Inverse Kinematics for a Serial Chain with Joints Under Distance Constraints

Inverse kinematics (IK) problems are important in the study of robotics and have found applications in other fields such as structural biology. The conventional formulation of IK in terms of joint parameters amounts to solving a system of nonlinear equations, which is considered to be very hard for general chains, especially for those with many links. In this paper, we study IK for a serial chain with joints under distance constraints, in particular, either a spatial chain with spherical joints, or a planar chain with revolute joints (in this paper we ignore other constraints such as joint limits and link collision-free constraints, a common approach in studies of inverse kinematics). We present a new set of geometric parameters, which are not joint angles, for such chains, and use a novel approach to formulate the inverse kinematics as a system of linear inequalities, which is an exact, not an approximate, formulation of the IK problem. It follows that the IK problem for such a chain with an arbitrary number of joints can be done efficiently in many ways. Under our new formulation, the set of solutions for an IK problem (as specified by the positions of the two end points of the last link), and more generally the set of solutions for all IK problems, is essentially piecewise convex. Our approach can also be generalized to other linkages such as those with prismatic joints sandwiched between rotational joints and with multiple loops that have a tree decomposition of triangles. The efficient algorithms and nice geometry entailed by piecewise convexity considerably simplify IK related problems, including motion planning, in the systems under study, and thus broaden the class of practical mechanisms at the disposal of robot designers.