Solving the Kinematics of the Most General Six- and Five-Degree-of-Freedom Manipulators by Continuation Methods

This paper presents a unique approach to the kinematic analysis of the most general six-degree-of-freedom, six-revolute-joint manipulators. Previously, the problem of computing all possible configurations of a manipulator corresponding to a given hand position was approached by reducing the problem to that of solving a high degree polynomial equation in one variable. In this paper it is shown that the problem can be reduced to that of solving a system of eight second-degree equations in eight unknowns. It is further demonstrated that this second-degree system can be routinely solved using a continuation algorithm. To complete the general analysis, a second numerical method—a continuation heuristic—is shown to generate partial solution sets quickly. Finally, in some special cases, closed form solutions were obtained for some commonly used industrial manipulators. The results can be applied to the analysis of both six and five-degree-of-freedom manipulators composed of mixed revolute and prismatic joints. The numerical stability of continuation on small second-degree systems opens the way for routine use in offline robot programming applications.