Optimal Control of a Rigid Body using Geometrically Exact Computations on SE(3)

Optimal control problems are formulated and efficient computational procedures are proposed for combined orbital and rotational maneuvers of a rigid body in three dimensions. The rigid body is assumed to act under the influence of forces and moments that arise from a potential and from control forces and moments. The key features of this paper are its use of computational procedures that are guaranteed to preserve the geometry of the optimal solutions. The theoretical basis for the computational procedures is summarized, and examples of optimal spacecraft maneuvers are presented.Comment: IEEE Conference on Decision and Control, 2006. 6 pages, 19 figure

Rigid Body Constrained Motion Optimization and Control on Lie Groups and Their Tangent Bundles

Rigid body motion requires formulations where rotational and translational motion are accounted for appropriately. Two Lie groups, the special orthogonal group SO(3) and the space of quaternions H, are commonly used to represent attitude. When considering rigid body pose, that is spacecraft position and attitude, the special Euclidean group SE(3) and the space of dual quaternions DH are frequently utilized. All these groups are Lie groups and Riemannian manifolds, and these identifications have profound implications for dynamics and controls. The trajectory optimization and optimal control problem on Riemannian manifolds presents significant opportunities for theoretical development. Riemannian optimization is an attractive approach to tackling these problems because it does not require the imposition of additional space-preserving constraints on the solver. Rather, these constraints are accounted for in the optimization algorithm. As such, implementing these solvers in trajectory optimization and optimal controls problems through direct transcription offers a reduction in the number of constraints imposed on the problem. The on-manifold optimization methodologies are applied to the Lie groups listed above. Then, the direct transcription of the optimal control and trajectory optimization problems on these Riemannian optimization is presented and applied. These trajectories are utilized in an unscented Kalman filter to show how these generated reference trajectories interface with state estimation. Finally, the fundamental equation of mechanics is utilized for generating initial guess trajectories which satisfy path and terminal time constraints. The results contained herein show, for the first time, that direct transcription of trajectory optimization and optimal control problems on Riemannian manifolds may be effectively conducted with on-manifold optimization techniques using relatively simple optimization algorithms