A Unified Geometric Approach for Inverse Kinematics of a Spatial Chain with Spherical Joints

Conventionally, joint angles are used as parameters for a spatial chain with spherical joints, where they serve very well for the study of forward kinematics (FK). However, the inverse kinematics (IK) problem is very difficult to solve directly using these angular parameters, on which complex nonlinear loop closure constraints are imposed by required end effector configurations. In a recent paper, our newly developed anchored triangle parameters were presented and shown to be well suited for the study of IK problems in many broad classes of linkages. The focus of that paper was the parameterization of non-singular solutions; among many specific types of IK problems, only one, that of a spatial chain with spherical joints imposing 5 dimensional constraints, was developed in detail. Here we present a unified approach to the solutions of that and two other types of IK problems. The critical concepts in our approach - the geometric formulation in anchored triangle parameters, and the application of loop deformation spaces - are general for all IK problems, and especially useful for redundant systems. For the three IK problems addressed in this paper, we demonstrate convexity properties of the set of IK solutions. We also give detailed descriptions of the parameterization of singular deformations. Similar ideas apply readily to linkages involving multiple loops.