Using Physics-Informed Neural Networks for Shape Estimation of Concentric Tube Robots

Abstract

Accurately modeling the mechanics of concentric tube robots (CTRs) remains challenging due to nonlinear curvature coupling, torsional interactions, and sensitivity to material and geometric parameters. Physics-informed neural networks (PINNs) offer a flexible alternative by embedding governing equations directly into the training process. In this work, we develop a PINN for modeling the free-tip kinematics of a three-tube CTR with a max length of 250 mm, incorporating geometric compatibility, constitutive relations, and equilibrium equations based on Cosserat rod theory into the training pipeline. We additionally introduce a Fourier-based joint encoding to represent the actuation rotations. Along 20 equally spaced points for each backbone configuration, the PINN achieved mean and maximum errors of 0.80<inline-formula><tex-math notation="LaTeX">$\pm$</tex-math></inline-formula> 0.42mm and 1.71<inline-formula><tex-math notation="LaTeX">$\pm$</tex-math></inline-formula>1.12 mm, outperforming the analytical Cosserat model and demonstrating that physics-informed learning can approximate nonlinear CTR mechanics. A dataset-size ablation against a standard neural network showed that the two networks perform comparably when the training data densely cover the workspace, whereas the PINN was substantially more accurate under sparse coverage, retaining error at or below the analytical model with as little as <inline-formula><tex-math notation="LaTeX">$5\%$</tex-math></inline-formula> of the data.