Learning-Based Forward Kinematics and Its Inverse Kinematics Implementation for Concentric-Tube Robots

Concentric-tube robots (CTRs) are promising for minimally invasive surgery but their complex kinematics present a significant challenge. While recent learning-based forward kinematics (FK) methods offer high accuracy, they often result in nondifferentiable models, which hinders efficient computation. Addressing these issues, we propose a precise, explicit, and model-free kinematic framework for CTRs, combining a refined learning-based FK approach with an optimization-based IK method that uses the learned FK functions. First, we introduce a boundary-dense sampling strategy to acquire real-world data. Then, using fully connected feedforward neural networks with tangent hyperbolic activation functions, we learn and extract explicit FK functions. The IK problem is formulated as a constrained nonlinear optimization problem, complemented by an initialization strategy that provides high-quality initial guesses. Comprehensive experiments validate the proposed FK method’s high accuracy, with a tip position error of only 0.498 \pm 0.010% of the total robot length, and its fast computational speed of up to 162 367 \pm 7145 Hz. Furthermore, we demonstrate the IK method’s effectiveness in precise and efficient IK computation, accurate path following, and successfully handling the multisolutions problem.