(1) C(2) ARS No. 7 "-"A(2) B(2) C(]) {i ARS No. 8 = A(2) B(2) C(2) i, ARS No. 9 -- A(3) B(1) C(1) ARS No. i0 -A(3) B(1) C(2) _ ARS No. Ii = A(_',) B(2) C(1) " !. ARS No. 12 = A(3) B(2) C(2) i_ • _ The follow.xng discussion defines the functions that must be performed '_n the !- cosine computer and attitude error and body rate computer sections of the systems defined above. l These comput,er definitions serve as the starting point for preliminary decign of the ARS computer. The objective of the prehminaz T computer design is to provide size, weight, power consumption, and rel._,tive reliability estinaates for the computer portion of the ARS. 20113-FR 1 I 1965011367-064 - 55 - ;.RSNo. 1 -A(1) B(1) C(i) '{he flo_, chart of the computer fu:_ctions for thi., ARS configuration is shown .n Figure 3-7 and I_-8. Due to accuracy and speed requ',.rements for thi_ configuration the :o . purer design is characterized by the maximum number of functions to 0e solved. This system is the first to be analy,',.ed in tile A RS cc, mputer parametric study. ARS :,._o.2 -A(1) B(_)C(_) For this configuration it is assumed that gyro pickoff non-orthogonality, body axes pickoff misalignrnents and gyro spin axes non-o.,_.hogonality do not contri- bute more than 0.1 degree error at th_ output. .j" The computer flow chart is sho_n in Figures 3-7 and 3-8 with the following modificat b:,ns: • The storage of the 30 "C" constants is eliminated and the operations in blocks 9 through 13 become: a) Eliminate blocks 9_ 11, and 13 since these constants are zero. ] i b) ffhe blocks 10 and 12 are unity multipliers. enmma_e and the operations• The storage of the 18 "A" constants is "' " '*_' ir._blo':ks 20 through 28 become: a) Brooks 20, 27, ar_-d 25 are unity multipliers. i] b) Blocks 23, 26, 21, 24, 22, 28, havo zero constants. 2O: 13-F31 i I 1965011367-065 56- ) ,t With the above definJ._on of the "A f' c.onstant locations the computer cont'_.'ol logic nmst do the following, ga.tir_g of the direction cosines: _'i is gated to bloc_ 20 o"2 is gated to block 27 _3 is gatcd to _.._-'_,"i-:25 .$ In figure 3-8 blocks 41, 42 and z.3 are elimir_ated. Ttte information f:'om bieck 40 to block 47 is tra"_s_:litted to satisfy the follr,'xj:_g idenlity: _. ARS l'lO.3 - A(1) B(2) C(1) The computer flow cha:_t for this ARS confJgurat:on is shown in Figures 3-5 and 3-6 _,,ith the following modific,_'Aon: • In b igure 3-8 only "triode A2 _' _s selected (switch is in the position A':P _! • The block 36 and 38, are ehrr.inated, i,j The required c_mputations are esse;-.L_ally the same as for AHS No. 1, however high speed rate col_.putations are no_ required, / gi ARS No. 4 - _(t) ._{_') (:¢2) I" The computer flow chart for thi_, $.RS configuration is shown in Figures 3-7 and _ 3-8 .: ,t_ the fol[owing modificatioI_s: _, f • lncor_:orate all modlfica:ions as i_ AH£ No, 2. $ Incorporate all modifi,.e.tions as in AI_S No. 3, _L 20113-FR1 1965011367-066 I 57- ARS No,5- A(2)B(_)C(I) The computer flow cha_ for this ARS configuration is shown in Figures 3-7 and 3-8 wi%,h the following modJfication,._: • In Figure 3-8 blocks 4l, 42 and 43 are eliminated, • The information from block 40 to block 47 is transmitted to satisfy the f...,.owmg identity: Y_ = Y'' = "5( ARS _+o.6 -A(,_)B(1)C(2) Due to _ssumpt_ons introduced for ARS No. 2, ARS No. 6 computer configura- ticn is ic_enticalto configurationof the ARS No. 2. ARS No. 7 - A(2) B(2) C(1) The computer flow chart is the same as for A(I), B(2), C(1) configuration with the following modification: • In Figure 3-5 blocks 41, 42, a_d 43 are eliminated. • The informa_ion for block 40 to block 47 is transmitted to sat_'_fy the }i following identity:yv = _,, = y ARS No. 8 - A(2), B(2) C12) II The computer flow chart for this ARS conhguration is iaentical to AI_$ A(I) l i 1965011367-067 - 58- i // ,. , ARS No. _ A(3) B(I) C(1) It appears at this time that this system configuration will requ._re the same r_mputex, capability as ,t,Rg N,o, 5. Additional study of the utilit/of the I [, prer, ision torq_dng capability in the _'ro may _ter this conclusion. 1 t" i RP,S No. 10 A(3) B{1) C121 i Th_ same comments made under ARS No. _ also apply to this system andARS No. 6. AR_ No. 11 A(3) B(21 C(I) This system corJiRuration will use t_he same computer capability as ARS No. 7 pending further study of the utility of the precisio_ gTro torquing. ARS No. 12 A(3) B(2) C121 The same cor_mentsmade.under ARS No. IIalsoapplyto thissystem and ARS No. 8. i / ¢ ¢_ _ _ 2CI13-FRI 1965011367-068 l .! •- 59- PFR FOR MAN (_ ANAL'iS IS INTRODU CTION 'the SDIvLEG-A_5 periormance described in this section contains discussion of the ABS error sours-ca and shows how these, errers affect ARS performance i in the attitude and ":ate modes of the ARS operation. J The discussion and evaluation of the error sources is based primarily on the j.j findings arrived _.t on varlous SDMEG development programs (e. g., JPL Con- tract No. 950607,. Phases A and B). iJ SUM_u'RY IJ ' In the ARS parametric study, the ARS contlgurations were designed onthe basis If of varying: $ Gyro corr.,plexity i l Gyro defined under Contract No. 950607 Gyro with caging capability i Gyro with torquing capability o System outputs i j Three-axis attitude error and body rate Three..axis attitude error only l_ . System accu:_, acy High accuracy Low accuracy For the _RS performance analysis, gyro complexity variation can be disre- garded _ me t_.is variation introduces 0nly _-;econd-order error in the perform- H ante equations. '[1 1965011367-069 - 60 - From the sole point of view cf ARS design, the gyro caging and torquing cap- i ability provides the advantage that the gyro spin axes can be maintained in an o_hogona! atti:ucle. However, the gyros designed under Contract No. 950607 [ ca_u be started up bo that ',he non-orthogunality in the spin axes causes only a second-order error in the performance equations. Therefore, the main ad- vantages of thc gyro torquin_ and caging c_pabihty is Ju ARS operational procedures. The types of system •*_"-oa, ,,._selectedare primarily a reflectionof cemputel sampling :'aterequir_.rr,ents and data filteringrequiremPnts for computing body rate. For the AP3 performance analysis,thisARS variationmeans only that the errors must be estimated for both the attitudeand rate modes of the ARS. Ir the estimate of body rate error, ".heerror reductiondue to data filtersis not inc._uded;however, itis pointedout what portion of the to_R!rate error ca_ be i subject, ed to _iltering. i The levelof system accuracy (highor low) is reflectedinA RS design in that the high-accu_'acy system can compensate for the gyro pickoff axes misalign- : merit and misahgnment of the gyro assembly with the vehicle _ody axes. There- ; fore, for the ARS performance analysis, the misalignment numbers used for tbo i high-accuracy system are those which reflect state-of-the-art of ahgnmer_t J measurements and stability. The low-accuracy system will reflect the s_ate-of| i the-a_ _lignment ofgyro pickoffsxes and r.lignmen_,to the vehiclebody axes. r. ! Consequently, the 12 AIRS configurations are g_ouped_ for the performance esti- l mat_" as fo]]owv:I t • AP_ performance in attitud_ mode -High accuracy Low accuracy _ • ARS performance in rate mode l High accuracy :' _ Low accuracy !. " _)ll3-FR1 1965011367-070 - 61 - Since t} e basic function of the ARS is to align or "point" some axis of the spacec_ af alor T a line in sl, ace0 the attitude performance of the ARS is expres_ ed in term_ of the error that could occur in th_,_ operation. In the _ttitudemode, Table 3-7 gives the vehiclethrust axis alignment uncer- tainty !ollowin[_a maneuver consistingof initialARS alignment, vehicle orients_ionand de!ta-V thrustings. Itis assumed the above maneuver consumes one ho,!r. Table 3-7. ARS Performance in Attitude Mode ! Erl_)r Source Vehicle Pointing Error in Degrees High Accuracy i Low Accuracy I Data Sampl_ng Errors < 0.004 < 0.004 Readout Errors 0.018 0.u16 :! Aiignmen_ E:rors 0.006 0.054 ,. Gyro Drift Errors 0. 028 0. 028 Computer Errors 0. 017 0. 017 RSS 0.036 0.066 T tLie 3-8 gives the error in computed body rate in the rate mode. Note that tl,e distinction between high and low accuracy in the ARS rate output is lost in } the presence of the high noise level on the gyro readout. J (_ Table 3-8. ARS Performance in Rate Mode :i , J_ , ,, : :) E_._zorSource Rate E_ror in,Deg/Sec ;f., High Accuracy ix)WAccuracy I. Gyro','-ndl_eadoutNoise 2.42 2"42fI ( I 2. Gyro Drift Rate O. 036 0. 036 3. All ARS errors not 0.005 0. 017 '-]"( including1 and 2 _.bove RSS 2.42 2.42 1965011367-071 "/ - 62- DISCUSSION OF ERRC'R SOURCES _u,mmary The ST_Ik4_t'--&1_-_-T,_, ._n.T,_R _T']_ fl_om _T')I_TE_ innn_,-f_rt]nn_, a D_ i_nis- ! aiignments and data processing of the gyro readouts. These error sources can i be classifiedintofour groups: • SDMEG errors • Physical misalignments or uncertainfies in alignment measurement_ • Data sampling errors • Computer errors The SDMEG errors include the g_'ro drift mechanizism and the errors in the readout mechanization whose output are the shaped pulses used for gating the i counters in the I/O logic )f the ARS mechanization. These errors have been ! ex,.ensiveiy analyzed and evaluated by test under various SDMEG development programs (e. g., JPL CoJltract No. 950607). This discussion will hlclude ! general descriptiJn and evaluation of the SDMEG errors per the SDMEG de- velopmen+ findings, i The ARS errors due to r lisalignments deal with the relative alignment among pickoff axes, %he two-gyro spin axes non-orthogonahty and the alignment of the gyro assembly with the vehicle body axes. Depending on the configuration, the ARS performance will be affected in two ways by the alignment errors. In the low acc_racy ARS configuration the ARS performance will be limited by the state of the art of physical alignments without the computer previsions to compensate for the misalignment errors. In the high accuracy ARS configura- tion ARS performance will be limited by the measurement uucer_aint_ee of the mis',lignment_, The computer will compensate for the misalignme_ts to a aegree of their measurement uncertainty. _ ' 2,0113-FRI 9650 367-072 - 63- The data sampllng errors resultf:_omfinitesampling rate and time sharing of the equipment used to read the gyros. The magnitude of the data san%pling error is the functionof SDMEG rotor speed, counting rate of the:counters and the vehicle rate. These errors can be _inimized by the computer capability _nd will be equal to zero for the case of no vebic]e motion. The computer errors, by definition in this study, are only the roundoff (word length), and algorithm errors. The system errors which logically corJd be classifed as computer error_ (e. g., pulse shaping, etc. ) are treated as SDMEG errors or data sarnpli_.errors. The oummary of error sour(-es and the corresponding errnr magnitudem are shown in Table 3-9. Detailed discussion of the error sources can be found in the remainder of this section. SDMEG Errors The SDMEG _rrors are attributed to gyro drift and the SDMEG readout errors. Both of these types are discussed below. Gyro Drift -- Drift estimates used in the _D_EG-ARS performance study are based on an _nalysi_ published in Hone_vell Report 1726-FR1 (15 April 1964). The assumed SDMEG parameters are: Size (OD of rotor) 2 inches Maximum electric field 990v/rail (high power) 170 v/mil (low power) Rotor-to-electrodegap 0.0015 inch Average rotor wall thickness 0.055 inch Rotor ma_s 20 gm '_...."_-;"_otor inertia 9( gm-cm 2 Acceleration capability 30 g.(high power) 1 g (low power) Operational rotor speed I00-200 rev/sec 20AI3_FI_I I 1965011367-073 T_bl? 3-9. ARS Error Sources (ARS Error Model is Shown Ln Figure 3-15) i i Sl_InAxle* S_inAxis"" Attitude Rate ( SDkg._G-AR$ Error Sources E. cor in Alignment Error Error B,._V r_Irection Error-s Vector Vectnr H:,t,. Cosine | (RAD) (RAD) (RADI_ee) ,]AI)/._ .... . - i ml _ i' r_ ,, • -_Trorm 1 _ • Axta! Mass Unbsk4tee 6S-MDI iO._00006 o Tor, i°;EDIJ F EI O. 000424 O. G0042 • Mlcgnetic Field Torque _j4J _MI i O.00000-=, 0. 000005 _" Readout Ev,"_rs . • Bpin Axis (Tncert.ainty 6°s0, 000033S 0. 034n "_,_ • Noise Error _N 0, 000141 • Trigger Uncertslnty _T _ c oooolo_ "_ • Pattern Uneerta/nty /'_p O.OOt'_855 • RiSe Tir._ _ 5_RT O.0000034 I .... ,., .......... |Alignment Errors 60 a _ • Pickoff Aliijnments O.0006 _. 000064 i • O:n'o to Vehicle All.merit Pa O.O00B i " O.000084 i ; • Oyro Spin Ax/s Non-ortholfonallty 0. 00244 Data Sarnplinll Errors _,' i i • Due tO Readout ICqustlon < 10.5 _- • Synchronlsation.......,,, Error < 0. 00004 ) .......... {' Com_:ter Errors I 4;c _c• n 000303 0.00t)o_ _ ! i •Cslcu_ted _or ARE sl/pment, vehtol,) orientation an_ delta-V thruetin_ (1O,000 fl/sec at I/_ g) assumln/ I hour for manmsver completi_.,__. _ **Calculsted _or 30 1 ifyro oapabil/ty in o-g epaoe oavironn_nt, { • '2, ] 9650] ] 367-074 65- For thisgyro, 172(;-Y)._Igives the followingone-sigma driftfunctions: w ° A_ialr.-_s_u_ba1_noe(l6_!)driftr_t_ind_g/hr • _le_tric_orqu_(i_l)driftroteindeg/h_ , , + 0.262 ao • Magnetic torque ( 6S M ) drift rate m deg/l_r 1=o. 0.o i where a .= vehicle ecceleration in g's a o = SDMEG acceleration capability m g'_ f :, rotor" speed in rev/sec B = magnetic field in gauss 5S = rate of change of gyro spin ve,:tor in inertial space To evaluate these d_ift functions, some assumptions mus_ re made about the vehicle acce]e_-a_ion profile, the mode of SDMEG operat,ion (high- or low- power mode), the rotor speed, and _,he magnetic field which exi.,'ts at theSDMEG rotor. 20113_FRI i 1965011367-075 - 65- In general, the vehicle and the AHS can be in three different n_ode._ of _peration: • Mode _ -- Cruise: V_alcle accelerations at zero; control jet pulsing to maintair, ; a prescriked limit cyc)e can be neglected SDMEG operating in a low power mode, ao = 1 g _l SDMEC rotor speed at 200 rev/sec, f = 200 1 Magnetic fields at the rotor are due only to residual fields after SDMEG degaussing, hence B = 0.01 :. -_ • Mode 2 -- Midcourse delta-V cor._ecti-ms, consisting of the ARS . alignment mode, vehicle reorient_tion and delta-V thrusting mode: 1 During tile delta-V thrusting the vehicle accel,_ration may be about 1/4 g to impart a delta-V of 70 ft/sec ,,_ SDMEG operating in the high-po_er mode, a ° 30 g's i SDMEG speed at 200 rev/sec, f = 200 • Magnetic field at 0.01 gauss, B = 0.01 • Mode 3 -- Delta-V correction to orbit consisting of the ARS alignment & mode, v_hicle reorientation and delta-V thrusting: . Durt_:lg the delta-V correction the vehicle accelera_ion may be at an _,,erage of 0.5 g's to impart a delta-V of 7,000-t0, 000 R/see | SDMEG operating in the high-power mode, a o _ 30 _'s I;! -i SDMEG speed at 200 :'ev/sec. f -- 200 , Maguetic field at 0.01 g_uss, B = 0.0l :j , ,- 20113-FN_ [ 1965011367-076 - 67 - It can be assumed that Modes 2 and 3 may take up tc, two hours for execution, including the two veorientations and thrust_:.ng. Uherefore, the one-sigma gyro drllt equations for the cruise mode are: F 16_CI = O.O00B deg/hr i = i.4 x 10 .5 tad/see _SM! = 0.00028 deg/hr } = 0.49 x 10 -5 rad/sec and, the 1RSS gyro drift _'ate dur_ng cruise is j V + _.- 16SM = C.00084 deg/hrI _ , -5 = 1.5 X 10 rad/sec During ARS alignment and the _eorientation maneuver gyro drift rates are I- 0 I: 00 4 " -5 42 x 10 rad/sec 16SIvLI= 0.00028 deg/hr -5 iI_ "-:0.49 x ]C rad/sec I! RSS _-" 0. 0242 aeg/hr_5 = 42.2 x 10 rad/sec [l_. The IRSS value of 0.0242 deg/hr for gyro drift occurring during the ARS align- ment and the vehic]e reorientation maneuve_ _ are used as estimated gyro drift I] _n the ARS performance equations. I 20113-FR1 1965011367-077 6-3- During the delta-V thrusting At 1/4 g, gyro drift equations _re 15Sg = O. OOlO deg/hr ="V_ + <0. 2 + _0.000118 = O. 024dcg/hr IL,_L• = 0. 00028 deg/hr r RSS =- 0. 024 deg/hr -_ 42 x 10 .5 rad/sec i_ During d(,l_a-V thrusting at t/2 g, gyro drift equations are _ 1 18i i = 0.0021 deg/hr . | I_i_1--.li'o.o,.,=I=+(o.oo,.,=i'-+{-_.ooo,,;,,.O. 0242deg/hr ! RSS -- 0.0244 deg/hr -- 42.5x 10-5 rad/sec i} T_ Assume that the mzdcourse delt_a-V correction co)_sumes one hour to a,ign the } ARS and to oriel, t for delta-V correction. The time taken for thrustu_ i_ 1 _ 70 = 0,00242 hr =_00_).:= Therefore, the gyro error at the end c_ the delta-V thrusting is (0,0242 x ,) + 0.024 x 0.002_2 = 0 0242 + 0.000058 t = O. 0243 ,_ ,,' .... :: 4.24 x 10 -`4 radi_ns I. i tI= _ 20113-FR1 1965011367-078 69- Assuming one hcur for the ARS alignment and orientation for the delta-V correction for an orbit injection, the gyro ei-ror at the end of delta-V thrusting is 0.0042 x t + 6.0244 :_ -,- 10 t000 , 00 = 0.0242 + 0.0042,. t. = 0.0284 degree = 4.96 x 10 -4 radians The attitude error due to gyro drift of 6. 0284 degree as estimated aoove will be. used in the ARS performance equations. Readout Erro_r.__s-- The SDMEG reedout errors ,_re: • Sp_n axis uncertainty 6._S • Signal noise, rotor noise and counter error 6a N • T rl,gge r unce _-tainty 5a T • Pa,_tern uncertainty 6_p • Rise time 6aRT j. Numerica2 estimates of the readou',, errors for the SDMEG-ARS are based on _,he analysis and development of the SDMEG component in Phases ]A and IB. In Phase IB, one of the major task,=_ included breadboarding of the SDMEG readout logic to be used for the c',,eckout of the readout performance. Spin Axis and P.attern U.n_¢,e_Ftaiiq/Ly_,-- Patterniug the SDMEG rotor in,,olves determination of the preferred inertia axis aboat which the rotor wi!] exhibit a stable spin. After the spin axis i_ determined, the rotor patte -n can be apn]ied relative to thi,_ axlu su as to satisfy a suitab!e mathematical relati._nsbip between the rotor polar coordi- nate angles, The mathematical relationship between the polar _, 20113-FRI 1965011367-079 t_ I - 70 - coordinates of the ro_c,rprovides the means of reading out the spin axis orientate.onrelativeto the gyro case. The er._v.:sattributedto the patterning,r,_,cessw_.!_,res':_frc,,m: • inabilityto specify exactlythe spin ,xislocationin the i rotor (spinaxis uncert_ii_ty) _ • Pattern lint dev_'ation from a ruathematical configuration "_ (_atternuncertainty) -I The corresponding direction cosine errors of the SDMEG readot_t i have been estimated as: 6aS O. 000C325 (i_) - spin _xis uncertainty : _p _ 0.0000855 (I¢_)- pattern uncertpinty The _S and _p errors can be classifiedas s:_tematic errors, meaning thv.they are eithe:,bias or slowly varying errors wRtl the a_gle betw_.enspin axis and a pic_offaxis. Noize Error -- Noise erz_)rs arise from three sources: • Signal noise, described by the sigr_al-_o-r_ise ratio ! of thereadoutJ i • Rotor noise, resultingfrc.mmicroscopic d_via_ions Of the Pattern line I • Counter uncertainty i_i The readout scatter c_used by these errors will occur at the frequ_.ucy_ o_ the computer sampling ra_e; therefore, these errors Will toner btlt,_ to the lmcertalnty in, the derived body rate signal. :_ J 9650 367-080 - 71- i During the Phase IB, all three noise erro,.s were experimentally determined, and these findings are being used in the SDMEG-At_,S i performance analysis. The one-sigma values of these errors a,-e: ® Direction cosine error due to signal noise = 0, 0000917 (lo) • Direction cosine error due to rotor noise - 0.000033 (le) The counter erzx)r is attributed only to uncertainty of the last count. q:kerefore, the contents of t_ie counter will be in error by a maximum of one count. However, if a uniform distribution for this error is assumed, it can be shown that the one-sigma of the uniform distribution will be ,--11.-of one count unit. v; 3 The counter error of _ o¢. one count unit can be converted to the corresponding error ':'i_3tl,,ereadout direction cosine as fol.tows: The SDMEG readout mechanization equation is: r a - 4.,_- 2 (see system equation_) where -" readout direction cosine P = partial count T = total coun_ From the above, the error in a due to errors in P and T iu: (_)_ = !_4,2 l..i.!2 /.p..2 IT, (6P) 2 + - , TJ lTi if a 5-mc counter and 200 rev/sec for the rotor speed is a_sumed, then 5_ .1,06 T = --------200 __ 2.5 x 104 c0unts/rev ] ,, 20113-FRI 1965011367-081 - 72 - Due to _tor pattc__.,geometry 0.31 __ P/T -_ 0.69 Therefore, let (P/T) = 0.50, I_ 6P = 6T, then (6a)2 = (4/T)2 (6P)2 1.25 or 6a = (4/T) 5P (1.12) = "_.54x .1..04_v_-l- (1.12) = 1.03 x 10 -4 Therefore, the RSS {$_N ) of the *.hree noise errors is de,N = V(0.917)2_*+ (0.33) 2 _'(I.03) 2 x 10 -4 = 1.41x t---.= 10 -4 Since the directioncosine;rate is computed inthe SDMEG-ARS by forming directioncosine differencesbetween directioncosine values • t on success:_ve computer cycles, j dent and have normal distributions,then )!t -%', .r _¢/. . _ , - 2011$-IP1RI ,) 1965011367-082 .................. . ............... w ...................... l ! i - 73 - I 1 i Therefore, ff- ,i (6aN} K = SaN.._T-_2 ., 1.4ix0.0110-4"*f2 " O. 620rad/sec 1 14 deg/secI I ! il l'i, whpre At = time interval for one computer cycle. I _', If 163NI is the error in the magnitude of the gyro spin axis rate due! to re_.dout noise iit then = 0.°2. = 1. 975 deg/secl I _Errorin Counter Triggering logic -- The error which will introduce an uncertaintywhen the counters are started and stopped can be ai'_ri-. I buted to the following sources: I • Reflectancevariation • Rise time • Triggering unce_amty • II?umination varia'.ion and field-of-view irregularity N In general, the readout electronic_ whicti pro-Jlde the siop l_d _rt I,I pulses to the c_mnters accomplish lhe fuilct$ons of ptflSe discrlmlr_tlo_, pu]_ e shaping en_ gating of the train of pulses int_ the stop a_ _art _ pulses for the cou_ers. Conseque_ly, the particular rosin circuitI _ • I , s_,ctions are: the pulse-width discrimin_itor (PWD); het_A"tr_flive trigger (HIT)_ and start-stop gating. fl 2¢I I S..'lr_lll 1965011367-083 !I t - 74- it Consider a general pulse (Figure 3-3) occurring at the preamplifier ! output of a pickoffo ._' i_ PICKOFF i_ PRE-AMPUFIER GUTPUT TIME lr : Figure 3-9. Pulse Shape from a Pickoff Preamplifier : f ; In Figure 3-9 five different PUlSe shapes are shown which differ from I one another either in height, slop (rise time), pulse _,idth, or irregular ! slope of the pulse. These pulse shapes are attribt_ted to reflectance i variation and changes in pickoff lamp output, response time (rise _ime) i of the pickoff, illumination variation, and fiel-i-of-v_e,v irregularity. i The fact that the rotor surface has arc marks may- produce pulses. These pulse_, _._ever, will exhibit much smaller pulse widths than the pulses due. to Pattern lines. :1 Since any pulse s_Lapein Figure 3-9 may occur atthe gyro readout,the readout electronics must interpret the pulses as to minimize the time errors when commanding the counters to stop and _art. / The pulse-width _iscriminator (PWD) circuit compares the widths of allpulses to a fixedpercentage of the gyro rotor period. Ifthe input ,/ _i pulse is narrower than the fixed percentage of rotor period, the PWD •'J haSelin!inated..nooutput. Thus, the noise pulses due to rotor arc marks are 1965011367-084 - 75 - Since the electronic trigger circuits are voltage-sensitive, a HIT circuit is used which enables the trigger device to be sensitive to a fixed percentage of pulse height rather than to a fixed voltage. The test results of this circult indicate a large degree of error compensation due to reflection and trigger level variation. With the electrcnics as described, the readout/errcrs were evaluated by controlled experiments to evaluate: • Trigger uncertainty (6_) _ • Reflecta_3ce and rise time variation (6_RT) I Illuminatien and field-of-view irregularity (_'T) The one-sigma error values in the direction cosines were determined to be: 5cz_ = 0.0000034 6a_ = 0.0000100 5_RT = 0.0000034 :i The refore, ! r q(0•= . 34)2 "}"(0.I)2 O.0000105i 6oT I0-4 x"( and 6aRT = 0.0000034 t _1 Alignment Errors Pickoff Ali_nment_ -- The routine physieal alignment of the pickoff axes will °1 result in a misalignment of 2 arc minutes per alignment. The misalignments can be measured to an accuracy of 10 seconds of arc. + : 20113-FRI i 1965011367-085 - 76 - :_i _. The aligned pi_.koff assembly _,as subjected to an environmental test, Follow- i) ' Jug the exposure to _he shock, vibr-_tion and temperature environments the i J pickoff alignment_ remained constant within 3 arc seconds. • The environmental tesL. was as follows" • Temperature cycling 140 to 170 ° for 3 hours Shock +200 g's with rise time of 1.5 milliseconds !_ : Vibration i White noise, 28 g-rms, 15-2000 cps, for i8 seconds :, White noise, 10 g-rms, 15-2000 cps plu_ 4 g--rms sinusoid between J._i 15 and 40 cps ": White noise, 10 g-rms, 15-2000 cps plus 18 g-rms sinusoid bet_veen l_l 40-2000 cps. The frequency of the sinusoid is swept from 40 to 2000 and back to 15 cps in a *,otal time of ¢_ f_10 minutes at a rate proportional to frequency. Each gyro has three pickoff axes arranged in an orthogonal orientation with I_' respect to each other. However: due to the above mentioned n_isalignments t. they will differ from the o_thogonal orientation° If 60 is the non-orthogonalit3" between _ny _wo axes, then |" 60 - /ii20) 2 * (8) 20ar -seood a = 6 x 10 -4 radians _. ) /: The _neasurement uncertainty in the 60 a is !- _'_'-_-_(8) 2 = 12.8 arc-seconds I' -- _ 4 x 10 5 radians 4 i?:1 2ol 1_-F_1 1965011367-086 • l ,i _-_ - 77 - _',, Therefore, in the ARS Computer d_:signswhich have provisions for mis- _ alignment compensation, 6.4 x 10-5 rad pickoffmisalignment willbe used i'i in the ARS performance estimate. In the computer which has no misalig_.-menl compensation the misalignment is 6 x 10-4 rad. i_I Gyro Alignment to Vehicle Axes -- Assume thatitis desired to alignthe three orthogonal pickoffaxes with the vehicleboclyaxes. The misalignment _ between the two sets of axes canbe defined three Euler rotations throughby small misalignment angles which can be e_pressed in a matrix equation as ] follows: 1 P3 -P2 |I -p3 I pl P2 -P I 1[i !1 The 0i, 02, and p3 are the small Ruler rotations. ( _] For the error analysis it will be assumed that Pl = P2 = 93 = Pa' Further-more, it is assumed that the physical alignment can be accompl'mhed to Pa = 6 x 10 .4 rad. and this misalignment can be measured to an accuracy of _] 6.4 10-5 rad. x i] Gyro Spin Axis Non-Orthogonality -- An all-attitude ARS requires that _ts two attitude sensor-sensitive azes subtend an angle which is approximately equal II' to 90 degrees. One or two cc,-linear vectors can define only two d,;grees ofattitude freedom. Reasonable deviations from orthogcnality can be accounted ._ for in the ARS reachanization. However, since some ARS desig_'_s in the pre~_ent study do not have _he provision_ to comper:sate for the gy_.o spin, _is non- or_.hogonality, errors ,rill result. i i 1965011367-087 _" 78,. During the SDMEG development program (JPL C_ntract No. 950607) spin- vector caging repeatability was determined to be w_.*hin 0.10 degrees. The results of this tes_ can be fouled in _he Honeywell final report {1726-FR1 April 15, 1964} to Lhe JPL. Therefore, for error analysis purpose, it will be assumed tlmt _he gyro spin ._xis non-orthogonality will be within 0.14 degrees. _Data Sampling Errors The data samp!ing errors result from finite sampling rate and time sharing o_ the equipment used to read the g-vros. The gyfo readout concept uses the relative motion of _.he gyro rotor az_d the case, and from this information - deduces the spin vector attitude with respect to gyro case. Using this read- out concept the readout will be in error due to vehicle motion. There'ore, _he following discussion, i,'_ presented to describe the errors due to v_hicle motion and equipment time sh_rf_ - i_ terms of the chosen SDMEG readou'_ mechanization. SDMEG Readout Mechanization -- Figure 3-10 illustrates the rotor pattern geometry involved in reaAout of the spin vector direction cosines. The ',ocus of poin*.s occupied by the rotor pattern is given by: cos 8 = 2(1 - -_) (6) where 8 and $ are the angular coordinates of a ._Lnt on the curved pattern. 20113-FR1 i 1965011367-088 - 79- Figure 3-10. Modified Cosine Pattern Figure 3-11 illustrates what might be a typical traversal over the rotor surface by one of ",.he pickoffs due to vehicle angular motion. In terms of the angles _1 and _2' which are rneasured in the constant latitude plane and are shown in Figure ;3-1 i, the readout equation is: 4_ ! 2{_ 1 - _2 ) 2(2v - 2¢_2) ¢2 cos 0 - 2 = : : 2(1 - _-) (7) q)l+_)2 4)1 _22_ The readc:ut logic deducee the angles ¢1 and _2" tt_.gh speed counters, if g_ted by the pulses emitted from the pickoffs, will accumulate counts which will be proportional zo the angles _! and _2" Figure 3-12 illustrates a sequence of pulses emitted and what the counters _ measu_-e, The illus'tration is for a four-counter I/O in_.erface logic and _ typic:, for one pickoff readout. _ e _t 20113-FR1 i_. f 965011367-089 - ,_0 - _ SPIN AXIS ._ _,_ _>LOCUSOF A PICKOFF •_ _ _ _ * '--t - OVEROEIEROTORREVOLUTION /""/ "_ "'_' /" (LOCUSIS WAVY [;dE TO/// /! .......,__,- L, _,qBITRARYVEHICLE MOTION) /Z L/!, _Y'", - , [ Figure 3-11. Locus of Picko£fLTnderArbitrary Vehicle Mot._on i, VOLTAGE , ,. 1__._ ._ ) OLITPII; I J ' " I ! In T -• I t''_ i it'''z _t""a I I I I I cou,,TS_ _!c2COUNTS.EREI_Lc2 f ", COUI_ITEEC2 -|- HI'RE _ r- _ ! I I , I f ' I I czcoum'rs} FzcouN'r_ _",) PULSE DUETO COSINE PATTER/JLINE _,,_ PULSE DUETO LOGICLPtJE I_ PUt _,EDUETO REFERENCEi.INIE Figure 5-12,. Counter Timing 2GI13-FRI 2' t ' 1965011367-090 - 81 -. The counter' C 1 value is: Cl(tn) f(tn " tn-1 ) (8) Ci(tn+ 2) = f(tn+ 2 - tn._l) The counter C 2 value is: t ' = f(tn." -t n 2 )C2( n-1J 1 - (9) C2(tn+ 1) f(tn+ I - tn) where f is a fixed count frequency in pulses per second. Hence, the value of the cos O at the dlscrete _ime tn of cosine line crossing is: 4CI(tn) 4_l(tn) co_ e(t n) =- -2 ....... 2 C.(t )+ " t ) 4,l+(h 2I n C2( n, I 4f(tn - tn--1 )r : -2 (to) J f(t n- t .1 ) + f(tn_t " tn) 4(tn - tn-] } (tn+ 1 - tn-. 1) Note that if there is no vehicle motion the cos e value as compute_ from _he _i counts in Equation 10 is exact. However, if the vehicle has all arbitrary motion the reaoout will, in general, be in error, i 20113 -._IU. 1965011367-091 i - 82 - C A possible wavy locus oi a pickoff due tc an arbitrary vehicle motiozz is iPustra_.ed in Figu re _-t 1. The above discussJ on (-_ the readout equation may have implied thatt_,elocus _.ustclose _d rema,n at a cocmtantlatitude t _:hi[ethe d_.tais accumulating t_ be used ._.nthe readout equazion. However, upon ft'.rther analysis of the problem it can be deduced that the non-closing -_:_-d_on-constant latitude locii do not necessarily decrement the ._ validity of the readout equation, i The wavy locus re,_ultsfrom the vehicleangul_r :'etes and acceleration_ about the gyro spin axis and about directionswhict,are perpendicular to the gyro spin axis. R wiU firstbe shown thatany a._gularrate (lesst_n 13 rev/sec) or accel- erationabout any axis of rotation which is pezi_endiculRr to *.he gyro spin f a_is does not introducean error in the readout e luation. Itwillthen be. shown hc_vthe angular accelera_ons about the s7_inaxis resultin the read- out ez r._r. The constant rate about the spin _ j does not introduce any ._ -_rrorsin the readout. ! Although the vehiclemotion about an axi_ perp,zndicularto the spin axis does nat introducean error in the readout equationan error may resu1_due to insufficientlyaccurate computer mechanizatic._. Since three pickoffs"read" the spi. vcctor directioncosines at different._mes, unless thisis pr_-perly interpreted,thiscan result._ndynamic err,_rsin the de_er_ir__tionof the direciioncosine matrix. Because of this,_e t:ming of the directioncosine readout rnustbe zynchronized agahtstthe c')mputer corr_pu_tionalthne base. In the,present computer des!g_ for '_heSDI_L_G-ARS the synchron£zationis neglected; therefor-e, the ARS wilt sit*: _r ¢rcm tb_ _yu-_l_ro_ization error az_.oflg _ the six readouzs of the two gyros, l!_. ,:,.'_ -, i_ can be shown that for a real- _istic vebicle motion these errors are ce-_.'._l.tersbly s_atier than the:other ARS err:)rs; theZ-efore, lack of computer capabili_/to account for the hynehroni'za,tion errors is justified_ _ . - _ ZOl l _FRI 1965011367-092 l t -- 83 - -_!otion Normal t_o Spin Axis -° In interpreting whatError From Vehicle happens to the gyro readout when the vehicle motion is perpendicular gyr_ Q spin aXl,_refer to Figure 3-!_, below. SP_NAXIS f\_.OSINEPATTERN 3 o g I i[i " ,I Ii! • ||I U REFERENCE LII,JI ...J, [? Figure 3-13. Pickoff Locus Under Vehicle Mo_iox; First asstune that a counter is counting from a pulse due to refero.nce line to a pulse due to reference Line when it passes under the pickoff again. Thec ntents of the count will be the total count (T). Since the referenPe line is the Gre3t Circle which contains the gyro s,_in axis it is obvious that the _' pickoff motion (vehicle motion) perpendicular to the gyro spin axis w£11 not affect _e _otal count T. n U Investigste paths AD. BD, and CD and conclude that it takes the same amount> _ of time for the pickoff to traverse these distances _)rovided, of course, thatthe _as_ rate does, not ctmngs P.bou+_the spin axis. 1;t I 1965011367-093 - 84 - Fhe cos 0D for no vehiclemoron is: _ 4CAD -_ _, ccs 0D - 2 CAD count on path AD If the vehicle is moving a_Id pickoff goes through poLnl I_ Lhen the ..direction cosine of the spin _.xis is c_s 8D at the time when piekoff passes through D. If the pickoff paths were BD or CD, then the directl.on cosine at the point D is: ? 4CAD 4CBD 4CcD cos OD - -2 = -2 - 2 t T T T Since = CAD = CBD = CCD Therefore, using the proposed readout equa_on and synchronizing the readout •_ with the time when the pickoff crosses the cosine pa_ern the direction cosine can be deter_nined exactly under the vehicle motion if it is perpendicular to _, the gyro spin axis. L Error From Vehicle Motion About the Spin Axis -- The _gyro readout is i based on the relative motion of the gyro case and rotor about the gyro spin.. axis, If the relative rate is a constant, t.here will be no error due t the readout equation. AngULar accelerations in the relative rate will result in readout error. However, due to relatively small angular accelerations of the vehicle these errors can be neglected. The order of magnitt-de for the readout error due _o acceleration can be obtained as follows:451 (tn ) C O(tn) = -=---------2 (s_e Equation I0) 21" } : 20113-FR! ....... :_-___a:_: ._.._._ ......._ ._,_..,,_,,___._,_'...... -- '_- ..... r - t, m_m i ] 9650] ] 367-094 il 85 - !J _ the_,for,_,,ialo_cos_to)i_. w Iftheaveragevehicle_c-celerationduringthetime intervaltn - in_I is '_ 'JV(tn)_hen: _.} ,-', W(tn) itn - tn_i]2!.1 _l(t_,) -- 2 L-. Then theerror inreadoutis: assume that £V(tn) = 1 RAD/scc 2 1 , (tn" tn-t) = 20-'_see (200RPS rotorspeed) ii' Then the error in the r'eadout equation due to the vehicle acceleration of 1 RADIsec 2 .is: A osO(tn) j -- (1) i =, 2_ x 10-5 _r 5 n n I) 2o_la-F._l ,. ,. _Z%_,_ ! i 1965011367-095 - 86- Synchroni_,ation Error -- The previous section described the readout mech- -' ,_nism and it was shown that vehicle angular motion does not appreciably affect the readout accuracy. However, the three pickoffs in general "read" the spin w_.ctor direction cosines at different times and, _uless this is properly interpreted, can resu1,: in dynamic error. In the present computer design for the SDMEG-ARS the readou_ synchronization is neglected for the I, 2, 3, 4, and 6 counter mechanizations. The wor._tsynchronizationerror wil!resultifthe I/O logicis using only one cotmter0 and it can be shown that the error for this mechanization is on the order of 0. 00075 deg. if the vehicle limit cycle i_ate is at 0.1 deg/sec. The one counter timing is illustratedin Figure 3-14. t ? NO. 1 NO. 2 I I I _ n I ,I-], 111-1 I_n I',FI I__rl l ln !OUTPUT ,_ 'n** 'rn ".++ ;rr.*z 'rn+3 r_.4 "n+_ Tn,-4 'n.+.;' "n+s 'rn+t 'in+m,'rn*. T,.,+,_ .