Lie Group Formulation and Sensitivity Analysis for Shape Sensing of Variable Curvature Continuum Robots with General String Encoder Routing

TL;DR

This paper introduces a Lie group-based shape sensing method for continuum robots using general string routing, optimizing sensor placement, and demonstrating high accuracy through simulations and experiments.

Contribution

It presents a novel Lie group formulation for shape sensing with general string routing, including optimization of routing paths and validation through simulations and experiments.

Findings

Minimal number of strings/tendons needed for accurate shape reconstruction.

Achieved less than 2% end-effector position error in experiments.

Validated approach for general deflections and string routings in simulations.

Abstract

This paper considers a combination of actuation tendons and measurement strings to achieve accurate shape sensing and direct kinematics of continuum robots. Assuming general string routing, a methodical Lie group formulation for the shape sensing of these robots is presented. The shape kinematics is expressed using arc-length-dependent curvature distributions parameterized by modal functions, and the Magnus expansion for Lie group integration is used to express the shape as a product of exponentials. The tendon and string length kinematic constraints are solved for the modal coefficients and the configuration space and body Jacobian are derived. The noise amplification index for the shape reconstruction problem is defined and used for optimizing the string/tendon routing paths, and a planar simulation study shows the minimal number of strings/tendons needed for accurate shape reconstruction. A torsionally stiff continuum segment is used for experimental evaluation, demonstrating mean (maximal) end-effector absolute position error of less than 2% (5%) of total length. Finally, a simulation study of a torsionally compliant segment demonstrates the approach for general deflections and string routings. We believe that the methods of this paper can benefit the design process, sensing and control of continuum and soft robots.