Suboptimal Asymmetric Spacecraft Maneuvers Using Two Controls

Abstract

E XTENSIVE literature exists for controlling the attitude motion of rigid and flexible spacecraft. For a fully functioning spacecraft, it is assumed that all actuators required for completing the control task are available.Many different control strategies have been proposed for handling the nominal three-axis control case [1–5]. More specialized literature has considered off-nominal cases where actuator failures have occurred. For example, Tsiotras and Longuski [6] have considered the case designing control strategies for handling situations in which sensor and actuator failures limit the control options available for carrying out the original mission objectives. Kerai [7] has considered a more extreme case where only a single control actuator is available, but this case is shown to be uncontrollable, which is intuitively reasonable. Brockett [8] has shown that two controls can be made asymptotically stable about the origin. Tsiotras et al. [9–15] and Shen and Tsiotras [16] have further addressed the problem of stabilization of asymmetric spacecraft including tracking control laws. Kim et al. [17–19] have introduced sequential control concepts when the actuator failure is detected by monitoring residuals.Morin et al. [20] andCoron andKerai [21] have presented approximate strategies that switch between two different control laws. Much of the later work has considered complex mathematical approaches for overcoming the underactuated spacecraft control problem. This work addresses the formulation and solution of a rigorous nonlinear optimal control problem for handling spacecraft maneuvers, where actuator failures limit the number of control inputs to two axes, which is solved by completing three sequential submaneuvers. An asymmetric rigid spacecraft math model with only two available control torque inputs is assumed. The control design objective is to avoid the axis where the actuator failure has occurred. A spacecraft maneuver strategy is designed by formulating an optimal control problem. With only two control inputs available, the suggested strategy is as follows: 1) transform the given initial and final attitudes to specific attitudes using the Euler angle transformation that is selected to avoid the failed axis; 2) define three sequential submaneuvers; 3) define new attitude commands at switch times for each submaneuvers; 4) solve for optimal switch times for starting and ending each submaneuver; and 5) perform the defined single-axis submaneuvers. With three maneuver periods to be defined, two unknown switch times must be found. This work only considers the case of ideal control input commands. This approach is successful but leads to discontinuous control solutions because of jump conditions for Lagrange multipliers at switch times. Three issues make the calculation of an optimal control solution challenging: 1) unknown switch times must be determined to change from one submaneuver to the next submaneuver; 2) the number of constraints is high; and 3) the switch times introduce jump conditions on the necessary conditions that must be iteratively refined to generate the desired solution. The nonlinear necessary conditions are handled by introducing a multiple shooting method [22], which enforces both the end and interior points that define the optimal solution. Fixed final time maneuver strategies are developed and demonstrated. No simplifications are introduced into the spacecraft dynamics problem or the resulting control formulation. The most challenging part of the algorithm involves solving for the highly sensitive optimal switch times.