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A variational approach to geometric mechanics for undulating robotic locomotion

Sean Even, Patrick S. Martinez, Cora Keogh, Oliver Gross, Yasemin Ozkan-Aydin, Peter Schröder

TL;DR

This work addresses the challenge of connecting geometric mechanics with real-world validation for undulating locomotion. It introduces a variational integrator that relies on a dissipation metric, approximated by a Resistive Force Theory–like model, to map serpenoid gait shapes in ${\mathcal{S}}_{\mathrm{serp}}$ to world trajectories and compares simulations to experiments on a snake-like robot. Despite using a highly simplified energy model, the framework captures average trajectory trends and enables gait optimization that improves energy efficiency (Cost of Transport) by up to about 23%. The approach offers a practical bridge between theory and lab validation, with potential extensions to reinforcement learning and real-time gait adaptation in varied environments.

Abstract

Limbless organisms of all sizes use undulating patterns of self-deformation to locomote. Geometric mechanics, which maps deformations to motions, provides a powerful framework to formalize and investigate the theoretical properties and limitations of such modes of locomotion. However, the inherent level of abstraction poses a challenge when bridging the gap between theory or simulations and laboratory experiments. We investigate the challenges of modeling motion trajectories of an undulating robotic locomotor by comparing experiments and simulations performed with a variational integrator. Despite the extensive simplifications that the model based on a geometric variation principle entails, the simulations show good agreement on average. Notably, our approach merely requires the knowledge of the \emph{dissipation metric} -- a Riemannian metric on the configuration space, which can in practice be approximated by means closely resembling \emph{resistive force theory}.

A variational approach to geometric mechanics for undulating robotic locomotion

TL;DR

This work addresses the challenge of connecting geometric mechanics with real-world validation for undulating locomotion. It introduces a variational integrator that relies on a dissipation metric, approximated by a Resistive Force Theory–like model, to map serpenoid gait shapes in to world trajectories and compares simulations to experiments on a snake-like robot. Despite using a highly simplified energy model, the framework captures average trajectory trends and enables gait optimization that improves energy efficiency (Cost of Transport) by up to about 23%. The approach offers a practical bridge between theory and lab validation, with potential extensions to reinforcement learning and real-time gait adaptation in varied environments.

Abstract

Limbless organisms of all sizes use undulating patterns of self-deformation to locomote. Geometric mechanics, which maps deformations to motions, provides a powerful framework to formalize and investigate the theoretical properties and limitations of such modes of locomotion. However, the inherent level of abstraction poses a challenge when bridging the gap between theory or simulations and laboratory experiments. We investigate the challenges of modeling motion trajectories of an undulating robotic locomotor by comparing experiments and simulations performed with a variational integrator. Despite the extensive simplifications that the model based on a geometric variation principle entails, the simulations show good agreement on average. Notably, our approach merely requires the knowledge of the \emph{dissipation metric} -- a Riemannian metric on the configuration space, which can in practice be approximated by means closely resembling \emph{resistive force theory}.
Paper Structure (18 sections, 15 equations, 9 figures, 1 algorithm)

This paper contains 18 sections, 15 equations, 9 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Geometric locomotion
  4. A variational approach
  5. Discretization and local dissipation metric
  6. Variational energy
  7. Variational integrator
  8. Shape space and gait design
  9. Gaits
  10. Experimental protocol
  11. Snake robot
  12. Experimental setup
  13. Fitting of material parameters
  14. Evaluation
  15. Mapping performance differences
  16. ...and 3 more sections

Figures (9)

  • Figure 1: Geometric mechanics maps gaits of undulatory robotic locomotors to motion trajectories (blue) in world coordinates. The accuracy of this map compared to laboratory experiments (green) depends on the choice of model parameters, for which the Riemannian metric on the configuration space provides a natural description.
  • Figure 2: We represent the robotic system comprised of linked elements as a polygonal curve. Displacements of vertices anisotropically dissipate energy to the environment which is represented by local dissipation tensors (see Sec. \ref{['sec:VariationalIntegrator']})
  • Figure 3: The ellipses corresponding to two prototypical gaits within the shape space, accompanied by the shapes of the sequence and motion trajectory.
  • Figure 4: Description of our snake robot which includes ten planar actuators (Dynamixel XL430-W250-T) and rubber wheels that introduce frictional anisotropy. The robot also contains reflective markers at the rotational axis of the motors (marked in red) as well as on the head and tail of the robot which allows each segment to be tracked in real time using the Optitrack Motion Capture system.
  • Figure 5: Overlay of three lab trials given the same input data which correspond to the experiment in the center of Fig. \ref{['fig:TrajectoryComparison']}.
  • ...and 4 more figures