State-Switching kinematics of notched continuum robots

Continuum robots offer unique advantages for minimally invasive surgery in confined spaces due to their high compliance and dexterity. However, accurate kinematic modeling of cable-driven continuum robots is challenging due to friction-induced hysteresis and motion-state-dependent behavior. This paper presents a statics-based kinematic modeling framework for notched continuum robots that spans from high-fidelity physical modeling to real-time computation. A friction-aware static kinematic model (SKM) is developed for four motion states— MoveForward, MoveBackward, SwitchForward , and SwitchBackward —in which the state-switching process is explicitly resolved as a sequential beam-by-beam transition. A cable displacement compensation model is introduced to correct nonlinear slack-state behavior in the low-actuation region, yielding a compensated model (CSKM). A simplified model (SCSKM) is then derived via constrained polynomial fitting that preserves monotonicity while achieving computation rates exceeding 280 kHz. Experimental validation demonstrates end-effector position RMSE within 0.126–0.256 mm radially and 0.155–0.314 mm axially across all four motion states, confirming sub-millimeter accuracy suitable for minimally invasive surgical applications.