An optimal regulator for stabilization of multi-joint reaching movements under DOF redundancy: a challenge to the Bernstein problem from a control-theoretic viewpoint

Abstract

An optimal regulator problem for multi-joint reaching movements of redundant manipulators is solved from the non-linear control-theoretic viewpoint. Given a target point that the manipulator endpoint should reach, a task-space position feedback with joint damping is introduced, which enables stabilization of reaching movements even if the degrees-of-freedom of the manipulator is greater than the dimension of task space and the inverse kinematics is ill-posed. Usually, the speed of convergence is slow, depending on choice of feedback gains for joint damping. Hence, to speed up the convergence without incurring further energy consumption, an optimal control design for minimizing the integral of a quadratic form of task-space control input and velocity output for an interval [ t0, t1] plus a quadratic function at t = t1 are introduced. This leads to the derivation of a Hamilton–Jacobi–Bellman equation that is solvable in control variables. It is shown that, in the case of infinite time horizon [ t0, ∞), the optimal control reduces to a task-space velocity feedback, and the Hamilton–Jacobi equation becomes solvable in an explicit quadratic form. Finally, a functional relationship of the optimal regulator with the passivity of the closed-loop dynamics is presented.