On Modeling and Control of Continuum Robots for Minimally Invasive Medical Interventions

Over the past several decades, there has been rapid advancement in technology and techniques for minimally invasive medical interventions (MIMI), particularly in minimally invasive surgery (MIS). This progress has highlighted various challenges stemming from the limitations of the current standard of care. For instance, most existing MIS systems utilize rigid laparoscopic instruments, which impose constraints on accessible workspace, intuitive control, and ergonomics. These limitations are especially restrictive for emerging techniques aiming to access internal anatomy with minimal or no incisions. To address these challenges, applications involving continuum robots have recently emerged. Leveraging flexible structures, these robots offer the capability to navigate complex, winding pathways within the body, with the potential to expand the scope of minimally invasive procedures. Motivated by two specific MIMI applications - percutaneous nephrolithotomy (PCNL) for managing large and staghorn kidney stones, and catheter-based cardiac ablation for treating arrhythmias - this thesis explores continuum robotic systems to address these clinical needs. The thesis proposes a handheld concentric-tube robot (CTR), developed as an assistive tool for PCNL. Using an ellipsoidal approximation for the geometry of the renal calculi, it presents an image-guided, patient-specific approach to CTR design for PCNL. The method integrates an anatomically constrained, differential kinematics-based control scheme with an optimization framework to devise PCNL surgical plans and CTR designs tailored to the anatomical specificity of each patient. The surgical plans specify both the skin puncture location and an entry point at a suitable renal pyramid, thereby defining the percutaneous access tract for the procedure. The proposed method is validated through simulations in virtual environments, created from clinical data of a cohort of PCNL patients, as well as phantom-based experiments that demonstrate the efficacy of the approach and the optimality of the surgical plans. Furthermore, this thesis addresses the modeling and control of the continuum dynamics of tendon-driven continuum robots (TDCRs), focusing on tendon-driven catheters used in cardiac ablation. The thesis introduces a Cosserat-based dynamic model of catheters that incorporates a dynamic friction model which captures tendon-sheath interactions in a geometrically exact manner. This approach accounts for complex, motion history-dependent dynamics, and models tension losses even in cases involving out-of-plane catheter curvatures. Additionally, it presents a robotic-actuated tendon-driven catheter and a real-time control framework. This framework combines a model-free, learning-based approach to model catheter dynamics using Koopman theory with model-based control techniques. Real-time control at interactive rates is demonstrated through experiments in which two optimal controllers, a linear quadratic regulator (LQR) and a robust H-infinity controller, achieve closed-loop frequencies of 250 Hz.