--J-'L START PULSE +- ,I ST+,+PULSE ,+ i Figure 3-14. One Counter Timing _, ?0113-FRt I 1965011367-096 - 87 - J From Figure 3-14 the readout equatiop_ may be written as: _L 4Pi(t n) !J a l(tn) - 2 T(Tn+ 5) "1 i_ 4P2(_n+2 ) a 2(_.n+2 ) = -2 {.l T(Tn+5) The errors in partial (P) m_ld total count (T) due to vehicle motion during time intervals t n - t_l; _n+2 - _n+l; and _n+5 - _n+3 can be neglected as de*.ermined i_ in the previous, sections. {1 Assuming rates of 0. I deE/sec, and 200 rev/sec, for the vehicle and the rotor,3 t_ respectively,then al(tn) is out of synchronism with a2(_n+2)by ,0.,)%,.,°, 0.000 o,,eo. Therefore, the error due to lack of synchronization while reP.ding one gyro i_ is O.00075 degrees. :.J Similarly, ifboth gyros read with one counter the firstreadout on the.first I_' gyro may be out of synchronism with lastreadout on the second gyro by: i}. (0. 00075)x (3) = 0. 00225 degrees I_ The 0.0025 degrees error includes the reaoout logic "waiting period" while] switchingfrom firstto second gyro. {I {} fl t -" i 1965011367-097 88- i 4 Computer Errors By definition, the ct, rnputer errors arise only frc.m the limited word length and tke type of algorithm selected m the computer de,_zign. The computers are being r'esigned for 0.01-degree accuracy per one direction co._ine computation. Ti:erefore, the error m a gyro 3pln ,rector attitude, [ 6Sc[, due to c:)raputer errors is 0.01 -z, " _.= :aS c : _3 _57_3 := 3.02 x it raaians. The body rate is computed to ti,e resoluAc:, of 0. 002 deg/sec about each vehicle a._:is; therefore, the tctal rate error, i 6_ c ' due to computer elrors is i i 6_ c I = q _ 0.002 = 0.0046 deg/s_e " = 0.8 x 1C-4 radlsec : M A'FH EMAT ICA L CONSIDERA TIONS Assumptions Since the SDMEG-ARS outputs are the vehicle attitude and rate. its perfor- '_ manta will be specified in terms of the uncer_.ainties of the attitude and rate outputs. 7 If 50' and 5_ are the attitude and body rate uncertainties then, i_ general, ti_e i SDMEG performance equations are: _i _ : fOl_V¢2 '''¢n ) i _N : Ywlel, ¢2''" ¢_? : ! where el' ¢2"" eu are the errors as l!sted k 20113-FHI ,t - ---; ........ ' ............... ,"'"_'_w_"_';?"....... , ....... .'_'_ "_"_" _,._',_,,:..... %_" ,_,_' ,; ___ece _:_';-' ] 9650] ] 367-098 - 39 - J J The vector functions_:8 and'ft_are extremely non-linear _nd demanding in theirderivationand use. Therefore, simplifyingassumptic,_Ismust be intro- duc,_o. I assumptions thatw._.llsimplifythe mathematical detail A listof reason_.ble and still result in a good estimate of the ARS performance is: I 1. Theterms which contain e of second or higher order can be neglected. 2. The system errors can be entered into the error equation as shown I graphically in Figure 3-I5. I 3. The influence coefficients (or error gain coefficients) in the errorequal:ion for the cosine computer can be evaluated for the condition that the gyro pickoff axes are _rthogonal and aligned to the vehicle I body axes. I 4_ The influence coefficien'_s in the error equation for the attitude and body rate computer can be evaluated for the condition that the gyro I spin axes are orthogonal. Item 1 is the commonly used assumptton in the error analysis; therefore, it does not require special explanation. Nominally, the gyro pickoff axes are orthogonal and aligned to vehicle a_es; they depart from the condition only ,flueto misalignmen% errors. T erefore, the error equati ns can be expand¢'d ;_nd linearized about this condition. ! Similar reasoning holds for the justificatlm of assumption 4. Assumption 2 deals with defining the SDMEG-ARS error model. "[his model is illustrated graphically in Figure 3-15. In the model, the blocks numbered1 ¢h_'ough 7 represent the same data processing step_ as explained in the system sectioncJfthisreport. The extraneous inputs(e.g. readout errors, alignmenterrors, et:)represent allof the ARS errors an they were defined in the section i 1965011367-099 %, -91- on the error sources. The summing points ((3)) in the diagram a.,'e interpreted _s the root-sum-square (RSS) operations. Specifically, assumption 2 deals with "where" in the system eauations a oarti- cular error is entered. Due to linearization of the error equations (assumption 1) the error terms, t with proper gain factors., can be entered at any desired point _f the sy_Ltem; thus, ass,,mptton 2 is justified. t Error Sens_t%vity Under assumptions o_. this aection it was pointed out that the er:-or analysis will establish the ARS performance by the estimate of the attitl_de, 60, a".d body rate, 5_, =ncertamties. ! The meaning ,-:.ndderivation of the attitude uncertatity, 60, i_ not obvious without I further clarification. l First, the attitude error will have different mean_.ngs, depending _n whether theARS is used in the closed vehicle attitude control loop or used in an open loop as l an attitudemonitor-. Second, if the ARS is used in the ciosed attitude control loot,, not aii of the ARS errors should be u_ed in the AHS performance estimate. If the A_S is u_d as an attttud_ mc.,ritor, its performance will be specified by deter:_lining the magnitude of 5S (see Figure 3-15). In the case of closed loop operation the spin vector e,._or, b_, is only a partLal answer since, in thiscase, the error in the vehicle attitude _s of prime i_.nportance and xrill appear l! as an error in _(seeFig,,re3-15}. [i i 1965011367-101 92 "' To estimete the errc, r in _ cne must differentiate uc_.ween tbe ARS errors occurring in the forwm_d and feedback portions of the ioop, since they have different effects on the loop performance. The appropriate block diagrara showing the dosed locp and the error entries appears in Figure 3-16. In Figure 3 _i6 the two modes of the ARS ope_:'ation are shown by switch positions A (e_titude mc.te) and W (_ ate mode). The SDMEG-AR_ includes block,_ numbered {1), (2), (4), (5), and (6). The block '3) contains the autn- pilot and lhe vehicle which complete the attitude control loop. In the attitude control loop (ARS in attitude mode) the autopilot will drive the errors (AS 1 and AS"2) to zero independen',ly of the non-bias type errors in ',he ARS computer block (2}. The block {2} serves the purpose _f making a trans- formation of t.he attitude signals to torque =ommand signals about the three vehicle control axes. "£his transforn_.ation can bc regarded as a gaiu factor although of extremely non-linear nature. Therefore, since the transformation : is essentially a gain factor, the errors in this block c_n be neglected in the system performa:,ee analysis _,s an attitude reference. However, the anal-ysis of the system's performance a.s an a_titude rate senso:- rr ust consider the errors occurring in block (2). The block 2 is m:L_ected to two types of errors: gyro non-orthogonality; trans- ^ formatim_ error' due to error i_l S I and S 2 (5S-1 and. 6S2). These errors do not contribute to the error in 0 ; therefore, when deriving the 5_-error cax e must be taken to avoid including these errors. ARS PERFORMANCE IN ATTITUDE MODE So far, it has been established that vehicle attitude e_'roc {6_) will be a function of the readout, alignment, gyro drift and data sampling errors, and the errors introduced in block 2 of Figure 3-16 do not contr._bute to the error in e. Next, ; it will be shown how to arrive at an e_.press.:on _hich relates the _bove errors [ "" ; 20113-FRI i } i i 1965011367-102 93- / >- iF} F3 o I_,!_i" 4 __ t "",i = I '_! _ M E ''i _, },L,Z] kL.! o _ _,., 01 _ ,r 6 _' . -'. '-' 14 . m,.. w "-j ,ii. q,m .ms,, ., dl 71 ''_ I _ I L-..---_I _ ! '_ _ i_,_.r_ iu'_ • I _' I "" I I wt_,- ' _' i _ _-w + • I _ I I UJI_'_ I ,.u_ , r_iI IP' OOI I _i • . _ ,_ _._, _ , . _,_r--.... ._ + _ t I ( 20LI3-PT_I 1965011367-103 ; 94- : It can be seen in. :?igure 3-16 that the re_dout, alignment, gyro drift and data ;' sampling errors make up the error in gyL-o spin ,axis attitude 16S). Therefore, _ o ; the desired error equation must establish th o relationship between 6S 1, 6S 2 and 60 This relationship can be derived as follows: i i In the section defining the ARS syste_l equations it was shown that for the orthogonal gyro spin vectors (S ! S 2 = 0) • A __ A A = _zxs I - s:l_s2- s zxs 2) }i 1 Differentiate the equation foz 3 and obtain A _i 60 = ML_.SI) x S 1 S1 (S'AS2) _1 '2 ) q ' + 5._1 x 6S1 - 6S 1 (_2 " $1 x $2 ) ,o ' - s_ (A,_• g s2 _; 2 6 1 x + 2 _1 x 6S 2 ) The first two terms on the right side of the equation represent the error in 60 due to error_ in 1_ 1 and AS2. The remaining terms, enclc_ed in the _equence ; bracket, represent the error in 60 due to errers in the transformation {error in block 2 in Figure 3-t6) and must br reglected for the abowe reasons. Therelore, the error in 5_ is •_ 6_ = _(_z)xs 1 - sz (_(,,-s_)• sz xso) [ .} ,1 :_ 2OlI3-FR1 :i f ,| 1965011367-104 95 " _ ll'LCe _'j( _ 651 6(_)---6& 6o -- 6_1 x- (_2 x g The magnitude of 60 !s by " "Stnce definition the S, and S 2 _re unit vectors, 5-,_ S 1 = 0 and 6So,, S2 = 0[i Then it follows that the sine and cosine _-alues in the above equations are 216 fit)i sin SI' = 1 0 _ cos_16g2,sl x ;21-_1 Let the argument of the cosine be equal to _; then the expected value of the _-t cosine is _r fj E(cos21., = l f 2cos r d_ = 1/2 o l There_ore, the magnitude of the vehicle attitude error is Then l] .. . --" ' _..._:.'..'_lmr_...'_ _---,,,p."_e-_..__' _ .... : ,,.;1, ' 1965011367-105 , ;0 - 96 - The equationfor the vehiclepointingerror is deri_-edusing the following mathematical definitions. Let _-c_be an ex'ror vector about w_ch the vehicle is rotated from the commanJed ¢ttitade through an angle 150-_ The 5_ ve,:tor is a function of the ARS error. Now, if X, _', Z are the unit vectors alon&" the vehicle roll, pitch, and yaw _xes, then the m_glllar errors about the roll, pitch, c,nd yaw 2xes are 5_.X. 50" Y, and 5_' Z, respectively Since the X, .v,Z is an crthogonal triad i ¢6-_._2+(6_./f)2.¢6_ 2)2 _-18612 ; And the expected angular error about any one of the three vehicteaxes is (see Figure 3-17). attitudeerror/vehicle axis = r3 i tN,.j ) _9o f' !_t t _C----/_AN_D ATYITUK POlk LI_ P_ITING ._XgS Figure 3-17. _'ehiclePointing Error Geometry -20113-FR', 1965011367-106 II ° 97 !_. Therefore, the vehicle pointing error is n The equations which relate the I _S ! erro:: to the system errors are gkvenI below. The derivations of *_,e_e equations will be _hown later in this report. ,_ 5P 2 = 6P i i = 1 or 2 (pickof£ number) (11) / • i_! 6,, - o..s8(6.2 . _._). _._o6ea2 ¢_5_ kJ _ z " 1_" (t"O I) 18s12 -- t5_'12 * '5"DMI + 16sD' * 15SDMI2"+ 16_ci_ q I 20113-FR1 1965011367-107 - o 8 - The s_ulmneous solution of Equatioas Ill) through (17) yields: !i Substitution of the error term values from Table 3-_ yields: 'i .J + (1.41) 2 + (0.034) 2} x 10 -8 +(3.05>2_lo-8+ 2_a+ o.5_0_ _] For high-accuracy ARS design Pa - 0 = (0.000064)2 = 0.41 X I0"8 radians_ '_ Therefore, !,5S I for high-accuracy ARS is t._ °_ I_Nt =-/38.72 + _..s x oo4! x lO"4 If. = 6.3 x lJ "4 radians i = 0.0360 degree !_ and i" J8,__ " O.036 de_ree = vehiclepointingerror for hlgh-accuracy ARS - |, 20113- FRI ,.I ! 1965011367-108 sI - 99 - For low-accuracy ARS i Therefore, the vehiclepointingerror*for the low-accuracy ARS is: 15SI = -_/36.72 + (36) 2.5 x 10-4 i = 11.35 x 10-4 radian_ 1 = 0.066 degree I 3 • ! ,i J J i !I ._" _I 20113-FRI 1965011367-109 - 100 - ARS PERFORMANCE IN THE RATE MODE In the ADMEG-ARS the body rate is computed by two operations (see Figure 3-15): • Differentiation of six direction cosines (block 7 in Figure 3-15). • Transformation of direction cosine rates to vehicle body rates rcsclved along the body axes (block 6 in Figure 3-15). The quality of the six signals, which represent the six direction cosine_, and i the errors in the above two operations, 'affect the quality of the computed ' body rate. i The directioncosine rate error resultsfrom the gyro readout noise, differ- ":: entiatlon process and gyro drift. " The transformation error resultsfrom the factthattt',_gyro spin axes atti- _- lude._ are not absolutely known. Therefore, the ._omputed body rate coro- t po:mn_s may be along some other frame than f . true body axes. ¢, i The body ra_e error equation is derived lat,_ _ this ,'eport and is given here L in vector form as follows: i ", = S 1 5S 1 - co • 582 • SI + x • _, 5"_1 5S1 x + _ " S! ' 1 _, 2 where ! . _ 5_I ---body rate error vector (does no_ include computer error) f -8S 1 = error- in _1 vector _,_ S 1 = first gyro true spin axis attitude t_= body rate vector 20113-FI_I -'_ ! 1965011367-110 - I01 - l e_ 6S"1 = error in S 1 vector) S 2 = second gyro true spin axis attitude ^ ._ 5S2 --error in82 ! " _2' I,S2 = error _,n vector i By examinat_.on of the expression for 5_ l it can be deduced that the vectors ; _i x SI" 5_I: and S1 are an orthogonal set;therefore, the magnitude of the i 5_ 1 error vector can be written as follows: I In order to evaluatethe above ecuationcertain assumptions must be made about tho relative_,t_itudeof the vectors appearing on the rightside of the i_I ' equation. *i Therefore. let _.S 1 -- _.S 2 = , The RSS value of the thirdterm on the rightside of the equation can II be showntobe i! C_ 20113-FRI I"i ' ' ,_i....i'__'_'..,-. .. .>,i" " ,, _9650__367-___ - I02- ;i Therefore, the maf..tude of thr 5u 1 is i -I5_: 2: 6g,' _I_-Ji_6_"1 + - 2 2 2 2 / t 18_512 __32i6_12 + _ --',, 16ill + 16_C (19) The vehi_ " : rate error about pitch, yaw or roll azis ie: r ,ate error per axis = _ (20) {i ,B f The d ived body rate error I 6_ I can be obtained by sin',ultaneous solutio_a of Ec, *inn q (II) through (19) and substitution of the error term values in f Tab _ , 3-9. S_ .dtaneous solution of Equations (1i) through (19) yields = - 12+ 16_ 12 ii 18_12 1 -_ ' i6S i2 + 16_"¢ NI L + o. 75 _2 I,_ !:_+ 18_,c;I_ i • 1.s 18_NI" _. . ! ._. ,_ 20113-FRI ,L m__ i, lit LILt I ........... Illi_ ,h ,"_'_'_ " ' ............................. ' ' ........ 1965011367-112 - 103- Substitution of the error term v_iues in Tabte 3-9 and assumption of a vehicle rate _J of 20 deg/sec yields - } i2o12 2 + 0.75 151.31 18gl + (0.8) 2 X 10 -8 + 1.5 x (345)2 x I0"8 _- 1.5 x (345}2 x I0"8 iSul = 0.0422 rad/se,_ = 2.42 deg/sec Therefore. the rate error per control axis is _ 2.4__22= 1.4 deg/sec In the expression for 15_12 it can be seen that the contribution of the noise errer term 15SNI is about 70 times larger than the RSS of all other error terms. Therefore, the ARS performance fn the rate mode depeneds entirely on the abilityof _be data filtersto reduce the 1.4-deg/sec computed rate uncertainty. Note that t,%e distinction between h_.gh and low accuracy in the rate output of the AIRS is lost in the presence of the high noise level on the gyro readout. 20113-Y RI 1865011367-113 DERIV5 EIONS OF ERROR EQUATIONS Thiz section contains the deri,,ations of the error equations which are listed wi;,hout proof in Figure 3-15. The equations which are derived here are: 5_ 1 = (4/T) _5P 2 o,! + tPI/T )2 5T_ _21 = (4/T)45p 22 + (P2/T)2 5T 2 5c_3 = .366(6ot + 5 + O. 50 a • +v i %1 2 The expression for16 Iwas derived in the section "ARS Performance in Attitude Mode". 1 1 Derivation of 6a I and 6_ 2 The gyro readout equation whicb define the direction cosines of the twc reading pickoffs with respect to the gy-ro spin axis are 4P 1 a I = ---:- -2T 4P 2 :. a 2 =_ -2T !_ if &_l and 6cy2 are the errors in _l and a,2, respectively, then by differentiatton_! • . I 20113-FR1 "19650"1"1367-'1"14 105 - 4P2_ Formirg the RSS of the two terms on the right side of each equation yields: 1 . + I-¢--I 6T_601 = 6_2 v. P2" 12 ;, Deri_,ation of 5a 3 i The a3 is the third missing direction cosi :e whose value is determined from(. the knowledge of 0 i, _2 ard the relative alignment among the three pickoff axes. _Ihe expression for o 3 is : w..... [ ]"_'_'2c_3 "D2°I B2c_2 ± 022 - E2_I - A202 The constants A2, B 2, C 2, D 2, E 2 are determined from the pickoff alignment date. the expressions for these constants are: { J /151 x 1_2 [ _1_1 . 152) ain012 cos012 eos012 ctn012 ! P1 )2...... __ ,, _f "4"2 ; 1 (P2 1 cos 2 = ------ -"- . - 012 sin012 (!1 /_lXl_2/(t;1 . I_2t; 1. t;3-1_31 1_2) t ctnei2cor, e -csc0 cos02 ,! '1 :I 201i3-FR1 I 1965011367-115 - lOC- /2 " C2 = /151x 152, = 1 sin.912 t ' (P_ P2 P3 ) X D2 = /151 x I52 i (.151 . 132 152 . b 3 - t5! . 153) ='1 (ctn012 c'c,sO23 - CSCO12 cos01 .....(Pl P2 3) 1- 2 X i i - 107- Therefore. the Jifferential eq,:_t!on for c_3 becomes : 6t /_2) a_ + 6a3 -- -6D2a. 1 ,B 2a 2 / I , ± -2a 2 5a 2 - 2a 1 [SE 2_1 " 5A2_2 + 5all 2 _3 i The differentials for the constants A, B, C, D, E can be evaluated in terms of errors in O... If'_he error in O.. in 50.. then 1j U' D 8A 2 = 8 {ctn 012 } = -CSC 2 012 8012 = -6 0i2 Similarly: !.I 6B 2 = 8 823 l"i 6ivc2 I: -6x 5 D 2 = 8 813 5E 2 = 0 !-i 'rbe SX is also a function of alig.rmment measurements and it can be neglec_ed If since itis of second order. The proof for thisstatement goes as follows:Ii . It it_ given that (21) • X = 1_1 P2xP3ij -i , ° 20113-F_1 .,-. _ . . _ ,, ,:__,.._._._/:7_,__, _., ..,.. 9650 367- 108- Dif'eret_t:ate Equation 121) ._X = _151 • P2XP3 + Pl " 5152XP3 _ P1 " P2X815 3 (22) I t - 109- • t ) and let .J , @' i l e23 = _ + _823 • }7 813 _13 Subs+..!tu4.e these expressions into Equation (24) and obtain sx = -_e12 [(_2 -_) " "_3] I ! _5e:3 L(g1 - _3) • 1_2_ (25) !i -Se2S [('_S " _2 ) " P1 ] [ i! Since the _ts are of the same order a8 5O's, the 5X quantity is file second order error; therefore, it can be ellminated from further consideration. l ,i " ' Therefore, the differential equation [or a 3 becomes I 6_ 3 = - 6 813_1 - 6 823_2 (26) I ( _2 _1 }i 4- "_"3- 6'_2 + 6ttl + t_l_2 _ el2 lj_1 The 6O's are angular errors tn the relative aligm_.ent among the pickoff axes whose expected values are the 8arne for ali alignments; therefore, let :i.l 8812 = 8813 = 8823 "-- 88 a Then the RSS of the Equation (26) is [] '? 201,13-FRI F- I 1965011367-119 - !I0 - I_ carl be deduced +.hat the valup, for 6a 2 will be maximum if _3 is small and2 2 _- CCi --_2" ? ; The a3, being the missing cosine, will ha_c values ranging between 9.76 to 1.0; • therefore, evaluate it at 0.76. And since 2 2 2 ctI +cL 2 +a 3 = 1 (,he value ofoa 3 :. 0.50 58 + 0.3e6 (Sa + 6a ) D er}va.tionof 5S1 1 ' i t. :_ Wri*.e _._,,;direction cosine row me trix in pickoff a_.es [ct] and the transformatian -_[ matrix from pickoff to body ay..es [A] the_,- the cosine computer output is }- IS.I-- [a] [A]r ! and the error in the cosine computer outputis [6s,] = [6a] + r s The |6_1 is the row matrix whose elemen-'.sare the previously derived errors _n a 1, _2' and a 3, namely k i h ;.ominally,the pJ,;koffaxes are aligned along the vehicle body axes, then [ the aDgnment mazrix [A] is the unit matrix. _. ; I ! I - 111 - i The mstrL_ _A eens_sts of elements which are small miealignment angles be'_ween piekoff axes and vehicle body axes. For the error analysis pur- } poses write this matrix as followe. (see Alignment Errors): f , 0 03 -02 [6A] = "'P3 0 Pl " 02 -Pl 0 i _ Therefore, r - 0 p3 "P2 ! .6Si = 6a 1 b_,.,,,6a3 + .al o:2 a3 j "P3 0 pl I ! P2 "Pl 0 rSSi _ = al " a2 P3 +'3021' 6a2 +a 103 " a3Pl} 5_.3- alP2 +a2 Pl) The expression on the, right side of the equ_tlon is a row matrix whose ele- ments define the error vector 5S11:i - 1 Forming the R_ of each element and solving for the magnitude of 6 S l ,i , t! and letting Pl = P2 = P3 = Pa tl 1 '_ 20113-FR1 ) , II ...... . .... ' .... --' --_ , " , .... , ............... p . _ i II '" I I 1965011367-121 o- 112- z,: ', J Derivation of I 6_ I 1 .. J The body rate error equation can be derived as follows: The time differenti_,!s of S 1 and 2 are , .} S1D = S1 + ,._x S 1 (28) : S2D = S2 + uJx S2 (29) where _- -;- { SII_ _;2D gyro drift rates ,: S 1, S2 = gyro spit axes rates with respect to the rotating body axes .:} , _ = body rate S, S2 -- unit ,:ectors along *_he two gyro spin axes. iL " ! 201i3-FRI ! ]9650] ]367-]22 - 113- Perturb Equations (28)and (29)about theirnomin81 values and subtractthe nominal solutions SID -_ = mx + 8m >= = _8S I (30) S2D- 8S : mx + amxS 2 -5S 2 (3i} The drift rates S1Damd S2D appear in Equations (30) and(31) because in the nominal solution their values 'lave been assumed to be _ero. Note that the rate erro;rs 5S 1 and 5S 2 are the sum total of gyro drift and gyro readout noise. Crossing Equation (30) and S, _ and Equation (o.)'_Iand $2 "bSIXSI = ('m" SI) 6"_I - _'_I ' SI m + 6m" SI SI - 8-m (32) If it is assumed that $1 and $2 remain unit vectors then - 8s2j 6S,' : 0 and • : 0 i _ 1 2 i Dotting Equation (32) by $2 and Equation (33) by S1 201Z3-FR1 ............. --............ ,_--- __--:" , ............. ,_ 1T_-_I_ 1965011367-123 - I14- ^ Solve *,he scalar Nquations (34) and (35) simultaneously and eiiminate 6w • S2 - -6's2x;2" ;i -- -_';2 _g2" ;I-5;. ;I (as) _sl "s21;" g__'_I"g2+ _;'si sl "s2+ 5sl x s1 • ;21 Solve Equation (36) _o_- 5-¢J" S1 and substitute this _.;pression into Equation (32) 5_ = 6S, X 1+_ • SI _ . _ .... : + " {_" S2 5_2 S1 + 5S2 ZS2" Sl + SI" $2 6:_1' $2 (37) 1 -(SI" S2 )2 S I •: - _ 2 5S_ X 1 _ Since S ! " S2 is of the s_rne order (see Error Sources) as the errors in S and m the rate of S the Equation (37) can be simplified by elimination of the ! i products of S 1 S 2 with either 6S or 5E ,s_'=6_I " '_16sl - s2 _s2 sl _ s2' XS 1 +_ +$1 _o - " +6 2,X "1_ " (38) :, ) And ', ?i 2....... ......... 2 The derivation steps leadlng from Equation (38) to Equation (39) are shown }' under "ARS Performance in the Rate Mode". 20113-FR1 ........... m ....... , ,.1 £ _._ " _. _ ......... .-- _:-I_ _,=_ :'llnll:..: -&_,,_ 1965011367-124 -- 115- ARS COMPUTER AND I/O LOG!C INTRODUCTION _,he objective of this study is to establish _.he characteristics of the digital computers necessary to satisfy the computations2 requirements of 12 different attitude reference systems, each utilizing two SDMEG'_. The attitude reference systems are intended to provide _hree-axis information for con- trol of a spacecraft similar _:o Voyager. The 12 systems result from varia- tions of the following p_ram eters: A SDMEG (gyro) configuration ,_ ] (1) The SDMEG developed ruder contract No. 950607 (2) SDMEG with caging caoability f3) SDMEG witn to:'quing capablity B The attitude reference system (ARS) outputs (1) Three-axis attitude error and attitude rate (2) Three-axi_ attitude error only C Accuracy j (1) High accuracy { _ (a) C,_mputer output error less than 0.01 degree _b) U' ,dating of attitude rate information 10C rimes pe_r _ second and attitude information 10 times per second (2) Lo-_ accuracy I I (s / Computer output error less than 0,1 degree /a) Updating of attitude rate information 10 times per I sec, or._. and onne per attitude information second ! i _' 20113-FRI 1965011367-125 - .1.16- i[avying the seven oarameters res_flts in the following 12 system configurations for which the ARS computer requirements are to be determined. ARS No. 1 --A(1) B(1) C(1) ARS No. 2 = A(1) B(1) C(2) ARS No. 3 = A(1) B(2) C(1) ARS No. 4 = A{1) B(2) C(2) ARS No, 5 -- A(2) B(1) C(1) ARS No, 6 = A(2) B(1) C(2) ABS No, 7 = A(2) B(2) C(1) ARS No. 8 = A(2) B(2) C(2) ARS No. 9 -- A(3) B(1) C(1) ARS No. I0 = A(3) B(1) C(2) ARSNo. 11 = A(3) B(2) C(I) ARS Ne. 12 = A(3) B(2)C(2} From the point of view :,f computer design, the two gyro configurations with the capability of cagin,,f or torquing are the same. On the basis of computer : design then. ARS configarations 5 through 8 have the same rpquirements as ARS configurations 9 through 1':. This .implificat!on reaves 8 ARS configura- tions. • ARS No. i - A(1) B(1) C¢1) - The flow charts of the functions tc be performed for this ARS are showni_ ;£igures 3-18 and 3-19. This system is the most complex, having the maximum number of functions to be solved in the least amoun_ of time with the greates_ accuracy. It is the system on which the choice of a computer design was ba_ed. ' 20113-FR1 ' i i 1965011367-126 - 117 - 20113-PRI 1965011367-127 1965011367-129 " C - 121 ,- • ARS No. 2 - A(_) B(1) C(2) -Here it is assumed that the gyro pickoff non-orthogonality, pickoff axes misalignments from body axes and gyr0 spin axe• non-orthogonality will have magnitudes such that the tots_ effect will contribute less than 0.19-degree error at the output. Tb _ following r_odz_.ca,,o..s can be made to the flow chart_ of, Figures 3-.I.8 and 3-19: Eliminate the constants in Blocks 9-13 Eliminate blocks 9, 11 and 13 (constants ffi0) Blocks 10 and 12 are unity multipliers Eliminate 18 constants (Blocks 20-28, Figure 3-18) Eliminate blocks 21, 22, 23, 24, 26, 28 Blocks 20, 25 and 27 have unity multipliers Eliminate blocks 41, 42, 43, 45 • ARS No. 3 - A(1) B(2) C(1) - Eliminate blocks 36 and 38 • ARS No. 4 - A(1) B(2) C(2) - Incorporate all modifications made in ARS 2 and 3 • ARS No. 5 - A(2) B(1) C(1) - Eliminate blocks 41, 42, 43,_ 45 • ARS No. 6 - A(2) B(1) C(2) - Using the assumption introduced for ARS No. 2, this configuration is the same as that for ARS No. 2 - A(1) B(1) C(2) • ARS No. 7 - A{2) B(2) C(1) - This configuration is the same as ARS No. 3; in addition eliminate blocks 41, 42. 43, 45 • ARS No. 8 - A(2) J:?(2) C(2) - The capability of torquing or caging is immaterial for a system of low accuracy. The computer flow chart for this configuration is tne same as ARS No. 4 - A(1) B(2) C(2). 20113-FR1 1965011367-130 . 122- i " On the basis of computer requirements, there are six di2ferentsystems: i • ARS No. 1 • ARSNo. 2, 6, I0 • ARS No. 3 • ARSN0. 4, 8, 12 • ARS No. 5. 9 L • ARS No. 7, II r I Using these six systems, the study was broken down into four areas of ! investigation: • Gyro interface logic. • Gyro startupprocedure M_ 'I • ARS computer design • Telemetry- computer interface The design of the ARS-SDMEG systems presented here is based on the follow- ing information for these four areas of investigation. Gyro h_terface Logic The basic information derived from the ESG is of an analog 1,ature, It is in the form of two time intervals which are related to the gyro attitude. The interfere logic transforms these analog data into binary ntumbers suitable for use in a digits2 computer. The acquisition of the two time intervals is complicated by the factthatthere are three intervalsavailablefrom the ESG and only two have validdata. The decision as to which pickoffshave valid data must be made in the interfacelogicor inthe computer. 20113-FRI 1965011367-131 - 123- The design of the gyro interface logic is also influenced by the rate at which data is required in the computer.. Data must be available from the SDMEG ; at least as fast as output da_a is required from the computer. Where data is required at much lower rates, an i/0 configuration can be chosen to minimize haz dwarf. . GEro Slartup Procedure Since t.h_, gyro probably will not be able to survive a launch while suspended, it i_ necessary to start the gyro in space. Assuming a cosine rotor pattern, this gyro is most accurate ff the startup ensures the correct rotation direction of the rotor. The startup procedure involves the transmission of a number of discrete commands (spin, damp, -reverse, etc., ) to the gyro which are condi- tional to past and present states of the gyro. This proc._dure can be accom- i plished by either a "black box H which generates the commands at preset "i intervals or by a stored routine in the computer. The AHS Comp,_ter :i Given the data from the ESG in the _orm of a binary count, the following :_i quantities are computed: i i • Three-direction cosines for each of the two SDMEG_s t • Vehicle attitude ei for '1: • Vehicle attitude rate '1 The computer takes the two valid digital words from the input-output logic ! interface and calculates two direction cosines. From these two cosines, a _] third direction cosine is gener._ted. Inthis manner three direction cosinesare obtained for each of the two SDMEG_s. The two s ts of direction co ines, ] 20113-FR1 1965011367-132 , - 124 - along with a set or six direction cosines received from the guidance and naviga- tion system, are used to determine the vehicle attitude error. Also, in some of the six system configurations, attitude rate is also computed from the two sets of three direction cosines and the past history of t:-e vehicle attitude rate Telemetry-Computer Interface The scope of this 'study does not allow for the examination of all the possible schemes for obtaining computer data for telemetry. However, three possible schemes have been selected for discussion here. These selections were made by assuming that only words in the computer memory would be required for telemetr:_. The main 7clue in discussing the problem here is to indicate the flexibility of the computer in dealing with the problem of telemetry. GYRO INTERFACE LOGIC Input-Output " The portion of the gyro interface logic concerned with the ARS computer is } called the input-e,,tput lo_ic (I/O}. The I/O deals with the time interval be- tween pulses fro.n the gyro and converts this time interval into a binary word ! by counting the pulses from a fixed-frequency source during the time between i pulses. It is necessary, then, to determine the frequency of the source and the magnitude of the maximum count necessary to attain the accuracy required by the system specifications. The following calculations are made to determine the length required in the (I/O) counters and the frequency required in the (I/O) _.ock. Two binary { numbers are generated by me input-output logic -- a partial count (P) and a ( total count (T). The computer uses the_e binary numbers in the equation I t Cos0 = 4 IPI - 2 ,1, [ to determine the spin axis direction cosine (cos 9). 20113-FR1 1965011367-13g - 125 - Neglectingsecond order errors, the error in the directioncosine as a function of error in (P) and (T_ may be expressed as where Acos 8 = error in cos 8 P = error inP AT = error in T ! The errors Ap and AT result from reading the counter immediately preceding the recording of a count. Therefore, i 5P = 5T = 1 count ' and A COS @ = _ 1 --_- (41) i } ! Since P o. 31 <-_ < 0.69 the maximum error in the direction cosine will be 4 A Cos8 = -_-(I - 0.31) I or (42)2.76 A COS 8 = ---- T I I ! 20113-FR1 1965011367-134 e: - 126- Since it is desired to have an _ccuracy in the computations equivalent _o _0, 01 degree at the system ou_t f,_r the high accuracy _ystems, and 0. 1 degree for the low accuracy systems, the tolerable error in the direction cosine may be expressed as follows: In terms of the error in 8 the error in the direction cosine is Z_ cos 0 = sin 8 Z_0 where Z_0 = the error in 0 There?ore, the maximum error in the direction cosine may be expressed as Z_ cos 0 = AO -i and for the two accuracy levels of O. 01 degree and 0, 1 degree 0.01 x 2_Z_ cos 8 = - 0. 000174 radian360 and 0.! x2_ cos 0 - - 0. 00174 radian360 Therefore, substituting these values f(.;r _. cos 0 in Equation (42), the following / values result for the total count during a rotor period, (T): l 2.76¢ 0. 000174 = -- ,_ T _, or _' 2.76 i T = = 15,900 countsb':'0V67_ ¢ tI Z0113-Ftl1 1965011367-135 - 127 - For the low accuracy system 2.760. OO174 - T or 2.76 T = 0.00174 = 1,590 counts Therefore, to hold 15,900 counts the high accuracy system will require,. counters with at least 14 bits; the low accuracy systems will require counters with at least 11 bits to ho!d 1,590 counts. J Since the minimum period of the gyro rotor is 5 milliseconds, the frequency of the clock for the high accuracy system must be at least 15,900 _ 3.18 x 106 cycles per secgnd 5 x 10 "3 For the low accuracy system the clock frequency must be at least 1,590 318,000 cycles/second 5 x 10 -3 = i Gyro rundown is another factor affecting counter length. Since it is aesired to maintain system ir_te_rity until the gyro rotor speed has decreased by 50 percent, the rotor p_riod will increase from 5 milliseconds to 10 milliseconds and the co_'uts during a period of the rotor will double. Therefore, to accommodate this -,mdition mmther bit must be added to the length of the _ounters, thus totaling 15 bits for the high-accuracy counters, and 12 bits for the low accuracy counters. 20113-FR1 i 1965011367-136 - 128- Gyro Readout Electronics In normal gyro c_eration, a readout is obtained from only two pickoffs while *.he third p_ckoff is in the "dead zone". For this reason, only two channels of readout electronics were to be provided. Thzs required that the electronics be switched from pickoff to pickoff as they went in and out of _e dead zone. In this way, both sets of electronics were always conn_.cted to pickoffs not in the dead zone. The sensing of the dead zone and swito_hing of electronics between pickoffs had to be done by the I/O logic. These logic decisions re- quire _ddition_d hardware in the I/O logic. Sin_e a general puz_ose computer was chosen for the ASS, it is best to use the computer as the decision maker wherever possible. For this reason, a third set of readout electronics is to be provided on the gyro, and decisiovs regarding the pickoff in the dead zone vJ_I1 be made in the computer. Gyro Output Rate The maximtun-accuracy system requires an outputrate of 10 attitude calc_,_tionsper second and 100 attituderate calculationsper second. To accomplish thisrequires 100 directioncosine calculationsper second. This in turn requires the acquisitionof 100 new sets (three.counts)of partialcounts frr _ each ball. (Tottdcounts are acquired at a much lower rate, perhaps one a second, and are stored in the computer for lateruse in the computations.) The maximum time for a partial count occurs for cos 40 degrees. cos 40 °ffi 4 Pmax _ 2 so thatPma): -- 0.0035 second _). 005 In the worst case, where Che read heads make equal angles witL the spin axis, allthree heads w_llpass the start line _n two-thirds of a ballrotation t 1965011367-137 - 129- So, to the maximum count time of 3.5 ms, is added two-thirds x 5 ms equals 3 1/3 ms to get a total of about 6.85 ms to complete the outputs of the three pickoffs, from one gyro. If a separate counter is provided for each pickoff, then the set of counts can be completed in a max-mum of about 6.85 milliseconds This would allow for the acquisition of over 146 sets of counts per second. This allows for a drop in ball speed to 7.3 milliseconds per revolution (137 rpm) to maintain the rate of acquisitior of 100 per second. This meets the requirement of 100 sets of partial counts but requires that both gyros have a set of three counters. So, for the maximum, information rate (100 direction cosines per second), a six-counter system is required, but the ball speed may not drop below 137 rpm. Having devised an I/O scheme for the system of maximum output rate (10C per second), it remains to devise one for those systems of lower output rates (10 per second or less). If a single counter is used to acquire counts from all six channels, one at a time, then the maximum total time required for the six acquisitions is 6 x 5 ms = 30 milliseconds. For the 10 acquisi- tions-per second needed in the systems of lower computation rate, there are 100 milliseconds per acquisition. This makes the one-counter system feasible. The requirement, then, is for only two I/O systems -- a six-counter system for the highest computation rates (100 per second) and a one-counter system for all other computation rates (10 per second or less). I/O System Operation The Six-Counter System-- Figure 3-20 is the logicdiagram of the channelcounter and logicfor one channel of the six-counterI/O nystem. One such logic con- figurationis used in the single-countersystem and six are used inthe six- counter system. 