Enhancing Safety in Uncertain Robotic Systems via Control Barrier Functions and PD Control

This paper establishes the practical input-to-state safety (ISSf) of the robotic system under input disturbances in the context of a trajectory tracking problem. A safe trajectory is generated through a quadratic program (QP) based on the control barrier function (CBF). Then, a model-free control approach, namely, the proportional-derivative (PD) controller, is adopted to track the safe trajectory. The practical input-to-state stability (ISS) of the disturbed robotic system under the PD controller is proved by deriving a practical ISS-Lyapunov function. Building upon the Lyapunov function, sufficient conditions for the practical ISSf of the system are provided in the presence of tracking errors and external disturbances, ensuring that the system state remains within a relaxed safe set. The theoretical results are demonstrated in experiments conducted on a 7-degree-of-freedom (DOF) Franka Emika Panda robot.