Discrete-Time Optimal Attitude Control of a Spacecraft with Momentum and Control Constraints

An algorithm is proposed to solve optimal control problems arising in attitude control of a spacecraft under state and control constraints. First, the discrete-time attitude dynamics are derived by employing discrete mechanics. Then, the orientation transfer, with initial and final values of the orientation and momentum and the time duration being specified, is posed as an energy-optimal control problem in discrete time, subject to momentum and control constraints. Using variational analysis directly on the Lie group SO(3) (the set of 3 x 3 special orthonormal matrices), first-order necessary conditions for optimality are derived, leading to a constrained two-point boundary value problem. This two-point boundary value problem is solved via a novel multiple shooting technique that employs a root-finding Newton algorithm. Robustness of the multiple shooting technique is demonstrated through a few representative numerical experiments