Optimal Actuator Failure Control Using a Homotopy Method

Most spacecraft are designed to be maneuvered to achieve pointing goals. This is generally accomplished by designing a three-axis control system, which can achieve arbitrary maneuvers, where the goal is to repoint the spacecraft and achieve a desired attitude and angular velocity at the end of the maneuver. New control laws are required, however, if one of the three-axis control actuators fails. The nonlinear failed actuator control problem is addressed by computing an optimal open-loop solution. This approach ensures that controllability is maintained throughout the three-dimensional maneuver. This work develops a Davidenko-like homotopy algorithm to achieve the optimal nonlinear maneuver strategies. The control torque is minimized by introducing quadratic torque penalties for large-angle three-axis spacecraft reorientation maneuvers. The innovation of the proposed approach is that the controllability of the spacecraft is continuously altered until, at the end of the homotopy method, only two of the three actuators provide control inputs. As a benchmark for the nominal case, the solution strategy first solves the three-axis control case when all three actuators are available. The failed actuator case is recovered by introducing a homotopy embedding parameter epsilon into the nonlinear dynamics equation, where the factor 1-epsilon multiplies the actuator control input that is assumed to fail. By sweeping the homotopy embedding parameter, a sequence of neighboring optimal control problems is solved, which starts with the original maneuver problem and arrives at the solution for the failed actuator case. As the homotopy embedding parameter approaches one, the designated actuator no longer provides the control input to the spacecraft, effectively modeling the failed actuator condition. This problem is complex for two reasons: 1)The governing equations are nonlinear and 2)the homotopy embedding parameter fundamentally alters the spacecraft's controllability. Given the strength of these nonlinearities, Davidenko's method is introduced for developing an ordinary differential equation for the costate variable as a function of the homotopy embedding parameter. For each value of the homotopy embedding parameter, the initial conditions for the costates are iteratively adjusted so that the terminal boundary conditions for the three-dimensional maneuver are achieved. Optimal control applications are presented for both rest-to-rest and motion-to-rest cases, which demonstrate the effectiveness of the proposed algorithm.