Deep Koopman Approach for Nonlinear Dynamics and Control of Tendon-Driven Continuum Robots

Tendon-driven continuum robots (TDCRs) have received widespread attention in the medical domain due to their slender shape and flexibility. Modeling the dynamics of TDCRs involves continuum mechanics that result in nonlinear and computationally intensive models posing challenges for real-time control. In this work, we propose efficient, and control-oriented modeling of the nonlinear dynamics of TDCRs leveraging the deep Koopman approach. This method transforms the states of the system into an intrinsic nonlinear manifold, where the autonomous dynamics are 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. 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, achieving a position tracking error of 1.64 mm (SD = 0.74) for multi-sinusoidal, and 0.60 mm (SD = 0.30) for sinusoidal (0.05 Hz) target trajectories. The results demonstrate the potential for applying the proposed approach for real-time control of a broad range of TDCRs.