Automated Formation of Closed Form Kinematic Displacement Polynomials for Open Loop Planar and Spatial Chains

Abstract

A new numerical approach for determining inverse kinematic polynomials of manipulators is presented in this paper. Let the inverse kinematic polynomial of a manipulator in one revolute joint variable θi be represented by gnTn + gn-1Tn-1 + gn-2Tn-2 + • + g1T + go = 0. T = tanθi/2 and go, g1...gn are polynomial type functions of hand position variables. The coefficients g are expressed in terms of undetermined coefficients and hand position variables. Then the undetermined coefficients are evaluated by using direct kinematics and the solutions of sets of linear equations, thus determining coefficients g and the inverse kinematic polynomial. The method is general and may be applied for determining inverse kinematic polynomials of any manipulator. However, the number of linear equations required in determining coefficients g become significantly larger as the number of links and the degrees of the manipulator increase. Numerical examples of 2R planar and 3R spatial manipulator are presented for illustration.