Inverse Kinematics for a Serial Chain with Fully Rotatable Joints

Abstract

Abstract-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.

Forward and Inverse Kinematic of Continuum Robot for Search and Rescue

“…A forward kinematic model [2][3] is then derived by product of exponentials formula [7][8]. The inverse kinematics [9][10] for the robot is carried out by a geometrical approach. Finally, the forward and inverse kinematic simulation is completed by Matlab.…”

Stratified Deformation Space and Path Planning for a Planar Closed Chain with Revolute Joints

“…1(a) illustrates a path between two deformations of a certain spatial 1000S loop with randomly chosen link lengths. The two ends of the path were generated by a method we call diagonal sweeping [8]. For each, we found a valid vector r of 997 positive inter-joint distances, and a vector s of 997 random angles in [0, 2π] specifying triangle orientations (all angles are valid), in 19 milliseconds with Matlab on a desktop computer.…”

“…In [9], we denoted the set r(DSpace) of feasible values of r by DStretch; we keep that notation, and also write DStretch + for r(NDD). Our proof of Theorem 1 shows that, for a given nR loop, a value of r is feasible (corresponds to some valid loop deformation) if and only if that value and the given link lengths allow the successful construction of the n−2 anchored triangles: if some entries of r are too big or too small, one or more anchored triangles will be impossible to construct.…”

“…Although there are as yet no theoretical bounds on the worst-case running time of LP , in practice [19] LP is considered a mature field with many efficient algorithms. We have also developed efficient methods other than LP (described in [8]) that take advantage of the kinematics of a planar nR loop and the particular simplicity of the constraints. Our fastest generation methods have linear time complexity Θ(n), which is optimal.…”

“…For a planar nR loop, we need at most n − 2 singular deformations on codimension-1 strata, one for each triangle, so our path will be piecewise linear with at most n − 1 segments, each on the closure of one codimension-0 stratum. With this approach, the running time of our algorithm is determined by the time needed to generate O(n) singular deformations of depth 1, and has an upper bound of O(n 2 ) when using deformation generation methods with linear running time [8].…”