1965011367-138 - 130- 20113-FR1 1965011367-139 - 131 - The following sequence occurs for the acquisition of counts. The decoding of the appropriate I/O instruction sets a flip-flop in the computer, causing fi:e"!oad e_.able"lineto go to a "one" stateand the computer prograin resumes operation. The "load enable:'line,being at a "one", opens the gates to the counter and the interrupt. The I/O is now ready to send data to the computer. For acquisition of partial counts, the "total count enable" line is a zero. A start pulse from the gyro sets the "couu% enable" flip-flop which gates the i clock into the counter. The counter continues counting until a stop pulse clears • ) the "count enable" flip-flop. H the pickoff is near the dead zone so that no ; stop pulse occurs, the counter counts through to zero, stops, and generates a pulse to clear the "count enable" flip-flop. (If the pickoff is not in the dead zone the count will be finite, ff the pickoff is in or on the edge of the dead zone, the count will be zero. ) The "dear pulse to the "count enable" flip-flop also sets the "interrupt" flip-flop. On receiving an interrupt, the computer completes its current instruction. The interrupt causes the computer to enter a short subroutine which loads the counter contents into the computer memory. The computer then clears the interrupt £1Ip-flop and resumes operation in the m',._n program. The _,ext start pulse from the gyro starts ;. the cycle over a_r:_in. , ._ " t For acquisition of total counts, operation of I/O logic is the same as that [ / for partial counts except for the following: The 'ttotal count enable" line is a "one" as a result of the computer. The "count enable _' flip-flop will be {i} set by a start pulse and will be cleared by the next start pulse so that the co_mter counts for a full revolution of the bail. ! It was shown previously that an extra bit was necessary in the counter to i l, accommodate the large counts resulting from gyro rundown but not necessary • } to retain the extra bit in the computer calculations. The "range indication" flip-flop is included to determine whether the n_ost significant bits or the i; 1965011367-140 - 132- least significant bits of the count are loaded into the computer. When total counts are required and the "total count enabl e" line is a "one" the "range indication" flip-flop will be set when the most ._ignificant bit of the counter is a one. When the range signal to the computer is a one , the most significant bits of the total count and all subsequent partial counts will be '',.¢ ''loaded into the computer. When the range signal is a ,.ero , the least significant bits will be loaded. The "range indicator" ,./p-flop will be cleared whei, a new total count is taken and the first start count occurs. The One-Counter System -- Where as the six-counter system is made up of six of the configurations :lust described, the one-counter sys*_cm (Figure 3-21) uses only one of these co_igurations, the main difference being that each of the six sets of readout channels must be connected in turn to the "chamml counter and logic" preriously described. This is done by having the inter- rupt signal increment a modulo six counter. The new value of the count connects a new chain]el to the "channel counter and logic" through the "channel select logic". The new channel is counted up and an interrupt is generated to increment the modulo six counter and continue the process. The six channels will continue generating counts, in turn, as long as the "count enable" line is a "one". As before, the state of the "total colmt enable" line determines whether the ou_uts of the I/O are total or partial counts. GYROSTARTUP Given a general purpose computer, the gyro startup procedure can be accom- plished with a stored program. This works particularly well for startup be- cause the computer will not be doing anything else. A programmed startup procedure requires the low-speed real-time clock included in the computer, to time intervals up to one hour. Figure 3-22 is a program flow diagram for the gyro startup routine. It is estirnated to require about 75 instructions to mechanize. 20113-FR1 1965011367-141 - 133 CHANNEL 1 SDMEG COUNT ENABLE NO. 1 CHANNEL 2 CHANNEL i TOTALAND : SELECT COUNT ENABLE ELECTRONICS CHANNEL 3 LOGIC i I INTERRUPT CLEAR ,_ ER ARS COMPUTER L C / INTERRUPTi B | CHANNEL 1 SDMEG --" NO.2 AND CHANNEL 2 ; ELECTRONICS CHANNEL 3 i i TLsIoouLo}SIX-COUNTERii Figure 3-21. One-Counter System of Gyro-Computer InterfaceLogic 20113-FR1 1965011367-142 134 - INITIATE GvRO START t SUSPENSION JPOWE_ ON J J POWERONABOUT 2 MIN. _ORWARDI SP_NPOW{_ROFF J DAMP FOR J iii lA3OUT 10 MIN JI ! _EVERSE[ ( ORIENTATION _ SPIN POWERON oK? j [ ABOUT4MIN. 1TM _ ._ _ rORW.I.RD REVERSE SPIN POWER SPiN POWEROFF ON . DAMP ABOUT ABOUT 6 MIN. 10 MIN. APPLY DAMP OF SPIN COILS ON _ i ORIENTATION AmUT10M,N. A_OUT6M,N. _ OK" _,/ ES I LORIENIATION _ IS POWEROFF OK? _/ COMPLETE DAMP ABOUT10 MIN. r Figure 3-22. Flow Diagram For Gyro Sta_up Program 20!13-FR1 1965011367-143 - 135 - THE ARS COMPUTER !n the time availablefor thisinvestigation,itwas not possible1:ostudy each of the si_ differentsystems and determine an optimunl computer configura- tionfor each. Instead,a computer was designed for the maximum complexity syste,7.,and portions of thiscomputer were modified or removed as neces- sary to satisfythe requirements of each of the other systems. In the to]lowing discussion, most of the material pertains to the maximum-comp]exity system. Selection of Computer Type The computer for the maximum-complexity ARS could take any one of several forms. The three computer types that might be considered for hhis applica- tion are (1) special-pro'pose incremental, (2) special-purpose total-value, and (3) general-purpose, stored-progran= total-value. An approximate com- parison of the complexity of each of these elree mechanizations under the constraints imposed by this problem is made in the following paragraphs. Each of the classes of computers mentioned usually is applied to a specific class of prob!ems. General-purpose, total-value computers are used when the computing rates are not excessively high and when a wide variety of F,rob-- lems must be solved. Special-purpose computers are usually employed in situations where a specific type of problem must be solved repeatedly, partic- ularly when it must be solved at a moderately high rate. Increment.a] com- puters can b,.• used when the inputs are smoothly varying functions a'd are particularly useful when extremely high output rates are requ_:red. Special- purpose comFuters as a class increase in complexity vdth the size of the job and require considerable amounts of hardware for sequencing a_,d controlling the arithmetic operations when a large number of computations must be per- formed. Also, wh,-n a new computational task is required of a specia]-purpose computer, it usually results in a significant increase in the amount of hardware. On the other hand, in a general-purpose computer, if there is sufficient time remaining in the computational cycle and enough extra memory capacity, a4- ditional tasks can often be added without increasing the hardware requirements. 20113-FR 1 1965011367-144 - 136- A total of approximately i20 computations is required in the maximum com- plexlty ARS, Of these, approximately 63 are long operatlons, such as multi- plications and divisions, and appr,_ximately 57 are short operatic,ns, such as additions and subtt'actions. Some of these computations are perf(Jrmed only once, ,jl.hersare repeated I0 times, and the remaining computations must be repeated 11 times during each computer iteration period. This ;-esu:4tsin a total of about 1250 computations that must be made during each computer iteration. Since the computer iteration rate is ]0 per secor.d, a total of approximately 12, 500 computations must be made each second. Of these, approximately 7000 are long computations, and the remaining 5500 are short computations. If we multiply the number of long computations times 16 (l 6 bits pe_ _ word) and add to this the number of short computations, we obtain a figure of 117,500. This is roughly the equivalent number of short computa- tions that must be performed by the computer each second. Thus, we have approximately 8.5 microseconds to do each equivalent short computation if they are done sequentially by a total-value computer. This should be more than adequate for either a general-purpose or special-purpose mechanization. Incremental computation techniques might also be used, since the inputs are continuous functions of time. For a maximum input rate of approximately _even radians per second and with 14-bit direction uosines, the required iteration rate of an incremental computer employed in the solution of these equatlon_ can be computed as follows: 1 increment 7 Minimum Increment Rate : x 2-14 second 115,000 increments/second Thus, tbe maximum input and output rate of each computation would be ap- pr'oxlmately 115,000 increments per second, which corresponds to the : mlnimurn iteration rate. Assuming serial/parallel bit processing and a I. 5-megacycle clock rate, each computat_jn would require a minimum of 20113-FRI 1965011367-145 ,° - 137 - four microseconds. The computer operates in a 4-bitescr'i,, _,4-bLt _,d,,,]_,.l fashion, requiring 2-2/3 microseconds for processing 16 blts, and I micro- second for memory access cf the operand, totaling about 4 microseconds for the operation. Therefore, at most, two computations could be performedJ sequentially during each iteration by each arithmetic unit, and a total of about 6000 arithmetic units wouid be required. Thus, the number of components re- quired by an incremental computer would be many times the number required of a general-purpose computer that performs the same job. Even ifspecla] techniques such as variable or multiple increment operation were ased, many :,. arithmetic units wo_ld be required, and each of them would be more co nplex l i _.han those of a single-increment computer. ! i Incremental computation is therefore not practical, leaving a choice between2 the two types of total-value computation -- general-purpose and specia]-purpose. [ ] A special-purpose computer to perform these computations would re_mre a ',c_y large and complicated control section to sequence the operations prope_.'ly, l,_ addition, other tasks, such as the startup routine, would require either additional i_,l hardware or additional control section complexity. It would therefore require less hardware if a general-purpose computer were employed, since the. startup :1 _.i instructions could be stored in a portion of the memory, and a small control section could be included for interpreting them. A general.-purpose total-value mechanization has been selected for the folt_)wiug if! _ reasons: !'l • It requires less hardware than an incremental mechanizatmn in i ) this particular case. li} • It requires somewhat less hardware than a total-.value special- purpose mechanization and is much more flexible, allowing for _! the addition of new tasks at a later date. i:, _} _| 20113-Fttl 1965011367-146 - 138- A simple serial]parallel total-value mechanization is permissible with the computation rates required of this problem, yet a large portion of the computation cycle remains available for addi- tional tasks, The arithmetic unit can be time-shared to perform apprL, priate :-metions in diffet'ent modes. For example, during stactup the ARS computations need not be made, and the entire computation cycle is available for the stertup program• _ Description of Comp'lter The block diagram of the proposed general purpose computer for the maxi- mum complexity ARS is shown in Figure 3-23. It consists of three main sections -- memory, arithmetic, and control -- along with input and output registers and a source of computer chock. The memory and control sections are of relatively conventional design. The a_-:_:_u_-t_: uT_t, which operates m a serial/parallel mode, consists of t'z,'_. se_'lai/parailel registers, a se._lal/ parallel adder-subtractor, and the necessary logic for performing the tinle- dependen_ interconnections of the,_e a,_its. The input _-egisters (:,:_da:r, 16 data bits along with address bits ana an inte_'rupt bit• They are loaded s:_ynchro- nously from outside the con_put_" and transferred to the computer ,_ ,_nory by halting the normal opera_ion of fi_# cnmputer as _oon as possible ,_f_:, the interrupt bit becomes a "1". The c,utput registers are loaded frou_ ';.e accumu- lator by instructior, s stored in the r,_emory. The number of inp_ :::_ output registers depend on the requirements c.: each of the 12 diffever.i _ ; _tems. Memory -- Three classes of memories were c,:,nsidered tc:;:, " _.,_ARS computer: • The fixed memory, such as prewired plugboard0 t:ore ropes, or the class of reactive element coupling devices in which the information is stored in the ab._ence or presence of a material deposited on a substrate. 2,_ i 3-FRI 1965011367-147 - 139- Figure 3-2'- Co.nputer Block Diagram 20113-FR1 1965011367-148 - 140 - • Nondestructive reado,_ d%'DRO) memories in which information contained in memoir is unaltered when read. The NDRO merr, e,_y is generally composed of two-hole _. storage de,flees in which information is written in the device _hrough one hold and read from the other hole. • Destructive readout (DRO) memories in which the reading of a word destroys the memor._ conter," so that it must be rewr_.tten after each readout This type of memory is generally composed of tiny *.oroids (one for each bit). It ,.q the memory most commonly used in large ground-based computers. The fixed memory was elimirated from consideration because it is necessary to have the capability of altering the contents of some memory locations in , the ARS computer programs. The fixed memory does not have *,his capability. It would be possible to have fixed memory for the fixed portions of th_ pro- gram and a variable memory for the variable portions of the program. Th._s alternative was eliminated because of the additional hardware required to operate two indepenaent memories. The choice, then, is between the NDRO and the DRO memories. The DRO memory dissipates about twice as much power as £he NDRO memory. The difference is due to the fact that rc__U and write currents of equal magni- tude are necessary to read or write in the DRO memory. In the ND}_O memory, only a read or write Current is necessary. Therefore on the basi_ of power considerations the NDRO memory was selected for the ARS com- puter and the hardware requiremenzs of the ARS computer memory will be based on the NDRO configuration. The memory., will contain up to 1024 words. The number will vary with the computationa) neede of the 12 attitude reference systems. Each word wi.]:l be 17 b_ts long. Sixteen bits are for an instruction or data and one bit is t for parity. It is generally considered advisable to do error checking.at .! t J 20113-FRI i 1965011367-149 }L - - the computer memory because itis the least reliableportion of the I computer. Although itis probably desirable to do some form of error detectionin other portionsof the computer, the scope of this study does not allow for a fullinvestigationof thes_ needs. Memory speed requirements are not very severe in thisapplication.Therefore a conservative s_ate-of-theoartNDB/D memory with a write time of 4 microseconds and a read time of 1 microsecond will be postulated. Itwillbe seen laterthatthe_e memory speeds are adequate for the maximum complexity ARS. A block diagram of the memory sectionis shown in Figure 3-23. The address registercontainsthe address of the memory word currently in use. This address is decoded and used to elmble th_ proper driver- I switch pair which in turn c_uses eithera read or a write to occur upon receipt of the appropriate pulse from the control section. The memory register (M) serves several functions: During a read opera- i tion it holds the output word after it _s read out of the memory. If the output word is an instruction, the instruction code field is sent to the instructionregisterand the address fieldis sent to the address register, both of these transfers being effectedinparallel. Ifthe outputword is i an operand, itis used in serial/parallelform by the arithmetic section. i Arithmetic Section -- The arithmetic sectionfor the _aximum complexityARS containstwo registers -- A and Q, The A register is the accumulator which always holds the resultof the last operations. For simple operationsi ' such as adds, subtractsand shifts,iti_ the only one of these registers used. During add (subtract)operations, the contents of M is added to i (subtractedfrom) the contentsof A directlywithoutany intermediatetransfer of the contents of M. For the multip!yoperation, the multiplicandJs held i in A, the multlplleris held in M and the double lengthproduct forms in A and Q. For the simpler systems ir_which a programmed multiply is used, +) the Q register is omitted since it is not necessary. 20113-FR i 1965011367-150 - 142- The adder-subtractor is a log'c network that performs serial/parallel addition, subtraction and counting on the input words. The ar;.thmetic timing and control logic implements the subcommands inherent in each operation by making a suitably interconnection of the registers and the adder-subtractor. The required timing is obtained by con- trolling the shift pulses. y Control Section -- The instruction register of the controi section re- ceives the instruction code bits from the M register. They are then decoded, and the appropriate signals are formed and sent to the arith- metic section to perform the required operation. For mtdtistep operrtions, such as multiply, the iteration counter is used to count and identify each step in the operation. The P counter holds the address of the cu:'rent instruction, and, at the end of each operation, it is stepped by one count : to obtain the address of the next ins_.ruction. The control section also : controls the reading and writing operations in the memory by means of pulses transmitted at the appropriate times. Interrupt signals are re- ceived by the control unit which, in turn, causes the required actions to occur. : Computer Parameters Some important computer parameter_ have been omitted in the discussion to th,:s point. These consist of clock rate, memory size, data word length, • instruction list and instruction word format. The factors that affected the • choice of these parameters and the resulting selections are discussed in the following subsections. _- Data Word Length -- The number of binary digits or the length of the com- puter data word is determined by the accuracy required in the computer out- i put. The desired accuracy in the outputof the maximum accuracy computer is 0.01 degree or approximately 0.00017 radian. To establishthe relation- ship betwcen the computer output accuracy and the tolerable error in the round off of the data words, the computer output is assumed to be the product of 12 data words. _, 20113-FRI 1965011367-151 - 143- Therefore, the error in the product may be related to the roundoff error in the data words as -_(X 1 X 2) • . . X12) -- &X I(X 2X3. . . X!2) + AX 2(X 1 X 3 .... _12) + • " " +_Xl2(Xl X2" " " Xll ) Since, in the product, the factors will be direction cosines, the factors will have a maximum value of unity, a_.d /'(X 1 X 2. . . X12)< AXI + AX 2+. . . _X12 From statistics, the variance of a sum of random variables is the sum of the variances for the individual variables. _erefore ap2 _ + a22 + + 2Ol 2 . . . a12 where 2 ap = variance of the error in the product 2 a i = variance of the error in the individual data words of the product Since the error in the data words is due to roundoff arxt the words are di- rection cosines rounded to th_ same number of digits, it is reanonable to assume the errors will have equal statistics. Therefore 2 2 Op = 120 where a 2 • ai 2 i • I, 2 . . . 12 20113,-FR1 1965011367-152 - 144 - Assuming the roundoff error of the data words in the product is unfformily distributed on {o0.5_, +0.5_) and, from the definition of variance, we obtain 2 = f+O. 5{ I z _-o. 5_ (ax)2 _ d (AX) whure = scaling of least significant bit in the data word or L- J2i = 3g 1-0.5g 2 g2 12 Therefore 2 _2 P Cp = - Since the error in the product is made up of'12 error_ with u_form distri- °_ bustions, the probability distribution of the error in the product will be approximately Gaussian. Therefore, the error in the product will be less than g 68 percent of time, iess than 2 g 95 percent of the time, and less than 3 g 99 percent of the time. It is desired to have the error in the computer output, or product, less than O. 00017 radian. If it is sa,_i_factory to have i this occur 68 percent of the time, then the scaling of the least significant ,_ bit in the data word may be O. 00017. Since the d_rection cosines in the " i .' 20113-FR1 1965011367-153 145- product have a maximum value of unity, the data word length must have a range of 1/0. 00017 or 5880. This will require a word length of 13 bits, but, to avoid noninteger scaling within the computer, the word length is selected to have 15 bits plus sign or 16 bits. Clock Rate and Memor_ Size -- To obtain the data essential to estab- lishing clock rate, memory size, and instruction requirements, it was necessary to formulate an approximate computer program for the ARS computations bated on a preliminary list of instructions. The instruc- tior, s in the program were then divided into two groups, the first con- sisting of those containing long operations (multiplications} and the second consisting of those containing short operations (additions and subtractions}. The instructions in the second group that contained operations requiring a "write" operation in memory were also tabulated in a third group. The divide and square root operations appear oniy once, and, since the direction cosines are continuous functions of time, it is possible to re- place these operations by iterative subroutines such as the following Divide: _n+l -- _n(2 - _n b) where 1 _n"_ as n-. ® where X n ¥ - t t ")N 4 1 ;I.'R'1_'1 , 1965011367-154 146 - The magnitude of n need not be very great in continuous, real-time situations such as this, since we can use the p2eviously computed value of F or x as the initial value for each new iteration. In fact, worst-case calculation_ have shown that from 3 to 5 iterations of these formulas are sufficient to achieve the required accuracy, depending on the computer iteration rate of the parti- cular system configuration. Thus, the proper number of multiples of the operations required to implement the iterative formulas above are included in the instruction count rather than the two operations. For the maximum-complexity ARS, a total count of 2958 short operations, 890 "w--ire" operations, and 773 long operations was obtained. Since with a 16-bit word length, a long opecation takes approximately 18 times as long to execute as does a short operation, we obtain 131890 _ + 2958 + 773 (16) = 16,661 equivalent short operations For a 16-bit word length and a I. 5-megacycle clock rate, a serialparallel word time (which corresponds to the execution of a short operation) is 4 micro- seconds. Thus, the operations themselves required 16,661 (4) -- 66,644 micro- seconds. Since I00,000 microseconds are available during each computer cycle for the maxim.Am complexity ARS, approximately one-third of the computer, time is available for processing I/O interrupts and any other computations that may be necessary. 20113-FR1 1965011367-155 147 - The actual number of instructions that must be stored in the memory is about 300. In addition, approximately 145 instructions are required for the _tartup program, subroutines, bookkeeping and input-output. Ap- proximately 100 memory locations are required for variable or scratchpad storage and 50 for storage of constants. Thus, a total of at least 595 storage locations are required for the maximum-complexity ARS computer. A memory size of 1024 words will be assumed, since this represents the next largest power of two. A 10-bit address field is required' to specify a memory of this size. Instruction Word Format -- The AI:tS computer word, based o,_ the ac- curacy requirements of the more accurate systems, contains 15 binary digits {bits), numbered 0 through 15 starting at the left: 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 • _ • k , "-v Instruction Code Address Field, Y Bits 5 through 15 are used to determine he memory address referred to by the instruction code. The ll-bit field can address up to 2048 words in memory. The five bite, 0 through 4, are used to represent 32 different instructions (add, subtract, jump, etc.). Since the required tasks can be accomplished in the ARS computer with fewer than 32 instructions, not all combinations of the five bits are used. In the simpler systems, a ] 2-bit word is used and is broken down _xlLoa nine-bit address field, capable of addressing 512 words, and a three-bit instruction code representing eight different i,_structions. 1965011367-156 148 - InstructionRepertoire Following is a list of the instructions that are available for programming the ARS computer. Along with each is a brzef description of the instruction. Ex- cept _or the instruction whzch deals with some of the special requirements of input-output equipment, the list is composed of instructions commonly used in digital computers. Some of these instructions will not be used in the simpler systems. (1) ADD (A) + (M} -_ A (Add) -- Contents o_ the addressed cell in memory are added to the contents of the accumulator; the sum being placed in the accumulator. Execution time is 4 microseconds. (2) SUB (A) - (M) -_ A (Subtract}-- Contents of the addressed cell in memory are subtracted from the contents of the accumulator, the difference being placed in the accumulator. Execution time is 4 microseconds. (3} CAD (M) -, A (Clear and Add) -- Contents of the addressed cell in memory are transferred to the accumulator. Execution time is _ microseconds. (4) CSB -(M) -* A (Clear and Subtract} -- The (2's) complement of the addressed cell in memory is transferred to the accumulator. Execution time is 4 microseconds. (5) NEG --(A) -, A (Negate) -- The (2's) complement of the accumulator is formed and placed in the accumulator. Execution time ia 4 micro- seconds. (6} WRT (A) -, M (Write} -- Contents of the accumulator are trans- ferred to the addressed cell in the scratch pad "memory. Execution time is 6 microseconds. 20113-FRI 1965011367-157 - t49 - (7) MPY (A) × (M) -_ A,Q (Multiply) -- Contents of the addressed cell in memory are mu!tiplied by the contents of the accumulator. The most significant half of Lbe product is formed in the accumulator, and the least significant half is formed in the Q register. Execution time is 65 microseconds. (8) SM¥ 11AI × I1 M} -_A (ShortMultiply) -- Contents of the most significanthalf of the addressed cellin memory are nlultipliedby the contentsof the most significantha!f of the accumulator; the product being formed in the accumulator. Execution time is 17 microseconds. (9) LQP CA) A (M) -. A (Logicat Product) -- The logical bit-by-bit product of the contents of the addressed cell in memory and the contents of the accumulator are formed in the accumulator, Execu- tion time is 4 microseconds. (10) LQS CA)V (M) -, A (Logical Sum) -- The logical bit-by-bit sm-n of the contents of the add2essed cell in memory and the contents of +.he accumulator are formed in the accumulator. Execution _.me is 4 microseconds. Cll) SHR (A) x 2"N -_A {Right Shift) -- Contents of the accumulator are shifted to the right N bits specified by the operand address portion of the instruction ward. The sign is extended in the high order end of the accumulator. Execution time is 2 + 2/3N microseconds. (12) SHL (A) × 2N - A CLeft Shift) -- Contents of the accumulator are shifted to the left N bits specified by .*-heoperand address portion of the instruction ward, Zeros are inserted into the low order end of the acctunulator. Execution time is 2 + 213N microseconds. 20113-FR1 1965011367-158 - 150 - (13)ATQ (A)_--_(Q) (A-Q exchange) -- Contents of the accumulator and the Q register _.reexcha,lged. Execution time is 2 microseconds. (l,i)BPA (Addr.) -_ifl"(A) _0 (Branch ifPositive) -- Ifthe cnntcnts of the accumulator are positive,the contents of the address register are writtenintothe program counter. Execution time is 2 micro- seconds. ; (15)BZA (Addr.) -.P iff(A) = 0 (Branch ifZero) -- Ifthe contents • of the accumulator are zero, the contents of the address register are _;' written into the program counter. Execution time is 2 microseconds. " (16)BMA (Addr.) -'P iff(A) < 0 (Branch ifMinus) -- Ifthe contents :_ of the accumulator are negative, the contents of the address register are written into the program counter. Execution time is 2 rn_croseconds. L (1'7) JMP (Addr.) -_ P (Branch Unconditionally) -- Contents of tl_e oddress register are written into the program counter. Execution time is 2 microseconds. (18) OUT (Output) =- This instruction is used when it zs necessary to transfer data from the computer to the output register. This instruc- tion differs from the other instructions in that the operand address field is a "function code. The "function code" is decv,iled in the out- ,, put register to determine the piece of output equipment being dealt with. In this manner _ single instruction can ,ileal with the output needs of i, all the, equipment which receives information from the computer in thi_ system. ,# "_ ,_ '_011 ",l,,,,'z'_'-_ 1 1965011367-159 - 151 - Interrut_zation -- Interruptoperation allows the computer program to operate normally excel3tfoc the sho_ time ittakes to load the inputregister intothe computer memory. As the main computer program is running, a piece of auxiliary equipn_ent becomes ready to transmit a word so the computer. Operation is as follows: The auxiliary equipment sends an interrupt signal to the computer (a single line going to _ "one"). On receiving the interrupt, the computer stops executing the main program upon completion of the current instruc- tion. A particular memory address, determined by the interrupt, is read out. (Each interrupt line has associated with it one paz_icular merrory location. ) The word read out of this address is a jump instruction to a short subroutine wh_.ch will load the input data word in a manner prescribed by the suhroutine, Upon completion of the subroutine a jump is made back to the main program. Computer Descriptions for 12 ARS Earlier in the _udy, 12 ARS systems were synthesized on th_ basis of _rie variation of certain system parameters. It was also shown that, _n the basis of the computer requirements, there are only six different systems. In this subsection it is shown that three different systems of computer hardware are adequate to satisfy the computer requirements of the six s:mtems. In other words, some of tho six systems require identical hard- ware even though they do c_,mputing jobs of different complexity. Instruction Cou:_ -- As previously stated, an approximate program was written for the ARS computations. Not all of these computations are required of all s_x sy_0tems. The number of computati,_ns for each of the systems is grouped by computing function in Table 3-10. The letters A1, "42 through F at the extreme left of Table 3-10 serve as a k_y between the instruct"" count of Table 3-10 and th_ calculations indicated in Figure 3-18 and 3-19. qf_11 ".1 ""_'l"tl 1965011367-160 - ]52 - 20113-FRI 1965011367-161 - 153 - Some of the systems provide attitude and rate outputs while others provide only attitude. For each of those that provide only attitude, the figures on the left in Table 3-10 are multiplied by one. _or those that provide both attitude and rate, some of the figures are multiplied by 11, some by 10 and others by one. This is necessary because rate must be computed ten times as fast as attitude. Note that computational block A 2 has three sets of figures. The divide and square root opera- tions make up this gr¢_up. Since they are implemented by programmed subroutines, the number of iterations for a given accurac: depend on the computing rate; the lower speed systems ,.equire more iterations than the higher speed systems since the subroutine begins with a less accurate initial vaiue. %gain, three categories have been provided for the instructions -- one for !ong instructions, one for short instructions, and one for instruc- tions requiring a writt operation {these are all short instructions). A11 of the instructions ! ave been totaled at the bottom of the table. Also included are the tO*_Llnumber of constants that are required by the computations for each system. The results of this tabulation have been used to determine memory size and instruction complement, thereby cstablishing the word length. Memory Size -- The number of instructions in the pr._gram for the ARS computations required by each of the six systems is given in Table 3-11 along with the number of constants thatmu_tbe stored. The startup program would require about 75 instructions and subroutines, bookkeeping, and input-outputwould requ're about 70. Tile require- men_s for short-term or scratchpad storage vary between systems from about 40 words to about 100. These figures are tabulated ana totaled in Table 3-11. The actual memory size for each system (next highest power of two) and the number of bits required in the address field are also included in this table. Only two different memory sizes result, and fortunately the systems which can utilize the smaller memory are the less accurate system in which we can afford the smaller word length. 20113-FRI 1965011367-162 - 154 - Table 3-11. Storage Requirements J, __ ...... 1 _. 6, lb 3 I 4.8.12 5°9 7,11 _ mn nr l_&rl_ctions ARS Computers 300 175 2?5 100 275 250 Startup 75 75 75 75 75 75 Sub_ouUnes 20 20 20 30 20 20 fl<_okkeepin8 50 ,',0 50 50 50 50 Data Varttble 100 60 85 40 l O0 80 Constama 50 0 50 6 50 50 Total 5o_ 380 555 285 570 525 ,, , , Actual Number 1024 51: 1024 5i2 1024 _ 1024 in Memory I ! Address Word 9 1o I o j 1o Jo10Length (bits) I i Computer Speed and Instruction List -- It was previously shown that the ARS computations could be performed at the maximum required ra ! provided certain assumptions were made concerning clock rate, memo cycle tinies, and the list of built-in _,etructtons. It will now be shown that, for some of the _ystem requirements, multiplication and some of the other instructions can be programmed, r_ulting in a simpler ! control section in these cases. The following aasumptions were made to calculate the ti_-_e required tc perform the ARS computation for each of the six systems: • A 1.5-megacycle clock rate was assumed, resulting in a fJerial parallel word time (for a 16-bit word) of 4 micro- seconds. t 20113-FR 1 1965011367-163 - .155 -- • It was assumed that a nondestructive readout (NDRO) memory would be employed with a read-cnly time of about 1 micro- second and a write time of 4 microseconds. These are the times for a complete cycle. • Square root and divide computations wottld be programmed razlzer than built in, and multiply would also be programmed if ther-e we:,e sufficient time available. The number of computations and memory operations required for each of the six systems are listed in Table 3-12. Each of these are then multiplied by a number corresponding to the execution time oZ the operation and the total cycle time is obtained. The total time available is also given. Programmed multiplication is permissable in four of the six systems. Since there is still addi_onal time available, some of the other instruc- tions can also be programmed. It appears to be possible to reduce the number of instructions to eight in these cases, thereby reducing _e number of bits in the instruction code to three, The list of instruc- tions for the simpler systems would probably include the following instructions from the list previously given: 1, 2, 3, 6, ll, 12, 16, and 18. For programmed multiplication, the wired-in multiplication instruc- tion is omitted from the computer, which amounts to removing the _-register and a small amount of control !ogic. Computer Systems -- Combining the resultsfrom Tables 3-10, 3-11,and 3-12, the sixsets of dfflerentcomputer requirements canbe satisfiedby three computer configurations. The characteristicsof each of these configurationsand a listof the originalARS system numbers whose requirements are met by each computer configurationis given in Table 3-13. The appropriate gyro interZacesystem is also specified in each case. 20113-FR1 1965011367-164 - 156- Table 3-12. Computer Speed and Instruction Requirements ARS1 ARS2, 6, lO ARS_ ARS4, 8o12 A.qs5, 9 APS .. ',-.t ,gperat/_B (_ - 11) (_ - i2) (N- 16) (N - IZ) (N • lS} Oi ; lS)| II J..o.'