Lie Algebra-Based Probabilistic Shape Estimation of Flexible Continuum Robots From Sparse Pose Measurements

Estimating the three-dimensional (3D) shape of tendon-driven continuum robots (TDCRs) under sparse pose measurements and unknown tip forces remains a significant challenge. This paper presents a novel Lie algebra-based probabilistic framework for continuous-arclength shape reconstruction of variable-curvature TDCRs subjected to unknown tip loads. By unifying Cosserat-rod and Cosserat-string theories, a Lie algebra-coupled static model is established that explicitly represents tendon actuation as distributed wrenches along the robot’s backbone. A sparse Gaussian Process prior in se(3) is derived via linearization to encode actuation-coupled shape distributions. This prior is fused with sparse, noisy pose measurements from electromagnetic sensors in a continuous-arclength batch estimation framework, enabling uncertainty-aware reconstruction along the backbone. Simulations and experiments on single-segment prototypes under diverse tip-loading conditions validate the method’s efficacy. Results demonstrate that the proposed approach achieves accurate shape estimation, with average shape (tip) errors of 1.6% (2.5%) of the total length, surpassing state-of-the-art methods. Comparable precision on helical-routing TDCR further confirms the framework’s generalizability.