Quadrupedal Passive Locomotion - DLAR Lab

Paper: “Breaking Symmetries Leads to Diverse Quadrupedal Gaits”

Authors: Jiayu Ding and Zhenyu Gan
Journal: IEEE Robotics and Automation Letters, Vol. 9, No. 5, May 2024

This research investigates how formal symmetry concepts from group theory can explain and predict the diversity of gaits in quadrupedal locomotion. Using a simplified quadrupedal spring-mass model, the authors present a unified framework for defining, analyzing, and classifying the symmetries of different gait patterns. By systematically breaking temporal, spatial, and morphological symmetries, the model reveals a spectrum of natural quadrupedal gaits, including pronking, two types of bounding, four forms of half-bounding, and two forms of galloping. These gaits are not isolated motions but exist as interconnected branches in the solution space of a hybrid dynamical system. The study demonstrates that each symmetry-breaking event at a parameter bifurcation point leads to the emergence of new, distinct footfall patterns.

The visualization shows the simplified quadrupedal spring-mass model and demonstrates how breaking different types of symmetry gives rise to a variety of quadrupedal gaits. The figures illustrate the branching of gaits through bifurcation diagrams, show how changes in the torso’s center of mass influence bounding patterns, and provide solution branches and keyframes for distinct half-bounding and galloping gaits as they emerge in the model.

Dimensionless parameter analysis and systematic numerical continuation reveal how changes in the system, such as the position of the torso’s center of mass, drive transitions between gaits. The work provides both a comprehensive method for generating all possible quadrupedal gaits in a given mechanical model and a mechanical explanation for why transitions between gaits occur as speed or morphology changes. This unified approach gives new insight into the mechanics underlying gait diversity and transitions in animals and robots, and provides tools for reference gait generation across a wide range of robot designs.

Reference

@article{ding2024breaking,
title={Breaking Symmetries Leads to Diverse Quadrupedal Gaits},
author={Ding, Jiayu and Gan, Zhenyu},
journal={IEEE Robotics and Automation Letters},
volume={9},
number={5},
pages={4782–4789},
year={2024},
doi={10.1109/LRA.2024.3384908}
}

Paper: “A Passive Dynamic Quadruped That Moves in a Large Variety of Gaits”

Authors: Zhenyu Gan and C. David Remy
Conference: IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS), 2014

This research introduces a conceptual passive dynamic quadruped model with elastic, massless legs, showing that a wide range of quadrupedal gaits can emerge purely from the system’s mechanics. The model incorporates three phases for each leg: stance, swing, and waiting for touchdown, and generates footfall sequences based solely on the initial conditions of the system. By varying these starting parameters, the authors identify trotting, pacing, walking, tölting, bounding, and galloping within the same model. Each of these gaits displays distinct ground contact patterns and stability characteristics.

The study uses numerical shooting and hybrid dynamical simulation to discover periodic motions and assess their stability. The results show that symmetrical gaits, such as trotting and pacing, arise naturally for certain parameter choices, while asymmetrical gaits, like bounding and galloping, require individualized leg parameters and initial conditions. Each gait is characterized by its footfall pattern, ground reaction force profile, and speed, and the research includes comparisons to real animal locomotion, such as horse gaits. Although all identified gaits in this fully passive system are dynamically unstable, the findings illustrate that mechanical structure and passive elasticity alone can generate the diverse gait repertoire observed in quadrupedal animals.

This work provides fundamental insights for the design of energy-efficient quadrupedal robots, highlighting how natural dynamics and compliance can be exploited for gait diversity without the need for active control.

Code and Source Files

The source code of our framework is available on GitHub: https://github.com/DLARlab/BreakingSymmetryLeadstoDiverseGaits.git

Reference

@inproceedings{gan2014passive,
title={A passive dynamic quadruped that moves in a large variety of gaits},
author={Gan, Zhenyu and Remy, C. David},
booktitle={2014 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS)},
pages={4876–4883},
year={2014},
organization={IEEE},
doi={10.1109/IROS.2014.6943192}
}