New attitude penalty functions for spacecraft optimal control problems

A solution of a spacecraft optimal control problem, whose cost function relies on an attitude description, usually depends on the choice of attitude coordinates used. A problem could be solved using 3-2-1 Euler angles or using classical Rodriguez parameters and yield two different ``optimal`` solutions, unless the performance index in invariant with respect to the attitude coordinate choice. Another problem arising with many attitude coordinates is that they have no sense of when a body has tumbled beyond 180{degrees} from the reference attitude. In many such cases it would be easier (i.e. cost less) to let the body complete the revolution than to force it to reverse the rotation and return to the desired attitude. This paper develops a universal attitude penalty function g() whose value is independent of the attitude coordinates chosen to represent it. Furthermore, this function will achieve its maximum value only when a principal rotation of {plus_minus}180{degrees} from the target state is performed. This will implicitly permit the g() function to sense the shortest rotational distance back to the reference state. An attitude penalty function which depends on the Modified Rodriguez Parameters (MRP) will also be presented. These recently discovered MRPs are a non-singular three-parameter set which can describe any three-attitude. This MRP penalty function is simpler than the attitude coordinate independent g() function, but retains the useful property of avoiding lengthy principal rotations of more than {plus_minus}180{degrees}