用曲率多项式与虚功原理，实现连续体机器人形变与受力的高精度联合估计。
Integrated Shape-Force Estimation for Continuum Robots: A Virtual-Work and Polynomial-Curvature Framework
- 基于曲率空间的多项式模态系数作为广义坐标，构建虚拟功静态模型。
- 二阶多项式模型在仿真与硬件实验中均实现形变误差小于3.2%、受力估计误差低于4.1%。
- 无需迭代、计算轻量，适合传感器稀疏条件下的实时控制与手术机器人应用。
缆绳驱动连续体机器人（CDCRs）广泛应用于需要在狭小空间内灵巧操作的手术和检测任务。现有基于模型的估计方法或假设恒定曲率，或依赖几何空间插值，均难以在大变形和稀疏传感条件下保持精度。本文提出一种集成形变-受力估计框架，结合缆绳张力测量与末端位姿数据，同步重建主干形状并估计外部末端受力。该框架采用多项式曲率运动学（PCK）与基于虚功的静力学公式，直接在曲率空间中表达，以多项式模态系数作为广义坐标。通过基于柯塞拉杆（Cosserat-rod）的仿真及带扭矩传感器的CDCR原型硬件实验验证，结果表明：二阶PCK模型在形变重建与受力估计上均表现优异，兼具轻量化的形状优化与闭式、无迭代的受力估计，为先前恒定曲率与几何空间方法提供了紧凑且鲁棒的替代方案。
原文摘要 · Abstract (English)
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.
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