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 due to continuous deformability, while sensing provides only discrete measurements. Recently, a dynamic state estimation method known as a boundary observer was introduced, which uses Cosserat rod theory to recover all states from tip velocity measurements. 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 approach offers a key practical advantage: it requires only a force/torque (FT) sensor at the base and eliminates the need for external motion capture systems. Both observer types are inspired by energy dissipation principles and can be combined to enhance performance. We conduct a Lyapunov-based analysis to study convergence and reveal a useful property: as observer gains increase, the convergence rate first improves and then degrades. This convex trend enables efficient gain tuning. We also identify cases where linear and angular states are fully determined by each other, further relaxing sensing requirements. In summary, this work achieves dynamic infinite-dimensional state estimation with minimal sensing requirements and systematic parameter tuning, which has not been demonstrated in existing approaches. Simulation and experimental studies using a tendon-driven continuum robot validate convergence under fast dynamic motions, the existence of optimal gains, robustness to external forces, measurement noise, and model uncertainty, and real-time computational performance.