State equations for maneuvering and control of flexible bodies usingquasimomenta

Abstract

This paper is concerned with the derivation of state equations suitable for the task of designing controls for the maneuvers of flexible bodies in space while controlling the elastic vibration. The equations are in terms of quasimomenta, defined as momenta corresponding to quasicoordina tes, and are simpler in form than the equations in terms of quasicoordina tes. A perturbation approach permits design of a dual-level control, a high-authority control for the rigid-body maneuvering and a low-authority control for the suppression of the elastic vibration. PROBLEM of current interest is concerned with the maneuvering and control of flexible bodies. This problem arises in space applications and in robotics. The equations of motion for flexible bodies in space constitute a hybrid set of differential equations,1 in the sense that the equations for the translation and rotation of the body as a whole are ordinary differential equations and the equations for the elastic motions are partial differential equations. The equations can be derived by the ordinary Lagrangian approach2 or by Lagrange's equations in terms of quasicoordinates.1 The latter have the advantage that they are in terms of components along the system body axes, which are more suitable for the design of feedback control. Lagrange's equations in terms of quasicoordina tes are augmented in Ref. 1 by appropriate kinematical relations to obtain a set of hybrid state equations of motion/The state consists of rigid-body displacements in terms of inertia! components and velocities in terms of body axes components. The elastic motions are all in terms of body axes components. This paper presents an alternative approach to that of Ref. 1 in the sense that the state uses quasimomenta instead of quasicoordinates. The equations in terms of quasimomenta are based on the equations in terms of quasicoordina tes but they are simpler in form and lend themselves more readily to integration. The hybrid state equations in terms of quasimomenta are then cast in a form suitable for the task of designing controls for the maneuvers of the flexible bodies while controlling the vibration. To this end, a perturbation approach is used to divide the problem into one of design of a high-authority control for the rigid-body maneuver of the body and a low-authority control for the vibration suppression in a manner akin to that used in Ref. 3. For practical reasons, the hybrid perturbation equations for the low-authority control are discretized in space, resulting in a set of first-order differential equations with time-varying coefficients.