Fast NMPC on SO(3) for Spacecraft Control in the Presence of Reaction Wheel Fault and Disturbance Torque

The nonlinear model predictive control (NMPC) of an underactuated spacecraft equipped with reaction wheels on special orthogonal (Lie) group SO(3) is discussed regarding numerical efficiency. Due to off-centricity in its structure, the spacecraft is subjected to an inner disturbance torque. The system has to deal with disturbance torques while one of its reaction wheels is out of order and, thus, is considered underactuated. Failure may occur from the beginning or in the middle of the assigned maneuver. Our goal is to design an appropriate control method that is able to compensate for the underactuation, especially when the system is subject to disturbance torques. This can be dealt with by using NMPC due to its capability of considering dynamic constraints. The NMPC is designed based on the discrete form of the equations of motion of the spacecraft, which are established using Lie group variational integrators permitting us to take advantage of Lie group properties such as being symplectic and momentum preserving while keeping the geometric structure of the system. In addition, the issue of underactuation is addressed in the MPC design by imposing a zero-torque constraint on the failed reaction wheel. For the sake of calculation reduction for enhancing the practical performance of the algorithm, the NMPC-related two-point boundary value problem is solved using an iterative Newton-like procedure. The calculation trick is based on ignoring nonessential nonlinear constraint-related parts of the sensitivity derivative equations in the repetitive process of estimating the initial conditions for the Lagrange multipliers. The proposed method is applied to the model of a spacecraft attitude control simulator through simulations. The obtained results confirm that the control system is able to bring the underactuated spacecraft to the desired position in absence of any disturbance. However, the disturbance rejection performance depends on whether the disturbance lies in the actuation span space and also on the time when the fault occurs.