_, (L) 775 618 95 82 718 ?-0 L x N 12, 368 7, 416 I, 520 984 II, 488 I, 440 Short (S) 2, 958 2, 928 356 348 2.848 346 w,,__ .0 e_6 .o . .4e . _)-,-_v{.].)-A 16..1 .. 28._ 2.o. 1.41, s,.m ...5S 8$4A (microseconds) 66, 644 45, 83_ 0,944 6,475 92, 420 7, 960 Prog mulUply timo 296.832 177, 984 39,489 23,616 275, 712 34, 560Lx Nz 6 x4 To'.al(microseconds)witbpro t mult_l._, 3e3, 476 223° 116 44, 524 I 30, 092 338, 132 42,220 iTime available(microseconds) 100,000 1, 00O, 000 10|}, 000 1, 0C0, 000 100, 000 100o 000.. ; Mult_oly Built in Program Program Program Built i_ Pro_an. ! : Table 3 -13. Final Computer Configurations LI Word Memory Computer System ,_ize Size Mult/pl'y Gyro i Configuration Numbers _ (._its) (Words) Interface _ A I, 5, 9 It' 1024 Built in 6 counters i- B 2, 6, I0 12 512 Program 1 counter C 3_ 7, I 1 16 1024 Program 1 counter ]: _ ii i i | T 20113-FRI t "t 1965011367-165 - 157 - TELEMETRY-COMPUTER INTERFACE The following schemes for obtaining useful ARS operating data via teleme,*xy link are only intended to show the flexibility of adoptin_ the ARS computer to the proble:u. Such unknown factors as available bandwidth, mission length, number of data points to be taken, etc., would determine the choice of a telemetry system. In each case data is obtaLned by reading certain selected memory locations. Telemetry Data Points by Program This _s the simplest scheme but requires that all memory locations to be read out must be selected before launch. In this method the main program within the computer runs in a normal manner. A jump instruction is placed at points within the main program at fairly regular interva] s (about every 10 seconds, for examp: 2). The jump will be to a short subroutine which controls the loading of the contents of one of a set of predetermined memory locations into telemetry equipment. A jump from the subroutine to the main program is made to :esume normal operation after loading is complete. This method of obtaining data for telemetry require_ additignal program space but no added hardware. Telemetry Data Points by Local Exerciser In addition to the need for selecting memory locations before launching this scheme also reqJires some additional hardware over the previous method. The block diagram of thisdata acquisition system along with the associated computer hardware is shown in Figure 3-24. The additional required hardware consists of the pulse generator and counter. (The telemetry register is con- sidered part of the computer because it is necessary for any of the telemetry data acquisition techniques. ) Operation is as follows: The pulse generator generates a pulse at the desired acquisition rate (pezhaps one pulse every 10 seconds}. This pulse increments the counter and also goes to the computer, where it is interpreted as an interrupt. Th_ ir.terrupt causes the computer 20113-FR1 1965011367-166 - 158 - MEMORY •¢, OUTPUT REGISTER | I .... I, L_ TELEiETRY I BUFFER ]REGISTER INTERRUPT j. I"! _ GENERATOR Figure 3-24. Telemetry Data by Local Exerciser II II I I I ,...u_' MEMORY " MEMORY_ OUTPUT ' i ; I I REGISTER Ti ...... I ' I I IMEMORY• ADDRESS TELEMETRY;_ REGISTER BUFFER " li TM Figure 3-25. Telemetry Data by Remote Interrogation L 20113-FR1 2 1965011367-167 - 159 - to enter a short subroutine which loads the counter into the memory address register, reads the contents of that address and transfers them into the telemetry buffer for transmission to the ground. The counter, then, is used as the address of words to be read out at a rate determined by the pulse generator frequency. The counter length determines whether all or just a portion of the memory will be transmitted. Telemetry Data Points by Remote Interrogation In this scheme it is intended that the address of the data to be transmitted is sent to the computer by telemetry link. Figure 3-25 is the block diagram for this method. The memory address of the desired data is transmitted to the vehicle. The address is then loaded into the telemetry buffer. This action causes an interrupt to be sent to the computer. The computer enters a short subrc_tine which transfers the address from the telemetry buffer register to the memory address, register. The data word at that address is read into the memory out- put reF, ister andis then transferred to the telemetry buffer. Finally, _:',aedata wo_-d is fed into the telemetry equipment for transmission. As in the other telemetry schemes, these functions are under control of a short subroutine. When the subroutine is comp]ete the main program resumes operation. 20113-FR 1 1965011367-168 - 160 - VEHICLE BODY RATE FROM THE END AXIS ROTOR PATTERN INT I_ODU 3T ION The purpose of this investigation is to determine if it z,_ feasible to derive a signal proportional to vehicle body rate from the output signal of the end axie rotor pattern. Previous study indicates that the noise present on the readout from the cosine pattern results in a derived rate of approximately 1. 4 degrees per second, Since it is desired to measure the vehicl.e body rate v_hile the attitude control system is holding attitude in a limit cycle, it is necessary to measure rates of the order of 0,001 to 0, 0001 de_rees per second, In the event the end axis readout is suitable for m3a.-uring vehicle body rates of this magnitude_ it may be possible t_ use an end axis readout during attitude hold operation and use the cosine pattern readout _ during attitude maneuvers i::volving t_:e nigh body rates. [. } EFFECT OF ATTITUDE ON DERWE RATE SIGb A._T Section IV describes the operation of the readout F,ystem using the end axis rotor pattern and indicates that the effective protion of the readout signal from the polar pickoff is <} [4b ]A 1 =_cos _ sin _rr sin (wt +4) (43) The quantities a and b are defined in Figure 3-26° r is the radius of the pickoff field of view, w is the rotor angular velocity, t is time and is the phase angle determined by the dir£2tion of Lhe pickoff displace- ment from the null position. For the purpose of th_s investigation both a and _bwill be assumed zero, which results in cansiderable mathematical simplification and does not affect the general nature of the conclusions. { 20113-FR1 1965011367-169 - 161 - SPtNAXiS A ,- ENDAXIS ._____, PATTERN /" _.-" -- --_r'_ ,- ROTOR, R.g...E_/ /. \ ,% v ,,--,_ _ ,_ lab _OTORPATTERNGEOMETRY (b) ENDAXIS GEOMETRY Figure 3-26oConvemional End-Axis Pattern and Pickoff Geometry 20113-FRI 1965011367-170 - 162 - Under these assumptions Equation (_3) becomes 1 _ot! (4i_A]! = _sin[_4--br sin . In Equation (44) the term b is p':oportional to the tangent of the angle of body rotation and is therefore proportional to the angle with an error of less than 1 percent for angles of 10 degrees or !ess. The output of fl_e synchronous demodulator is essentially the time average of Equation (44) over one-half cycle of_he square wave used to switch the demodulator. That is, ¢0 ( ! 4b ¢°ti A1 (4[-_ j 1 sir, _- sin dt : 0 The integrand in Equation (45) may be expanded in an infinite series ef Bessel functions as 1/2 sin (m sin wt) = __ J2n-1 (m) sin (2 n-l) _0t (4( n=l ; where ' 4b :" m - _r i Substituting EquaLion (46) in Equation (45) and integrating AI 2 _ J2n-1 (m)i: = -_, --(2n-!) (4_ i' n=l g [ 20113-FR1 1965011367-171 - 163 - Expanding the Bessel functions in powers of m the first terms of the series in Equation (47) becomes lira m3 m5 i1 = _" - _ + 1200 "'" _48) Differentiating Equation (48) with respect to time dA1 - 1 dm I m2 m4 1d't TT dt 1 4 + _-_ -... (49) From Equation (49) it is apparent that t,hc _ime rate of change of A1 is a function of the attitude (m) as well as tJ rate of chauge of attitude "1dm}-d_ . If "'_dt is to represent the attitude rate.._with an error of l percent or less, 2 m T_; 0.01 or m _; 9.2 40 since m - wr then ) , That is to say the attitude angle must not exceed 0.15 of the angle subtended ) by the radius of the pickoff field of view if an error of !ess than 1 percent is to be maintained in the attitude+rate slgnai. In the evpnt the attitude angle is equal to the angle subtended by the radius of the pickoff field of view the error in the .,-ate signal would be approximately 40 percent. In the event . 20113-FR1 1965011367-172 - 164 - the attitude angle exceeds the angle subtended by the radius of me pickoff ficld of vlcw, but does not exceed the limits of the rotor pattern, the attitude signal, (_A1), saturates; and any rat.e signal derivecl from it would indicate zero body rate. If the attitude angle excecds the limits of the rotor pattern tne readout system would give no irdorma_ion of either" attitude or body ,'ate. PICKOFF FIELD OF VIEW Fren, the above discussion it is apparent that the usefulness of both the a_titude and attitude rate signals obtained from the end axis rotor pattern is limited by the p!ckoff field of view. In turn the pickoff field of view is iimited by the sensitive area on the picKoff sensor. Sensors currently available for this application have a useable area with a diameter of : about 0.050 inches. The pickoffs as cur.'ently designed have a magnifica- I ., lion of 5 and therefore the diameter of the fie]d at the rotor is about 0. 010 inch. This subtends an angle of about 0.6 of a degree. Therefore, the radius oe the field of view subtends about 0.3 degree. Since the end axis ; pattern will be used for attitude hold type operation _'ather than for maneuvers, and the limit cycle amplitudes is typically +0.1 degree, this field of view would be sufficiently large to ensure presence of the rate signal. At the extreme of the limit cycle the ratio of the attitude angle to the angle sub- tended by the radius of the _ie,.d ef view, t_)' is 1/3 and m ': -"4- = _ " ,2 2 . m__._ _- 0.04414 f t ! • .] 20113-FR1 ,t! 1965011367-173 - 165 - On the basis of Equation (49) this represehts an error of about 4.4 percent in the attitude rate signal. Since the limit cycle rates are typically 10 -4 deg/sec the absolute error is approximately 4.4 (10 -6) deg/sec. In the event it is necessary to ha'_e the rate slgnai over a range cf attitude greater than ±0.3 deg, or if greater accuracy is required in the rate signal, it may be possible to redesign the pickoffs to have an o; tical system with a magnification of one. This would result in a field of view at the rotor equal to the sensitive area of the optical sensor -- 0. 050 inch in diameter -- which would permit an attitude change from *.he null position of ±1.4 degrees before saturation of the attitude signal. Therefore, the rate signal could be derived . in this range. At the pxtremes of a limit cycle with an amplitude of +0.1 " degree, the erro_ in the rate signal due to attitude would be about 0.2 per- cent. The absolute error for a limit cycle rate of 10 -4 de_/sec would be ' 20.0 -7) deg]sec. EFFECT OF NOISE ON DERIVE RATE SIGNAL The relationship in Equation (50) will be used to estimate the effect of readout noise on the attitude rate signal derived from the end axis pattern readout. _' n J where _n (_0) = power spectral density of readout noise 1 _R (w) = power spectral density of the noise on the derived rate G {_) -- Fourier transform of the system used to derive rate t Test information on polar pickoffs used in the ESG Monitor program indicate a noise level of the order of 5.0 arc seconds (rms). Assuming this noise is"white" its power spectral density will be a constant equal to the mean square of the noise. ! 20113-FR1 1965011367-174 - 166 - That is 2 ¢ = n (51) n ¢ = (5.0) 2 arc sec/rad/sec (52) n 2 n = meazl squared readout noise Assuming the rate signal is obtained by differentiation G (_) = j_ and (53) IG( )i2 = 2 From statistics, the average square of the noise on the rate signal will be given by: - /R: = 2_1 _R (w) dw (54) where R 2 = mean squared rate noise n Substituting from Equations (50), (51) and (53) into Equat!on (54) +m --2 n 2 f ¢2R n = _ dr0 (55) -¢O Obviously, the above integral is infinite if the entire frequency spectrum is . considered. However, for attitude control of a space vehicle only noise in i the bandwidth of the control system is of interest. If the bandwidth of the control sys*_em is _c/2 radians per second, evaluation of the integral in Equation (55) between the limits +¢0c and +w c will provide a reasonable estimate of the mean square noise on the rate signal. That i J, 20113-FR 1 i ¢ 1965011367-175 - 167 - +W n n2/°2= u du (56) -_1 C and +w -- -2 3 c2 n Rn = _ 3 (57) -W C or -_ --_ _J (58)Rn _ 3 Assuming the bandwidth of the control system is 5 cps, _ 60 rad/sec C and from Equaiions (51) and (52) "2 n = (5.0) 2 substituting in Equation (58) R = 57.4 x (arc sec]sec) 22 104 n R: = 4.5 X 10 -2 (deg/sec) 2 Rn -_ 0.2 deg/sec Obviously, the above noise level of 0.2 deg/sec will make it impossible to detect rates of the desired level of 0. 001 to 0.0001 deg/sec. In order to detect a rate of 0,001 degrees per second or greater Rn < 0.001 deg/sec 20113-FR 1 1965011367-176 - 168 - or R < 3.6 ar; sec/sec n and R-2 < 13.6)2 r! From Equation (58) .._ 3 rr _"_n_ 13 - 3 C substituting for B _ and n c -_ 3_ (3.6)2 n -- (60)3 "2 n = 5.75 (10 -4 ) n = 0. 024 arc sec (rms) This indicates that the noise on the readout must be less than 0.024 arc seconds if vehicle body rates as low as 0.001 degree per second are to be detected by differentiation of the signal from the end axis readout. CONCLUSIONS The principal conclusion of this investigation is that the level of noise on the readout limits usefulness of the rate signal derived from the end axis readout system. Also, the investigation reveals that the noise level must be reduced by a factor of approxima.tely 200 if vehicle rates of 0. 001 degree per second are to be detected. : Further, the investigation indicated that the optical pickof_ has an adequate _ field of view for deriving the vehicle rate signal during the attitude hold ! limit cycle. ?0113-FRI 1965011367-177 - 169 - Gk'RO CAGING AND TgRQUING CAPABILITY CAGING One objective of the SDMFG-ARS parameter study was to establish the utilityof SDMEG caging capability. ("Caging" means precisely aligning the gyro spin aTis with a line in the gyro case. ) When the attitude reference system study progranl was conceived itwa_ feltthat a caging capability in the SDMEG would permit simp!if!'_.'ionof the data processing required to obtain information useful to the autopilot or attitude control system. This is true to some extent; however, there is littleto be gained from a caging precision greater than that available from the use of the damping coils on the current gyro design. Theoretically, the caging capability will align the gyro spin vectors orthogonal with each other. Since the proposed calculation for the attitude error signal to the autopilot is _ --Q_*_l + I;_*;21_ %" ;' _;_(__";_)"Q2";1_s2 ;_(59_ where = attitude error Q1 and Q2 -- commanded positions of spin vectors in body coordinates Sl and $2 - observed positlons of spin vectors in body coordinates 20113-FR1 1965011367-178 - 170- The calculation is simplified to when the spin vectors are orthogozml and sl s2 -- o andI;1 x ;21 = 1 In the even_ the spin vectors are assumed orthogonal and the signal for the autopilot is calculated from Equation (60) {when in fact the sp_n axes are not orthogonal) the error in the calculated signal ib _e difference between Equations (59) and (60). That is - { -80 -- Igl x;212 : (61) If the angle between $1 and $2 is a, t then sin 2 I" 1 - ----2-- 2 1 2 1_In (62) Since i ISll--IB21 "_I_11--lq21- 1 i _. 20113-FR 1 ,_,_ 1965011367-179 - 171 - The quantities in the parenthesis approach cos a as the system approaches a null. Therefore, 2 I_1 _- cos _ cos_ + cosa (_3) sin _ 2sin Report 1726-FR1 on JPL Contract No. 950607 indicates in Figure 12-3 that gyro spin axis can be aligned to a line in the case within 0. I degree, using the damping coils of the current gyro design. Therefore, error in the ortho- gonality of the spin axes of the two gyros (due to caging error in the present I g'jro des_,_n) would be of the order of +0.2 degree. } Using a = (90 ° ± 0.2 =) in Equation (63) then I_ t m 12 x 10 -6radians This would obviously have negligible effect on the accuracy of pointing the vehi cle. Since the above is the error associated with the caging capability of _e current gyro design, it does not appear further development of caging will be required. Also, since the ca_ing is accomplished by the gyro damping coils, no further system additions will be required. When initially spun up the gyros are damped as a normal part of the start-up procedure. They will therefore be initially caged to the axis of their damping coils. If it can be assumed the spacecraft is held to the sun-Canopus reference frame by optical sensors during cruise, with the gyros used for attitude mar euvers from the sun-Canopus reference, the damping coils should be energized prior to such attitude maneuver to cage the gyros and remove any gyro drift that may have occurred. 20113-FR1 / ] 96501 ] 367-180 - 172 - TORQUING Itwas originally thought that having capability to precisely position the gyro _pin vectors in any selected direction {torquing) would simplify system cal- culations, and thus reduce system size, Results of the study did not bear this out. If gyro torquing were to simplify system calculations it would occur as a result of initialalignment of the gyro spin vectors coincident with two of the vectors defining the master or navigation reference frame. To do this it is necessary to have on board the orientation of the masLer reference system because the gyro torquing mechanism is fixed to the gyro case and thus the vehicle. This information is not generally available. In general the orienta- tion between the master and vehicle references is known only when the sun sensor is locked on the sun and the star tracker is locked on Canopus, with vehicle attitude being held by the outputs of these sensors. Under these con- ditions, gyro torquing is not necessary for initialalignment. Proper mount- i ing of the gyros enables alignment of the spin vectors with the master reference with the damping coils of the gyro as presently designed. The result of such an alignment of the gyro spin vectors is to reduce two direction cosines for each spin vector to zero. This saves four storage locations plus the time required for 12 multiplications and 12 additions, which is not a large or sig- i nificant saving. In the ARS configuration sele=ted for the parameter study, i tnese savings would occur in the ground-based equipment. ! : Another tentative application of gyro torquing was directed at the solution of i: a noise problem encountered during the parameter study. :_ In the course of the parameter study it was determined that the noise on the gyro _ readout from the coaine patterned rotor produced a threshold of about 1.4 degrees } per second on the derived vehicle body rate signal. Since itis desired to measure body rates of the order of O. 001 degree per second, this is excessive. However, it _ was thought that a cleaner rate signal might be derived from the end axis pattern readout. If this were the case, a gyro torquing capability would provide the means fo } I 20113-FRI 1965011367-181 173 - positioning the end axis pattern unc.er the pickoff at such tiraes a_ necessary to m,easurc the lower rates. Unfortunately an analysis of the process of deriving vehicle body rat,,from the end axis pattern readout did not reveal the desired improvement. This analysis is presented under a separate heading in Section Illof this report, Although the readout noise from the end axis pattern is somewtlat reduced, the rate signal threshold is stilltoo great to measu1;e the desired low body rates encountered during the limit cycle of attitude hold operations. The calculated,threshold due to noise was 0.2 deg/sec. CON CLUSIONS In view ot"the results discussed above, it was concluded that additional caging and torquing capability beyond that provided by the damping coils of the current gyro design does not produce a simplification of the ARS considered in this s_dy. 20113-FR1 1965011367-182 - 174 SIZE. WEIGHT, POWER AND RELIABILITY } The SDMEG attitude reference systems considered in the parameter study _ were divided into three m_jor subsystems to determine system size, weight, power, and reliability. The subsystems are: • Gyros • Compute r • Power supply The gyro subsystem includes the gyro mechanical assembly, the vacuum electronics, the suspension electronics, and the readout electronics that produce the start and stop pulses, The computer subsystem has a digital computer for processing the gyro out- puts to obtain vehicle attitude error and vehicle body raee signals suitable foe the autopilot. The computer also controls the gyros during startup The computer subsystem also has the interface electronics which condition the gyro outputs to be compatible with the digital computer input. ']"he power supply subsystem includes the equipment necessary to convert the primary power (2400 cycle, square wave} to the voltages and frequencies needed for operating the gyros and computer. i, SIZE, WEIGHT, POWER AND RELIABILITY OF GYROS Estimates of the size, weight, power and reliability of the gyro subsystem are based on the results of the study program co,:.tracted under JPL Contract No. 950607 and published in the reports 1726_FR1, 1726-FR2 and 1726-SR1. The estimates of the gyro electronicsassume pottedwelded module construc- ! tiontechniques. Microminiaturizaticn ofthe electronicswould probably re- sultin smaller weight and size. In order to accommodate the microminiature circuits,the present circuitdesigns would have to be modified or redesigned. However, under the ground rules of the system parameter study the SDMEG i is to be used as currentlydefined. 2011$-FRI 1965011367-183 - 175 - Since itwas concluded thattorquing and caging capabilitybeyond thatavail- able with the damping coilswillnot simplifythe system, no provision has 'beenmade made ".ortocquing capabilityin the size, weight, and power es- timates of the system. However, Table 3-15 does present the estimates of size, weight, and power of the torouinghardware foc information. The gyro size, weight and reliabilityestimates are summarized in Table 3-14. Table 3-16 summarizes g_'o power requirements. Tttble3-14. Summary of Gyro Size, Weight and Reliability Size Weight Reliability (in3)_ Obs) {_ Tailures/I000 hrs)i "_Mechanical Assembly 65 5.0 i.4704 Suspension and Readout Electronics 60 5.5 1. 5399 Vaouum Electronics 8 0.7 0. 0678 Total for one Gyro 133 11.2 3. 0781 Total for two Gyros 266 22.4 6. 1562 i- Ullq m. i I Table 3-15. Sizing Estimates for Gyro Torquing Hardware i i i, J i i i i ,. ............ ; ........ m i i , Size Weight [ Power:, Coars..e Drivers 7.4 m 0.74 lbs. 3 wattsFine _rivers 18.7 in 3 1.9 lbs. 8 atts 3 Torquing Amps 50 in 3 10 lbs. 30 watts TOTALS 76 in 3 13 lbs. 41 watts (6 VDC) ......... , _ __ i i i1| 20113-FRI 1965011367-184 - 176- f c m Table 3-16. Power inputs Required for SD MEG per Gyro Function Power Form One Gyro Two GyrosWatts Watts! ,= • l Suspension _22.5 vdc; 500 ma; l_ reg; t 22 44 : 0.I_ripple ._ High g Mode • . _ Suspension _6 vdc; 300 ma; l_reg; 3.6 ? 2 0.1 rippze ': Low g Mode t Spin 50 v; 1 a; 2400 cps; 20 40 Variable level; 2 phase Damp 6 vdc; 1 a; 104 ripple; 6 12 variable level ! PO Lamps 4 vdc; 420 ma; variable 2 4 level current; i; O. 05_reg Preamps :_22.5 vdc, 20 ma; 0.5 1 : l_reg, 0. l_ripple i Vacuum Pump 3 vdc; 1.5 a; controlled 4.5 9 I Filaments by grid current Vacuum Pump +180 vdc; I0 ma; I_ ripple; I. 8 3.6 t Grid 10_reg; controls filament curr_nt Rotor Cagi.g Same as damping 6 12 Maximum 38.4 76_ 8 Load . ip • i i i ..... i ,, J A J 1965011367-185 - 177- COMPUTER SIZE, WEIGHT, POWER AND RELIABILITY In the discussion of the ARS computer, three computers A, B and C were defined. It was indicated that these three computer designs would satisfy the data processing requirements of the 12 attitude reference systems being considered in the SDMEG-ARS parameter study. Component Counts Approximate component counts _ave been _nade for each of the three computer subsystems. It was assumed that integrated circuit elements would be used wherever possible. At the present time, all computer circuits are available in integrated circuit form, except for memory drive circuits. Thus, inte- grated circuits were assumed for the logic elements, flip-flop, and sense amplifiers, and con_.,entional, discrete element circuits for the memory drivers. Although identicsl circuits were r,ssumed for each system, this need not be the case. System B could be opera'_.ed at a lower clock rate than the others, since there is considerable extra drne remaining in the computer cycle. The lower speed circuits that could be er_ploYed as a result of lowering the c2ock rate would cut the pov, er requirements considerably and possibly increase I the permissable packing density i The component counts are given in Table 3-17. Each section of the computer hasbeen divided into sub units, each consisting of several stages of similar logical design such that the number of components in each sub unit can be more readily i estimated. Flip-flops and special elements (such as sense amplifiers) have been separated from the loj_ic elements because their power consumption generally differs. It is also possible that shift registers may beco_ne available with several stages on a single ¢l_ip. This would reduce the size and increase the reliability of the computer. 20113-FRl 1965011367-186 - 178 - 30113+F1_1 1965011367-187 - 179- The total component counts show that systems B and C do not differ greatly : in complexity, while system A is approximately twice as complex as the others. _Ihese results are about what might be expected. Elimination of thei t builtin multiply (and other in_trnctions) results in considerable hardware savings in the arithmetic and control sections. Systems B an_i C differ only in word length and in the size of the memory. Only one input-ou_l_ut register was included in these totals. It was assumed that the remaining regi_ters would be associated with the equipment with which they communicate. '_bis approach was taken because the number o_' registers and their structure wo-Jd depend on ',he design of the associated equipment. In order to compare these systems with similar systems constructed from c_.nventional discrete component circuits, the Table 3-17 figures were translated into equivalent transistors, diodes, and resistors. The resultin_ figures are given in Table 3-18. Although a few additional capacitors and. o*,her components may be required, those listed in +he table constitute most of the system com- ponents. The relative complexities of the three system remain about the same as in Table 3-17. Table 3-i8.Component Count for Mechanization with Discrete Components m t Transistors Diodes Re:_istors 1 System A 3. 346 14, 120 8.'898 System B 1,378 5,644 3, 726 I System C 1. 644 6, 568 4, 4':16 t't :i ,] -J '1 -: 20113-FR 1 1965011367-188 - 180 - Si_ze, Weight, and Powe r CompariBons Size, weight, and power estimates were made based on current state-of-the- art packaging schemes for integrated circuits, drivers, and memory. These figures, given in Tables 3-19 and 3-20 do not include power supplies. Table 3-19. Power Requ_ements for Computer Configurations 50 vdc 10_Duty Cycle Computer 4 vdc 12 vdc (amp) (amp) (amp) Peak Avg Computer A 3. 0 3.6 1.0 O. I Computer B 1.0 2.7 1.0 0.1 ! Computer C 1.5 2.7 1.0 0.1 T f i Table 3-20. Size, Weight, and Power Comparisons ' ' SyStem A Syst_n .B 1 System C ' System $|_e '' Weight Power $ise WeIghl Power I "Slze Weight Power Components . l (cu. in. ) (Ibl) (watts) (cu. in. ) 111_1 (watt._) I lets, in. ) fibs) (watts) L Integrated C_reult 66 I. 33 12 26 0.52 4 30 0. _ 6 Elements ._ Tran,.lstor Drl_re 117 2 i U 8B 1, 5 37 117 2 38 Circuits i: Memory 30 l 20 O. 78 30 Ii Case, Connectors, 175 8 125 6.3 150 7 etc. - - 1Totals 388 12.33 [ 60 259 9,1 41 32? 10.6 t 'i 1965011367-189 I: 181 - The ratios between the figures for the three system_ are much closer in terms ; of size, weight, and power thsn they are in terms of complexity. Reason for this difference is that the figures for the driver circuits, memory, and case (where appropriate} are much greater than those for the integrated circuit elements and their supporting cards. As a result, the integrated circuit elements, while varying in number between systems more than any otner part of the system, contribute only about 10 percent to the size and weight figures and about 25 to 50 perce,t to the power, figures. Reliability System reliability is expected _o increase considerably with greater use of integrated circuits. Each integrated circuit element may replace 20 to 50 discrete components and their interconnecting leads and will be much more reliable than the circuit it replaces. In the future an integrated circuit may have several shift register stages ov a single chip and the system reliability should increase still further. All of these expectations depend on an ultra- reliable method of interconnecting the circuit elements. Althouth methods have been proposed to achieve these goals, none have been tested under _ctual "in service" conditions. In calculating the reliability for these three systems, failure rates given in the current literature as estimates of those expected from integrated circuits were used. Each driver transistor was assumed to have a failure rate about three times greater than a logic transistor or an integrated circuit because driver transistors operate at higher voltage and current levels, The driver li circuits have additional elements that result in a further increase in the over- all failure rate. For the majority of the interconnections, a specia, technique, l1 such as the Honeywell MICPAK, was assumed. The composite failure rate , for each of the three computer configurations is summarized in Table 3-21. I.i 11 20113-FR1 1965011367-190 - 182 - Table 3-21. Reliability Comparisons Failure To_'Failure" ' - TOtal _,_qt,u-e J IRate I,_teNo. of Urdt_ (_/X, Hrs.) No, of Umts Rate Failure katr i• (_/_. Hrs. ) No. of Unit e {_/K. Hrs. )(_,IK. I|rs. ) ! V i Inlcgrated Circuit 0.002 678 1.352 254 _. _,08 | 307 O. 81.I ["}t n_PP.tS D, :,cr T ranslslor 0.008 184 1.472 132 1.056 184 1.472 Clr';'uJl5 ( onn..c lion-; 0.00005 BS00 0.440 3800 0. 190 4600 0.230 Coluf,uf, r F.ly, lu,-c 3.264 1. 754 2.31 (J ILIt_" ( '/K.Hr_. } _[ i ii POWER SUPPLY SIZE, WEIGHT, POWER AND RELIABILITY ESTIMATE The estimate of size and weight for the SDMEG power supply is based on the use of _ separate supply for each voltage required. They will _" be packaged in a singlehousing, includingthe a.-cspin supply_ : All the supplies operate continuously;only loading r_quirements vary. Therefore, the _witching logic could be placed on the outputs of the supplies. Re_uLationwould have to be such as to maintain constant _ voltagethroughout the load range of each supply. The 6 cdc to the damping circuit requires a voltage build-up and tail-off. This can easily be accomplished with z_n R-C network on the suppty's output. ,V i The +4 vdc to the p'_ck-off lamps must be variable through some range, necessitating a variable output from the regulator. _ The spin suppl.y will use the 2400 cps input power, which will be fed into _ a wave shaper i:o produce a sine wave from the input square wave. It _ will then be fed through an amplifier into a phase splitter. The outputs 20113-FRI 1965011367-191 - 183 - are fed into power amplifiers and thus to the spin coils. The buildup. and decay of spin power can be accomplished with an R-C circ,_t in the t biasing of the input amplifier. The input to the supplies will be through a transformer. Each voltage required will l'.ave its own secondary windings. The transformer _!11 suffice for both isolation and voltage cond/tiordng. The three voltages needed for the computer will be non regulated. Input voltage is regulated and tolerance on these supplies is 10 percent. There- fore, no regulators should be necessary. It is estimated the power supply will weigh 12 pounds and have a volume of 240 cubic inches. Although the three computer configurations use slightly 0ifferent amounts of power, the difference is not sufficient to make a significant change in the power supply size, Also, it is estimated the _-_ power supply will have a failure rate of 1 percent per 1000 hours. The estimate includes the following: ±22.5-vdc supply for gyro high G suspension and pickoff preamps; a ±6-vdc supply fo_" gyro low G suspension and damping; s precision 4-vdc supply for pickoff lamp filaments; a 180-vdc supply for the vacuum pump grids; a 3-vdc supply for the vacuum pump filament, a 50-volt, 2400-cycle, two-phase supply for gyro spinup; and 4-, 12-, and 50-vdc supplies for the computer. SUMMARY Although the parameter study began with 12 different configurations for the _'I'} attitude reference system, it was concluded thet gyros designed with precision torquing and caging ca_,ability did not permit syslem simplifi- ! !_ cation. Therefore, it was assumed all attitude reference systems would use the same _'os, rather than three different gyros. Thu_0 the i original!2 :-)'stemconfigurationswere reduced to four. Itwas then concluded ,I 201 !3-FR 1J 1965011367-192 - 184:. t}l,it Hl:'(.(- COml)utcr configurationswould satisfythe requizements of _he v(:,11ai,,i,,gsystems. Therefore, three differentattitudereference systems r(sul1(.d,allusing *he same gyros and gyro power supplies, but differing i,_the computer and computer power supply. The size., weight, power and reliability estimates for the gyro, computer, and power supply subsystems were presented in Tables 3-14, 3-16, 3-19, and 3-20. The information from these tables has been combined in Tables 3-22, 3-23, and 3-24. Table 3-22. Summary of Sizing Estimates , i i i i I Size Weight Input Power Reliability (in. 3) (pounds} (watts) (_/1000 hours) Standby Maximum ,, , ,, , , ,, , I'rli - -- i G>r'os " 266 22.4 7.2 77 6. 1562 t'(,,,q)utcv A 388 12. 3 55 50 3. _.640 Con_puter B 259 9. 1 36 41 1. 7540 ! 327 10.6 49 54 2. 3160(',,,nl)utt'v C i H,)w('v Supply ! 240 12.0 1. 0000 -- i ,H i ll,i i i ( ,,t,,I)ining the above information the estimates for the resulting three attitude i I'('['('l'['|lCe systems are: t Table 3-23. Sizing Estimate for Total Systemsi i i ,.. .,,, , i il ] InputPower ] • • -Size Weight (watts) Rehabllity i Standby Maxi ,, , , :::_:__ :.... , ...,,37,, , ,,i ........ S,,._t(,,nA I 8_8 46.7 62 I " ! 10.4202 ._.,._le,n B I 765 43.5 43 ] 118 { 8.9102 ._ystemC 833 45.0 56 I 1211 9. 4720I,HW I I,,,,I ......... ,n j I I [ , i _01 _8*,lilB 1 1965011367-193 - 185 - Table 3-24. Sun:mary of Size, Weight, Power and Reliability for AR._ ..... I Size Weight Input Power Reliability System I (zn. 3) (pounds) (watts) (_/1000 hours) ,, , Standby Maximum ARS No. 1 898 46.7 69 151 10. 4202 ARSNo. 2 765 43.5 48 130 8. 9102 {, ARS No. 3 833 45.0 62 144 3.4720 ARSNo. 4 _65 43.5 48 130 8. 9102 ARS No. 5 898 46.7 69 151 10.4202 J ARSNo, 6 765 43.5 48 130 8.9102 AHSNo. 7 833 45.0 62 144 9. 412 _ ARSNo. 8 765 43.5 48 130 8. 9102 J ARSNo. 9 898 46.7 69 151 10.4202 ARS No. 10 765 43,5 48 130 8. 9102 ARSNo. 11 833 45.0 62 144 9. 4720 ARSNo. 12 765 43.5 48 130 8. 9102 i, .J Table 3-22 summarizes the sizing estir:lates for the gyros, computer and l power supplies. Table 3-23 presents the sizing estimates for the three distinct ARS configurations resultint; from the study. Table 3-24 indicates ! system sizing estimates relative to the original selection of system con-'1 figurations. 