Symbolic differentiation of the velocity mapping for a serial kinematic chain

Abstract

This paper symbolically calculates the derivatives of a serial kinematic chain's velocity mapping. The derivatives are taken with respect to a change in one of the joint angles, as well as with respect to time. The result is a linear mapping, whose explicit form depends on the mathematical representation chosen for the Cartesian twist of the end effector. Three of the most common representations are treated: the “inertial”, “body-fixed” and “hybrid” representations. Numerical calculations illustrate the correctness of the presented results.