I. Introduction
Nowadays, spacecrafts have vital importance in human’s life due to their ability to provide information in different fields such as agriculture, weather forecasting as well as telecommunication applications. Therefore, studying their structure and optimal methods of their flight control in a way that includes all physical and environmental limitations is of special importance. Nonlinear model predictive control “NMPC” is a special kind of control method that provide the optimal control while considering all the system’s constraints [1], and therefore, it has been receiving special attention in the literature [2],[3], and [4]. The configuration manifold of spacecrafts is not the Euclidean space but the Riemannian manifolds and Lie groups. Therefore, the equations of motion and the formulation of the control method have to be expressed considering the real configuration manifold of the system [5]. The NMPC on manifolds has been the subject of several studies in the recent years [6], [7], and [8]. Exploiting the geometric structure of the equations and Lie group methods, especially Lie group variational integrators “LGVI” lead to have special properties in the obtained necessary conditions for optimality such as avoiding singularities and ambiguities and preserving the geometric structure of the equations [9] and [10]. In addition, LGVI are symplectic and momentum preserving structurally [11]. The necessary conditions of optimality form a two-point boundary value problem “TPBVP” where a fast solver based on indirect shooting methods is used to solve it [12] and [13]. For the sake of time reduction in performing the computations, it is possible to remove some non-essential nonlinear terms related to the constraints of the system in sensitivity equations, where applying constraints in the process of solving the optimal control equations is guaranteed [14]. The discrete equations of motion of a spacecraft and the NMPC equations on Lie groups become more complicated when the system of a spacecraft is equipped with reaction wheels as its actuators to generate driving torque. The complexity increases when the spacecraft body is exposed to external disturbances [15] and [16]. However, considering the effects of momentum exchanges between the wheels and spacecraft body and the mentioned disturbances make the model more like reality. In the present work, such systems are considered as the subject of study for driving the model and designing the NMPC to bring the system to the zero point on SO(3). Our simulations are performed based on modeling of an actual spacecraft simulator located in Mechatronic Lab at Isfahan university. The simulator has three reactions wheels which are responsible for controlling the Euler angles. The main issue related to this device is the presence of off-center in z-direction of the body coordinate system, which makes the extraction of its equations of motion and the NMPC controller much more difficult due to the effect of external disturbance torque applied to the system because of the off-center. Our goal here is to provide an efficient control method in a way that is able to bring the system to the zero point even in the presence of structural-related disturbance. The results obtained through simulations confirm the achievement of our desired goal.