Approximate optimal trajectory tracking and dynamic obstacle avoidance for affine system via online learning

• Published in: Journal of the Franklin Institute Vol. 362; no. 16; p. 108037
• Main Authors: Basheer, Ambreen, Li, Man, Qin, Jiahu
• Format: Journal Article
• Language: English
• Published: Elsevier Inc 15.10.2025
• Subjects: Barrier-like cost; Dynamic obstacle; Input constraint; Online learning; Safe trajectory tracking; Safe trajectory tracking; Input constraint; Barrier-like cost; Online learning; Dynamic obstacle
• ISSN: 0016-0032
• DOI: 10.1016/j.jfranklin.2025.108037

This work studies the optimal trajectory tracking problem for the affine nonlinear system subject to input constraints, with the consideration of dynamic obstacle avoidance. An augmented dynamical system is constructed in term of the tracking error system, desired trajectory system, and the locally designed obstacle system. Then, a novel cost function, which captures all control objectives, is designed such that the safe trajectory tracking problem can be formulated as an unconstrained optimal control problem over the infinite-time horizon. The safety is ensured by a novel barrier-like cost function, that penalizes the system trajectory only when it gets close to the dynamic obstacle. This cost function also has the advantages of ensuring the positive definiteness of the optimal value function and the continuous differentiability of the resulting Hamilton-Jacobi-Bellman (HJB) equation. Under the designed cost function, a continuous controller is developed, which avoids the possible performance reduction caused by the switch of the controllers. Considering the difficulties of solving the resulting HJB equation analytically, we propose an online learning based safe tracking control algorithm to approximate the optimal controller, by using the state-following kernel approach for value function approximation via kernel sparsification to reduces the computational resources and using the least square approach to minimize the resulting residual errors. The Lyapunov based stability analysis is provided to show both safety and ultimate boundedness of tracking errors. Finally, some simulation results are provided to verify the effectiveness of the proposed online learning-based safe tracking control algorithm.