On-line Shape Estimation for Hysteresis Compensation in Tendon-Sheath Mechanisms Using Endoscopic Camera

The tendon-sheath mechanism (TSM) has significantly advanced both robotic systems and minimally invasive surgery (MIS) by enabling flexible and precise movement through narrow and tortuous paths. However, the inherent flexibility of TSM introduces nonlinear behaviors which depend on its geometrical shape and applied forces, making accurate control challenging. Furthermore, the shape dependency becomes critical in endoscopic robots, where the geometrical shape varies and is not directly visible, limiting the applicability of existing distal sensorless compensation methods. To address the geometry identification problem of TSM, this paper proposes an approach that utilizes real-time visual input from an endoscopic camera for on-line calibration of the TSM's physical model. By introducing the concept of the ‘Equivalent Circle,’ complex shapes of TSMs are simplified, enabling the estimation of their equivalent geometry without direct observation or measurement. Simulation results validate the equivalent circle model, demonstrating minimal deadband percentage errors despite larger discrepancies in equivalent radii across varied configurations. On-line calibration experiments achieved a percent error of 1.38% (±2.92%) for accumulated curve angles and 2.32% (±3.08%) for equivalent radii, demonstrating the method's reliability in shape estimation across varying conditions. In prediction and feedforward experiments, leveraging the equivalent circle to compensate for deadband in arbitrarily shaped TSMs resulted in a maximum trajectory error of 0.25 mm and an RMSE of 0.09 mm. This approach advances distal sensorless control, improving the operational accuracy and feasibility of endoscopic surgical robots under varying geometrical and force conditions.