Efficient dynamic modeling and real-time control of the planar variable-length hard-magnetic concentric tube robots

Recently, emerging hard-magnetic concentric tube robots (HMCTR) have shown great potential in applications such as tumor-ablation surgery. However, their development is greatly limited by complex dynamics due to geometric nonlinearity from large deformations, time-varying free segment lengths, and the complexity of magnetoelastic behavior, which also makes real-time, accurate control difficult. In this work, an efficient modeling and nonlinear model predictive control (NMPC) framework is proposed for the planar variable-length HMCTRs. An efficient global angular parameterization method (GAPM) is first developed, which features pre-integrable and concise inertial forces and accurately captures the large deformations of continuum robots using only a small number of degrees of freedom. A nonlinear model predictive control (NMPC) scheme that explicitly enforces actuator limits, measurement disturbances, and minimum safety-distance constraints. Simulation results demonstrate robust trajectory tracking and safe-distance navigation under both uniform and non-uniform magnetic fields, with near-real-time performance. These findings underscore the framework's computational efficiency and control accuracy, highlighting its potential for clinical translation in HMCTR navigation and tracking.