In the pursuit of autonomous spacecraft proximity maneuvers and docking(PMD), we introduce a novel Bayesian actor-critic reinforcement learning algorithm to learn a control policy with the stability guarantee. The PMD task is formulated as a Markov decision process that reflects the relative dynamic model, the docking cone and the cost function. Drawing from the principles of Lyapunov theory, we frame the temporal difference learning as a constrained Gaussian process regression problem. This innovative approach allows the state-value function to be expressed as a Lyapunov function, leveraging the Gaussian process and deep kernel learning. We develop a novel Bayesian quadrature policy optimization procedure to analytically compute the policy gradient while integrating Lyapunov-based stability constraints. This integration is pivotal in satisfying the rigorous safety demands of spaceflight missions. The proposed algorithm has been experimentally evaluated on a spacecraft air-bearing testbed and shows impressive and promising performance.

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