Optimal control allocation for a multi-engine overactuated spacecraft

This paper addresses the control allocation problem for an over-actuated system with a nonlinear relation between the control inputs and the realized force and torque vectors. Specifically, we consider a multi-engine rocket example where the attained force and torque vectors are nonlinear functions of the thrust magnitudes and engine gimbal angles. Since the system is over-actuated, the surface defining this nonlinear relationship represents a non-unique map between the control variables and the resultant force and torque vectors. In this work, our goal is to command the actuators to produce desired force and torque vectors with minimal actuator reconfiguration effort, subject to actuator dynamics and constraints (e.g. slew-rates, maximum gimbal angles). In achieving this objective, the control space is traversed in a way that preserves the force and torque impulse requested from the control allocator, thus ensuring that the transient motion of the actuators produces the desired change in linear and angular momentum. With our problem formulation, we are able to express the original problem as a convex optimal control problem, which can then be solved onboard and in real-time by taking advantage of modern convex solvers. The advantages of our method are manyfold, providing a systematic method for solving the control allocation problem, enabling a wider flight envelope, and reducing actuator fatigue.