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Full State Estimation of Continuum Robots from Tip Velocities: A Cosserat-Theoretic Boundary Observer

Tongjia Zheng, Qing Han, Hai Lin

TL;DR

The work tackles state estimation for soft continuum robots with theoretically infinite degrees of freedom by introducing a boundary observer based on Cosserat rod PDEs that uses only tip velocity measurements. It shows local input-to-state stability (ISS) of the estimation error and proves a discretization-agnostic property: the observer can be implemented with any Cosserat-rod numerical method while preserving convergence guarantees. Extensive simulations demonstrate robustness to tip-velocity noise and certain model errors, and identify actuator-model accuracy as an important practical consideration. The approach offers a principled, PDE-based pathway to enabling feedback control of continuum robots under realistic sensing and modeling uncertainties.

Abstract

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.

Full State Estimation of Continuum Robots from Tip Velocities: A Cosserat-Theoretic Boundary Observer

TL;DR

The work tackles state estimation for soft continuum robots with theoretically infinite degrees of freedom by introducing a boundary observer based on Cosserat rod PDEs that uses only tip velocity measurements. It shows local input-to-state stability (ISS) of the estimation error and proves a discretization-agnostic property: the observer can be implemented with any Cosserat-rod numerical method while preserving convergence guarantees. Extensive simulations demonstrate robustness to tip-velocity noise and certain model errors, and identify actuator-model accuracy as an important practical consideration. The approach offers a principled, PDE-based pathway to enabling feedback control of continuum robots under realistic sensing and modeling uncertainties.

Abstract

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.
Paper Structure (20 sections, 4 theorems, 59 equations, 14 figures, 1 table)

This paper contains 20 sections, 4 theorems, 59 equations, 14 figures, 1 table.

Table of Contents

  1. Introduction
  2. Modeling and Problem Formulation
  3. Notations and Preliminaries
  4. Cosserat Rod Models for Continuum Robots
  5. Configuration
  6. Kinematics
  7. Dynamics
  8. Inputs
  9. Formulation of the Estimation Problem
  10. Design and Stability of the Boundary Observer
  11. Implementation
  12. Simulation study
  13. Study 1: Impact of Tip Measurement Noise
  14. Study 2: Impact of Robot Modeling Errors
  15. Study 3: Impact of actuator modeling errors
  16. ...and 5 more sections

Key Result

Theorem 1

Consider eq:compact error PDE. If $\Gamma$ is positive definite, then the estimation error $\|\tilde{y}(\cdot,t)\|_{H^1}$ is locally input-to-state stable in the sense that there exist constants $k_0,k_1,k_2,b,\lambda,\kappa_1,\kappa_2>0$ such that for all $\|\tilde{y}(\cdot,0)\|_{H^1}<k_0$, $\|y(\c

Figures (14)

  • Figure 1: A Cosserat rod.
  • Figure 2: A continuum robot subject to gravity, actuation of two tendons, and a tip load of $1~\mathrm{N}$ (not plotted).
  • Figure 3: Tension of the tendons.
  • Figure 4: Three initial conditions of the actual states (dark gray) and the observer (light gray).
  • Figure 5: Comparison of the estimated configuration (light gray) and actual configuration (dark gray) for initial condition 1. After 0.5 seconds, the estimated configuration converged to the actual configuration and thus they overlapped.
  • ...and 9 more figures

Theorems & Definitions (13)

  • Remark 1
  • Remark 2
  • Remark 3
  • Remark 4
  • Theorem 1
  • proof
  • Definition 1
  • Definition 2
  • Theorem 2: dashkovskiy2013inputzheng2021transporting
  • Remark 5
  • ...and 3 more