Discrete Mechanics and Optimal Control for Constrained Multibody Dynamics
Leyendecker S., S. Ober-Blöbaum, J.E. Marsden, and M. Ortiz
Proc. 6th Intern. Conf. on Multibody Systems, Nonlinear Dynamics, and Control, ASME, (2007), 1-10.
Abstract:
This paper formulates the dynamical equations of mechanics
subject to holonomic constraints in terms of the states and
controls using a constrained version of the Lagrange-d’Alembert
principle. Based on a discrete version of this principle, a structure
preserving time-stepping scheme is derived. It is shown that
this respect for the mechanical structure (such as a reliable computation
of the energy and momentum budget, without numerical
dissipation) is retained when the system is reduced to its minimal
dimension by the discrete null space method. Together with
initial and final conditions on the configuration and conjugate
momentum, the reduced time-stepping equations serve as nonlinear
equality constraints for the minimisation of a given cost
functional. The algorithm yields a sequence of discrete configurations
together with a sequence of actuating forces, optimally
guiding the system from the initial to the desired final state. The
resulting discrete optimal control algorithm is shown to have excellent
energy and momentum properties, which are illustrated
by two specific examples, namely reorientation and repositioning
of a rigid body subject to external forces and the reorientation of
a rigid body with internal momentum wheels.