On Integrating Friction Dynamics Into Cosserat Models for Tendon-Driven Continuum Manipulators

This article presents a dynamic modeling framework for tendon-driven continuum manipulators, with a focus on cardiac catheters. Tendon compliance and tendon-sheath friction produce nonuniform tensions and motion-history-dependent dynamics (hysteresis), which, if neglected, degrade model fidelity and control. We develop a Cosserat-based formulation that integrates tendon compliance and a dynamic LuGre friction law into the governing nonlinear PDEs. This allows geometrically exact spatiotemporal computation of tendon-tension transmission without assuming initial friction states. Experiments on two clinical catheters demonstrate that incorporating friction dynamics reduces distal-tip error from 1.62±1.01 mm (frictionless) to 0.99±0.65 mm (≈39% improvement), and achieves 2.31±0.91 mm mean absolute error for out-of-plane spatial actuation of a four-tendon ICE catheter. Comparative analysis against a Coulomb-based capstan model highlights the significance of dynamic friction, which produces rate-dependent tension transmission and more accurate modeling of hysteresis effects.