## Repeatable generalized inverse control strategies for kinematically redundant manipulators

##### Abstract

Kinematically redundant manipulators possess an infinite number of joint angle trajectories which satisfy a given desired end effector trajectory. The joint angle trajectories considered in this work are locally described by generalized inverses which satisfy the Jacobian equation relating the instantaneous joint angle velocities to the velocity of the end effector. One typically selects a solution from this set based on the local optimization of some desired physical property such as the minimization of the norm of the joint angle velocities, kinetic energy, etc. Unfortunately, this type of solution frequently does not possess the desirable property of repeatability in the sense that closed trajectories in the workspace are not necessarily mapped to closed trajectories in the joint space. In this work, the issue of generating a repeatable control strategy which possesses the desirable physical properties of a particular generalized inverse is addressed. The technique described is fully general and only requires a knowledge of the associated null space of the desired inverse. While an analytical representation of the null vector is desirable, ultimately the calculations are done numerically so that a numerical knowledge of the associated null vector is sufficient. This method first characterizes repeatable strategies using a set of orthonormal basis functions to describe the null space of these transformations. The optimal repeatable inverse is then obtained by projecting the null space of the desired generalized inverse onto each of these basis functions. The resulting inverse is guaranteed to be the closest repeatable inverse to the desired inverse, in an integral norm sense, from the set of all inverses spanned by the selected basis functions. This technique is illustrated for a planar, three degree- of-freedom manipulator and a seven degree-of-freedom spatial manipulator.