State estimation of robotic systems is essential to implementing feedback controllers which usually provide better robustness to modeling uncertainties than open-loop controllers. However, state estimation of soft robots is very challenging because soft robots have theoretically infinite degrees of freedom while existing sensors only provide a limited number of discrete measurements. In this paper, we design an observer for soft continuum robotic arms based on the well-known Cosserat rod theory which models continuum robotic arms by nonlinear partial differential equations (PDEs). The observer is able to estimate all the continuum (infinite-dimensional) robot states (poses, strains, and velocities) by only sensing the tip velocity of the continuum robot (and hence it is called a ``boundary'' observer). More importantly, the estimation error dynamics is formally proven to be locally input-to-state stable. The key idea is to inject sequential tip velocity measurements into the observer in a way that dissipates the energy of the estimation errors through the boundary. Furthermore, this boundary observer can be implemented by simply changing a boundary condition in any numerical solvers of Cosserat rod models. Extensive numerical studies are included and suggest that the domain of attraction is large and the observer is robust to uncertainties of tip velocity measurements and model parameters.

翻译：机器人系统的状态估计对于实现反馈控制器至关重要，相比开环控制器，反馈控制器通常对模型不确定性具有更强的鲁棒性。然而，软体机器人的状态估计极具挑战性，因为软体机器人具有理论上无限的自由度，而现有传感器仅能提供有限的离散测量值。本文基于著名的Cosserat杆理论设计了一种用于软体连续体机械臂的观测器，该理论通过非线性偏微分方程（PDE）对连续体机械臂进行建模。该观测器仅通过感知连续体机器人末端速度（因此称为"边界"观测器），即可估计所有连续（无限维）机器人状态（位姿、应变和速度）。更重要的是，估计误差动力学被严格证明具有局部输入-状态稳定性。其核心思想是通过边界耗散估计误差能量的方式，将连续末端速度测量值注入观测器。此外，该边界观测器只需在任意Cosserat杆模型数值求解器中修改边界条件即可实现。大量数值研究表明，吸引域范围较大，且该观测器对末端速度测量值和模型参数的不确定性具有鲁棒性。