Modeling and boundary robust control of flexible beveled puncture needles based on variable length Euler-Bernoulli beam

Flexible beveled puncture needles may undergo lateral deflection in soft tissue due to bevel-induced tip forces, which deteriorates targeting accuracy. To accurately characterize the needle motion during insertion, a coupled ordinary and partial differential equations (ODE-PDE) dynamic model is developed based on the variable length Euler-Bernoulli beam (VL-EB) theory. For the axial insertion subsystem, a nonlinear extended state observer (NESO) with a backstepping controller is developed to regulate insertion in the presence of unknown disturbances. To handle boundary uncertainties caused by the needle's unique structure and other factors, a boundary disturbance observer is proposed for the PDE subsystem. An improved boundary sliding mode controller (SMC) is then constructed for the coupled system, where an auxiliary system is employed to accommodate saturation-induced performance degradation. Rigorous Lyapunov analysis establishes the stability and convergence of the closed-loop system. Finally, simulation results demonstrate that the proposed method effectively suppresses needle bending and ensures robust and effective closed-loop performance under disturbances and input saturation.