2"z_!s table shows that as a result of the conclusions, original configurations 1, 5, and 9 are the same size; 2, 4, 6, 8, 10 and 12 are 'J the same size and 3, 7, and 11 are the same size. ,1 I 20113-FR1 1965011367-194 - 186 - FILTERS FOR BODY RATE ouTPuT INTRODU CTION Use of the SDMEG for spacecraft attitude reference and stabilization requires deriving the body rates by differentiating the available body attitude signals or using a pseudo-rate type control law. In any practical application, both types of control would probably be most effective. Any spacecraft maneuver requiring constant rates, typical of acquisition search patterns, necessitates deriving or approximating the spacecraft body rates. Since most spacecraft have severe fuel limitations, the_,c rates must be controlled at all times. The nature of the SDMEG readout and the proposed following computational electronics indicate that, for large body rates, satisfactory rate information results using the straightforward difference equation: ' -i] 1 - z ,,,,, [£1 T whe;-e ' [X ] = a belayattitudecohunn vector ' st Z = e T :, sampling period As _he body rates decrease to the point where only a few of the least signi- ficant bit_ of the input buffer counters differ, the r_te computation (Equation 7) becomes highly sensitive to the input counter noise caused by ball pattern ,:trots, etc., and some other filter is required to provide, or avproxi- !L }_ ma+,e, the actual body rates. For instance, if the act_l body rate is zero 1965011367-195 - 187 - N _, 2n differed inand the input counter (z eprP.senting a binary number a n n =0 the LSB during successive sampl:ng periods, the indicated rate wou]d be 2A where ! A is the scale factor of the LSB I If A is 36 arc-seconds and T is 0.01 second X. = 7200 arc-seconds/second = 2 deg/seci 1 i Obviously this introduction of noise is intolerable in any spacecraft applica- I tion where the body rates are on tb_ order of < 0. _t - 1.0 deg/sec. The following preliminary analysis is intended to provide relative tradeoff 1 data on the abzliLies and requirements necessary to provide an adequate rate I signal from a SDMEG. Comparative data ultimately to be established includes rate limitations and computational requirements fo=: I ® Digital pseudo rate • Derived rate using Optimum differentiatir, g filter !; Averaged rate computations Differenced - lagged attitude signal DIG!TAL PSEUDO RATE Stable low rate limit cycle operation is possible using a pseudo,-rate app. oath in much the same fashion as in continuous systems.,* The flow diagram shown ,i i, i i i n i *Nicldas, J, C. ; Vivian, H. C. ; Derived-Rate Increment Stabflization.JPL TR 32-69, 31 July 1_6l. Cronquist_ 1t. G. ; We!p, D.W. ; An _nalysis of Compensation Techniques and Sensor Nonlinearities in a Mass Explusion S_agle Axis Satellite Att:tude Conzrol System, TheLAs GGC/EE/64.-6, USAF-IT; June ]964. 1965011367-196 w _ v f in Fig_2re- 3_ 27 pro_des bot h. Stabili_zation-in the large, an d-convergence...t0 limit cycles wRh-rates on theorder of .0,.001 deg]$ec, Again this Computation, 17 ! which compares succeseive values of a filtered input attitude.signal, will be_ i h_iuenced by .counternoise and .w_:d prob_bly ff_c_on bctter v_h a lower ,. : .sampling rate. The low rate limitcycle is achieved by comP&r[ng-csucees_ive differences to a prestored number and, in the event R is exceeded, enabling the reaction system and updating the prestored number as shown. If.the-re- action system is not enabledduring a sample period,, the_prestored number ._ is replaced by its original value, Figure 3-28 is a typic_ phase plane por- I trait oi a limi.t cy,_le acqu_itiov, using 'this control law,. A1thoug h this com- putation will not provide rate--information exp]icitJ,y,- it permits satisfactory i control- signals for both hold _nd maneuver modes. ; DERIVATION OF BODY ._RATES Typical Inpu_ Buffer Noise J The SDMEG outpu_ buI'fer counter -will accumu_te pulses to represent a number _ 2n. In the event that the output attitude is drifting at a low constanta n n_0 rate, the absolute value of *,he counter contents will increase or decrease at the spacecraft body rate. If the body rate is near •zero the buffer counter LSB varies randomly due to system defficiences with equal probability of T'ecording a o = +1, 0, -1. The input attitude signal appears as shown in Figure 3-29.This noise signal n(t) is represented by an auto-correlation function. ¢ (_) = n(t) n(t+_) -- nn = 3_ (T- IT]) x <_ T; -166), = 0 V >-%', i 1965011367-198 2- 189 - | lr"_'°*,=._<"1"""'- -- 1 _lll' _j y,,il, Jill .i YES .- _ f' Jim 1 : Q _lVlill>TailiI_O YE_"x - K1 > 1 13 11 /__. p(n)K2 . I Lp--,,n,.. , I/r,.._ _ ].,-.c__ -2.A -A_ 0 J_ 2_ T T T T n --_ SPACE Figure 3-35. Effect of Finite Memory Averaging Rate Filter on Quantized Noise Filter 20115-FRI 1965011367-213 - 2q4 - where A and B. are specific constants1 1 ST z = e T = Sample period S = Laplace transform variable The operational requirements of the above filter are clear when the nature of the delay operator z -1 is considered. From Laplace transform caiculus r eccl. _ that _C[f(t-T)] = e-sTF(S) = z-IF(S) Thus, multiplying the function F(S) by z -n implies that the time function fit) has been delayed by nT. For a simple case where A 0 + A1 z -I G(z)= -i = 1 + B 1 z e i Thus -I -i e0 = A 0 ei+A 1 ei z _ B I e0 z or e0(nT) = A 0 ei(nT) +A 1 eii(n-l)T] - B 1 e0[(n-l)T] 20113..It RI 1965011367-214 - 205- From this above expression it is clear that implementing this first-order transfer function requires: • Storing three permanent constants (A0, A 1, B 1) • Storing two temporary data points _e i [ (n - 1) T] , 90 [ (n - 1; T] multiplications I(A0. ei), [A 1 • e i(n-1)], [B 1 • e 0(n-l)] I Three • Two additions to form the desired result e0 (nT) • Miscellaneous program instructions to accomplish the above operaticn A sample computer program fo, a first-ocde:" transfer function that uses an in- struction repertoire inherent to most spaceborne computers is shown be__ow - Instruction Comment LDA e. e. => A 1 1 MULT A 1 A 1 • (g) = A 1 e: => A STA M (A) = Ae. -->M a . a ADD M 3 (A) +A 2 • eiz-I -_>A SUB M 2 (A) - B 2 e0 z-I = e0 => A LOR R i (/Q = e 0 => R i (This is the transfer function, filter, output. ) 20113-FR1 1965011367-215 - 206 - Instruction Comment MULT B 2 B 2 • e 0 => A = => M2STA _2 (,a) B 2 el) LDA M a (M a) = A 1 ci => A MULT A2/A 1 A2/A 1 • (A) = A 2 ei STA M 3 (A} = A 2 e i => M 3 Continue !t can be seen that rather than storing both A1 and A2 in the permanent memory, A 1 and A2/A 1 are stored and multiplied times A 1 e i at the appropriate time° This approach enables use of fewer program insturctions to accomplish the de- _l_ed result° In addition, for higher order transfer functions, it will reduce the _emporary memor T locations required by about a factor of two, This stmpie example illustrates the techniques required to implement a sampled data difference equation. The advantage of this approach over a standard nu- merical method is that the impulse-modulated difference equations (z transforms) permit exact representation of the continuous function at the sample periods. More important, all the analysis and synthesis tools available for frequency domain studies also apply to the sampled system studies. Continuous system analysis techniques are made applicable for the sampled system by elementary conformal mapping techniques. By letting the new complex variable ST : 2o is-F 1 1965011367-216 207 - The entire complex S plane is mapped into a new complex z plat;e, as shown in Figure 4. The entire left half of the S plane is included w_thit: the unit circle 0f the z plane. All S plane root locus techr:iques apply in the _ dnmain analogous to the S plane. If the z plane is mapped to another complex plane by the bilinear transformation l+w Z - 1 -W The z plane transfers to the w plane, as shown in Figure 5. The w domain is generally used for frequency response (bode plots) analysis. The advantr _,e of using the w-z domain for synthesis and analysis is that the transformation is linear between them. Thus, individual compensation blocks can be manipulated without influencing the entire transfer function, as in the S -. z plane case. This elementary discussion of sampled data analysis and app:lication is intended to serve as a brief introduction. For a more thorough discussion, the interested reader is referred to the works of Ragazzini and Franklin, Tou, Kuo, Truxal, Monroe, etc. i 1965011367-217 208 - Im $ _m Figure 3-36. S _ z Plane Map Im z ]m Figure 3-37. Z _ w Plane Map 1965011367-218 209 .-- CHE CKCUT O£' SDMEG-ARS INTR GDU CT iON /elemetry fur the monitoripg of the gyro operation was discussed in Report No. 1726-FR2, on JPL Contract No. 958607. The following discussion is concerned wt_h telemetry for the monitoring of the attitude reference system computer and the gyro-computer interface logic. The SDMEG-ARS parameter study is principally concerned with unmanne(l spacecraft apphcatlons; therefore, any verification of the system operation must be through ,*.Memetered data. However, since the telemetry capacity available for t).t,_ function is quite limited, it is essential that the minimum amount of da:.a necessary t'._ establish the status of the system operation be transmitte.:i Ideally, the telemetry between the spacecraft and the earth-based facility would provide the capability f:,r: • Detection of failare • Isolatmn and analysis of failure • Elimination of failure Although it would be desirable to provide all the above capabilities for the SDMEG-ARS computer, the elimination of a fail.ace requires redundancy in the system. Since the current study assume_ a non-redundant system, the data celected for. telemetry from the ARS computer will be selected to provide the first two capabilities. 2011 _-I,'R 1 1965011367-219 - 210 - CHECKOUT AND MONITORING OF ARS COMPUTER Itis proposed that two methods be used for deiection of a failure in the ARS computer. Since the ARS will not be used continuc_usly:but will be put in service at such t_mes as it is necessary to change the spacecr-,_ftattitude from the attitude provided by the Sun and Canopus sensors, the computer should D_: put through a checkout routine before each use of the ARS. Also, during operatlon of the ARS in control of the spacecraft, data should be telemetered to verify that the computer and control system are functioning properly. The computer checkout routine would be the same problem the computer solves in normal operation. Data would be stored in the computer memory to simu- late the 9u_put of the interface logic. On command from the central controller and sequencer, the program for computation of the gyro spin vector direction cosines would process the simulated data, and the results would be te)e- metered to the ground facility. On the ground, the correct results for the test problem would be compared with the _elemetered data to establish whether or not the computer was properly performing the direction cosine calculation. A similar procedu,'e could be used for checking the attitudeerror and body rate con'nutatio_. However, httle or no additional infolmation would be obtained about computer operation. The same computer subsystems are used fer the direction cosine computations, the attitude error computation, and the body rate computation. It is extermely unlikely that the computer can fail on any one of these programs without failing on all three; therefore, it should only be neces- sary to process mforma_ion through one of the programs to ensure that the computer .is functioning properly. Te monitor the actual operation of the ARS, it is suggeated that the outputs of the direct)on cosine computations, the attitude error computations, and the body rate computation be telemetered to the ground facility. The frequency with which this information is transmitted is not critical; however, it should be several times the limit cycle frequency of the attitude control system, and syn- chronization with the limit cycle should l,e avoided. i I 20113-FRI } 1965011367-220 - 211 - The output of the direction cosine computation w!ll be six binary numbers repr_.senting the direction cosines of the two gyro spin vectors in body coordinates. This :nformation could be checked on the ground in two ways: The sum of the squares of the directio,_ cosir, es for each spin vector should be equal to unity, and, in the steady state, the measured spin vectors should equal the commanded spin vectors. These checks should be made with some tolerance. The magnitude of the tolerance will require further investigation but will be a function of the accuracy of the ARS and telemetry, The output of the attitude error calculation will be three binary numbers. These numbers represent commands to the autopilot or attitude control system and ideally in the steady state are zero. Since it is anticipated that the spacecraft steady-state condition will be a limit cycle, the attitude error signal will oscillate, and a check of the telemetered output of the attitude error calculation should show this oscillation. The output of the body rate calculation will be three binary numbers repre- senting the components cn the body axes of the angular body rate in inertial space. Since the spacecraft attitude control system will establish a limit cycle about the commanded attitude, the body rate output of the ARS should reflect the limit cycle rate and frequency after the system has had sufficient time to respond to any commands. These characteristics may be used to check the vahdity of _he body rate calculation on the ground. CHECKOUT OF GYRO-COMPUTER INTERFACE LOGIC The gyro-computer interface can be checked by providing simulated start- stop pulses at the input and observing the resulting output from the counters in the interface logic. If the computer is checked out prior _o the interface ]og_c, it may generate the start-stop pulses and provide thE; interface with the telemetry system for transmitting the counter outputs to the ground facility. Since the time between the eimulated pulses is known, the correct counter outputs are known and can be compared with the data received on the telemetry at the ground facility. 20i13-FR1 i 1965011367-221 - 212 - SUMMARY The preceding paragraphs suggest that the checkout of the ARS computer and gyro-computer interface logic be accomplished by telei:_eteringthe result of a checkout routine to t|_eground facility. Also, it is suggested that the computer operation in the contrul system be monitored by telemeterir_g the outputs of the dlrection cosine cgmputation, the attitudeerror computation and the body rate computation. The suggested checkout and monitoring schemes are feasible and can readily be programmed on the AR[_ computer described in this report. The actual details of the checkout program, _uch as initiation,termination, etc., will require further definition of the interface° of the computer with the telemetry system and the command system. i 2OlI3-FRI i 1965011367-222 213 - SE CTION IV SDMEG ARS F(R A CAPSULE LANDEF( INTRODUCTION AND SUMMARY Task II of the SDMEG ARS study contract requires the synthesis of an attitude reference system for a specific mission. This mission is that of a landing capsule which is separated from a parent space vehicle. After separation, the capsule is two-axis stabilized (pitch and yaw) to the constant ESG spin axis position with roll rate stabilization only. Rate signals for autopilot s_abili_ation are also required in the pitch and yaw channels. The major requirements for this system, as specified by JPL, are presented in Table 4-1. This section of the report discusses the synthe,_is and preliminary design of a system to meet the given specifications. The forrn this system assumes is shown in Figure 4-I. The gyro ,s the same as the one presently being developed for JPL, with two minor modifications. Two pickoffs would be required instead of three and a more easily applied rotor pattern would be used. The gyro electronics (suspension ,_ld vacuum) are also the same as the SDMEG electronics presently being developed. Only the readout electronics have been extensively revised to meet the ARS requirements specified by JPL. The selected readout electronics are comprehensively described as part of the system description. The projected physical characteristics and major performance parameters of the selected gystem are presented in Table 4-2. The system was synthesized primarily to meet the attitude output requirements of Table 4-1, and comparison of the two tables shows that this goal was accomplished. (In Lhe absence of direction, the performance specifications of Table 4-1 were assumed the equivalent of three sigma values, and the values listed for comparison in Table 4-2 were also calculated as three sigma. If the performar._e values of 20113-FRI i 1965011367-223 - 214- Table 4-I. Specified Requirements for a Capsule Lander ARS Time perxods of the mission Delivery to earth launch - 6 months to I year Earth launch to gyro spin up - 200 uays Gyro spin up to capsule aeparatio. - ,3 hours mac,mum Cal_.sule separation to retrofire ignition - I0 to 30 minutes Retro firing time - 10 minutes maximum End of retrofire to ena of control requi, ements - 5 to 20 days Expected environments Acceleration - separation - 0. Ig retrofire - 0.5g Rates - separation - 2 deg/sec retrofire - 2 deg/sec ARS attitude output requirements {pitch, yaw} Forrrl - Imalog Accuracy i 0. I del to end of retr_Ire ± _ deg after retrofire Threshold 0. 05 deg Linear Range ± 3 de_[ Usable Range {Saturated silInal, but _Ign sense :_.tained) ± 10 deg Bandwidth 15 cycles/r.*cond Quality (L e. allowable peak to peak noise over frequency range) 0.05 deg up to 30 cpS 0. I deg 20 cpS and over ARS rate output requirements (pitch, yaw} Separate signal desired Form - analog Accuracy ± 0. 05 deg/sec Threshold 0. 02 deg/sec Linear Range + I.5 deg/sec Bandwidth 5 cycles/second Quality 0.02 ¢_eg/sec up to 20 cps 0. 05 deg/sec 20 cps and over ARS rate output ._equirements {rol 1) Form - analog ; Accurac.y I Provlde signal with s_gn information Linear Bange (nr, magnitude required) whev-ver Thresho!d roll rate excd.eds I. 0 ± 0.3 rad/secUsable Range BandwidLh Quality if aralog General ARS requirements Power Source 2400 _ O.Of%cycles/second square wave Rise and fall times 5 ± 4 microseconds Amvlltude controllea to produce 50 volts RMS + 37.absolute Spikes 5 ,,c,lts maximun" amplitude 5 microseconds maximum duration Packaging Any assumed technique must be of standard practice by January 1966. The assumed technique must be capable of surviving the oterlhzatlon and launch environment requirements given in JPL Spec Nos. GMO_.50198-ETS-A, XSO-$02?5-TST-A, and $0350B, respectively. i I 20113-FRI 1965011367-224 215 - l ii I L AUTOMATIC GYRO I_ STARTING _- i ELECTRONICS ELECTRONICS ' ' / ROLL RATE COMPUTER ROLLRATE i i ii DIGITAL _ ANALOG L PITCH CONTROL ." ATTITUDE - ERROR LOGIC • COMPUTER i 'r S_TRA'-'--" RATE PITCH COMPUTER RATE PPED I DOWN ., . COMPUTER RATE II i i"t -DIGITAL : ANALOG I YAWCONTROL ATTITUDE | :_LOGIC _ COMPUTER1 ERRORii I Figure 4-1. Two-Axis ESG-ARS Block Diagram 20113-FR1 1965011367-225 - 216 - "_able 4-1 were intended as one sigma the same comparison can be made by dividing the projected performance shown in Table 4-2 by three.) However, me accuracy and quality of the rate outputs are not as good as desired. As shown in the performance analysis, this is a consistent result; for the type of noise in the SDMEG ARS the attitude and rate performance requirements of "Fable 4-l are not compatible. The tollowing discussion is divided into these four major topics: • The analysis and reasoning leading to the selection of a particular system configuration is presented • Tile system's general requirements are translated into specific hardware requirements; both the required gyro modifications and readout electronics are described o A performance analysis is used to estimate the performance of the selected system • The selected systems-'s reiiabihty, size, weight, and power requirements are estimated, and along with the basis for these estimates are presented and discussed. 20113-FRI 1965011367-226 217 - Table 4-2. Summary Cbaracteristics for Selected ARS Physical Charactecistic_._ 3 Size Gyro Assembly 104 in. Electronics ..t_'sernbly 160 in 3 Weight Gyro Assembly 6.0 lbs. Electronics Assembly 8. 5 lbs. Power Requirements Steady State 11.9 watts Peak Power 53.7 watts Reliability 0. 9814 (5 day descent) 0. 9523 (20 day descent) Performance Parameters (3a) Pitch and Yaw Attitude Outputs Accuracy End of retro _:0. 074 deg. 5 days ±1.44 deg. 20 days +,5.76 deg. Qual_ly less than 20 cps 0. 027 deg. over 20 cps 0. 002 deg. Pitch and Yaw Rate Outputs A(curacy +0. 19 deg/sec Quality less than 20 cps 0.37 deg/sec over 20 cps 0. 048 deg/sec R:)ll Rale Outpu_ The reqmred .+.0 3 rad/sec roll rate accuracy can h _. supplied for only the first 24 hours after separation. SYSTEM CONFIGURATION 1w() bas_, types of systems can be considered for this application, with the _ystem _.ype dictated by the basic gyro configuration. The first type, employing a ¢on:,entlona] SDMEG, is basically a digital system s:nce the conventional g_ro reqmres exlens]ve digital data processing to provide intelligence. Digital 20113-FR1 1965011367-227 - 218- data processing could tben be used to c_culate error signals, end a digital-to- analog converter would be necessary to obtain the required analog outputs. A functional bluck diagram of such an ARS system is shown in Figure _-2a, The second type of ARS assumes a modified SDMEG readout system witl_ either a direct analog readout (e. g., end axis pattern) or a pulsed readout system which could be used *o directly control an analog error angle calculation. $ functional block diagram Jf an AIRS which assumes the latter type of modified MEG readcut system is shown in Figure 4-2b, The functional blocks of Figure 4-2 which change significantly between the two systems are so indicate_ Even a cursory examir,ation and comparison of the two types of systems indicates that the synthesis of a system with the desired simplicity depends in turn on using a simplified ESG readout system. The following paragraphs present the analysis used to select a particular ESG readout scheme. The discussion is divided into two parts - end axis patterns, and equatorial patterns. END-AXIS PATTERNS One ESG readout mechanization which has been copsidered for application in the subject attitude reference system is the end-axis pattern. The following p_:ragraphs discuss both conventional and nonconventional end-axis readout te chnique s. The geoxaetry of a conventional end-axis readout pattern and pickoff are shown in t_._gure 4-3. Let A be defined as the ratio of the high reflectivity rotor surface in the pickoff field of view to the total field of view area. 1 X VI_ X 2 1 sin-I XA =-_ + ........_r 2 + , r ' r _ X <_r (84) r 20113-FR 1 i 1965011367-228 - 219- *'_fll '1q.,"_-+ I 1965011367-229 - 220 - bPIN AXIS ENDAXIS PATTERN DEMODULATOR _OTOR, REFERENCE MARK5 .,.._*'" "t f 1 1_ ROTORPATTERNGF.OMETRY PICKOFFF|E.D / OF VIEW "_ X PATTERNEDGE \ / VIEWLENTER "" _' SPINAXIS (b) EIMDAXISGEOMETRY Figure 4-3, Conventional End-Axzs Pattern and Pickoff Geometry !. 1965011367-230 - 221 - where r = pickoff field-of-vlew radius For a uniformly responsive pickoff, the pickoff output is proportional to A. Considerable analytical simplification results ifthis function is approxi- m:,ted by the center slope matched sinusoid. [ IA1 = "2" .1 + sin _ ; -r < X < r (85) The difference b=tween Equations (84) and (85) is shown in Figure _i-4. Since the rounding of the sinusoidal function representing A 1 (relative to A) near I:_1 = r approximates the expected effects of diffraction and residual aberration at the edge of the pickoff field of view, the response represented by A 1 is likely to be closer to reality than the exact expression. The variation of X with time will be X = a + b sin (tot+ _b) (36) where a, b = (shown in Figure 4-3) to = rotor speed $ = arbitrary phase angle dete,-mined by direction of pickoff displacerc, cnt Substituting Equation (86) into Equation (85) then yields: z lr,--" l_'_ sin (tot + + cos _'7 sin sin (tot + _ (87) - j 20113-FRI i 1965011367-231 - 222 - 1.0 _.i __1" • - -_ LIN.V._,,_OUTPUT -_ / t i 09 -- ..... / _'- • _ 1 x " 1 -1 l i I i /I i [ 1 'Io._ ..... , I , o.:_ o.4 o.6 o._ ,.o l., ,., _.6 Figure 4-4. Pickoff Output for Conventional End-Axis Readout A simple and effective method to get the desired information from this signal : is the use of a square wave switched synchronous demodulater. A block • diagram of such a system is shown in Figure 4-5. The reference square t : waves mu.sL of course, be generated from the output of a second optical pickoff's viewing reference marks on the rotor (eee Figure 4-3) so that this square wave is of the same frequency as the polar pickoff signal. Since this synchronous demodulator rejects all d-c and even harmonics, and a term of the form coe[K sin (_t + _)]consists of only even harmonics, the it effective portion of the signal represented in Equation (87) may be written as: ]- -_ cos • r L;'_it/ 20113-FRI ! 1965011367-232 - 223 - Figure 4-5. End-Axis Pickoff Readout Electronics The zero-degre _se and 90-degree phase components of this signal (outputsj of the synchronour --,. nodulators) wouid then be directly proportional t() tUei ! pitch and yaw error a " desired for the attitude reference system. It .should be noted that the din,.nsion a , which is caused by improper pattern ] applicatien, does net affect null position or signal phasing, but mereiy result._1 in a slight change in sensitivity. An indicatiou of the linear range of the polar pickoff readout scheme can b(_ 1 obtained by examining Figure _-4, More accurately, the ideal linear output J may be defined by the slope of Figure 4-4 at X = 0 as t 1 2x ; A2 = g +--rrr (89) and the linear range, as a function of percent linearity P, is then found by solving } 2X l 4X A2 - A1 nr 2 .sin n'-rP(lO -2) = - 2X (90) i AI n-'_ .! 20113-FRI i i 1965011367-233 - 224 - For 10 percent linearity 1:he solution yields X/r equal to 0.62. Since present pickoff technology limits the optical pickoff field of view to about 10 nil-inches, this result indicates a 10 percent linear range of about ±0. 175 degree for a 1-inch radius rotor and a Lotal range in which the signal output is at all pro- poz'tlonal to displacement of +0. 286 degree (equal to pickoff field of view). These extreme limitations on linear range make the conventional end-axis ,:eadouZ scheme unsuitable for the attitude reference system being studied and have led to a consideration of other end-axis readout schemes - particularly mechanizations in which the linear range of the readout system is not limited by the pickoff field of view. However, no completely applicable readout scheme based on end-axis patterns has been envisaged. As an example of the difficulties encourtered, the readout m'..chanization associated with the modified end-axis pattern shown in Figure 4- 6 is briefly discussed below. The e,_d-axis pattern of Figure 4-6 is applied so that the relationship between the rotational angle ¢ and the error angle 7 (total pickoff displacement) is given by = A + K 7 (91) This simple readout equation would allow a corresponding simple readout mec':_aniza_ion using a combination of analog and digital circuitry. The actual mec hanization of such a readout system would be similar to that presented in the d[scus3ion of Equatorial Patterns (following subsection). The _hortcoming of such a readout system is obvious from an examination of Figure 4-6b,which i_ drawn approximately to scale for A = _/2, K = 18. The finite line width of the pattern [iine widths of about 8(10 -3) inches ere presently used] causes a proportional readout null uncertainty. For a 1-inch radius rotor, the cor- ;.i responding null uncertainty of nearly 10.5 degree is excessive for' the attitude _ reference system under consideration. 20113-FRI 1965011367-234 - 2_.._ - SPINAXIS I (a) ROTORPATTERNGEOMETRY PlCKOFFPATH "-/ (b) ENDAXIS GEOMETRY I J -, Figure 4-6. Modified End-Axis Pgttern Geometry I i l 2011_-FRI I 1965011367-235 - 226 - The possibility stillremains that a dual range end-axis readout system -- usinR the conventional end-axis pattern to provide the required null accurac_ and tilemodified end-axis pattern to provide the required linear range -- could be mechanized. However, the apparent complexity of such ", _ystem, the difficulty of matching scale factors in the two ranges, etc., die-rates investigating alternate approaches first. EQUATORIAL PATTERNS l'attern and pickoff geometry for a very attractive ESG readout mechaniza- tion employing an equatorial pattern is shown in Figure 4-7. This pattern appears desirable for the ARS under consideration because of the relatively simple readout equation. The assumed patterning equation is: _ = _r(i - -_) ; -B _'_ B (92) _PlH AXI& --_"_UATOR_PICKOF NO. 2 CONSTANT LAT ITtIDI[ LOCUS L PATT(RN } Figure 4-7. "Constant Slope" Equatorial Pattern Geometry 2011S-FB, I J • i i 1965011367-236 - 227 - Variation of the parameter B in Equation (92 trades cff sensitivity (scale factor) for linear range. Since the deriv_t: -e of the included rotational angle ¢ with respect to the pickoff displa,.ement angle _ is a constant, this type of pattern may be named a "constant sloI_e" pattern. The two pickoffs shown in Figure 4-7 wouldbe mounted parallel to vehicle pitch and yaw axes; hence, angle _ sensed by each pickoff would be a direct measure of the re- quired a_titude reference system error angles. For the rotor pattern of Equation(92), linear range and usable range are the same and are equal to approximately B (ignoring linewidth effects). Selection of the constant B requires consideration of not only the desired readout range, but also the manner in which the readout errors are affected by this constant. Since the derivative of 7 with respect to @is directly proportional to B. and ¢ is the angle which is actually sensed in the readout scheme, any increase in B will cause a proportional increase in the readout angle error's caused by inaccuracies in determining ¢. It is therefore desirable from the standpoint of accuracy and sensitivity to keep B as small as possible consistent with the system linear range requirements. The readout equation to be mechanized in the readout system is readily obtained by solv!ngEquation (92) for y. Figure 4-8 demonstrates a conceptual (single-axis) readout electronics n_ech- anization which could be used to solve this readout equation. Since, in terms of the pickoff output of Figure 4-8, T 1 ¢ _ 2_ -}- (94) where _ = rotor period Equation (93) may be rewritten as 7 = B I'I" 2T 1 T (95)l 201 ] 3-FR1 i 1965011367-237 - 228 - (a) PICKOFF OUTPUT (b) COMPUTING ELECTRONICS Figure 4-8. Conceptual Readout Mechanization Equation (95) can be solved by a computing mechanism such as shown in Figure 4-8. The switches(digital controls) are closed for times T 1 and ",', respectively. The outputs of the integrators, when operated upon as shown, then yield the requited error angle 7. Two such computing channels, operating on alternate rotor revolutions and discharged between each operating cycle, would provide a continuous analog attitude signal for one channel of the ARS under investigation. The extreme simplicity of the ESG readout equation and the resulting simplicity of the readout mechanization has been obtained, at least in part, at the expense of a relatively complex rotor pattern. The simplest ESG rotor pattern to apply is a great circle pattern since it requires rotating the ESG rotor {or patterning tool) about a single axis. Any pattern which is not a great circle requires (at least theoretically) simultaneous rotations about two axes. The difficulties associated with the precision centrol of two simultaneous rotations greatly increases the complexity of the rotor patterning process. Fortunately, the 20113-FRI 1965011367-238 - 229 - "cons*_ant slope" rotor pattern of Figure 4-7 can be very accurately approxi- mated, at least in the critical region near _ _ 0, by a great circle pattern of low inclination, The appropriate readout equation for a great circle pattern is: _1 = tan -1 [tan i cos ,_1 (96) where i = great circle inclination angle For the type of attitude reference system under considecation here, where the primary concern is accuracy near the null (_ = 0 or ¢ = _), the "best" approximation is obtained by matching slopes at the critical point. From Equations (93) and (96) d_ B d_ 1 / 1 Ld-'_ = _ - de _,_ 2 tan i or -1 2B ,',97)i = tan 17 The accuracy with which the great circie pattern of Equations (96) and (97) approximates the "constant slope" pattern of Equation (93) may readily be calculated as: AY = Vl- _/ = tan-1 <_- COS _}-BI1 -¢ ) (98, This error is plotted in Figure 4-9 as a function of Vl(the actual readout angle) for several different values of B. The 10 percent error line iucluded in Figure 4-9 as an aid in interpretation demonstrates the extreme nonlinearity of the readout error contributed from this source. As expected, increased linear range (by increasing B or i) can be obtained at the expense of scale 20113-FR1 i 1965011367-239 - 230 - " r T ! - o,---4 1 o,_ }---q/-..... 0.4 " -- O.! "! J 0 1 J 4 $ 4 _I (_ZG.R[[S) Figure 4-9. Error Contribution - Great Circle Approx- imation of "Constant Slope" factor sensitivity. The errors involved are such, however, _hat for the attitude reference system under consideration reasonable pattern linearity can be pro-- vided for the total required usable system ra,_e (±10 degrees). This would be accomplished by selecting a great circle of suitable inclination (approxi- mately 7.5 degrees} and patterning the rotor in the critical area near ? = 0 with this great circle. When the systematic error of the great circle pattern approaches that which could be easily obtained (requiring no expensive pattern development or patterning equipment or procedures) with the more complex "constant slope" pattern - about 0.1 degree - the rotor pattern would then be extended to approximate the "constant slope" equatorial pattern. Even a piece- wise linear (continuous) approximation to t,he "constant slope" rotor pattern would be adequate. These considerations lead to the type of pattern shown in Figure 4-10. In addi- tion to the combined great circle -cons*.ant slope rotor pattern, a reference line (pattern line along a rotor merldian) has been added to this pa_+.ern. 2011S-FRI 1965011367-240 - 231 - CONSTANT5L0Plk PATTERN _%-_ 12") (''7"6°} [ H GREATI_IRCbEPATTEL _--_. ? _ ) ] _u"...___ -'_/ :o,,._[t,,_oPE*T'rE,,_ / \ i I (al TOPVIEW _ "/ (b) FRONTVIEW It) SlOEVIEW Figure 4-10. Combined Great Circle - Constant Slope Pattern 20115-FR1 1965011367-241 - 232- This reference lineis usefulin decoding the information in the readout pulses (eogo, w_thoutthe reference line,how can the pulse at which the two-tirne • t'" " _' measurements ofFigure 4-8 are initiatedbe iden.lfled,i. Careful study of the pattern represented _n Figure 4-10 reveals one major deficiency - it does not exhibit the required type of symmetry. Since the gyro will have to be s_arted remotely, by an automatic starting system, it is de- slrab]e that the pattern be equivalent (fulfill the same readout equation) when the rotor is spun in either of the two possible orientations. Further explanation of this consideration, is contained in MH Report 172G_PRI0. If the pattern line from which the times of Figure 4-8 are measured i_ defined as '_bat pattern line immediately following the reference line (the reference line can oe detected, for example, by pulse width discrimination), the pattern represented in Figure 4-10 then fulfills Equation (93)in the orientation shown. However, in the other orientation the readout equation would be This is effectively a 180-degree phase change in the indicated output (error) angles. The:'e are two immediately obvious methods which can be used to remedy this deficiency. First, enough intelligence could be included in the rotor pattern _o sense rotor orientation (e. g., two reference lines - one wiaer than the other) and the readout angles can then be switched to reflect the rotor ori entation. Second, the rotor pattern could be modified so that it exhibits the desired sym- metry. A pattern which, with no additional patterning difficulty, exhibits this symmetry is shown in Figure 4-11.The two. pattern halves are again combination great circle -cor.stant slope patterns, with one pattern half rotated by an angle 2i _bout a line through its center. 