Redundant Manipulator Kinematics and Dynamics on Differentiable Manifolds
A recently published treatment of nonredundant manipulator kinematics and dynamics on differentiable manifolds is extended to kinematically redundant manipulators. Analysis at the configuration level shows that forward kinematics and dynamics of redundant manipulators are identical to that for nonredundant manipulators. The manifold-based inverse kinematics formulation that is presented for redundant manipulators, in contrast, yields parameterizations of set-valued inverse kinematic mappings at the configuration level, where sharper results are obtained than those presented in the literature using velocity formulations. Explicit expressions are derived for set-valued inverse kinematic mappings for both serial and nonserial (called compound) kinematically redundant manipulators, as functions of vectors of arbitrary parameters. Parameterizations are presented for both manipulator regular configuration manifolds and self-motion manifolds, the latter comprised of sets of inputs that map into the same output. It is shown that kinematically redundant configuration manifolds and self-motion differentiable manifolds are distinctly different and play complementary roles in redundant manipulator kinematics. Computational methods are presented for evaluation of set-valued inverse kinematic mappings, without problem-dependent ad hoc analytical manipulations. Redundant serial and compound manipulator examples are presented to illustrate computation of set-valued inverse kinematic mappings and the use of self-motion manifold mappings in obstacle avoidance applications. Differentiation of configuration level inverse mappings yields inverse velocity and acceleration mappings as functions of time-dependent arbitrary parameters that play a central role in manipulator dynamics and control.