Manipulator Kinematics and Dynamics on Differentiable Manifolds: Part II Dynamics

Abstract

The manipulator differentiable manifold kinematics formulation presented in Part I of this paper is employed for forward and inverse kinematic analysis and formulation of ordinary differential equations of motion on disjoint, maximal, singularity free, path-connected components of regular configuration space, for a broad spectrum of manipulators. Existence of forward and inverse configuration mappings throughout maximal singularity free manifold components yields analytical forward and inverse velocity mappings. Efficient computational algorithms for forward and inverse configuration and velocity analysis on a time grid are derived for each of three manipulator categories. Manifold parameterizations are used to transform variational equations of motion in manipulator coordinates to second-order ordinary differential equations of manipulator dynamics, with both input and output coordinates as state variables, eliminating the need for ad hoc derivation of equations of motion. Criteria that define manipulator differentiable manifolds are shown to guarantee that the equations of motion derived are well-posed on maximal singularity free components of manipulator configuration space. This process is illustrated by presenting terms required for the evaluation of equations of motion for three model manipulators. It is shown that computation involved in the evaluation of equations of manipulator kinematics and dynamics can be carried out in real-time on modern microprocessors, supporting the in-line implementation of modern methods of manipulator control.