Full State Estimation of Continuum Robots from Tip Velocities: A Cosserat-Theoretic Boundary Observer (2303.06130v4)

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 propertyone 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.

• The paper introduces a boundary observer leveraging tip velocity measurements to efficiently estimate the full state of continuum robots using Cosserat rod theory.
• It provides local input-to-state stability proofs for error dynamics, demonstrating robustness against model uncertainties and sensor noise.
• Extensive numerical experiments confirm the observer’s accuracy over a wide domain and its easy integration into existing Cosserat rod solvers.

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

The paper "Full State Estimation of Continuum Robots From Tip Velocities: A Cosserat-Theoretic Boundary Observer" introduces a novel approach to state estimation for continuum robots, leveraging the Cosserat rod theory. Continuum robots, unlike their rigid counterparts, exhibit high flexibility with theoretically infinite degrees of freedom, posing significant challenges in state estimation when using conventional discrete sensors. The paper addresses these challenges by proposing a boundary observer that can estimate the entire state of continuum robots by solely relying on tip velocity measurements.

Core Contributions

1. Boundary Observer Design: The paper presents an observer with a unique boundary-based approach for soft continuum robotic arms, utilizing Cosserat rod theory. This method involves modeling continuum arms through a series of nonlinear partial differential equations (PDEs). By sensing only the tip velocity, the observer can estimate the full state, including poses, strains, and velocities, making it particularly resource-efficient.
2. Stability Proofs: A significant theoretical contribution is the local input-to-state stability of the estimation error dynamics. The observer integrates sequential tip velocity inputs in a manner that dissipates estimation errors' energy at the system's boundary, enabling a robust state estimation even with uncertainties.
3. Implementation Feasibility: The design of this observer allows for straightforward implementation by modifying boundary conditions within numerical solvers for Cosserat rod models. This adaptability ensures that existing computational frameworks for continuum robots can incorporate this observer with minimal adjustments, primarily acting as virtual wrenches at the boundary.

Numerical Validation and Results

The paper conducts extensive numerical studies to validate the observer's effectiveness. Notably, these studies reveal:

• Robustness to Uncertainties: The boundary observer demonstrates substantial robustness against variations in model parameters, sensor noise, and input uncertainties. This feature is critical given the variable conditions under which soft robots typically operate.
• Wide Domain of Attraction: Numerical experiments suggest that the observer operates effectively over a large domain of initial conditions, surpassing typical limitations associated with local stability analysis.
• Comparison with Ground Truth: During simulations, the estimated robot configurations closely matched actual trajectories under dynamic conditions, underscoring the observer's accuracy.

Practical and Theoretical Implications

This boundary observer has profound implications for both the practical deployment of continuum robots and theoretical advancements in PDE-based robotic control systems:

• Practical Applications: In fields like medical robotics or underwater manipulation, where continuum robots' adaptability is leveraged, this observer can provide more precise control without extensive sensor arrays, reducing costs and complexity.
• Theoretical Frameworks: By extending control and estimation frameworks to encompass PDEs, this paper opens up new research directions in robotics. The observer's stability proof can potentially be adapted to other nonlinear, infinite-dimensional systems.

Future Developments

Looking ahead, several research pathways emerge from this work:

• Enhanced Robustness Analysis: Further exploration into the robustness of the observer against higher degrees of model uncertainties and sensor errors could solidify its application in uncontrolled environments.
• Integration with Control Strategies: Combining this state estimation approach with advanced control strategies such as model predictive control could improve the adaptability and precision of continuum robots in complex tasks.
• Physical Deployment: Testing the observer in real-world scenarios, beyond simulation, will be crucial to assess its practicality and gauge real-time computational demands.

In summary, this paper provides a robust framework for continuum robot state estimation using minimal sensor data, marking a step forward in the efficient deployment of soft robots while addressing fundamental challenges in infinite-dimensional system estimation.