A New and Efficient Algorithm for the Inverse Kinematics of a General Serial 6R

Abstract

In this paper a new and very efficient algorithm to compute the inverse kinematics of a general 6R serial kinematic chain is presented. The main idea is to make use of classical multidimensional geometry to structure the problem and to use the geometric information before starting the elimination process. For the geometric preprocessing we use the Study model of Euclidean displacements, sometimes called kinematic image, which identifies a displacement with a point on a six dimensional quadric S62 in seven dimensional projective space P7. The 6R chain is broken up in the middle to form two open 3R chains. The kinematic image of a 3R chain turns out to be a Segre-manifold consisting of a one parameter set of 3-spaces. The intersection of two Segre-manifolds and S62 yields 16 points which are the kinematic images representing the 16 solutions of the inverse kinematics. Algebraically this procedure means that we have to solve a system of seven linear equations and one resultant to arrive at the univariate 16 degree polynomial. From this step in the algorithm we get two out of the six joint angles and the remaining 4 angles are obtained straight forward by solving the inverse kinematics of two 2R chains.