Accurate control of continuum robots is challenging, especially in the presence of external disturbances. To address this issue, reinforcement learning (RL) has been increasingly investigated in continuum robot control due to its online policy updating capability. However, the gap of Sim2Real transfer caused by inaccurate modeling results in RL implementation a dilemma. In this article, we propose a safety-critical Real2Sim2Real online RL framework, where the Real2Sim transfer is firstly achieved by the identification of a continuum robot using the Koopman operator. We further improve the training efficiency by introducing an imperfect demonstration into the RL framework. The offline policy is trained in simulations and then tested on a real continuum robot platform. During tests, the tracking performance is influenced by the hysteresis effect, which cannot be captured by the Koopman operator. This results in a millimeter-level tracking root mean square error (RMSE). To address this issue, we online update the policy through continuous learning, and the RMSE of the online controller outperforms the offline controller by 89.16% in free space and 85.70% under external payload, respectively.

Continuum robots, with their slender, flexible structures, are increasingly utilized in delicate tasks. These robots must navigate confined spaces with high precision, often employing follow-the-leader (FTL) motion to ensure minimal trauma. The baseline FTL motion planning algorithm can not adjust the continuum robot's configuration using all the segments, especially the tail of the robot. Furthermore, it has limitations in dynamic, obstacle-rich environments, where real-time re-planning and obstacle avoidance are essential but often not sufficiently addressed. To address the above gaps, we propose a novel FTL motion planning framework inspired by snake hole-digging motions that can configure its optimal shape during the insertion task. Our approach outperforms the baseline FTL by 75.36%. The framework incorporates real-time obstacle avoidance by integrating Control Barrier Functions (CBF) and Quadratic Programming (QP) into the motion re-planning process. To avoid deviations caused by obstacle avoidance, the Control Lyapunov Function (CLF) is also integrated, forming a CLF-CBF-QP framework for online motion re-planning. The target reaching error of using CLF-CBF-QP decreases by 69.15% compared to using only CBF-QP. We also provide the open-source code of this work online.