Discrete mechanics and optimal control for constrained systems

Authors

  • S. Leyendecker,

    Corresponding author
    1. Aeronautics and Control and Dynamical Systems, California Institute of Technology, 1200 E. California Boulevard, Mail Code 107-81, Pasadena, CA 91125, U.S.A.
    • Aeronautics and Control and Dynamical Systems, California Institute of Technology, 1200 E. California Boulevard, Mail Code 107-81, Pasadena, CA 91125, U.S.A.
    Search for more papers by this author
  • S. Ober-Blöbaum,

    1. Control and Dynamical Systems, California Institute of Technology, 1200 E. California Boulevard, Mail Code 205-45, Pasadena, CA 91125, U.S.A.
    Search for more papers by this author
  • J. E. Marsden,

    1. Control and Dynamical Systems, California Institute of Technology, 1200 E. California Boulevard, Mail Code 205-45, Pasadena, CA 91125, U.S.A.
    Search for more papers by this author
  • M. Ortiz

    1. Aeronautics and Mechanical Engineering, California Institute of Technology, 1200 E. California Boulevard, Mail Code 205-45, Pasadena, CA 91125, U.S.A.
    Search for more papers by this author

Abstract

The equations of motion of a controlled mechanical system subject to holonomic constraints may be formulated in terms of the states and controls by applying a constrained version of the Lagrange-d'Alembert principle. This paper derives a structure-preserving scheme for the optimal control of such systems using, as one of the key ingredients, a discrete analogue of that principle. This property is inherited 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 discrete equations serve as nonlinear equality constraints for the minimization of a given objective 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. In particular, for the optimal control of multibody systems, a force formulation consistent with the joint constraints is introduced. This enables one to prove the consistency of the evolution of momentum maps. Using a two-link pendulum, the method is compared with existing methods. Further, it is applied to a satellite reorientation maneuver and a biomotion problem. Copyright © 2009 John Wiley & Sons, Ltd.

Ancillary