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. This work focuses on soft robotic manipulators, also known as continuum robots. We design an observer algorithm based on the well-known Cosserat rod theory, which models continuum robots by nonlinear partial differential equations (PDEs) evolving in geometric Lie groups. The observer can estimate all infinite-dimensional continuum robot states, including 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. The distinct advantage of this PDE-based design is that it can be implemented using any existing numerical implementation for Cosserat rod models. All theoretical convergence guarantees will be preserved, regardless of the discretization method. We call this property ``one design for any discretization''. 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，反馈控制器通常比开环控制器对建模不确定性具有更好的鲁棒性。然而，软体机器人的状态估计极具挑战性，因为软体机器人理论上具有无限自由度，而现有传感器仅能提供有限数量的离散测量值。本研究聚焦于软体机器人操作器，即连续体机器人。我们基于著名的Cosserat杆理论设计了一种观测器算法，该理论通过演化于几何李群中的非线性偏微分方程（PDE）对连续体机器人进行建模。该观测器仅通过感知连续体机器人的末端速度，即可估计所有无限维的连续体机器人状态（包括位姿、应变和速度），因此被称为“边界”观测器。更重要的是，我们严格证明了估计误差动力学具有局部输入-状态稳定性。其核心思想是以通过边界耗散估计误差能量的方式，将连续的末端速度测量值注入观测器。这种基于PDE的设计方法具有独特优势：它可以利用任何现有的Cosserat杆模型数值实现方案进行部署。无论采用何种离散化方法，所有理论收敛性保证都将得以保持。我们将此特性称为“一种设计适用于任意离散化方案”。大量数值研究表明，该观测器的吸引域范围广阔，且对末端速度测量不确定性和模型参数不确定性具有良好的鲁棒性。