Optimal attitude control using a global attitude representation on SO(3) for deep space exploration under multiple constraints

Spacecraft attitude maneuvers, particularly in deep space exploration scenarios such as asteroid missions, are frequently subject to stringent pointing constraints, which arise from operational requirements such as avoiding bright celestial bodies to protect sensitive optical payloads and maintaining continuous antenna pointing toward ground stations to ensure communication links. This paper addresses the problem of optimal full-attitude stabilization under such mixed attitude constraints and proposes a novel optimal control strategy based on the Successive Galerkin Approximation (SGA). The attitude kinematics and dynamics are formulated on the special orthogonal group SO(3), providing a globally valid and singularity-free framework for deriving the associated error system. The constraints are rigorously described using geometric vector inequalities, distinguishing between Attitude-Forbidden Regions (AFRs) and an Attitude-Mandatory Region (AMR). To enforce the constraints from arbitrary initial attitudes, a composite potential function is constructed. This function incorporates an exponential barrier term that penalizes constraint violations, thus effectively guiding the trajectory back into the feasible region. The stabilization objective is then modeled within an infinite-horizon optimal control framework, giving rise to the Hamilton–Jacobi–Bellman (HJB) equation. Given that an exact analytical solution for this nonlinear partial differential equation is generally unavailable, the SGA algorithm is implemented to compute a near-optimal numerical approximation of the value function, facilitating the development of the SGA-based optimal controller. To preserve the geometric structure during numerical integration, a geometric Runge–Kutta method is utilized, ensuring that all state trajectories evolve on the SO(3) manifold. Finally, numerical simulations across multiple scenarios with mixed constraints verify the robustness and superiority of the SGA-based controller in terms of constraint satisfaction and optimality.