Geometric Optimal Control of Rigid Bodies

This paper treats the geometric formulation of optimal control problems for rigid bodies and it presents computational procedures based on this geometric formulation that can be used for numerical solution of 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 presentation of results for several different optimal control problems for a single rigid body and for multiple interacting rigid bodies. Although the emphasis in this work is on computational results, the philosophy and the approach follow the work on optimal control problems for rigid body kinematics by Roger Brockett and his colleagues in the 1970s. This version of the paper provides an overview of the concepts and contributions; the final version of the paper will include additional details on geometric optimal control of rigid bodies.