Real-Time Adaptive Safety-Critical Control with Gaussian Processes in High-Order Uncertain Models

This paper presents an adaptive online learning framework for systems with uncertain parameters to ensure safety-critical control in non-stationary environments. Our approach consists of two phases. The initial phase is centered on a novel sparse Gaussian process (GP) framework. We first integrate a forgetting factor to refine a variational sparse GP algorithm, thus enhancing its adaptability. Subsequently, the hyperparameters of the Gaussian model are trained with a specially compound kernel, and the Gaussian model's online inferential capability and computational efficiency are strengthened by updating a solitary inducing point derived from new samples, in conjunction with the learned hyperparameters. In the second phase, we propose a safety filter based on high-order control barrier functions (HOCBFs), synergized with the previously trained learning model. By leveraging the compound kernel from the first phase, we effectively address the inherent limitations of GPs in handling high-dimensional problems for real-time applications. The derived controller ensures a rigorous lower bound on the probability of satisfying the safety specification. Finally, the efficacy of our proposed algorithm is demonstrated through real-time obstacle avoidance experiments executed using both a simulation platform and a real-world 7-DOF robot.