Factor Graph-Based Shape Estimation for Continuum Robots via Magnus Expansion

2026-04-16

AI Summary

This work addresses the trade-off between infinite-dimensional modeling, compact state representation, and uncertainty quantification in reconstructing the shape of continuum robots from sparse, noisy sensor data. The authors propose a low-dimensional state estimation approach based on factor graphs, leveraging geometrically varying strain (GVS) to parameterize the strain field. For the first time, closed-form kinematic constraints derived via the Magnus expansion are incorporated as factors into the graphical model, establishing a geometric prior—grounded in Cosserat rod theory—between strain and pose. This enables compact, probabilistic, and modular state inference. In simulations on a 0.4-meter tendon-driven continuum robot, the method achieves average positional errors below 2 mm across three sensing configurations; when using only position measurements, it reduces orientation error by a factor of six compared to a Gaussian process regression baseline.

Technology Category

• Intelligent Robots: State Estimation
• Reasoning under Uncertainty: Relational Probabilistic Models
• Machine Learning: Probabilistic Circuits and Graphical Models

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Search and Retrieval-Augmented AI: Web query analysis, representation and understanding

Abstract

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.

Problem

• continuum robots
• shape estimation
• factor graphs
• Magnus expansion
• sensor data

Innovation

• Factor Graph
• Magnus Expansion
• Geometric Variable Strain
• Continuum Robot
• Shape Estimation