A model reduction method for the control of rigid mechanisms

This paper proposes a control strategy for flexible mechanisms. Our starting-point is a classical collocated PID control of the joint actuators designed for the equivalent rigid mechanism. An additional state feedback is implemented to control the flexible modes. On the basis of a few vibration measurements, it generates an additional command for the joint actuators. This non-collocated control scheme is designed according to the H procedure in order to have robust performances and stability with respect to configuration changes. In this paper, a new reduction methodology is presented to build a linear low-order and sufficiently accurate model of the mechanism with the PID feedback, which is suitable for the design of the H controller. First, a detailed Finite Element model of the mechanism is elaborated including the initial PID compensator. This set of nonlinear differential and algebraic equations is then linearized around a chosen reference configuration and a reduction technique is developed to extract a compact set of ordinaray differential equations. The retained degrees of freedom are the joint coordinates and a few modal coordinates representing the deformation of the whole mechanism. The kinematic description is thus decomposed into two parts, a rigid body motion described by the joint coordinates and a flexible motion for which shape functions have to be selected. For this selection, a modal analysis of the controlled mechanism is performed and the first few modes are kept, as in the Craig-Bampton or McNeal-Rubin reduction techniques. For robust performance specifications, the variations of the model with respect to the configuration changes should be estimated. Thus, the reduction procedure is realized for a few values of the joint coordinates, and the reduced models are compared. As the set of shape functions changes with the configuration, the physical meaning of the modal coordinates should be reinterpreted in each model, which is one difficulty of the approach. To illustrate the method, the case of a two-link flexible manipulator is presented. Simulation of the complete nonlinear Finite-Element model with the global control scheme is performed in order to assess the final performances of the closed-loop system