Canonical Subproblems for Robot Inverse Kinematics

The inverse kinematics (IK) problem for many common robot manipulators may be decomposed into canonical subproblems which are solved by finding the angles on circles where they intersect with other geometric objects. We present new algebraic solutions and geometric interpretations for six subproblems using a linear algebra approach, and we demonstrate significant computational performance improvements over existing IK methods. We show that IK for any 6-dof all revolute (6R) robot with three intersecting or parallel joint axes may be solved in closed form using subproblem decomposition. For any other 6R robot, subproblem decomposition reduces finding all IK solutions to a search over one or two joint angles. The first three subproblems, called the Paden-Kahan subproblems, are Subproblem 1: Circle and Point, Subproblem 2: Two Circles, and Subproblem 3: Circle and Sphere. The other three subproblems, which have not been extensively covered in the literature, are Subproblem 4: Circle and Plane, Subproblem 5: Three Circles, and Subproblem 6: Four Circles. Our approach also finds the least-squares solutions for Subproblems 1-4 when the exact solution does not exist.