This paper addresses the challenge of modeling distributed tendon tensions and friction losses in tendon-driven continuum manipulators, with a particular focus on cardiac ablation catheters. Tendon compliance and friction between the tendons and the inner channels of the catheter's sheath introduce several undesired effects, such as motion history-dependent frictional dynamics (friction hysteresis) and non-uniform tension profiles along the catheter's backbone. These factors adversely affect model prediction accuracy, motion precision, and control performance. To tackle these issues, we employ Cosserat theory to model the dynamics of tendon-driven ablation catheters, and incorporate the LuGre friction model and tendon compliance into the nonlinear partial differential equations (PDEs) that govern catheter behavior. We present a comprehensive framework for simulating catheter dynamics, including geometrically exact calculations of dynamic friction, along with associated tendon tensions as functions of arc length, without relying on prior assumptions about the model states. In actuation-based experiments, our friction model achieved a 38.89% improvement in distal end positioning prediction over the frictionless model. Validation through simulation and empirical experiments demonstrates that our model effectively replicates the complex, motion history-dependent frictional dynamics observed during the loading and unloading actuation phases that are inherent to the class of tendon-driven robotic systems.

Tendon-driven continuum robots (TDCRs) have received widespread attention, particularly in the medical domain, due to their slender shape and flexibility. Modeling the dynamics of TDCRs is inherently complex and involves continuum mechanics that result in nonlinear and computationally intensive models. Consequently, current modeling approaches pose challenges for real-time control, essential for practical implementations. In this work, we propose a novel method for efficient and control-oriented modeling of the nonlinear dynamics of TDCRs using an intrinsic bilinear model leveraging the Deep Koopman approach. This method transforms the states of the system into an intrinsic nonlinear manifold, identified via deep learning, where the autonomous dynamics can be approximated linearly and the actuation input enters the system with a bilinear term. The proposed model captures the nonlinearities including space-dependent variations in the system spectrum. Task-space position control is implemented using a linear quadratic controller, leveraging the linear nature of the Koopman operator. The accuracy of the proposed method is experimentally validated using a dual-tendon robotic steerable catheter with a bending section of 55 mm, achieving a position tracking error of 1.79±1.26 mm with a control loop frequency of 250 Hz. The results demonstrate the potential for applying the proposed approach for real-time control of a broad range of TDCRs.