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

TL;DR

This work addresses the challenge of estimating backbone shape and external tip force for cable-driven continuum robots under sparse sensing. It introduces a curvature-space framework that uses a second-order polynomial curvature model (PCK2) with state $\mathcal{S}=[m0,m1,m2,\delta]^T$ to represent the backbone, and develops mappings among joint, shape, and task spaces together with a real-time solver. A virtual-work-based force estimator computes the distal wrench from cable tensions by solving a redundancy-resolved static equilibrium, yielding a closed-form solution in the curvature representation. Validation through Cosserat-rod simulations and hardware experiments demonstrates that PCK2 provides superior shape fidelity and force accuracy, achieving robust and real-time integrated shape–force estimation suitable for applications in constrained robotic manipulation and surgery. The approach offers a compact, scalable alternative to constant-curvature and geometry-space methods, with potential for extension to full 3D SE(3) configurations and real-time control integration.

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.

Figures (7)

• Figure 1: Integrated shape–force sensing framework. Tip pose $\!\rightarrow\!$ PCK shape $\mathbf{\mathcal{S}}=[m_0,m_1,m_2,\delta]^{\top}$; with tensions $\tau$ and model Jacobians, the virtual-work balance yields $F^{*}$.
• Figure 2: Schematic of the kinematics for a cable-driven continuum instrument.
• Figure 3: Shape estimation across fifteen external force configurations arranged as a $3\times 5$ grid (rows: $F_z\in\{+1,0,-1\}\,\mathrm{N}$; columns: $F_x\in\{-2,-1,0,1,2\}\,\mathrm{N}$) under actuation tensions $\boldsymbol{\tau}=[5,\,5,\,0,\,0]\,\mathrm{N}$ with measurement noise. The ground truth backbone is shown by black markers with a thin dashed construction line, and PCK0, PCK1, and PCK2 reconstructions are overlaid as solid red, green, and blue curves. A purple arrow at each tip indicates the applied force vector. Tip pose inputs are perturbed with zero mean noise $(\sigma_x=\sigma_z=0.5\,\mathrm{mm},\ \sigma_\theta=0.005\,\mathrm{rad})$, and actuator tensions include zero mean Gaussian noise $(\sigma_\tau=0.02\,\mathrm{N})$ and a small per trial bias drift. Each cell shows the mean over five trials. Axes are in meters.
• Figure 4: Correlation between shape and force estimation errors across fifteen loading conditions. Each subplot compares the force error magnitude $e_F$ with shape or tip errors for PCK0–2. Fitted regression lines and Pearson coefficients ($\rho$) quantify how strongly the two errors vary together. Higher $\rho$ indicates stronger coupling.
• Figure 5: Experimental setup for validating shape and tip-force estimation. An ELP HD USB camera (1024$\times$768 px, 30 Hz) tracks the tip pose, while a miniature load cell ($\pm{}1\;\text{N}$ range) records contact forces. The continuum instrument tested here has backbone length $L = 40\;\text{mm}$, distance between central backbone and pulling wires $r = 1.8\;\text{mm}$, Young’s modulus $E = 65\;\text{GPa}$, and second moment of area $I = 4.83 \times 10^{-15}\;\text{m}^4$.