20113-FRI i 1965011367-242 - 233 - 0 ,.4 / \ I _ _' +.. 0 r_ 4.1 r/l % °,-4 °o-I I1) I _. I _In I \ / N J rli_l 1 1 *) _,11"_ 1 1965011367-243 - 234- i 7 if the parameters associated with the pickoff output when using the pattern of Figure 4-!1 are defined _.s in Figure 4-14, the readout equation to be solved may i be shown to be ?b (tOO) This is exectlyof the same form as the previously developed reaaout equation IEquation (95)]. Hence, an identical (conceptual) computing mechanism can be used to effect the required solution. Some of the practical patterning details associated with the limited range equatorial patterns are also presented in Figure 4-11.First, the line width of the "constant slope" pattern is represented as 6(10 -3 ) inches. This width has been specified somewhat narrower than the usua_ pattern line width of about 8(10 -3) inches in order to minimize the "effective" width of the pattern -- the rotor rotational angle included between the constant lati'ude points on opposite sides of the pattern line. For a 1-inch radius rotor the 6(10 -3) inch pattern width still has an "effective width" of about 5. 15 degrees. Therefore, in order that the rotor pattern satisfy Equation (i00)ineither of the two possible orientations, it is necessary that the trailing edges of the two reference meridians (trailing edge triggering in the electronics is assumed) be separated by 5.15 de- .grees. The 6(10 -3) inch line width of the constant slope pattern is considered near the minimum of what can be tolerated without encountering severe sensitivity to noise (particularly rotor blemishes) in the readout system. Even this line width supplies only about 70 percent of the maximum possible output of an optical p_ckoff with a ] 0(10 -3) inch field of view, thereby making use of pulse width discrimination in the readout system desirable (th_,s type of discrimination minimize_ the sensitivity to noise spikes}. A preliminary inves*igation of the complexity of the computing electronics neces- sary to decode the pickoff signals from the rotor patterns of Figures 4-10 and 4-11 showed very little tradeoff on this basis, The necessary electronics for the 20113-FR 1 1965011367-244 - 235 - two-rotor patterns is very nearly the same. The pattern of Figure 4-11, however, has two very definite advantages. First, a data rate of twice per revolution (instead of once per revolution for the pattern of Figure 4-i0) can be provided. Second, the rr_a:dmum t_me delay in the basic readout system is limited to the ESG rotor period (instead of twice the rotor period for the pattern of Figure 4-10),These two advantages have subsequent desirable effects on the system's frequency response, the ease of differentiating the attitude data to obtain rate information, aria the ease of designing an adequate attitude control system for the carrying vehicle. Because of these advantages, and because the pattern of Figure 4-11 appears to provide more than adequate accuracy for the application being considered (this point will be further investi- gated in a performance analysis), it is the one which has been selected for further development. 20tI3-FHI 1965011367-245 236 - SYSTEM DESCRIPTION GYRO MODIFICATIONS The functional characteristics of the SDMEG for the subject ARS will necessarily differ from the standard SDMEG being developed for JPL (see Honeywell Reports 1726-FE1 and 1726-FR2) in only two ways: • Rotor Pattern • Optical Pickoff Configuration The selected rotor pattern was described in a previous paragraph of this report. The s implicity and low accuracy requi.rements of the selected rotor pattern should resul*, in a correspondingly small patterning development efforL As a matter of fact, substitution of this patter_, for the present cosine pattern repre- sents a significant gyro simplification. Furthermore, it is obvious that this pattern requires only two optical pickoffs (instead of the standard three) to accomplish the readout process. Hence, the additional gyro simplification of omitting one of the optical pickoffs would be uaec for this ARS. In addition to the above indicateU modifications, it might also be desirable to include provisions for mechanically adjusting and stabilizing the ESG damping coil alignment in the gyro design. 'rests have shown that the ESG spin axis repeatably cages to an orientation accuracy of about one arc minute during : damping. Since this one-_r _ minute accuracy is accurate enough for this ARS_ precision caging would probably be used for initial spin axis alignment. The ; above caging orientation adjustment would then be used to align the initial spin axis orientation parallel with the vehicle roll axis (normal to the plane of the two optical pickoffs). 20113-FR1 1965011367-246 - 237 - E LEC TR ONICS DESCRIPTION The operating principles of both the standard electronics associated with every ESG - suspension electronics, vacuum electronics, optical pickoffs - and some _f t},_ sp_cial purposc electronics necessary for this ARS - remote starting electren_.cs - are avmlable m other documents and wi[i not be repeated here. The ph}s_cel characteristics of all these electronics are included as part of the s_ze, weight, and power summary. The, follow.n R paragraphs present the operating principles of only the elec- _ron:cs which are ,_lque to this ARS configuration, the gyro readout electronics. The presentation _s d,vlded into two major headings, attitude calculations and rate c_Iculationso Altitude Calculations The gyro readout equation to obtain attitude information from the ESG was given in Equation (100), and a conceptual mechanization of the computation electronics which could be used to solve this equation was presented in Figure 4-8. The most difficult part of these electronics to mechanize -- at least with an analog mechanization -- is the indicated division. A considerable eiectromcs s_mphficatmn can be effected by recognizing that, for the period of t_me which the sub]ect ARS must operate (5 to 20 days), the rotor period in the' denominator of Equation (100) can be well approximated by a constant° If 7bt _s the assumed constant value, the readout equation may then be written as B 2Tl_i• ,%- Sinc.. _ the factcr B/Vbl is constant, the solution to this equation no longer re- quir._s dw:siono Typically, an ESO with the configuration of the JPL SDMEG wo,dd be expected to have a rundown rate of one-half rp_ per day or less, causing a dlfference of less than five percent between the actual and (assumed) 20113-FR1 I 1965011367-247 - 238 - constant rotor period for a nominal 200 rps operating speed. Furthermore, the approxfmation used tc obtain Equation (101)is seen to cause no nulloffset or error; itmerely resutts in a fivepercent sensitivity(gain)variation. The operational characteristics of the assumed electronics components used to synthesize a functional block diagram af the readout electronics are pre- sented in Figure 4-12. Both analog and digital circuits are shown. This figure is not meant to represent functioning of the circuits _hich would be used in an actual design; the represented building blocks are merely idealized logic ele- ments which aid the synthesis of a fun,:tional block diagram. An analog computing module which could advantageou31y be used to solve Equation (101) is presented in Figure 4- 13. The operation of this circuit may be readily understood by first considering only the upper one-half of the network and the corresponc/ing digital control waveforms of Figure 4-14. Assume the integrators have zero initial conditions,, Then, at a slight incre- ment of time after the reference pulse (wt = 0 + c in Figure 4-14) control signals _ and 1_ switch appropriate reference values to the inputs of the two integrators. Subsequently, with the same time lag at t = T 1 + c/w signal _ removes the input to one of the integrators and at t = Vb* _]b) signal 1_ removes the input to the remaining integrator. Also, at = 7b/_), signal Q turns on the summing amplifier so the difference of thet two integrator outputs is transmitted to the system output through an appro- .. priate gain constant. This output is maintained for one-half a rotor revolution. Then, at t = 360 ° /w, signal Q removes the integrator outputs from the system output, signal @ is used to discharge the integrators and restore zero initial conditions, and the cycle is repeated. The remaining half of the computing module of Figure 4-13 performs exactly the same function for the ....... buihalternate half cycles of rotor revolution. The lag e .,,l,,,.,_ i= into the sys- tem provides a means of discharging the integrators without introducing any significant ransientsintothe system output,and allows the entirecomputing job (forone readout channel) to be done with justtwo identicalcomputing cir- _ cuits. ) % 1965011367-248 - 239- ANALOGCOMPUTINGCIRCUIT SIGNALCHARACTERISTICS III II e0 eO ": i 0 r ec = "0"e2 ec _ I "t el ec " KL dr, = "0h • 0 = 0, ec = "1" • i • DIGITALLOGICELEMENT SIGNALCHARACTERISTICS "0", ei < ek; ei o-- SCHM;DT .-o •o e° = el<,THETRIGGERVOLTAGE, TRIGGER. "1", ei > ek IS VARIABLE I I I! 0 II i r ONE _ "1"s.o .Oo,o I--i I L__ llOII THE PARAMET'_R"rQIS VARIABLE AND_ATE _ i = = "11." • 1 o--.. "1", IF • 1 1 ANDe2 = "0" e2 o-- e° _o = "0", IF EITHER• 1 = "0" ORe2 ;1_,,,)-'_---I- - o.AND- "0 ORGATE "0", IF •1 e2 e_o.-, e° e°" "1", IF EITHER e1= "1" OR s2 = "1'' nll_ el _ T FLIP "0" -- - I I I I '2 _- RFLOP_ -"_'4 .,"1" V" _"'L_I iIIOU m e4 IS ThE COMPLEMENTOF e3 Figure 4-12. Pl_tform Components for Readout Electronics 20113-FRI I 1965011367-249 - 240 - THENUMBERSINCIRCLESPROVIDECROSS-REFERENCE TOFIGURES4-14-AND4-15 r >-l, , , ,] _,)@@ Figure 4-13. Analog Attitude Computing Module Functional Block Diagram i /-_111;3,,,1_'_1 I i 9650] i 367-250 - 241 - __._ T1 _'---_ b I I t _,l _ 1 L .. I ] r -7 , i 1 _-] I _ I i _ ! I ' • I i 11 --I I I L '13 ! ,, I l 15"_l I , t I i i I 1 0 100 20;J 3OO t (DEGI_E£S THE NUMBERS Ik CIRCLES PROVIDE CROS_-REIrER_NCE TO FIGURE8 4-13 AND 4-14 Figure 4-14. DigitalControl SignalWaveforms 20!I3-FRI 1965011367-251 -- 242- Altcroately, al_ a.alog attiiude computing module which employs only one- ,half of the computing circuitry of Figure 4-13. in conjunction with a "sample and h._,ld" type of circuit to provide a continuous analog output, co,_ld be: mechanized. A brief investigation of this concept indicated that calculations w(:u!d the,l be accomplished only on alternate cycles of the ESG readout signal.% and that n,) appreciable equipment savings would be effected. H_mce, the system of Figure 4-13 was adopted. rhe digital control logic ased to generate the necessary control signals for the analog attitude computing module is shown it. Figure 4-15. The operation of_his circuitry, when examined in conjunct_en with the waveforms of Figure 4- 14 and the individual element characteristics of Figure 4- 12, is l'elaliveiy straightforward. However, several special features built into this coWrol logk: deserve fr, r:her atten;:ion. First, a pair of one shots (output signals Q and Q )',:reused to provide the lag : discussed in th,_ pre- vious paragraphs. Second, the reset signals to the three flip-flops on the exlrcme right side of Figure 4-15 have no effect on the circuit during steady- state operation. These reset signals do, however, provide synchron:.zation when the circuit is fir._t started so that the initial acqt'isition process is entirely automatic. _hird, separate triggers for the reference and constant slope pattern pulses are provided so that different voltage triggering levels may be used. The Height [,asensitive Trigger (see MH Report 1726-FRi) for the constant slope pattern pulses is used in conjunction with a pulse width discri- minator to provide rejection of all but the desired pulse. The triggering level for this pulse may then be set at quite a low value without introducing noise susceptability, thereby allowing use of a narrow pattern line. For the (assumed) nearly constant rotor _peed operation of ESG and the pattern geometry of Figure 4-! 1, the pulse width discriminator of Figure 4-J,5 need be nothing more than a simple lag. The Schmidt trigger used for the reference pattern line would be set at a higher level; the circuits a_tached _o the ouLput of this Schml0t trigger , pro ¢:de very good noise rejecticn and will. operate properly regardless of whether or no: the incoming pulse_, from the constant slope pattern trigger the circuit. \ 20113-FRI 1965011367-252 - 243 - a _ m 1965011367-253 244 - Rate Cal¢ ulations Th_ required pitch and yaw rate signals are obtained quite simply by "differentiating''the output of the (corresponding) attitude computing module (Figure 4-13). The clrcuit chosen to approximate the required differentia- tion is shown in Figure 4-16. The _ransfer funcUon of this lead-lag clrcuit is simply N O _ E i _'(1+ K) _ S+ 1 ff-_-S + 1 HK K This circuit is used because it cap be adjusted to closely approximate an ideal differentiato:: over the frequency range of interest. Selection of the break points for this differentiating circuit i._ partially a matter of judgment. The rate signal has a required bandwidth of 5 cps. The magni!_: "'_ of the lead-lag circuit approximates the magnitude of the ideal differentiator at the 5 cps point to about one percent if the frequency ol the lag is chosen as 30 cps. A practical value for the (low frequency) lead is about 0. 01 cps (7 _ 16 seconds) indicating a required gain in the forward loop of about 3(103). The gain ,qt_,bility of this amplifier should be about one percent to achieve accuracy compatible with the differentiator approximation. The adequacy and p_rformance of this approximz_tion to the desired di;ferentiator is further dis- : cussed in the system performance analysis. For the third vehicle axis a signal with sign information (no magnitude) is required whenever the roll rate exceeds 1.0 + 0. 3 radians per second. Thi_ rate signal is ,much more difficult to derive than the other two. Two possible methods for obtaining this signal have been investigated, and neither method ts wholly satisfactory. The first method assumes that the speed of the ESG ! rotor is known; the vehicle roll rate is then obtained by comparing the Y, {, 2CI13-FR1 _'-----"----"" .: :i ...........-¢_._-_._z_.... _'.-_:, ,-_--:: _ ......... •.... ,_ _2_ : -_ -._-- _ -_i -_,_ ' 1965011367-254 - 245- (a) Rate Computer Block Diagram Eo - w1 , / i 'r I I i .... I __ log w i _i K T" w- RAD/SEC 7 (b) Rate Computer Transfer Function Figure 4-16. Pitch and Yaw Rate Computer 20113-FRI 1965011367-255 - 246- measured speed and the known speed. The second method ass-rues that the ESG is mounted as far as possible (more than ,,ae foot} from *.he roll axis of the capsule lander. Roll rate is then sensed by measuring the associated centripetal acceleration. The first method has an immediately evident limit&tion. It requires a kmow- ledge of the ESG rotor speed to the same accuracy as the rate calculation. The JPL SDMEG configuration is expected to have a rundown rate of about 0. 5 rps per day. When the effects of unknown magnetic fields, temperature variations, and possible pressure changes inside the F_G envelope are all considered it seems unlikely that the gyro rundown rate can be predicted better than about ten percent, or 0. 05 rps ;'or day. tlence, the precise rotor speed knowledge necessary to make the required roll rate measurement (0. 3 rad/sec = 0. 05 rps) would not be available for more tha,, about one day of ARS operation. Given that this one day of operation is adeq Jate, the rate calculation still per- mits a formidable mechanization task. The required measurement accuracy of ±0. 3 radians peL second requires measuring a nominal rotor speed of about 100 rps to a precision :_ 0• 05 percent. This requirement meano that analog techniques simi!ar to those previously employed cannot be used here, Also, the use of familiar demodulation techniques to detect the difference between a (for example) 100-cps reference signal and 100. 15-cps rotor speed signal cannot be used since they do not supply sign information. The bloc': diagram of a mechanization which could be usect for the rol! rate calculation is shown in Figure 4-17. • The _ariat_le frequency pulse generator ! and clock drive are pre-set to generate a pulse train pro_rtional to nominal i ESG rotor speed. The pulse fre4uency would be gi,,sn by: ! where: * (Oro initial rotor speed u = nominal rotor rundown rate. r 20t13-FR1 1965011367-256 - 247 - L "-L_ z - I " L, i i 0 C._ tJ_ o ,_ =i _ o I , ,_ _ " ,j _ ._ '_ • , _ I ._ Wa.Z L_J ,_, ° m "} 201 t3-FRI i 1965011367-257 - 248 - This pulse generator would typically consL_t of a voltage cL,ntrolled or capaci£ance controUed oscillator fol]o'_edby appropriate shapL-ig circuits. AND gates numbers 1, 2, and 3 and 1lie two flip-flops, are used as control ele- ments. The signals which provide Lhe con_;ro!function are J_ustrated in FigL,-e 4-18. Signals _ and Q a_'eboth one pulse per rotor revolution signals, but are 180 degrees out of phase. Signal Q is used to control the pulse counter so itcounts only on alternate rotor revolutions. During the first half of the rotor revolutions when the pulse,counter is not counting, signal Q enables AND gates 4 and 5 so that the remainder of their inputs may be examined. Then, signal Q resets the pulse counter and IrE No. 2 so the cycle may be repeated. The p,,lsecounter circuits and AND gates 4 and 5 are interconnected so that AND gate 4 is fulfilledifthe pulse count is less than _\TI,and AND gate 5 is fulfilledifthe pulse count is greater than N 2. For eitiherof these conditions, corresponding to roll rates outslde the accepted limits, a square wave output will be supplied from the appropriate AND gate. -_or a p_alsecount between N 1 and _72 the veLicle roll rate will be less than the specified amount, and zero outputs are provided. In order to further clarify the roll rate calculation, a numerical example is presented in this paragraph. Assurae an ESG with its rotor speed set at ex- actly 100 rps at the beginning of the mission and a nominal (pre-measured) rundown rate of 0. 5 rps per day. I'he variable frequency p-lse generator, with a nominal frequency of one inc., wiP. then be adjusted _o provide a pulse rate output varying from 106 pp_ at the beginning of the missmn to 0. 99(I06) pps at the end of the first day. This frequency is chosen high enough so the normal counter uncertainty of ±1 pulse causes an insignificant system error. 1 In the _ of a second rotor period 104 pulses will occur, hence a 13-stage counter with a total counting capacity of 16, 383 pulses is indicated, The ±1 radians per second roll rate limits correspond to an apparen_ rotor speed change of ±0. 159 rps, or a corresponding count deviation of ::15. 9 pulses per rotor revolution. Hence, the N 1 and N 2 valves wou2d be set at 9984 and 10, 016 respectively. 20113-FR1 i ! 1965011367-258 - 249 J ¢% ! ! ! I ! g'J | .-- tu o t._ ¢fi _a a_ z J _: ,-i 0 m b- he] , 0 0 • o 0 m ¢9 ! % • g °_ "q[- ®G@©®® 201!3-FRI 1965011367-259 - 250 - If the requirement of sign information in the roll rate signal can be omitted, a much simpler alternate rnecha:_ization can be used. Assume that _he SDMF.G is mounted so the gyro is separated from the roll axis of the vehicle, and oriented so the centripetal acceleration associated with a steady roll rate will be impressed on a single EflG suspension channel The error signal (direeQy proportional to rotor sag) which generates the suspension voltages necessary to balance this centripetal acceleration can then be directly used as an indica- tion of vehicle roll rate. The direction of the roll rate, however, is obviously missing from this information. The sensitivity of the JPL SDMEG suspension electronics is such that an error signal of about 0. 1 volts per g is available. Selecting a reasonable precision of 1 my measuring accuracy then indicates that the ESG should be mounted at least one foot from the vehicle roll axis. The 1 ± 0. 3 radians per second roll rate signal requirement then corresponds to sensing at 3 ± 1 mv suspension electronics error signal. The above considerations show that calculating vehicle rate about an axis parallel with the roll axis of a single strapped-down ESG is at best cumbersome and inaccurate. It is therefore recommended that this calculation not be _n- cluded as a part of the subject ARS. :£he state of tbe art of rate gyros is _such that a relatively inexpensive version of such a gyro could readily be used to provide the desired roll rate information with_ minimal size, weight, powe c, and reliability penalties to the SDMEG ARS. 20113_FRI 1965011367-260 I- 251- PERFORMANCE ANALYSIS INTRODUCTION AND SUMMARY The performance analysis presented below assumes an SDMEG-ARS as defined in the previous paragraphs. Specifically, a rotor with pattern as shown in Figure 4-ii is to be used. The pattern is assumed to have a 3 degree linear range and 12 degree total range. A nominal gyro operating speed of i00 rps is assumed. The individual error sources as considered in this performance analysis are presented in Figure 4-19. The effects of _hese errors on both the attitude and rate outputs a_'e discussed individua]ly below. A summary of the attitude output errors is presented in Table 4-3. The low frequency, or bias, errors are presented ir_ two different classes. First, errors at the end of the retrofire period (a maximum of 40 minutes after separation) are presented in terms of null errors and maximum errors. The null errors occur at tlle center of the linear range. The maximum er'rors occur at the three degree limits of the ARS linear range whece pattern and calculation approximations begin to contribute significant additional errors. These errors are nearly invariant with time. The second type of error (e. g., gyro drift) is an error which is time dependeut and increases with mission length. Additionally, high frequency or noise errors are also presented in Table 4-3. The. error functions and magnitudes presented in Table 4-3 are explained in the following discussion of error representations. These high frequency errors are used to calculate both the noise on the attitude output and the accuracy of the rate outputs. t 20113-FR1 .I ¢ i 1965011367-261 - 252 - 253 - - 254- A summary of the most Important system performance parameters _s presented in Table 4-4. The attitude accuracies and linear range are taken directly from Table 4-3 or from prevlcus d._scusslons, The calculations necessary to obtain the qua.]lty of the attitude out puts and the quality and accuracy of the rate out- puts are presented in the following discussion of error representations. Comparison of the required and projected performance parameters presented in Table 4-4 shows that all the specifications on the attitude signals can be me_, However, botn the accuracy and quality of the pitch and yaw rate signals are ]ess than desired. This comparison has been macie assuming that the stated ;_equirements are the equivalent of 3_ values. If the mo:_e lenient interpretation that the stated reqturements represent one sig:na values were used, the appro- priate numerical values for the projected performance column of Table 4-4 could be obtained by dividing the given values by three. This interpretatmn would enable meeting all the required performance parameters except the low frequency noise on the rate signal The inability to meet this noise requirement is not surprising. In fact, it might be expected, since the attitude and rate requirements of Table _t-4 are not entirely compatible. Assume that 1deal low pass filters (infinitely sharp cutoff) with banaw,dth eqtlal to the required system bandwidth are used for the systern of Figure 4-19 (this is as good as can be done for noise rejection purposes), A further reasonable assumption is that the spectrum of the no_se inside the system bandwidth resembles white noise (power spectral density a constant value)_ This second assumption is verified by the following error : analysis. The relationship between the no_se in the rate signal may then be : found as: i 3 _1/2 ._ "Y 3if I -J D3 t where: f2 = bandwidth of rate h,_er. : fl = bandwidth or" attitude filter. n = vms noise in system output. 20113-FR1 1965011367-264 - 256 - For the requ._rements of :Fable 4-4 (5 and 15 cps bandwidths), the multiplying factor in the above equation is about 6. Hence, if _he noise on the attitude signal is 0, J5 degrees "he minimum no_se on tb_ rate signal would be approxi- mately 0.3 degrees per second. The preliminary design of the attitude reference system described here has been achieved primarily to meet the specifications on attitude performance° It is doubtful, however, that the noise ceqmrement on the rate signal could be rne_ with less than a complete change in the mechanization concept. ERROR REPRESENTATIONS The error represenzat._cns and approxm_ations dcscmbed below will be useful m mathematically descmbing the performance of the subject ARS. Useful Relationships The analysis includes use of autoco • relation func,.ions and power spectral densities. The autocorrelation function of a process X(t) is, in general, define._ as an ensemble average° Rx(-) : x(t.,) X(tl 4 7) When the quantaty w_th which we are concerned is stationary this ensemble average becomes independent of the time t 1 R (_) = X(t) x(t + 'r) X ',_ The further assumption of ergodlcity allows the computation of the autocorre- : lation function by a time average. T lira TT_ X(t)X(t + "r)dlRx(_) = 2-+_ "-T 20113-FR i 1965011367-266 -- 257 - The power spec, ra] dens_y of the process X(t) is simply defined as the Fourier, ransform of ,be autoc'orrelatlon function. oO S (f) "-_ R _') c i2_ f._ d7 X 4 -o0 The mean square value r,f the process X(t) is slmply the autocorrelation funotlOh at zero 7. T ¢c,_ : 1,m _ X2(t , CrxRx "[ oo 2'I" ) dt = X2 2 .r r o - varuan_, e of X0) X X mean of X¢_ Altevna, eiv, the same quant_ly n_.ay be found from the power speclra] denslIv as' H C_.J S {f)df X "Random" Square Wave .} Assume eve,,tpc,_n*sequa]iy spaced at a perlod T The value of the process _ _s _,ns, m_, be'ween even, points, suceedmg values are uncorrela_ed and all 2 the F.Io_ess values f(,rm a gauss_an random process w1t.hx,arian¢ e _A The '.' au_o(or're]a_v,n f'.,_ ,._onmay ,hen be found as. .t 2 "r_ % 1 -- + : t i T _i, and _,e currespondmg power spectral density 1,_:, 2 ,_ SA ¢I) = _2 _'7) "_ clA2 T sin 7rfT ! {r,fT) 2 I _'i 201i3-FRI i 1965011367-267 l! - 258 -- The form of this process is shown in Figure 4-20a. For the previously f_ assumed p_rameters, of course, T is 1/200 of a second. This error function _ is designated as a type "A" function in the following error analysis. tt • • • f, i-t .Oetermmmtm Square Wave Consider the error process formed by switching alternately (and instantaneously} _(ll.between two independent, slowly varying, zero mean random processes. The switching is assumed to be done at a rate very much higher than the highest frequency in the processes. A small portion of such a process is shown in ,_',l Figure 4-20b. The expected average of this process over a time T 1 still much smaller than the period of the hSghest f_ equency in the random functions is: ' T1 /_ 2 2 ;.i _l 1 _ _/CrBl + aB2 aB_ _; :i _ j B(t) dt ..... ; " - 2,I. o t" where the aBi'S represent the variances of the two processes. As shown i_ Figure 4-20b, this average may be considered the low frequency (or dc) corn- r ponent of the error process. The expected (statistical} amplitude of the residual after this low frequency component is removed is simply: aB1 + aB2 E [B] = 2 .... aB This residual square wave may then be represented as a Fourier series, as shown in Figure 4-20b. _0 - 4-B cos 2_.__+ 11 _ t B(t) - B(t) = (2s--_+ 1) 7 T171=O and the Fourier transform is: aB _o 2a B r _ ' _ 61f} + ' -- ...... , B(f} _/2 ' /.., (2m I-I)_. l.i61f'.i_Ill,_-_12TI +6 f + ._tlEI.._._2T.I.: Eli_-O I [ _,01X3-,FBS 1965011367-268 - 259 - R (_) ACt) -_l Tt t .... I 72 ._.. TJ _ _ "RANDOMAMPLITUD("SQUAREW_sVE BiO _f) _ "°' !°,t , , t 1 1_°. _i:., --4 _- N, '_ ' .,o, __ __ j ,1 (bl !_ "D(TERMINIST|C"SQUAREWAVE i C(O /. C(f)ACTUAL/ * 'N ,_ i OUTPUT /__.j A_.E ...I -s -2 .s o _ J,. _ 4_ T T T T T T T T (a)SAMPLEANDItOLDKRROIt l Figure 4-20. SDMEG-ARS Error Functions 20II3-FRI 1965011367-269 260 - For a 200 per second sampling rat_ the process consistsof a low frequency component plus discrete sinusoids at 100, 300, 500, etc, cps. This error functionis referred to as a type "B" process in the following error analysis. S__aamplel and-Hold Effect The final error representation to be discussed here is an approximation to the samp!e and hold effect introduced by the attitude computing module. This error process may be approximated by assuming a constant vehicle body rate as shown in Figure 4-20c. The difference between the actual body angle and system output results in an effective saw tooth error (as shown) which may be z __presented as a Fourier series. 00 C(t)= + __, sinn?r T n=l The frequency spectrum o; this process is also shown in Figure 4-20c. For a 200 per second sampling rate the process consists of a low frequency com- ponent (average value) plus components at 200, 400, 600, etc. cps. This error function is fle_,ignated a type "C" process. Error Calculations Consider the system shown in the sketch below (the functions of frequency refer to Fourier transforms). The power spectral density of the output may be found as sy of) !mi2,f)l2'v., -" Sx {f) 1965011367-270 A- 261 - And the mean square value {power) of the output may again be found as the :.} integral of the output power spectral density. These relationships are first usef,_l in calculating the quality of the attitude signals. J For the attitude errors being considered in this analysis the H(i2_f) is simply I t a filter, and is assumed to have the form 1 I In f m "1_ Three types of noise are put into this filter. The first type, the power spectral density represented by the type "A" function, is well approximated by the value .-_ z...._n<<__.!. This is readilyof the function at zero frequency since fm 2_ evident by examining the form of the power spectral density function shown .!} in Figure 4-2!. ? Sx(f) = Sx(o) = aAi T) i ,._,, Hence, the totalmean square noise power inthe outputattitudesignaldue to , I the type "A" error is found as: 20 x. _ it0_2 Sx(o) df _ S (o), + (i_) + 1 i_ _m i mi _i Assuming: _! And using the rss of the (la)type "A" errors from Table 4-3 yields:I' "_A = :2_ (15} ( o. 0o84)22 (o. I) 200 - . = 0,0049 degrees11 1965011367-271 - 262 - '_'_ 0,9_ .......0+6 + \,,+! 0.2 i -" _-- SVST_U.A._,_r.i, i , _ 4 100 150 2_0 O _0 +_ "- +(¢PS) +. Figure 4-21. Power Suectral Density Function This type of error is almost entirely conta+.ned in the less than 20-cps fre- i' quency region and is the only noise error In this frequency region (the type "B" • i and "C" errors have a minimum frequency of 100 cps). If this attitude error is assumed normally distrib_,,ted, the above rms value corresponds to anne sigma value. The required peak-to-peak noise specification is taken as 6a (3a represents zero-to-peak} and this value is entered in 'Fable 4-4. The second and third types of noise are the previously described Fourier series which represent the type "B" and "C" error functions. These sinusoids are simply attenuated by the magnitude of the second order filter. Substituting the appropriate numerical values yields: 'l - ,{ i:_,I 201_a-FRl ] 9650] ] 367-272 - 263 - , 'B °B + ...J _ 44 .-5- --- 3(400)I J 2_ 4_ j Yc(t) : _ sm , T + sin T +i77 2i7i0).... i •_J The numbers 44. S. 177, 400, 710, . . are the attenuation of *,he filter at the i I00, 200. 300, frequencies. The indicated peak-to-peak noisecp$ may d therelore be appro×imated as: yp. p 2 4 3 a B I44.5 + -_--- = 0.0022 degreez1777r where: } 3 0. 0376 degrees from Table 4-3 -'1 _ _ 0. 0075 degrees} The threshold of the attitude signal specified in Table 4-4 has been quite " arbitrarily assigned at the unity signal-to-noise level. The circuit used to differentiate the pitch and yaw attitude outPUtS was presented - in the discussion of readout electronics. The projected accuracy of _} these rates is summarized in Table 4-5. The assumed one percent gain stability ..] in the differentiator circuits translates directly into a possible one percent mag- [_] re'rude error m the rate signals. One percent of the required linear range,±l. 5 degrees per second, yields the ±0. 015 degree per second error i dicated in Table 4-.5. A second rate calculation error occurs because of the lead-lag circuit used to "| approximate the desired differenttator. The lag frequency was selected as 30 cps to give adequate performance at the 5 cps bandwidth point. The difference in the magnitude of the transfer function of the ideal differentiator and the 20113-FR1 1965011367-273 r264 o selected approximation at a 5 cps frequency is only I. 4 percent. Similarly, the lead-lad approximates the differentiator to within one;percent at the low end of the spectrum for frequencies over 0.07 cps. Hence, a maximum bias error of one percent, or O.015 degreee per second, has been assumed from this source. The remaining rate errors are caused by the various types of noise in the pitch and yaw attitude signals. Analytically, the type "B" and "C" errors ar¢ i easiest to handle. For _he frequencies involved in these _loise signals the ! gain of the differentiatsr is constant at K = 200. A second order filter anala- gous to that used for the attitude signals, but with the undamped natural fre- quency at the required five cps bandwidth, is assumed. The noise in the rate output from these two noise errors is then well approximated by the first two terms of the appropriate series, / Table 4-5. Pitch and Yaw Rate Errors Error Source Bias Magnitude Noise Magnitude (3a - deg./second) (peak-to-peak} Gain Stability 0. 015 ..... Differentiator Approximation 0.015 ..... 2 Type "A" Attit,:d, _ Noise 0.19 0.37 Type "B" Attitude Noise ..... 0. 0479 1 1 Type "C" Attitude Noise ..... 0. 0006 TOTAL 0.19 ..... ..-- , .,,, , , , I, These errors are primarily high frequency errors. 2. The majority of this error occurs in the 0 to 20 cp$ frequency range. : 20115-FR1 ,.4 ak;I 1965011367-274 J- 285 - _Blt) = 200 4 aB cos -1 + + 40-b - 3(2700) .... / 4_2_ sin .... Yclt) = 200 _ sin --_.- + T 1600 _.(-'6400) Where 400, 1600, 2700, 6400, . . are the _ttenuaticn of the 5 cps second order filter at the 100, 200, 300, 400 .... cps frequencies. The indicated high fre- quency peak-to-peak noise in the pitch and yaw rates is then seen to be well approximated by: yp.p -_ 2C0 4 I 200 _ t = 0. 0485 degrees2', k-ii ,, _3%, + _o-o-- Tj ,. In calculating the effect of the type "A" nmse on the ARS rate outputs the pre- viously assumed approximation for the power spectral _iensity function is still i a valid one. Sx(f_-_ SX (o) The to_a] rms noise in the rate outputs due to the type "A" noise in the attitude signals is then found as: 3 YA = j Sx(0) i_0 df = S x(0) -® 2 .,. +l _W2rJ _2n [Y_i 112 _ 0. 