Factor Graph-Based Shape Estimation for Continuum Robots via Magnus Expansion
For continuum robotics, this work provides a compact, probabilistic shape estimation approach that improves accuracy and computational efficiency over existing methods.
This paper introduces a factor graph-based method for continuum robot shape estimation that combines a low-dimensional Geometric Variable Strain parameterization with a novel kinematic factor derived from the Magnus expansion. In simulations, the method achieves mean position errors below 2 mm and a sixfold reduction in orientation error compared to a Gaussian process regression baseline.
Reconstructing the shape of continuum manipulators from sparse, noisy sensor data is a challenging task, owing to the infinite-dimensional nature of such systems. Existing approaches broadly trade off between parametric methods that yield compact state representations but lack probabilistic structure, and Cosserat rod inference on factor graphs, which provides principled uncertainty quantification at the cost of a state dimension that grows with the spatial discretization. This letter combines the strength of both paradigms by estimating the coefficients of a low-dimensional Geometric Variable Strain (GVS) parameterization within a factor graph framework. A novel kinematic factor, derived from the Magnus expansion of the strain field, encodes the closed-form rod geometry as a prior constraint linking the GVS strain coefficients to the backbone pose variables. The resulting formulation yields a compact state vector directly amenable to model-based control, while retaining the modularity, probabilistic treatment and computational efficiency of factor graph inference. The proposed method is evaluated in simulation on a 0.4 m long tendon-driven continuum robot under three measurement configurations, achieving mean position errors below 2 mm for all three scenarios and demonstrating a sixfold reduction in orientation error compared to a Gaussian process regression baseline when only position measurements are available.