Distributed impedance control of an n DOF serial link manipulator using interpolation theory

In this paper, we consider the general problem of impedance control of an n degree-of-freedom (dof) serial link manipulator (redundant or nonredundant) with emphasis on the redundant manipulator case. Other approaches in the literature insist upon keeping the number of impedance dofs less than or equal to the number of manipulator dofs with a clear partitioning of self-motion & desired end effector motion. Our approach does not possess these limitations and is based on, and even encourages, the imposition of multiple conflicting impedance objectives. These desired impedances can be either `fictitious' or actual, `link' or `joint' based, and can be customized to a particular application of interest. For example, fictitious impedances can be artificially created by a collision avoidance field generated by an array of range sensors. Impedance interpolation is the key vehicle for resolving the multiple (and likely conflicting) impedance objectives to produce a blended or compromise solution. Since the interpolation criteria need not be unique, several types of interpolation are proposed. One, an `additive' interpolation method is based on adding the torques implied by all of the impedances. Another proposed interpolation method (least-squares interpolation) is based on minimizing a quadratic function that is a weighted measure of all of the impedance errors. The problem when only 1 additional `non-end-effector' impedance dof exists is first solved followed by the problem when a finite number of impedance dofs exist. Finally, the solution for the distributed (both 1-D and 2-D) impedance problem is developed as a limiting extension of the finite dof problem. This latter case is particularly relevant to the situation where the robot wraps around objects. In all cases, two interpolation solutions (i.e. additive and least-squares) are presented in parallel. We show (using singular value decomposition) that provided that a sufficient number (or distribution) and type of impedance dofs are specified (e.g. ≥n for the discrete case), a stable control law solution will always exist. Furthermore, singularity robustness of the control law solution can be increased by judiciously adding impedance dofs to the problem. Finally, the resultant local dynamics are studied and shown to be conditionally stable and can always be made unconditionally stable by introducing sufficiently high joint impedances.