Task and Configuration Space Compliance of Continuum Robots via Lie Group and Modal Shape Formulations

TL;DR

This work develops analytic local compliance models for continuum robots modeled as Kirchhoff (Cosserat) rods by combining a modal curvature basis with Lie group (Magnus) integration, enabling closed-form task-space and configuration-space compliance without relying on constant-curvature assumptions or finite-difference approximations. The authors first derive the compliance for a single Kirchhoff rod, then extend to tendon-actuated continuum segments, revealing how tendon routing and Jacobian derivatives influence static stiffness and deflection predictions. They demonstrate a clear trade-off between model fidelity and computation cost via modal order, and show that neglecting the Jacobian-derivative term can significantly degrade accuracy in compliant robots. Experimental validation on a tendon-actuated segment yields deflection prediction errors below 11.5% of the total arc length, confirming practical applicability for design, planning, and compliant control of variable-curvature continuum robots. Overall, the framework bridges constant-curvature configuration-space approaches and full geometrically exact task-space models, delivering efficient, accurate analytic tools for passive stiffness modulation and interaction with complex environments.

Abstract

Continuum robots suffer large deflections due to internal and external forces. Accurate modeling of their passive compliance is necessary for accurate environmental interaction, especially in scenarios where direct force sensing is not practical. This paper focuses on deriving analytic formulations for the compliance of continuum robots that can be modeled as Kirchhoff rods. Compared to prior works, the approach presented herein is not subject to the constant-curvature assumptions to derive the configuration space compliance, and we do not rely on computationally-expensive finite difference approximations to obtain the task space compliance. Using modal approximations over curvature space and Lie group integration, we obtain closed-form expressions for the task and configuration space compliance matrices of continuum robots, thereby bridging the gap between constant-curvature analytic formulations of configuration space compliance and variable curvature task space compliance. We first present an analytic expression for the compliance of a single Kirchhoff rod. We then extend this formulation for computing both the task space and configuration space compliance of a tendon-actuated continuum robot. We then use our formulation to study the tradeoffs between computation cost and modeling accuracy as well as the loss in accuracy from neglecting the Jacobian derivative term in the compliance model. Finally, we experimentally validate the model on a tendon-actuated continuum segment, demonstrating the model's ability to predict passive deflections with error below 11.5\% percent of total arc length.

Figures (4)

• Figure 1: Kinematic parameters used in our modal-shape compliance matrix model.
• Figure 2: (a) The absolute deflection error in tip deflection, with error bars showing the minimum and maximum error, predicted by our compliance matrix expression and (b) the computation time for different numbers of modal coefficients, showing a trade-off between speed and accuracy.
• Figure 3: The setup used to experimentally validate the compliance model: force/torque sensor, wire-rope, pulleys, three out of five configurations used in the experiments, and mass used to apply forces to the end disk. The end disk of the robot has four string encoders that measure the segment's configuration.
• Figure 4: Boxplot comparing deflection error to the norm of the force applied to the end disk. The central red line is the median, and the box covers the 25th and 75th percentiles. Small dots represent the experimental data.