Computationally-efficient kinematics of an in-parallel actuated wrist

The knowledge of machines’ kinematics is the first and fundamental step for their study and many different tools and approaches are available to the researcher or the technician to work out a mathematical model. However when machine’s kinematics must be solved and implemented in some programming language for the real-time control of the machine, it is important that the developed model is computationally efficient and written on fast rate hardware. This is actually the case encountered by the researchers of the Polytechnic University of Marche; in fact they had to realize the control of a spherical orienting device which is available in the Laboratory of the Department of Mechanics and is in-parallel actuated by three linear motors: the performances of such machine were potentially very good but a first implementation of the controller, developed by means of the Flexmotion card on the NI/PXI platform, put in evidence the need of a boost in the evaluation times of kinematics loops. To this aim the Authors pursued a twofold strategy: looking for a more efficient way of deriving the kinematic model of the machine and implementing it on high-rate control hardware. In the last years many researches focused their attention on the computational problems related to solving the complex kinematics of PKM’s. Since 1990, for instance, Innocenti and Parenti-Castelli [1] introduced a new efficient way of solving the direct kinematics of Gough-Stewart based machines, taking advantage of the polynomial nature of the problem. Many researchers in the following years improved their approach and specialized it to different specific cases and in the present work the Authors applied the method to work out the solution of the kinematics of the mentioned spherical parallel machine. The resulting model is very compact and allowed an easy implementation on a control card based on the FPGA (Field Programmable Gate Array) architecture. The main characteristics of such device are the flexibility, that makes it fully reprogrammable and suitable for the realization of logic functions, and its implicit parallel architecture that ensures high-speed computation capabilities, thus making it particularly suitable for real-time applications [2]. Nevertheless, it is important to remark that the implementation of FPGA based algorithms requires a fixed-point arithmetic representation and therefore a particular care in software development. Finally, the Authors are presently developing a simplified model of machines’ inverse dynamics in view of the development of model based control laws: in this case the compactness of the model is obtained through the application of the principle of virtual work [3] and the use of screw theory [4] to simplify the formulation of the differential kinematics