Geometric structure-preserving optimal control of a rigid body

Abstract

Abstract. In this paper we study a discrete variational optimal control problem for the rigid body. The cost to be minimized is the external torque applied to move the rigid body from an initial condition to a pre-specified terminal condition. Instead of discretizing the equations of motion, we use the discrete equations obtained from the discrete Lagrange-d'Alembert principle, a process that better approximates the equations of motion. Within the discrete-time setting, these two approaches are not equivalent in general. The kinematics are discretized using a natural Lie-algebraic formulation that guarantees that the flow remains on the Lie group SO(3) and its algebra so(3). We use Lagrange's method for constrained problems in the calculus of variations to derive the discrete-time necessary conditions. We give a numerical example for a three-dimensional rigid body maneuver.

On the formulation of cost functions for torque-optimized control of rigid bodies

“…These results enables a straightforward physical interpretation of the components of τ and shows how the geometric treatment proposed in [1] is relevant to other works such as [2,8].…”

“…It would be interesting to see how the optimal torques for a cost function τ 0 ||M o || 2 dt compare to those computed by the authors of [8] using (1). Indeed some recent results on this comparison can be found in Ghosh et al [3] which appeared while the present work was in review.…”

“…In a recent work Bloch et al [1] examined the optimal control of the rotations of a rigid body which minimized the cost function…”

“…It is of interest to see how J and V are equivalent even though the formulations of these functions appear at first glance to be very different. To establish the forthcoming results, we make extensive use of results fromŽefran and Kumar [7] and follow the notation of [1].…”

“…Alternative formulations of the cost function we propose can be found in a recent paper by Bloch et al [1] and a work by Ghosh et al [3] that appeared while the present paper was under review. With help from [7], we are able to provide additional physical insight into the cost function found in [1] and show its equivalence to the cost function being proposed.…”