Estimating Dynamic Soft Continuum Robot States From Boundaries

TL;DR

This paper advances dynamic state estimation for soft continuum robots by introducing a base-wrench boundary observer grounded in Cosserat rod PDEs, providing a dual alternative to the existing tip-observer. By leveraging energy dissipation and a Lyapunov framework, it derives convergence properties and optimal gain conditions, including a convex gain-convergence relationship and finite-time extinction under absorbing boundaries. The authors develop a shooting-based numerical implementation and validate the approach through simulations and tendon-driven experiments, demonstrating real-time performance, robustness to unknown inputs, and improved accuracy with a dual boundary setup. These results enable full-state estimation from minimal boundary sensing, with potential for closed-loop control in dynamic, unstructured environments.

Abstract

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.

Figures (9)

• Figure 1: Using Cosserat rod theory, the robot is modeled as a continuous set of rigid cross-sections stacked along a centerline parametrized by $s$ from the base ($s=0$) to the tip ($s=L$). $g(s,t)\in SE(3)$ represents the pose of the cross-section at location $s$ and time $t$.
• Figure 2: In the scale case, the fastest convergence rate $\mu_{\max}$ vs. $\Gamma_1$ (with $\Gamma_0=0$) and vs. $\Gamma_0$ (with $\Gamma_1=0$). When increasing $\Gamma_0$ or $\Gamma_1$, $\mu_{\max}$ increases to infinity at a singularity, and then decreases. System parameters: $L=1$, $M=1$, $K=3$.
• Figure 3: Left: Initial configurations of the ground truth and the estimated rod. The estimate is deliberately initialized with a deviated configuration to evaluate its convergence behavior. Middle: Ground truth and estimated tip position trajectories using the base observer when $\gamma=1$. Right: Settle times of the observers as functions of the gain scaling factor $\gamma$. As $\gamma$ increases, the settle times initially decrease, reaching a minimum near $\gamma = 1$, and then increase again.
• Figure 4: Left: The rod is released to induce free oscillations. Right: Initial configurations of the ground truth and the estimated rod. The estimate is deliberately initialized with a deviated configuration to evaluate its convergence behavior.
• Figure 5: Left: Ground truth and estimated tip position trajectories using the base observer when $\gamma=1$. Middle: Settle times of special cases of base observer as functions of the gain scaling factor $\gamma$. Right: Settle times of special cases of base observer as functions of the gain scaling factor $\gamma$.