A HOMOTOPY METHOD FOR OPTIMAL ACTUATOR FAILURE CONTROL
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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 existing state-of-the-art strategies do not provide efficient solutions of this underactuated nonlinear control problem. This paper develops a homotopy algorithm to achieve optimal nonlinear maneuver strategies minimizing quadratic torque for large-angle three-axis spacecraft reorientation maneuvers. 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, , into the nonlinear dynamics equation, where the factor, 1 . , 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 1, 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: (i) the governing equations are nonlinear and (ii) 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-torest and motion-to-rest cases, which demonstrate the effectiveness of the proposed algorithm.