  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.