0624 degrees/sec. This is again a low frequency error, and the above value corresponds to a one sigma value. Hence, using the same interpretations as previously, the possible 13_1 rate erz_or is three t_.mes the above value, or U. 19 degrees per second, and the possible peak-to-peak poise error is _ix times the above value or O. 3'1 degrees per second. These values have be_ entered into Table 4-4. 1965011367-275 - 266- The rate threshold specified in Table 4-4 has again arbitrarily been set at the unity signal-to-noise point. ER]_ OR SOURCES The source, form, and magnitude of the individual errors as shown in Figure 4-19 are discussed in the following paragraphs. Pattern Accuracy Readout errors caused by pattern accuracy may be considered in _.hree separate categories -- patern location errors, pattern line edge deviations, and patterning approximations. PatL_,-n Location Errors -- Patterr location errors result from the gross (average_ p,sitioning of the "constant slope" patterns w_._h respect to the referenc (meridian) pattern line. Present state-of-the-art rotor patterning techniques limit the p-_ttern !ocatiol, accuracy to about 10 -4 inches. This accuracy, however, is not required for _his application. A 5(10 "4) inch pattern location accuracy can be obtained much easier (and at a Lnore reason- able cost). For the pa_tern ._ssumecPhere the sensitivity of readout angle to p_,ttern location is seen to be: T "_ 1 rad y This type of error yields a type "Z_"functionwith a 3a value of about 0.029 degrees. ] 9650] ] 367-276 - 267 - Pattern Line Edge Deviations - Pattern lineedge deviationsrefer to localized deviationsof the actual llneedge locationfrom the average location. This type of error It.somewhat a functionof the technique used to scribe the pattern line (scribing,sand blasting,etc.) but can generally be held to slightlyless than the patternlo,._tionerrors. A 3# magnitude of about 2(10-4) inches is assumed for this error analysis. The time representationof these errors would approximate a type "A" function. The type "A" error functioncontributesnot only noise but also a probable rms (bias) component tothelow frequencyattitudeoutput. Thd magnitude of thisbias component, at the outputof the assumed second order filter,i6 about 60 percent of the noise magnitude, This quantityis therefore entered as a low frequency error in Table 4-3. Pattern Approximation Srr0rs -- Patterningapproximatio,_errors refer zo t_ factthata great circlepattern is being used to approximate the "constant slope" patternnear the null. This approximation contributesno error at the null (,/: 0 degrees). Within the linearrange the appro_irn_tlonerror is 1 m_Lximum at the linear range limit (T = 3 degrees) and amounts to about 0.05 degrees (see Figure 4-9). This maximum error is used as a 3a value for . thiserror analysis. The error if conBistentbetween patternhalves a.-,d therefore contributesonly a d-c component, _ L TZi_ggerlngLevel The sensitivity of the E_G readout syltem to deviations in the electronics triggering level may be obtained by referrlng _o Emlatlon (85_. A change in the electronics triaevit _ level means tlmt the time pu_ses are generated at other than the pre-set percent of uon-refl_ct_e area in th_ pickoff field of view. The cb_nl_e in readout angi fo_ an area cl_q_ may be found al: I 1965011367-277 - 268 - r P 6A ; {reference meridian)6_"1 = B--r-- r 572 _ 2T--FP 5A ," (constant slope pattern) where: r = pickoff field of view radiusP r = rotor radius A -- normalized pickoff field of view a_ea Assumed nominal trigger level at A = 50 per cent. Two g_ne,'al levels of triggering accuracy are available. If standard voltage : sensing triggering circuits are used, the triggering level variations might be as much as I0 percent (3a). This tr'ggering accuracy results in an excessive error from the second equation above. Alternate y, height-insensitive triggering techniques {e. g., see 1726-FR1, pp 65-77} can be employed to limit the trigger- ing !evel variations to less than one percent. The difference in the two error equations allows the use of a Schmidt trigger for the reference meridian pulses {.See Figm-e 4-1.5} with a correspondir_g error of 0.0._ 2 degrees. ThE triggering on the constant slope pattern {discriminated with pulsewidth discrimination} re- quires the more c, ,plex heigl_t insensitive triggering circuitry. The error is then 0.9(10 -2) degrees. The total (3a} error is the rss of these two, or 1.5{10- degrees. The time function associated with this error would be expected to re- semble the type "A" function In addition to these high frequency errors, long-term variations in triggering circuit gains, rotor it_-_mination and photosensor sensitivity are e_pected to contribute an additior_. _ error of about the same magnitude. The low frequency attitude error (rss of bias and filtered high frequency error} _ttr_butable to triggering level _s therefore at,out 1:65(10 "2) degrees. 201i3.FR1 1965011367-278 -- 269 - Integrator Discharge Noise Discharging the in_egraturu ia the analog attitudc computing module affects the attitude reference system readout accuracy in two ways. First, any resi- dual voltages on the integrators after the reset procedure are the equivalent of no_,-zero initial conditions and cause prop,-rtional readout errors. Second, the time used to reset the integrators is taken from the computing cycle (see Figure 4-14) thereby causing a systematic readout error. The sensitivity of the attitude reference system to the first type of readout error is readily found by examining the readout equation. _'b b An integrator reset error may be interpreted _._ _ time error (i. e., percent of integrator full-scale output), and tl.,e correspondh_g readout error is: 5_ 3 _ ,Bb _± 6_b _ 6Vb]'-'-_ _ B _b The required integrator reset accuracy is therefore about 0.1 percent o_ full scale, with a corresponding reedout error of 0. C17 degrees. This requirement is obtainable, but will demand particular attention to the design of the reset circu_s and an allowance of adequate reset time in the computing cycle. Ifthisreset time is any significantportionof a computing period (one-half rotor period) a further computational error appears. Let the reset time be T r. The computational error caused by keeping the integrator_off(forreset) during the firstT r seconds of the computing cycle is: T r 6_/,I v b i 1965011367-279 270 - This is, however, a predictable and repeatabl2 effect and can therefore he compensated. The compensation, which might consist of either gain adjust- ments or biasing, would be the equivalent of adding the quantity Tr to --B 71. U the calcu]ated output angle, and would therefore by wholly effectiv_ only at the null (_, = Q degrees). A residual null errGr (after compensation} of 0.01 de- grees and maximum error of 0.03 degr_.es (corresponding to Tr 2 )10" bare been assumed for this error analysis. Calc'_lation Approximations The most significant calculation approximation in the proposed readout scheme results frovr, the assumption of constant rotor speed (Equation 101) which was employed to avoid the indicated division. If _Vb is the difference between assumed and actual rotor period the resulting readout error may be found as: This error disappears at the nt;ll of the readout system, irrespective of the error in the assumed rotor speed. A reasonable numerical value for thi_ error can be obt_,ined by assuming an initial epeed within about one percent of the selected nominal and an SD MEG rundown rate of about o_e-h,_lf rps per day. These values yield th_ maximum low fl, equency error limted in Table 4-_. A_ addltiort_lc!tlculatF_.ngl_ro#imat_gt error `api_,srsfrownthe sample ,andhold effect on the attttue_ comptttiag modt_lem. Tbi_ appr_xim,at_n "-emults in a Type "_" error with mal_imum value e_u,al to the vehicle r_te m_tiplied by the mamp- _g time. A_Buming • rat# o_ !. _ de gree_ p_r _tl_l _tl_e required linear range o_, tile rate otttputs) yteld_ a Type "C" errev w_th !.I,+-FRI 9650 367-280 271 - _b i. 5 A = _max -2 -= _ = 0.0075°/sec. The average or bias value of this error, as shown in Table 4-3, is A/2. Gain Balance and Stability The accuracy of the ARS output is obviously a function of the gain balance and stability of the analog computing elements. Consider only one of the two parallet computing channels. The sensitivity of output angle to gain stability is well approximated by: where: 6I( = fractional gain stabilityK Assuming an equal (probabalistic) error in each of Lhe two channels would then yield an output noise error (type B) with 3u value of 2B times the fractional gain instability. This result demonstrates that overall gain stability of about 0.1 percent will be required; this gain stability can be achieved with high feed- back around active elements, since the overall gain require,nents are low. Gyro Drift Errors } The gyro drift performance assumed here is the same as has been previously used for SD MEG performance analyses (see Honeywell Report 20113_PR50 I pp16-2o). i 1965011367-281 - 272- The first portion of the flight, separation to end of retrofire, will last a may.imum of 40 _Fn.inllSP._. This 40 minutes includes an assumed 10 mtnutes of retro firing time with actual acceleration of 0.5 g's and the SD ESG sus- pension electronics at a 30 g level. Tile calcu: _ted (l_) gyro drift rate during power flight is 0.0127 degrees per hour. In the free fall condition of the remaining 30 minutes the suspension electronics pr "load is assumed I at one g and the corresponding SD MEG (la) drift rate is 0. 004 degrees p£rhour. The total (3a) drift angle for this 40 minute portion of the flight is then readily calculated as 0. 009 degrees. The second portion Of the flight, during which control must be maintained, has a specified duration of 5 to 20 days. Using the same (free fall) gyro drift rate yields a total error angle of 1.45 degrees af*er f_-Ie days or 5.8 . degrees after 20 days (0. 288 degrees per day 3_). Power Supply Variations _i The power supplies for the SD MEG ARS will provide adequate regulation and isolation from the vehicle power so that the power variations will cause only second order attitude errors. The ARS preliminary design includes features (e. g., height insensitive triggering, common reference source for attitude computing module integrators, low gain and high gain stability in parallel computing paths) intended to minimize power supply variation effects. A preliminary estimate indicates that one tc two percent power regulation will provide adequate performance. SO115-F11 i 1965011367-282 - 273 - G__ro Initia_ Conditions Gyro initialconditions are a ,unction of both the ESG mechan[z&L[oii, the ES(3 alignrne'Itin the cartS,rag vehicle, and the pointing accuracy of this vehicle during the Kvro starting procedure. Because of the obvious un- knowns in the gyro alignment and 8tRrtup process, this error has not been included [n the performance analysis. The ARS has been designed, how- ever, so that approximately 50 percent of the total allowable system error can be alloted for the initialerection process. The difference between the allowable error of 0. i degrees and the predicted errors at the end of retro- fire (on an rss basis) is 0. 067 degrees for the maximum error and 0.093 degrees for the null error. These error allowances are considered more than adequate for the initialgyro alignment and erection procedures. 1965011367-283 - 2;: - SYSTEM PHYSICAL CHARACTERISTICS Size, weight, power, and reliabi]ity estimates for the projected capsule lander ARS are presented in the paragraphs below. SIZE AND WEIGHT _II_e ARS would logically be divided into at least two separate packages. The sing!e SDMEG with its associated optical pickoffs, vacutun pvmp, spin and damping coils, and magnetic shielding, would comprise one of the two packages. The mounting provisions for this gyro assembly would have to include precision alignment capabilities. The gyro-ass_ciated electronics wcmld be placed in one or more separate packages. The size and weight of the electron_ca and mechanical assemblies necessary to perform the various functions are listed in Table 4-6. The physical param- eters listed in Table 4-6, with exception of the suspension electronics weight and the readout electronics size and weight, have all been published in other documents (see 1726-FR2). The parameters for the suspension electronics assume welded module packaging of presently e=isting circuits. The remainder of the electronics parameters were obtained assuming maximum use of pre- packaged integrated circuits (gates, flip-flops, operational amplifiers, each in a single TO-5 can or flat pack) with p]ug-in circuit card type construction. Thi_ type of construction would yield a unit suitable for airborne applications (reasonable size, weight, and power) with minimum development time and cost. Use of more advanced packaging techniques o- multilayer circuit boards or threeodimensional integrated circuit matrices -- would permit a 50 percent or more improvement in all Table 4-6 electronics parameters at the expense of considerably more _levelopmevt effort. The estimates presented in Table 4-6 do not inv/ude allowsnces for power supplies. If the required form of electrical power (specified in the next paragraph) .4s not available from a separate source, additional electronics 1965011367-284 •- 275 - Table 4-6. SDMEG ARS SiT,e and Weight Characteristic._ Size Weight item (in 3 ) (pounds} , ,.,. ,. Gyro Assembly 104 _. 0 Suspension Electronics 60 5.0 Remote Starting Electronics 40 1, 3 Readout ElectroniGs * 60 2.2 Totals Gyro Assembly 104 6.0 Electronics Assembly 160 8.5 • Add 20 in 3 and 0.7 pound for roll r,ate calculation allowance would have to be included in the size and weight parameters. Also, s indicated in the table, the electronics necessary for the roll rate calculations have not been included in the primary figures. Meeting this requirement (for only the first 24 hours of the flight) would require an additional 20 cubic inches and 0.7 pounds. POWER REQUIREMENTS The most recent estimates (or measured values) of the power which will be necessary to start and operate the SDMEG and associated electronics of this ARS are presented in Tal:le 4-7. The values in this table indicate that a total of about 61.6 watt-hours of energy will be required during the 1.6 hour starting procedure, with a peak power requirement of 53.7 watts. After the system is started, the steady-state power requirement will be 11.9 watts, except that 30.3 watts peak power is rcquire_t _uring retrofire when the suspension electronics are switched to the high g mode. 1965011367-285 - 276- [ i Table 4-7. SDMEG ARS Power Summary Function I Power ! Power Time EnergyFnrr_ Regulation (watts) (hours) (watt-hours) , | , i , Spin 800 ~ -- 20 0.13 2.6 2¢ Damp 22.5 vdc -- 6 1.63 9.8 Vacuum 180 vdc 5_ 0.2 continuous -- System +22.5 vdc -- continuous -- 2.5 vac -- 0.8 continuous -- Suspension 2.4 KC • 0.I_ .... continuous -- Electronics 1 g ±6 vdc 1_ 3.6 continuous -- 30 g ±22. 5 vdc 1_ 22 1.8 39.6 Starting ±22.5 vdc -- 4.7 1.6 7.5 Logic and Control Readout 6 vdc 1,_ I. 2 continuous -- Electronics • ±22.5 vdc 1_ 6.1 continuous -- Totals 11.9 wr,tts continuous 61.6 waR-hours for starting 30.3 watts peak power during retro • Add 1.5 watts for roll rate calculation Except for the readout electronics, most of the values shown in Table 4-7 are taken directly from 1726-FR2. The suspension electronics and spin and damp power levels have been modified to reflect the latest laboratory values, and most of the regulation requirements on the precision power have'been relaxed slightly. The conditions on the readout electronica _ower, the maximum we c_ integrated circuits, are the came as specified previously. As indicated, an additional 1.5 watts must be added to the readout ele_ontc a level ef Table 4-7 if the roll rate requirement I_ to be fRlfflled for the firet 34 ho!wB. i t' 1965011367-286 - 277 - RELIABILITY A summary of the reliability predictions for the various subassemblies included in an SDMEG ARS is presented in Table 4-,3. These predictions are based on component failure rates presently being used by Honeywell for the Apollo stabilization and control aystem, and ass_ne proper part application and derating. The mission reliabilitymay be calculatedusing the mission parameters, operationalsequence, and subassembly failurerates. During the first200 days of the flight,earth launch to gyro spin up, only the vacuum system will be operated. Hence, with a failurerate of 0.180 (assuming open loop vacuum pump operationwith a simple power supply) the reliability(probabilityof suc _,ss)for thisportion of the mission is -klt 1 -0.18 (10 -5) (200) (24)R. = ¢ _ ¢ = 0.9914 1 The failure rate during the system start-up phase is 5. 367 percent per 1000 hours; but this phase only las*_s for about 1.6 hours. The resulting 8.61 (10 -5) probability of failure during this portion of the flight is insignificant, The system failure, rate during the descent portion of the mission is 8. 402 percent per 1000 hours or 0.20!6 percent per day which indicates a reliability of 0.9900 for a five-day descent or 0.9604 for a twenty-day descent. The total probabiliiy cf success (product of individual reHabllities) may be fom_d as -0.1875 (102 ) TR h 0.9914 ¢ where T is descent time in days. 7 The total mission reliability ranges from 0.9814 for a 5-day descent to 0. 9523 for a _.0-day descent. i 1965011367-287 : - 278 - Toble 4-8. ReliabilityPredictions for SDMEG AIIS SubassemNies ,t Item. n (_/I000 hrs) nk _. _,T-;"-_7 . , ,,_T'_ 'r_'j.f , _ _,'_ .c= : _ i : :} __2 .... : - ;-=_:::_- .___'t'_'_"lm_L' Mechanical Assembly {withoutvacuum) 1 O.050 0,050 Vacuum Pump and Electronics I 0.Ig0 0,180 Suspension Electronics 1 2.988 2.988 Pickoff Assembly 2 O, 594 1. 188 A,_tomaticStartin_ Electronic 1 2.025 2.025 Readout Electronics 1 3.872 3.872 The above reliability summary and calculations indicate where major effort ' Should be expended to improve system reliability. It is apparent the major reliability offenders are the vacumn system, the su_ _ension electronics, and the read0utsystem (both pickoffs and electronics). Redundancy or simplificationin these three areas could .-'_.:--._';7"_npro_,ethe reli'ability of the 8PMEG ARS. l' ' 1965011367-288 - 273 - SECTION V SDMEG MODIFICATION8 READOUT IMPROVEMENT The study program reported in thisdocument indicatesthattheminiature electzostaticgyro's current design'.cansatisfythe spacecr_t strapped-down attitudereference system requirements with exceptionof the accuracy of vehiclebody rate outputs. To provide the necessary accuracy _;,the Vehicle body rate outputs itwould be desirableto investigatethe possibilityof re- ducing the noise on the gyro readout. The firststep would be to identifythe exact noise source; the second step would be to modify the gyro design to eliminatethe noise sour"_ or attenuateitseffectat the gyro output. Un;il the noise sources are identified,itwillbe difficulto predict the gyro modificationnecessary to reduce the noise effect. However, it ,rillno doubt invol--_.the rotor pattern and/or the opticalpickoffs. In est_.blishinga noise levelgoal to use in improving gyro readout noise, the gyro readout noi_e levelis assumed acceptable when itcatmes an RMS noise levelon the system rate output eqaal to the lowest vehicle rate itis deslred to detect° It will also be assumed thatthe lob,eatvehicle rate to be detected is 0.001 degree per second and thatthe rate o_put is obtained by differentiationof the gyro output. In sddltion,th_ noise on the gyro re.lout willbe considered "white" with an effectiveb_indwidthof z5 cps. Fr,_m these assumptions and th6 relationship i 1965011367-289 - 280 - where li(_) = Power spectr_ density ot_input _o(_) = Power spect.rai density of output G(t_) = Fourier transform of system impulse response it can be shown (see Section "Vehicle Body Rate from the End Axis Rotor Pattern") that the noise on the system rate output is related to the noise on the gyro readout by 'the following relatlonsh_p r_ ; -_ 37 Ra_ n - 3 _B i - where 2 n = Average _quare of noise on gyro readout in degrees l_n2 ffi Average square of noise on rate output in degrees per second wB = Effective bandwiath of the nerve in r_lians per second Substituting R 2 = (o.ool)2 n _B = 275 1965011367-290 - 28i - in the above expression 2 n = 0.35 x 10 -9 (deg) 2 or, n = 0. 067 arc-secor.d RMS : This is to say that the RMS level of noise on the gyro readout should be reduced to 0.067 are-second if vehicle body rates of the order of 0. 001 degree per second are to be detected. SHAPING ELECTRONICS In additiol_ to the modifications for alleviating the noise problem on the ARS _'ate output, ther" are several other desirable system modifications. One rather simple modification is addition of a third set of readout shaping electronics. The gyro has three optical pickoffs arranged so the optical axes of the pickof._s form an ortbogonal triad. The cosine rotor pattern used with the three pickoffs does not cover the entire surface of the rotor. It covers only enough of tile rotor surface so that two of tne pickoffs are active at any one time_ With this situation it was originally believed that two sets of shaping electronics would be supplied for tl_,e two active pickoffs. This would be a satisfactory arrangement if thcre is sufficient time to switch the shaping electronics between the pickoffs as they pass in and out of the ro:or pattern range. The study results indicated that for either/or both high vehicle body rates and high ARS output data rates _here was not enough time for the switching function. It is faster to read all three pickoffs and take the data into the com- puter, even though one readout is erroneous. 2011S-FRI 1965011367-291 - 282 - The computer can then make the decision aa to which readout ¢o ignore with considerable saving in timv. Addition of a third set of shaping electronics was assumed in arriving at system size and weight estimates. The shaping electronics add about four to five ounces and two to three cubic inches to the system. ADJUSTMENT FOR INITIAL ALIGNMENT Another useful modification is a means of adjusting the orientation of the magnetic axis of the gyro damping coils relative to the vehicle. This permits fine adjustment of the initial alignment of the gyro spin vectors relative to each other and to the vehicle coordinates. The damping coils will repeatedly align the gyro spin vector to the same case orientation with accuracy; but this case orientation mu_t be aligned to coincide with the vehicle coordinates by adjusting the gyro position in the vehicle. However, without some means of adjusting the gyro position in its mountings, this is very _ifficult to do accurate._.y. Another possible way to accomplish this alignment would be to adjust the orientation of the gyro spin coils relative to the case. Either arrangement will make it possible to adjust the initial aligr.ment of the gyro. MODIFICATIONS FOR SINGLE GYRO APPLICATION The preliminary design of the landing capsule ARS revealed that system re- quirements can be readily satisfied with a single gyro if two modificatio1_ are made. Since the landing capsule application requires limited freedom in only _,7o axes it is _ nstural application for a single slectrostattc gyro, basically a two-axis sensor. To take advantage of this unique characteristic and provide the linear range required for the landing capsule it is neceBsary _to place a "constant slope pattern" around the rotor equator and reduce the number of optical pickoffs from three to two. The pickoffs would be locate¢i on the case 90 degrees apart, both viewing the rotor pattern during Jymtem operation. 1965011367-292 - 283- Since it is desirable to keep the landing capsule AI_ as simple as possible, :hese modifications are in llne with the design and will permit a relatively simple analog instrumentaticr, of the associated readout system electronics. SIML_TANEOUS PICKOFF READOUT ARhough simultaneous readout of the gyro optical pickoffs is not necessary for the ARS concepts and condRions of this study, simultaneous readout will be desirable when high vehicle body rate exists or where the gyro outputs are used in navigation c_lculations. In the current readout system the pickoffs are read sequentially and the passage of time between readings contributes an error in determining the g3,ro spin vector if the vehicle has an angular velocity. Simultaneous reading of the pickoffs will eliminate this error and may permit i the gyro readout to be synchronized with the system digital computer clock.| This is a desirable feature since it greatly simplifies the gyro computer i interface.) _ Since there is no proven way to obtain simultaneous readout of the gyro pickoffs, this would actually be a development effort rather than a simple modification of an existing _yro. One approach to obtaining simultaneous readout of three ortho_onal pickoffs ) requires measuring the rotor surface velocity at each pickoff. The velocityI measurement will probably require *he major development effort, but theo- retically the velocity at the three pickoffs can be obtained simultaneously. There is a relationship between the three measured velocities, the rot,,r /secular velocity and radius such that: / vl " el= WX J ..$ = x (lo2) 3 = wXR3 20113-FR1 i 1985011387-293 - 284- VI' V2' V3 are the vector velocitiesof the rotor surface at the three piekoffs. HI' I_2"R3 are the positionvectors relativeto the center of tbe rotor of the three points of velocity measurement on the rotor surface. _= _R" _C where _TR is the angular velocityof the rotor and t_C is the angular velocity of the gyro case. From Equation (102) ! IV1] = I_I il_llsin01 ! ! IV2J = l_0lII_21sin O2 (103) 1 Iq31 = I_1 Ig31 sine 3 i ! I where O1, 02, O3 are the angles between the pickoff axle and _. If t_C has i no component normal to _R' then O1, b' 0_ are the angles between the pickoff-4 ; axes ar_ithe gyro spin vector. Since R I, R 2, R 3 are radius vectors of the rotor II_11 = Ig21 = 11_31 = R also let then L. V I = t0Rsin O1 V2 ffi _oR ein 82 (104) t V 3 - _R sin 03 B0113-FRI 1965011367-294 ! - 285 -) , and from Equation (104) and considering that the pickoff axes are mutually 1 orthogonal I / /-V 2+ + , cos 01 = ,,i V22 V32 .i _/ VI2 + V22 + V32 _I / -v _. v_. i cos 82 = VI2 2 ,.+ (105)1 + V22 V32 ") cos 03 = V_2 + V22 - v_2 / V12+V 2i_ 2 +V32 f li Equations (105)give the desired directioncosines of the _yro spin veczor as functionsof the three measured surface velocities. J i [ [i [7 20113-FR1 i 1965011367-295 '" ' 1965011367-296 - 287 - SE CTION VI RE COMMENDATIONS INTR ODU CTION The end objective for the SDMEG is its application as an attitude sensor per- Iorming attitude control, guidance, and navigation functions. To reach this objective, certain component and system development and testing must be performed. This system Study, which is being completed by this report, is one of many study and development tasks required before operational systen_s are in existence. I An attitude reference subsystem is an integral part of any control, guidance or navigation system. This study, an attitude reference system parameter I study, defines the significant characteristics of this basic integral subsystem. I Some of the results of this study identify areas, which together with the re- ) sults of the component study effort, conducted concurrently, gives insight ) and confidence in required follow-on efforts. It is the intent of this section to i:lentify and recommend contLuuing study and development wo, k considered i to be necessary to permit the development cycle to continue to the end objec- tive of use of SDMEG's as an attitude reference for control, guidance, and I navigation systems. A time table of the necessary events is shown in Figure! 6-I. I I I ] 2011_-FR1 i 1965011367-297 20113-FR1 1965011367-298 - 289 UMMARY An e_ht-mon_h ESG-AR$ study has now been completed with the objective of obtaining information _se_ui in making recommeL_dationS for utfilzatlon ar_ integration of the SDMEG _n spacecraft attitude reference systems. The study included two major tas._.s with the first task bein_ a trade-off study of the parameters of an attitude r_ference system (ARS) which uses an SDMF-_ as a major component. Task II was the conceptual and preliminary design of an attitude reference system utilizing the SDMF_ for a specific landing Capsule miosion_ The more significant conclusions reached in this Btudy are as follows: 1. An investigation of 12 different attitude refez'ence system con- figurations with system pointing accuracies rar_ing from 0.04 to 0.07 degree revealed that over this accuracy range the sys- tem size and weight varied less than 10 percent. The study also revealed that the system size varied less than 10 percent between systems that had outputs of attitude error and vehicle body rate and systems that hod only attitude error. 2. Accuracy to be expected does not require computation to com- pensate for gyro drift. The nominal 0.05 degree capability is assured for a minimum of one hour operation, which is con- sidered adequate for any AV velocity or navigational sighting maneuver. 3. The noise present on the gyro readout from both the end axis pattern and the cosine pattern limits the usefulness of body rate signal derived from the gyro output. 4. It was concluded that precision gyr_ torquing dtd not simplify or increase the capability of the ARS concepts investigated and that a caging capability beyond that available _n the damping coils of the curren_ fyro design did not simplify t_e system design. _20113-FR1 i 1965011367-299 - 290- 5, The preliminary design for the ARS concept _vestigated t_ th_ parameter study established the major characteristics of the digital controller. The controller is a general propose serial/ parallel machine with a 16-bi_ word. The arithmetic will be performc_] with 4 bits per Bite. The bit rate will be 1.5 mega- cycle_ and the memory will be 1024 words. The controller will weigh about 12 pounds and occupy 368 cubic inches. 6. The preliminary design af the landing capsule ARS shows that all the requirements except the rate output accuracy may be satisfied wi_h a system using a single ESG. The accuracy of the basic ESG is adequate for thiA application; however, the noise on the output does limit the accuracy of the derived rate output. The developmeut program on the sensor has just recently completed Phases IA and IB. Design feasib_lity has been shown for multl-mode suspension with a low-£ suspension power of approximately e_ven watts. Design stud:es have determined a start-up-_n_space automatic spin-up procedure with the sensor not suspended to withstand the 200-g shock launch environment. Studies have also indicated that the gyro can be designed to meet sterilization requirements and that performance can be maintained without temperature control. A breadboard sensor and suspension system have been fabricated and Pha_e II will consist of a test program on this assembly _ well aa the build and test of a second breadboard sensor and suspension system, Followin 8 Phase II will be a Phase HI prototype design and test program. The sensor design will reault in reduced si_. and weight with compliance to performance specifications demonstrated by a qualification test program. Phases IV and V will be the fabrication and qualification of operational sensors. The design of the sensor iB continuin_ on schedule and all m_e_tones of the over-all plan to achieve operational sensors in 1968 are being met. _'rll 1 3- 'lrR1 1965011367-300 - 291 - The next step in the development of SDMF_ systems is a study consisting of three specif,ctasks: 1. A simulatlon study of the ARS to firmly establishthe system concept and its parameters. 2. A sim_atlon study to res_ive the noise problem in deriving rateinformation. 3. A reliabilitystudy (Thi_ is a logicaIfoLlow-on program since itgives maximum support to the gyro development program and also willresolve key problem areas hl ARS s3fstemappllcatiorm.) The first task is a natural follow-on to the parameter study reported on ilJ this document. In order to ensure that the basic system concept is sound. the system should be simulated a'-_ the simulation ased to verify the concept and fix final system parameters. The third task is required in order to _._ tablish the relationship between redundancy and system :eliability. This is a trade-off relationship that was not covered Ly the parameter study jest completed. It is not known how system size and complexity might chang,, under the reliability requirements of future spacecraft applications, and therefore it is important _o carry out a study to obtain and have available. for future use the trade-off information between size and reliability. The, form of the study will be such as to produce the relationship ketwcen pr,,h-_- bility of mission success versus mission time for as many reliability models as possible. By general.trig sizing estlmates for the models it wi.li provide: the desired trade-off data between reliability and system siz,, and complexity. Recognizing that these L,udiss will more firmly demonstrate the feasibility of the stra_.oed-down SDMEG Attitude Reference System as well as pcovidin_ data that will be useful in the evaluation of future spacecraft appli¢,aHous, a recommended prograxn has been formulated and is discussed in detail in the background portion of this section. 2011:J-FBl i 1965011367-301 - 292 - Supportln_ this program are two Honeywell programs _hich will provide _irect inputs to the ARS digital computer simulation. The first program is a company study of prefer,-ed digital mechanization of various flight controllers. Th_ output of this study will be the rnecha-_zation of the digital controller. Many of the potential de6ign problems of the SDMEG Reference System will benefit from experience gained in the corg0truction of the digital controller. The second suppox_ing study is a specific program for the deveiopry_.ent of a digital controller for applications similar to tile requirements of the AOSO program. The development of this digital controller also will support the ESG-ARS development program. Following the ARS simulation and redundancy programs, further detailed mission studies are necessary prior to the preliminary deeignphase of either an ASS or Navigational System. These studies ._ill play a major role in i determining specific system requirements and early consideration cC missionstudies i_ recorameuded in order to meet the 1967-69 over-all program timing. L I The mission studies will be particularly important in establishing the navi- gation system requirements. The SDMEG being developed for the ARS m_ty also be use0 in a navigation system when combined with either an acceleration- i measuring capability or with optical _nstrumentation for star and planet sightings. The following items are system efforts that should be completed to derive maximum utilization from the SDMI_G _ork that JPL has funded to date. Most c_ the following tasks are applicable to unmanned interplanetary mis- sions; hc:tever, they are also relate_ to maimed space flights. "_e selection of the specific study _ssks is dependent upon the contemplated end use of the equipmel,t. 1 I _ 20113-FRI 1965011367-302 - 293 - A. Mechanization of navigation and guidance systems oriented around the SDMEG Attitude Reference System 1. Determination of guidance requirements placed on ARS Digital Controller. 2. Studies conducted in the area of operational use of the SDMEG ARS for guidance. 3. Hardware oriented studies to develop technique for incorporative accelerometer-deterrnined velocity information into SDMEG Attitude Reference S_ stem. B. Analysis of SDMEG navigation and guidance systems 1. Sirnalation of an ESG system for a typical deep space 1 mission to de_ermine effects of qua.,tization, compu-J tational error propagation, iteration rate and body rate on navigation operation. 