Robust Relative Space Motion Control of Underwater Vehicles Using Time Delay Estimation

This paper presents a robust trajectory-tracking control framework for underwater vehicles operating in a relative coordinate system. Unlike conventional methods that define trajectories in the world frame, the proposed approach formulates the control problem directly in a moving reference frame, enabling accurate motion control with respect to dynamic and drifting objects affected by environmental disturbances such as ocean currents and waves. This relative-space formulation is particularly advantageous for tasks including diver guidance, floating-object inspection, and docking, where the reference itself is nonstationary. A coordinate transformation is introduced to consistently express the vehicle dynamics in the relative frame. Based on the transformed dynamics, a Time Delay Control (TDC) law is applied to estimate unmodeled dynamics and external disturbances without requiring precise system parameters. Theoretical stability analysis shows that the stability condition of the proposed controller is consistent with that of conventional TDC, allowing similar gain-tuning procedures. Simulation results demonstrate that the proposed controller achieves robust and smooth trajectory tracking even when the reference frame undergoes motion induced by ocean currents.