Full State Estimation of Soft Robots From Tip Velocities: A Cosserat-Theoretic Boundary Observer

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 infinite degrees of freedom theoretically while existing sensors only provide a limited number of measurements. In this paper, we design an observer for soft robots based on the well-known Cosserat rod theory which models soft robots by nonlinear partial differential equations (PDEs). The observer is able to estimate all the continuous/infinite-dimensional robot states (poses, strains, and velocities) by only sensing the tip velocity of the 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.