When deploying autonomous systems in unknown and changing environments, it is critical that their motion planning and control algorithms are computationally efficient and can be reapplied online in real time, whilst providing theoretical safety guarantees in the presence of disturbances. The satisfaction of these objectives becomes more challenging when considering time-varying dynamics and disturbances, which arise in real-world contexts. We develop methods with the potential to address these issues by applying an offline-computed safety guaranteeing controller on a physical system, to track a virtual system that evolves through a trajectory that is replanned online, accounting for constraints updated online. The first method we propose is designed for general time-varying systems over a finite horizon. Our second method overcomes the finite horizon restriction for periodic systems. We simulate our algorithms on a case study of an autonomous underwater vehicle subject to wave disturbances.

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