3D Maneuvers For Asymmetric Under-Actuated Rigid Body
Kim, Dong Hoon
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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. This work explores new maneuver strategies when only two control inputs are available: (i) sequential single-axis maneuvers and (ii) three-dimensional (3D) coupled maneuvers. The sequential single-axis maneuver strategies are established for torque, time, and fuel minimization applications. The resulting control laws are more complicated than the equivalent results for three-axis control because of the highly nonlinear control switch-times. Classical control approaches lead to optimal, but discontinuous control profiles. This problem is overcome by introducing a torque-rate penalty for the torque minimization case. Alternative approaches are also considered for achieving smooth continuous control profiles by introducing a cubic polynomial multiplicative control switch smoother for the time and fuel minimization cases. Numerical and analytical results are presented to compare optimal maneuver strategies for both nominal and failed actuator cases. The 3D maneuver strategy introduces a homotopy algorithm to achieve optimal nonlinear maneuvers minimizing the torque. Two cases are considered: (i) one of the three-axis control actuators fails and (ii) two control actuators fail among four control actuators. The solution strategy first solves the case when all three actuators are available. Then, the failed actuator case is recovered by introducing a homotopy embedding parameter, ?, into the nonlinear dynamics equation. By sweeping ?, a sequence of neighboring optimal control problems is solved that starts with the original maneuver problem and arrives at the solution for the under-actuated case. As ? approaches 1, the designated actuator no longer provides control inputs to the spacecraft, effectively modeling the failed actuator condition. This problem is complex for two reasons: (i) the governing equations are nonlinear and (ii) ? fundamentally alters the spacecraft?s controllability. Davidenko?s method is introduced for developing an ordinary differential equation for the costate variable as a function of ?. For each value of ?, the costate initial conditions are iteratively adjusted so that the terminal boundary conditions for the 3D maneuver are achieved. Optimal control applications are presented for both rest-to-rest and motion-to-rest cases that demonstrate the effectiveness of the proposed algorithm.