[Paper Review] Efficient Force Estimation for Continuum Robot

Qingyu Xiao, Yue Chen|arXiv (Cornell University)|Sep 26, 2021

Soft Robotics and Applications28 references4 citations

TL;DR

This paper presents a fast and accurate curvature-based force estimation method for continuum robots using fiber Bragg grating sensors (FBGS) and a simplified Cosserat rod model. By leveraging least-squares optimization on measured local curvatures, the method estimates both magnitude and location of single or multiple external contact forces with errors of 5.25–12.87% (magnitude) and 1.02–2.19% (location), achieving computation speeds 29 to 101.6 times faster than conventional methods.

ABSTRACT

External contact force is one of the most significant information for the robots to model, control, and safely interact with external objects. For continuum robots, it is possible to estimate the contact force based on the measurements of robot configurations, which addresses the difficulty of implementing the force sensor feedback on the robot body with strict dimension constraints. In this paper, we use local curvatures measured from fiber Bragg grating sensors (FBGS) to estimate the magnitude and location of single or multiple external contact forces. A simplified mechanics model is derived from Cosserat rod theory to compute continuum robot curvatures. Least-square optimization is utilized to estimate the forces by minimizing errors between computed curvatures and measured curvatures. The results show that the proposed method is able to accurately estimate the contact force magnitude (error: 5.25\% -- 12.87\%) and locations (error: 1.02\% -- 2.19\%). The calculation speed of the proposed method is validated in MATLAB. The results indicate that our approach is 29.0 -- 101.6 times faster than the conventional methods. These results indicate that the proposed method is accurate and efficient for contact force estimations.

Motivation & Objective

• To address the challenge of estimating external contact forces on continuum robots without embedded force sensors due to strict size constraints.
• To overcome limitations of traditional force sensors that only detect forces at specific locations, such as the robot tip.
• To develop a computationally efficient method for estimating force magnitude and location along the robot body using curvature measurements.
• To enable real-time force estimation for safe and precise interaction in minimally invasive surgical applications.

Proposed method

• A simplified Cosserat rod model is derived by assuming zero external moments and axial forces, enabling fast curvature computation without integrating rotation matrices.
• Local curvatures are measured using fiber Bragg grating sensors (FBGS) along the robot's length, providing robust, rotation-invariant data.
• The force estimation problem is formulated as a least-squares optimization to minimize the error between computed and measured curvatures.
• The optimization seeks the force vector containing estimated magnitudes and locations by solving for the optimal force distribution that matches observed curvature profiles.
• The method avoids solving boundary value problems (BVPs) iteratively, significantly reducing computation time compared to Levenberg-Marquardt or derivative propagation methods.
• The model is validated on a 290 mm Nitinol tube under single, double, and triple force conditions with varying locations and magnitudes.

Experimental results

Research questions

• RQ1 Can curvature measurements alone enable accurate estimation of both force magnitude and location on a continuum robot?
• RQ2 What simplifying assumptions on the Cosserat rod model allow for fast curvature computation without sacrificing estimation accuracy?
• RQ3 How does the proposed method compare in speed and accuracy to conventional BVP-based optimization techniques like Levenberg-Marquardt and derivative propagation?
• RQ4 Can the method reliably estimate multiple contact forces simultaneously along the robot’s body?
• RQ5 What is the impact of sensor noise and measurement error on the accuracy of force estimation using curvature-based methods?

Key findings

• The proposed method achieves a mean force magnitude estimation error of 5.25–12.87% across single, double, and triple force cases.
• The mean error for force location estimation is 1.02–2.19%, indicating high spatial accuracy.
• The root mean square error (RMSE) for single, double, and triple force estimation is 0.084 ± 0.073 N, 0.115 ± 0.102 N, and 0.1090 ± 0.1173 N, respectively.
• Computation time for single, double, and triple force estimation is 0.134 s, 0.344 s, and 0.730 s in MATLAB, respectively.
• The method is 29.0 to 101.6 times faster than solving the boundary value problem with derivative propagation and Levenberg-Marquardt optimization.
• The simplified model, derived under conditions of no external moments and axial forces, enables fast curvature computation while preserving estimation accuracy.