  This paper formulates optimal control problems for rigid bodies in a
geometric manner and it presents computational procedures based on this
geometric formulation for numerically solving these optimal control problems.
The dynamics of each rigid body is viewed as evolving on a configuration
manifold that is a Lie group. Discrete-time dynamics of each rigid body are
developed that evolve on the configuration manifold according to a discrete
version of Hamilton's principle so that the computations preserve geometric
features of the dynamics and guarantee evolution on the configuration manifold;
these discrete-time dynamics are referred to as Lie group variational
integrators. Rigid body optimal control problems are formulated as
discrete-time optimization problems for discrete Lagrangian/Hamiltonian
dynamics, to which standard numerical optimization algorithms can be applied.
This general approach is illustrated by presenting results for several
different optimal control problems for a single rigid body and for multiple
interacting rigid bodies. The computational advantages of the approach, that
arise from correctly modeling the geometry, are discussed.
