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

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 tip pose with cable tension within a unified two-stage pipeline: the first stage reconstructs the backbone shape from tip-pose observations, and the second stage estimates the external tip wrench from actuator tensions using the reconstructed shape. 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.