Estimating Dynamic Soft Continuum Robot States From Boundaries

State estimation is one of the fundamental problems in robotics. For soft continuum robots, this task is particularly challenging because their states (poses, strains, internal wrenches, and velocities) are inherently infinite-dimensional functions due to their continuous deformability. Traditional sensing techniques, however, can only provide discrete measurements. Recently, a dynamic state estimation method known as a \textit{boundary observer} was introduced, which uses Cosserat rod theory to recover all infinite-dimensional states by measuring only the tip velocity. In this work, we present a dual design that instead relies on measuring the internal wrench at the robot's base. Despite the duality, this new approach offers a key practical advantage: it requires only a force/torque (FT) sensor embedded at the base and eliminates the need for external motion capture systems. Both observer types are inspired by principles of energy dissipation and can be naturally combined to enhance performance. We conduct a Lyapunov-based analysis to study the convergence rate of these boundary observers and reveal a useful property: as the observer gains increase, the convergence rate initially improves and then degrades. This convex trend enables efficient tuning of the observer gains. We also identify special cases where linear and angular states are fully determined by each other, which further relaxes sensing requirements. Experimental studies using a tendon-driven continuum robot validate the convergence of all observer variants under fast dynamic motions, the existence of optimal gains, robustness against unknown external forces, and the algorithm's real-time computational performance.