A Slow Manifold Approach to Feedback Control of Nonlinear Flexible Systems

In this paper we consider the control problem for a class of coupled, second-order singularly perturbed nonlinear dynamical systems. The problem has important application to flexible mechanical systems including robot manipulators with flexible joinra, where the singular perturbation parameter ¿ is the inverse of the joint stiffness. For this class of systems it is known that the reduced order model corresponding to the mechanical system under the assumption of perfect rigidity is globally linearizable via nonlinear state feedback, but that the full order flexible system is not, in general, linearizable. We utilize the concept of integral manifold to represent the dynamics of the slow subsystem, which reduces to the rigid model as the perturbation parameter tends to zero. We show that linearizability of the rigid model implies linearizability of the flexible system restricted to the integral manifold. Based on a power series expansion of the integral manifold around ¿ = 0 we show how to approximate the feedback linearizing control to any order in ¿. The result is a nonlinear feedback control scheme to "stiffen" the nonlinear flexible system. That is, the behavior of the closed loop flexible system is nearly that of the controlled rigid system.