2. Performance analysis of deep space mission to make i trade-off studies between use of auxiliary updating equipment and long-term unaided system performance. ] This also inc!udes the determination of critical system i parameters for various flight regimes. J BACKGROUND This section describes in detail the problems creating the need for the individual studies and the basic objectives of each study. The simulation and noise study is treated in a combined discussion because of the natural interface they have with each other. ] ] 1965011367-303 o 294 - SIMULATION AND NOISE STUDY The need for the noise study arises from the desire to calculatethe vehicle bcxlyrate from the ESG outputs. In the parameter study R was somewhat arbitrarilydecided thatthe high accuracy ARS provide body rate information 100 _imes per second. To satisfythisrequirement itis necessary to read or obtainnew information from the gyro at each revolutionof the rotor, and _ake differencesbetween successive readings. Since each reading may have ±36 arc-seconds of noise, the dfffererlcebetween successive readings may be 72 arc-seconds and the time intervalis the rotor period. The rotor period may be as short as 1/290 second. Therefore, the indicatedbody rate may be as high as 72_ - 1--_- = 14, 400 sec/sec = 4deglsec2O0 Obviously an error in the computed vehicle body rate of 4 deg/sec is not ac- ceptable. The solution to reducing the error m the calculated rate may be approached i.'_ two, ways: 1. Devise filteri,_g that will reduce the effect of the noise 2. In,prove the readout so ae to reduce the nois. _ level The system study willbe concerned with the firstsoluti,>.,while the component study willbe concerned with the second solution. A1:.hougnfilterir,g can be determined thatwillattenuatethe effectof the noise, the filtering will also affect the phase of the rate signal and thus the stability Qf the attitude control loop. It is therefore necessary that a simulation of the control system be available to assist in the selection of suitable levels of filtering consistent with control stability. 1965011367-304 - 295 - Itis anticipatedthatthe filteringwillbe instrumented as part of the ARS digitalcomputer, As such itmay be an averaging type of filterthatcom- putes the average of a number of gyro readings and then makes use of the average value to calculatethe attitudeerror and rate signals. An averaging filter##illreduce zhe RMS noise by the square .--octof the number of .arnplee or readings used in the average. It willalso produce an effectivephase shiftthatwillincrease with the number of readings included in the average. The control system simulation willpermit the se!ectio,tof a compromise between system stabilityand number of readi:igsincluded in the average. In addition to the avera_ing type filter the Z transform equivalent of high- pass, lead-lag, and other networks that promise to be useful in the stabiliza- tion o! the control system in the presence of no_._e can be simulated as part of the ARS computer simulati . The control system simulatiov will allow a rapid trod thorough evaluation of the usefulness of the _arious networks over a large range of the network parameters. The effort required to obtain the same inf:)rmation through the use of purely manual analysis would no doubt be excessive ff indeed the information could be obtained by manual analysis. In attacking the noise prob]em with the approach of tv:_Jrowng the gyro pickoff system, readout data scatter can be traced to two major sources -- signal noise 1eve1 and sign_J rise time. These errors can be controlled so that their effects upor. readout position errors for an attitude reference system are insignificant. .Tf, however, it is desired to derive vehicle rs._e information from the readout system, the presently encountered level of data scatter must be greatly reduced. It ts possible to significantly reduce the data scatter presently encountered in strapdown readoat s'ystems both by some redesign of the present components and circuits and by making trade-offs with existing readout operating parameterb. 20113-FRI 1965011367-305 - 296 - The existing pickoff preamplifier is considered state of the art in low noise amplifier design. Hence, reduction Jf the effect of signal noise must be aco complished by increasing the signal level and, consequently, the signal-to- ryise ratio. The present pickoff sensor signal levels may be increased by ir_rea._in_ t._ !_ht ener_y !eve1 at the pickoff field stop and/or by increasing the field stop size, The present pickofi design utilizes a lens and lamp to light the aperture. This light intensity can be increased by increasing the lamp filament temperature, hence reducing the operating life of the lamp and im:reasing the power consumption. The light intensity could also be raised by designing a more efficient light system which could gather the light from the lamp filament and focus it on the pickoff aperture. Increasing the field stop size results in decreased data scatter even though I the signal, rise time is increased. (Tt.e signal level of the pickoffs increases as the square of the field stop diameter. This increase more than offsets the " increased signal rise time which varies directly as the field stop diameter.) It is possible to decrease the signal rise time of the present optical pickoffs by modifying" the present preamplifiers. These preamplifiers exhibit a rise time of about three to five microseconds. It has been demonstrated that minor circuit modifications can reduce thiJ tu below one microsecond, which is compatible with present photosensors. As photosensor and transistor tech- nology continue to improve, it is anticipated that significant improvements in both signal-to-noise ratios and signal rise times will be realized. The S_mulat_.on of the attitude control loop will also make it possible to verify the basic ARS concept. The concept fo_"the parameter study was selected to [ minimize the data processing. In partic',flar, it was selected to svoid matrix i multiplications. In so doing the attitude error signal is i:ot a linear function ef attitude error and it would be appropriate to investigate the control system behavior for large error signals. ] i 20113-FR1 1965011367-306 - 297 - Figure 6-2 is a gross block diagram of the simulation of the attitude control system that would be used in the ve_ficatton of the ARS con,:ept. For this purpose the noise simulation would be omitted. • lgure 6-_ ie a g_apnlt;u._ i--ep.'esent'-tion vf .t_.._.... ,..,.,,,_.,,,.._,1"-_'*_'-"".1,.,+...._..., ,.,.,.1,4t,_..,v A A included as part of the ARS computer simuJation. In this figure Q1 and Q2 are commanded or desired positions of the gyro spin vectors in the vehicle = coordinates. _1 and -_2 are measuredor observed gyro spin vectors_ in the coordinate system. The signal E) _bviously has a null when S 1 '. Q1same and $2 = and for small differences between the Sis and _e it can be expected that the system will control in a reasonable manner. However, for large d._ferences in S and Q the performance is not so 9reaictable. For example, if the vehicle is comn_anded to change it_ attitude by 180 with respect to each of the gyro spi_ vectors, _I = "_I anddegrees Q2 s _ $2" With these values of Q and S _he attitude error _ignal is zero. This is an unstable null and any small departure from it will result in an error signal of the proper sign to cause the system to drive to a stable null but t' ,._ pat_ or trajectory followed should be determined for attitude commands in the vicinity of the unstable null. The simulation will also pe,'_mit the investigetion of the effect of the data con- troller pal'ameters on the control system pel'formanee. For example, simu- lation of various data rates it, the computer may z_esult in a maximum data rate much less than the arbitra._y chosen 100/see for the rate calculation. Since it requires a minimum of two s&mples per cycle _e determine a sinu- soidal component in a signal, the ,_amplivg rate of 100/sec will permit the determinatior, of stnu_oidal components with frequencies up to 5N cycles per second. This is no doubt beyond the bandwidth nf the control system. Since the control system bandwidth will probably be of the order of 5 cps, a bandwidth of 10 cps in the rate information _hould be adequate and therefore iteration rate of 20/see on the vehicle rate calculation would be satisfactory. _0113-FRI 1965011367-307 - 298 - 20113-FRI 1965011367-308 - 299 - 20113-FR 1 1965011367-309 - 300- This reduction in the calculation iteration rate will permit an averaging of fiTe gyro readings for each rate calculation. The result of this averaging is a reduction of the RMS value of any random noise on the T_ro readout by a factor of _. Also, the reduction in the iteration rate for tbe vehicle rate calculation wil! result in fewer basic calculations per second in the digital controller. Therefore, the computer can be designed for a longer add time and possibly save some size in the controller. Currently the 100/sec ite_'ation rate requires approximately 110, 500 additions per second. A reduction in the iteration rate for the vehicle rate calculation to 20/see will result in shout 30, 000 adaitions per second. Therefore, the add time can theoretically be 33 microseconds. A serial controller operating with a one megacycle clock and a 16-bit word could perform an addition in 16 micr_,_econds. This would permit adequate time for the fetch and store op=,.ations as we1/as other computer bookkeeping. A/so, the serial machine would be a simpli- ficat';on fro_t the currently proposed eerial/parallel machine. In addition to using the least amount of equipment the seria! machine has some advantage_ in the event it becomes necessary to add redundancy to the machine for the purl_es of reliab_l_*-- --_dab_ O RELIABILITY STUDY A reliability study is suggested sinc,._ the parameter study considered a non- redundant system and conventional military specification parts iv arriving at the reliability estimates. Reliability can be obtained by a number of ap- proaches and the objective of the stu_ly would bc to determine which aF.proach requires the least expenditure in system size and complexity for various levels of reliability, 20115-FRI 1965011367-310 301 - The attack on the problem would be to set up as many redundancy models for the SDMEO-ARS as posslble and determine the reliabillty versus s_ze, weight, cost, and power consumption for each. Models that could be ex- amined are of the following basic types: • Redundancy gyros • Redundant computer • R_dundancy within the gyro • Rc'du;_ancy within the computer • Redundant gyros and computer • High reliability components in gyros and computer In order to establish the preferred system redundancy, giving proper con- sideration to the various trade-off considerations tnvol*red, a state reliability i anatysis method will be employed. Tilts aaalystz tool which is a simple extension of basic probability theory, permtt_ direct digital computer analysis of the most complex redundancy c_x_tgurations. Usir_ computerized state probability techniq_es, it is possible to determine the impact on the system caused by grossly changing the system redundancy mode, or by changing a single piece part failure rate. Whereas the presen t. method of system re- liability calculation requires considerable approximation and laborious hand calculationj state probabilities and figure of merit reliabilities, for all de- sired system coafigurations, are made available v_thiv minute_ using the computerized model, This resultant ease and versatility enables total system trade-ells that previously have been prohibited because of the ex- treme amount of hand calctflatton time required, even for simpie cases. The system-redundancy analysis involves the_use of probability state analysis techniques where the probability of the system under consf, deration being in some con_ition, state (fatied, tmrtially failed or operational, etc. ), tw ex- pressible by a matrix differeuttal equation: _:011$-FRI _t 1965011367-311 - 302 - dt where _P = a coltLm__n_tate. probability vector [h] = a stochastic matrix whose elements are comprised of the system failure rates T = transpose For example, given the following types of redundant systems: A. Standby, k I Tl_s configuration has four states of operation (simplified): 1. E - initial state of no failures o 2. E 1 - switch failed end thus unable to switch to the standby element 3. E 2 - switch failed and in process switcheo to the standby element and is unable to switch back upon demand or primary vlement failed 4. E 3 - complete failure (neither system capable of continuing to operate 20113-FRI 1965011367-312 - 303 -, B. Paral:el I i i I --_ This configuration has four states of operation: 1. E - no failure O " 2. E 1 - upper channel failed, lower channel 0k_y d 3. E 2 - lower channel failed, upper channel ol:ay 4. E 3 - total failure ? C. Nonredundant This configuration has two states of operation: I. E - no failure o 2. E 1 - total failure 30115-FR1 1965011367-313 - 304 - D. Other Co_binatior_.of Elements Which Are Also ActuationDependent These woula not cause any analysis difficulty. The probability of being in state E I at the end of t is Pl{t +At) = Pl(t}+P (t}{kAt +0{At)]O .., or i , d P1 (t) dt = Po(t)k Similar equations can be obtained for states E 2 and E 3 as functions of the 1 other states. With thisin mind, obsezwing the statetransitiondiagram, a transitionmatrix can be obtaineddirectly: E o E 1 E 2 E 3 _-Upper Row LeftColumn -.E (v + * kS2) (k +o ×sl (×sl) Xs2) 0 E 1 0 -k 0 k : [^l E2 0 0 -k k E3 0 0 0 0 Where the elements are the transition rates in .going from the state indicated in the left column to that indicated in the upper row, inspection then reve,,ls that the transpose of this transition matrix is the coefficient matrix of a first- order differential equation with the state probabilities as functions of time or: ' P(t) : A P(t) ,J .:....._ 20113-FRI . I 1965011367-314 -_,305 - From these schematics, the following transition diagrame are constructed: For the redundent configuration (A), the following transition d_agrarn is app!icabie: E1 ,I -IJ E o E 3 E 2 where: ' :t k = failure rate of the _.omponent kS1 = failure rate of switch yielding state E 1 i XS2 = failure rate of switch yielding state E 2J Using the basic laws of probability = +,_t) =I (Probability of being in a particular state E at t to {Probability of being in the same state at t = to) x] {Probability of not changing state) + (Probability of being in some other state at t --to) x (The probability of making transition to E during time interval _t) 20113-FR1 I 1965011367-315 - 306 - Thtm, the probability of being in E at (t + At) - 0 Po(t + _t) = Polt) [I - (k + )_Sl+ )'82TM + O(_t)] or, passing to the limit: dPo(¢> ---dr = - (k+XSI + _2 ) Po(t) ( It is well known that the probability of system success is equal to the product i of the individu'al subsystem success probabilities. Thus: N = [1 - Pi(fail)]Psuccess i=_l where N = number of subsystems Pi(fafl) = probability of subsystem failure The above first-order differential equation is easily solved for extremely complex redundancy schemes using analog or digital computer techniques. The ease of obtaining the system reliability assessmeets using this approach suggests it as a valuable design tool. In addition to generating relative reliability data, relative system design merit data such as cost, size, switching requiremems, etc., can be obtained by solving the differentia _ equation with constraints imposed on the various variables and relative costs associated with them. t_ An important concluding observation is that the proposed reliability study will be very broad. As indicated, its basic foITn will be such that one can interpret the best model (from those considered) versu_ ttrf.e. This makes the study applicable for many future ultss_'ns havb_g large variations in mission time. • 20tl3-FR1 1965011367-316 - 307 - S UPPORTING PROGRAMS There are two programs currently in progress in the Space Flight Systems Group at Honeywell that will directly benefit the proposed studies. The first of these programs is 8 ,_tudy with the objective of establishing a basic digital controller configuration tha_, could be used for numerous spacecraft control applications. The following programs w_r_ reviewed for the purpose of de- termining if they would benefit from a digital mechanization of the flight controller and if so what type of digital comroller could satisfy the require- ments of all, The programs were: • AOSO • LPTV • VO Y AGER • APOLLO • MOL • LOP • AjrcraftAutopzlots • Boost Vehicle Autopilcts In all cases the indications are that a digital controller using a serial arithmetic I i_ unit with a 16-bit word or less and one to two megacycle bit rate will satisfyi hhe requirements of all the programs. It appears that the input/outpu _.to the t controller would require some modification or tailoring for each application in order to effect a proper interface with other equipment. However, the basic .! controller consisting of arithmetic unit, memory and control unit, would remain I the same _or all applications. It is anticipated that the inputs to the controller would be derived from optical sensors, inertial sensors, thermal senso,'s, communicatiorR _ystem, and actuator feedbacks. The controller will operate on the,_e inputs to perform 20113-FRI T_r,, roll 1965011367-317 t I t - 308 - t I various logicfunctions,signalconditioningand signalcompensation and willprovide outputs to actuator , set. rs communication system, mode switching | and possiblyother subsystems. In thisprogram a configurationwillbe selectedfor the controllerthat promises to satisiythe greatest number of _pac_ r_gf,.Lccu_u_ L_LIU_L_,_I_=, and itwillbe carried through logicaldesign and simulation on a large computer. The second program will extend the first ir_ that it will take the logic design generated in the first program and will fabricate and test a breadboard of the controller. The breadboard will have the capability of performing mode control, power switching, signal compensatior and other functions that the study shows to be desirable in addition to the data processing necessary for direct control and stabilizationof the vehicle. Part of the breadboard testwillbe to includethe controllerin a controlloop with a simulated vehicle. Also, the second program willmake a study of reliabilityin the controllerfor the purpose of es*.ablishing tentativelevelsof redundancy for the various controllercomponents. The emergence and acceptance of microelectronic design techniques makes redundan,w a much more attractivesolutionto the reliabilityproblem than itonce was. Itis anticipatedthatthe informat_.onresultingfrorr,these programs would be directlyusefulin the design of a digitalcontrollerto be used with the strapped- down ESGts. For example, the components and circuitsselectedduring the breadboard stage ofthese programs would no doubt be used in the ESG controller also, and itis quitelikelythatmany decisions made on these programs would be validfor tlzeESG controller. There are also several programs in progress at HoneywellSs Aero Florida Division,the resultsof wbich v,-[llsupport the recommended follow-on effort, This divisionhas the responsibility_,odesign, develop, testand deliver the inertialguidance systems for the Centaur high-energy space vehicle. In additionto the platform and platform electronics,the guidance equipment in- cludes the digitalcoupler, digitalcomputer and signalcondflioner. , 1 ! ! 20113-FRI 9650 36z-3 e - 309 Also, Honeywell-Florida currently has a contract with the Air Fozce Research and Technology Division for the design, build, and laboratory testing of a gim- baled ESG navigvtion system. Coordination with this program has and will continue to benefit the SD-ESG componene development and ARS system programs. PRELIMINARY VvORK STATEMENT FOR FOLLO',V-ON EFFORT The recommended tasks as discussed in detail above can be reduced to a set of discrete work statements. These tasks logically fall into _wo over-all groupings and are detailed below, along with time-phasing in Figure 6-4. i i iiii i i MONTHSFROMGOAHEAD i ,. , , COMPUTERSIMULATION - ] I VEHICLESIMULATION .... .ram, J , REACTIONJETSSIMULATION ATTITUDECONTROLOOPSIMULATION !i DETERMINEPERFORMANCE FORRANGEOFCOMPUTER i PARAMETERS NOISESIMULATION i laid : INVESTIGATIONOFSYSTEM STABILITYWITHklOISE ImESEm im m RELIABILITYSTUDY SELECTIONOFREL_BILITY MODELS GENERATETRADE-OFFDATA ..... I i - I REPORTING _ I A A A & A A AI= i _ ' lU _ I Figure 6.-4. Preliminary Schedule 20113-FRI 1965011367-319 - 310 - _ TASK I - SIMULATION AND NOISE STUDY I. Simulate the ARS computer defined in the parameter study of JPL Contract No. 950915. a. Write a program to simulate the A_q co_m_pt!ter on the digital ! computer of the hybrid computer installation. i I b. Conduct checkout of the simulation. i 2. Simulate the selectedspace vehicle. I a. Select the equations of motion, including bending modes, if any, and coefficients typical of an intraplaP.etary vehicle. i bo Program the selected equations on the analog ,__nputer.c, Conduct checkout of th simula'.tL._:,. i 3. Simulate the gyro outputfor inputI, ',RSdigitatcontrollersimulation. a. Derive the equations relati_.g e_..put jf spacp vehicle simulation and gyro outputpulses. ? i b. Program the equationsderived in (a)eitheranalog or digital as necessary. t c. Check out the simulation. I 4. Simulate the dynamics of the reaction jets. a. Select a jet configuration and thrust level typical of the intraplanetaryvehicle. b. Derive an analog or digitalsimulation as appropriate to operate with computer and vehicle simulations. c. Check out the simulation. 20113-FRI 1965011367-320 - 311 - 5. Simulate the attitude control loop, ao Interconnect the simulations of the vehicle, gyro controller and control jets. "_ b. Verify the scaling throughout the simulations and check out the simulations. 6. Using the control system simulation investigate system performance. a. Determine system response to commands over the range of data rates on both attitude error and attitude rate. b. Determine effect on system operation of various controller accuracies (work length). 7, Simulate gyro readout noise. a. Write a program to generate pseudo random numbers with a distribution_Irnilarto the gyro readout noise. bo Insert the simulated noise in the control system simulation arm determine the effect on system performance, i_' 8. Investigatemethods of obtainingcontrol system stabilizationin the presence of gyro readout noise. ao Include digital pseudo rate in the computer simulation and in- vestigate the ability of this method to stabilize the system° bo Include a simple averaging filter in the computer simulatior and determine the ability of the filter to stabilize the system. Co Investigate the effectiveness of digital filters of a lead-lag nature to effect stability in the control system. 20113-FRI i 1965011367-321 f - 312 - I TASK II - RELIABILITY STUDY 1. Selec_ the reliability mcdels. 2. Prepare computer programs. 3. Obtain reliability data from programs. 4. Summarize results (reliability versus system parameters) i ! r v | 20113-FR1 'I 1965011367-322 APPENDIX A DERIVATION OF EQUATIONS FOR CALCULATION OF THE THIRD SPIN AXIS DIRECTION COSINE 20113-FRI 1965011367-323 ,° APPFNDLX A DERIVATION OF EQUATIONS FOR CALCULATION OF THE THIRD SPIN AXIS DIRECTION COSINE INTRODUCTION "'] Since gyro readout can previde data from only two pickoffs at any one time, the direction cosine of the spin axis relative to the third pickoff must be corn- / puted from the direction cosines for the two active pickoffs and pickoff align- ment data measured durivg gyro assembly and installation. The following discussion pre_.ents a derivation of the equations ana relationship necessary to this calculation. J DEFINITIONS For _e purposes of th_s discussion, the gyro pickoff axes will be defined by ili the unit vectors P1, P2, I_3 which in generalhave a small departure from orthogonality. I The three direction cosires that locate the gyro spin axis relative to Pl 1_2 If P3 are defined by: t6 _3 = P3 S is a unit vector along the gyro spin axis. For this derivation, three orthogonal sets of reference frames are define U ]'s'in teems of the pickoff vectors PI' P2' | I 20113-FR1 i 1965011367-325 -A2 - The reference frames are defined as follows: I1 = P1 P3 f:'z1 3 l I_1 = PlX (P2 x P1 ) / 1_'3x P1 i = P3 - P1 " P3 I_I " I_3x Pll ._ t2 : P2 x (PI x _2) I I_I x _21 = PI - P2 " l_l _2 ' ! P1 x P2 l - #J2 : 2 K2 = b 1 x f_2/ IPI _P21 _3 = f._x f'3 / IP2 _ P2 ! °_3 : P3x (P2 x P3) /](P2 x P3 i = P2 - 153" P2 P3 l i% x P31 _3 -- P3 The. corresponding three matrices be.tweenthe picker axes and the orthogona[ reference frames are: ' : 20113-FR1 P 1965011367-326 - A3 - - _i] "I . m ,. ! PI i 0 0 I 1 ) .I ['2 = P2 Pi (_IP2P_) P2 i_-i_ i. _3_i. i_2 _i " Mi _1 i IP3xi'il li_xi_; I ,: I _ ] 1 '! [_ I"(_I._2_2" _i"_2 o f2 i /PlXP2/ b2 : 0 i 0 J2 " M2 32l , I ' i iI P3 PI" P3 - PI' P2 P2" P3 P3" P2 (1_1 P2 ]P3) tC2 I?C2 i ], j ri . . AiPl (l_l P2 P3 ) P!' P2- P2" P3 PI" _ PI" P3 13 !3 ,.) . I_ 2 X _ ,_ I --'l IP 2 x P3 l ]P2 = 0 1 "(P2" P312 P2" P_ J3 " M 3 J3 /i, 2 ,_f%/ ,I i where (P1 P2 P3 ) = PI" P2 x P3 } ( 20113-FR1 ] 9650] ] 367-327 - A4 - All elements of matrices M1, M2 and M 3 may be determined from proper alignment measurements during gyro assembly and installation. DE RIVATION The gyro spin axis may be expressed in the defined orthogonai reference frames as: :-"(S" in) in +(_' Jn)3n +{S" kn)kn n = I,2,3 (AI) and since S is a unitvector (S. in )2 +(S. _n )2 +(S. kn )2 "_ 1 (A2) ' By inverting the matrices, M1, M2, M3, (in, in" kn } may be expressed in terms of the pickoff axis 1_1, P2' P3" The inverse of the matrix M can be found using the following formula: [ ,..thp element of inverse - (-1)i+J M'.m°r of i_th element of MJ__[Determinant of M ]1 i Consider matrix M 1 fo," the illastration of the matrix inversion procedure. -1 If the inverted matrix M 1 is defined by the elements Mij, then 20113-FR1 i I IIII II L I 1965011367-328 - A5 - i "P2 I-(P " 3) P3' P2 "P3- "P3 PI "P2 M21 % _'a_ [_-_l" _>_] IP3 xP ! 12 i I 1 .. [ .2]' . ]PIP2f_31I-(i_z .Pa) 1 -1 The other seven elements of the M 1 matrix can be derived in a similar i fashion and are given in Equation A3. Equations A4 and A5 give the inverses of M2 and Ma, respectively. t ,! _1 ' o o b1 bl J'l II_sxl_ll _'1"_3)(1_2"_'S)" (t_l'ha) l_3_'f'11 ]b3xlt'll [(bl "b3)Cf'l"f'a)" (_:l'b3!] " _'l If'axbll (i'1 "ha; o tba_bll b3 b3 20.113-F}-_ 1 1965011367-329 ++ 1 - +46- 20113-F_1 i 1965011367-330 - A7 - t -1 -1 M2-1_ M 3 are redefined usingTo simplify the algebra, the elements of M 1 , the symbols An, Bn, Cn, D n and E n. The definitions are given in Equations A6, • AT, and A8: F,. 0 01 , {-Al 0 _1 } 1 E 2 -A 2 0 -I ., M 2 = O 1 0 (A7) 'i} D2 B 2L. _]C 3 D 3 B 3 Li -1 }) M 3 = 0 E 3 -A 3 (A8) { o o 1( - ! Using the exprc3sions for i, 2, k, defined in Equations A3 and A6 in Equation A2, result_ in the following expressions: ] [,_ " " ][ _I,i r^ "S 2 21 =LPl _ 1B1 +P2 C1 + P3D1 )' S 2 + (__IA1 +P3E1) ' (A9) ,( Let A , PI'S = a{ ^ A P2" S = a 2 3 S = _3 ,li ,!t 2Oll3-FR1 1965011367-331 - A8 - Then I = .a._2 _[Blal + Ci_2 +Dlc_3l 2 +[EI_ 3 - Alal] 2 (AI0) Similarly u._ing A4, A5, and A7, A8 in Equation A2, = (ala) i , i i i I _- where }' " /% _'z//'a "A, = x ' PIo. _. i - (PI " 153)z A2 = ii51 x 152 / {.4z . 152) , A _ A3 = !P2 x P3 / tP2'' ' P3 ) -(F2"P33)2 " /(i, z_2 /'a _', /'2B1 = /PaXPI I" "_a " - " ) 20113-FR1 1965011367-332 .j - A9 - • _ . P2 ) ](PIP2 P3 ) [I - (PI B3 - IP2 xl_ 3 / (l_ I . l_2 l_2 . l_3 - 131 . 133) (P,r.,P_ r,-(_2._3_2] C, : 1153 x i_i!2 {PIP2P3 )2 }, c2 -- /PIxP2/2 (PI P2 P3 ) C 3 : /i_ 2 x i_3/2i - L . L ,, D I = /IS3 x P1 / (D'_" P3 _I" I_2- P2 " I_3) D2 = /1_ 1 x 6 2 [ (1_1 . 1_2 1_2 . I_3 - P1 " P3 ) ^ ^ A A ^ D3 = '_2 x P..o/ (P2 " I_"PI ' P_.- P1 " I_2} if _:_: /_3xP_l I( E2 : i I_Ix6_..I il / " (PI" P2 )2 il 1965011367-333 - A10 - E3 _ /P2xP3/ Equations Aii. Ai2 and AI3 are the desired equations for the calculation of the third direction cosine. Equation All is used to calculate a in the event pickoffs 1 and 3 are providing gyro readout information. Equations A12 and A13 are used in a similar manner when the other pairs of pickoffs are pro- viding readout data. _.0113-FR1 1965011367-334 APPENDIX B DERIVATION OF EQUATIONS FOR ATTITUDE ERROR AND BODY RATE 201 ]3-1"_1 -- _ iRa ii Jr _ - I---- lID_l_li_'__ . Q i 1965011367-335 - BI - APPENDIX B DERIVATION OF EQUATIONS FOR ATTITUDE ERROR AND BODY P_%TE l _,,c_.11._._..pres_-*- +h_ deriva_tionand discussionof the euuations meeharized in the "AttitudeError Body Rate" sectionof the system computer. ATTITUDE ERROR EQ.bATIONS i It iq assumed that the stabilization and control system- -11w._ b_ abl_ to perform suitable reorientation and attitude hold maneuvers with an i inpu_ signal that is not a linear functionof the attitude error. It is also assumed that the input signal has its null coincident with the attitude i. ! error null. The assumption is also made that input signal and attitude error have the same sign ever?where. It is ._Iso a_sumed that reorientation of the ve.hicle will be commanded by F,-oviding the attitude reference system with the six direction cosines *.hat define the new desired posit._on of the two gy;,'o spin vectors in the vehicle body axis. In this derivation Q1 and Q2 arc two unJ* vectors representing the commanded position of the _ro spin vectors in the vehicle ,_ )dy coordinates. S1 and $2 are two unii vectors representh_g the observed position of the spin vectors in body coordinates resulting from the gyro readout and the data processing of the "Cosine Computer" il discJssion. The differencebeL_veenthe commanded mid observed vectors is: ! I 20113-FHI 1965011367-337 - B2 - 2hese diffprences may be related to attitude error, 8 by = X _1 _1 0 is a mlique rotatlor,vector which willnul} simultaneously AS 1 and _S 2 ifthe vehicleis commanded to rotateabout thisvector untilthe magnitude : of 0 is zero. _, Equating _he two expressions for _'S1 and _'$2 8 x 2 - 2 - 2 (B2) Cre3s S1 into (B1) and cross $2 into(B2) i 8" = Q1 xS1 + (_'' $1 ) $1 (B3) _ _ = Q2x_2 + (_' §2) _2 (B4; ! _' Dot $2 into(B3) and S1 into (B4) i _ " §_ " qz x sz §_+(_" §z) §z"§_ (B6) i Solve for (8". SI) - . 1 Q2_ ' Q_"s_ s_.(§l _) (B_) S. Sl /$1x§2/z _)2 Sl + " " [ Substitute(B7) into(B3) [ / [.' 1 _Q1 $1 x " 20113=FR 1 1965011367-338 - B3 - Equation (ES)is the desired expression for the attitudeerror. From thisequation itcan be concluded thatthe attitt,de error is a function of only commanded and observed positions of the spin vectors. Since the command is given in body coordinates and sincc the spin vectors are measured or observed in body coordinates, the attitude error computed from Equation (B8) with these inputs will also be in the body coordinates. BODY RATE EQUATIONS Assuming the gyro spin vectors are known in body coordinates as a function of time the body rates may be determined as follows: The rate of change of the spin vectors in inertialspace may be expressed aS: -- 1dt II dt ]B = --- + _ x S 2dt II dt B ' where S 1 and S2 = spin vectors = angular velocity vector of the body in inertial space - rate of change of S relative to inertial space I - rate of change of S relative to body coo_'_llnates B 20113-FR1 1965011367-339 - B4 - Since the gyro spin vectors are fixed r,_.lative to inertial space --. A4. and "7" = = x _ (B9) ' B L . = = x o (B10) B Crossing S1 intolBg.' and $2 into (B10) __ -r- i ,, = s1 x sl + (_ sl ):jl 1Rll) "--x -__. ._ "U = $2 $2 + ( 2) 2 (BI2) t l dotting $2 into(B11) ands ! into (B12) _7 $2 = S-_ x _I" $2 + _' $I) SI" $2 (B13) _ o s_,,_2"s_ +(_. ;2)?_' _ _-_4_, Solve for -6". Sl and substitute in (B1 I) 20113-FR I - i f 1965011367-340 - BS- Equation(B15)relates vehicle body ratu to spin vector rate relative to the .( bo(_'. ht usl_)gthe above equationto caicuiate"o-,the body rate, the computer •=t_==, ^ ^ will appraximate S i end S2 by finite d_ferences of readings of S 1 and S,,._ 20113-FI_I i [ 1965011367-341 I APPENDIX C { COORDINATE TRANSFORMATION BY "METHOD OF LEAST WORK" I i ¢ i -e- l ¢ :'! I '! 20113-FR1 I 1965011367-342 -C1 - APPENDIX C COORDINATE TRANSFORMATION BY "METHOD OF [,F,AST WORK" Presented here is a simplified method of obtaining the matrix elements of orthogonal transformations and of El,ler rate to body rate transfgrmations. Consider a rotation of a coordinate system about the Z-axis. Y1 YO S _ : _ X 0 *) This can be expressed as a matrix transforming the vectors XO, YO' ZO intoX l, YI' andZ 1. 0 -_, _ x, o- ,xol ' IiyY1 - -s_ _ 0 i o IE J 1 m ,._ o o, _'-oJ 20113-FRI 1965011367-343 - C2 - A figure representing a resolv_r can be drawn that wilt show this same transformation. .#J I YO -- I " " ' Y1 = -XoS_ + YO C-_ i # I ZO " I -" Z1 = ZO Following a signalthrough the resolver in a diagonal directiongives a sine function. A dot can indicatewhich sine functionrequires the negative sign, A signalgoing straightacross is multipliedby the cosine. A slightly modified picture is more useful. ¢, XO _1_ X 1 YO -'/ i _':-Y1 i Z 0 -- :Z 1 With this as the basic building block any number of additional rotations may be added. Consider the t,'ansformationfrom inertialcoordinates to body axis coordi- nates through the Euler angte_ ¢_, e, 4D. t:' 20113-FRI # 1965011367-344 - C3 - The first rotation is as pictured above. To this is added the blocks for rotationof 8 about YI' and d_about X 2. CJ xl e x_ ¢ Xb, / I YbYO - _ 1 _ 2 , I I z Zo-- : " ' I " _ .... b To tran_:forma vector in the inertialsystem to the body axis system trace allpossible paths from XO° YO* ZO to X b, Yb 0 and Z b. Thus the expression for X b willin general containXo, YO and Z O with appropria_,e coefficients, More than one path may be possible between two points. For example going fror_X O to Zb is possibleby C_ SOC_b and (-sv)(t)(-s,). r Each element of the matrix can be written by inspection of the diagram. xb (c¢co1 (-so) Xo ] zb (sos,+ c_soc,) (s,/soc, -c_s,) (coc#) 2CI 13-FRI I 1965011367-345 C,t - This inverse c_n be written by following all paths from Xb, Yb' Zb to XO" YO" an'J ZO° r×o - '[ (cec¢) (-c_ s¢+ s¢sec_) (c¢ ses_- s_@) -Xbl I YO I = (COS_) 1C¢_0#+ S#_SeS_) (C,SeS_ - S¢_0_) Yb l-SO) (S_CO) (COCO) Zbt" In this case (orthogonality) the inverse can be written immediately from e the first matrix by interchanging rows and columns. A special case of a nos-orthogonal matrix is the transformation of Euler angle rates to body rates. The original diagram can be modified to show these rate inputs. Xo" - _ ..... _ I.... "Xb'P/ ,I _ _ __ Zb, r ¢J The transformation to body rates follows the same rvles as the orthogonal transformation. Follow all poseible paths from each of the Euler rates to +.he body coordinate system. 2011 $- FRI 1965011367-346 I- C5 - I Thue: -I "7: p 1 0 -SG I r 0 -SO COCO 0 ,t The inverse of this non-orhAogonal matrix is not so straight forward. Some I rules can be stated that allow the inverse to be written by inspection. I The rules are: .( (_) Cut the single line that connects two of the Eu!er rates (_neglect- {_ ing the lines th_.t are unity). Thls _rzll always be a sine function in the middle gimbal rotation. (2) Divide (inr::eadof multiplying)by the lastfuncttomof the n_iddle el gimbal rotationbefore arriving at the desired Euler rate. Itmay be necessary to go through two functionsof thismiddle gimbal # I rotation to complete the path. The example will clarlfy the rules. • 7 _- s_sa c_o - • 6 = 0 C_ -S_ q c_0 _ . L _ E t I1 . 20113-FRI I] 1965011:367-:347

Advanced control moment gyro development

Abstract

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Advanced control moment gyro development

Abstract

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