New approach to the maneuvering and control of flexible multibody systems

Abstract

A mathematical formulation capable of treating the problem of maneuvering and control of flexible multibody systems is developed. The formulation is based on equations of motion in terms of quasicoordina tes derived independently for the individual substructures and on a consistent kinematical synthesis causing the substructures to act as a single structure. A perturbation approach permits the separation of the nonlinear high-dimensional system of equations into a zero-order, low-dimensional problem for the rigid-body maneuvering and a first-order, high-dimensional, time-varying problem for the elastic motions and the perturbations from the rigid-body motions. The formulation lends itself to ready computer implementation. A numerical example involving a three-beam system demonstrates the effectiveness of the new algorithm. OR a number of years there has been a persistent interest in the dynamics of flexible multibody systems. This type of system is encountered in flexible robots, rotorcraft, and spacecraft. Equations of motion for flexible multibody systems have been derived by a variety of approaches. 1'9 Recently, the interest has broadened so as to include maneuvering and control. This narrows the choice of formulations significantly, as the formulation must be consistent with the control task. A set of equations of motion suited for the control task can be formulated by means of Lagrangian equations for flexible bodies in terms of quasicoordina tes.10 The advantage of this approach is that it yields equations in terms of body axes, which are the same axes as those used to express control forces and torques. In using the approach of Ref. 10 to derive equations of motion for a chain of flexible multibody systems, it is convenient to adopt a kinematic procedure permitting the expression of the velocity vector of a nominal point in a typical body in terms of the velocity vector of the preceding body in the chain. The resulting differential equations are nonlinear and hybrid,11 where the term implies that the equations for the rigid-body translations and rotations are ordinary differential equations and those for the elastic motions are partial differential equations. Because maneuvering and control design in terms of hybrid equations is not feasible, the partial differential equations must be transformed into sets of ordinary differential equations by means of a discretizationin-space procedure, such as the finite element method12 or a Rayleigh-Ritz-based substructure synthesis.13 The resulting formulation consists of a high-order set of nonlinear ordinary differential equations. A common approach to control design requires the solution of a two-point boundary-value problem. However, this is not feasible for high-order systems; hence, a different approach is advisable. The nonlinearity enters into the differential equations through the rigid-body motions. Indeed, the elastic motions tend to be small. In view of this, it appears natural to conceive of a perturbation approach whereby the rigid-body motions