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.