Integrated Shape-Force Estimation for Continuum Robots: A Virtual-Work and Polynomial-Curvature Framework

TL;DR

This paper presents an integrated shape-force estimation framework for cable-driven continuum robots that improves accuracy under large deformations by combining cable tension and tip pose data using polynomial curvature kinematics and virtual work principles.

Contribution

It introduces a novel polynomial curvature and virtual-work-based framework that accurately estimates shape and external forces simultaneously in continuum robots.

Findings

Second-order polynomial curvature model outperforms simpler models in accuracy.

The method provides a lightweight, closed-form force estimation approach.

Validation shows improved shape and force estimation in simulations and hardware experiments.

Abstract

Cable-driven continuum robots (CDCRs) are widely used in surgical and inspection tasks that require dexterous manipulation in confined spaces. Existing model-based estimation methods either assume constant curvature or rely on geometry-space interpolants, both of which struggle with accuracy under large deformations and sparse sensing. This letter introduces an integrated shape-force estimation framework that combines cable-tension measurements with tip-pose data to reconstruct backbone shape and estimate external tip force simultaneously. The framework employs polynomial curvature kinematics (PCK) and a virtual-work-based static formulation expressed directly in curvature space, where polynomial modal coefficients serve as generalized coordinates. The proposed method is validated through Cosserat-rod-based simulations and hardware experiments on a torque-cell-enabled CDCR prototype. Results show that the second-order PCK model achieves superior shape and force accuracy, combining a lightweight shape optimization with a closed-form, iteration-free force estimation, offering a compact and robust alternative to prior constant-curvature and geometry-space approaches.