Breaking Symmetries Leads to Diverse Quadrupedal Gaits

Symmetry manifests itself in legged locomotion in a variety of ways: A legged system can maintain consistent gaits from any spatial starting point, exhibiting the same leg movements on either side of the torso in phase, and some even demonstrate forward and backward movements so similar they seem to reverse time. This work aims to generalize these phenomena and proposes formal definitions of symmetries in legged locomotion using terminology from group theory. In this research, we uncovered an intrinsic connection among a broad spectrum of quadrupedal gaits, which can be systematically identified via numerical continuations and distinguished by elements within a symmetry group. These gaits, within the hybrid dynamical system, are not merely isolated movements but part of a continuum, seamlessly transitioning from one to another at precise parameter bifurcation points. Altering specific symmetries at these junctures leads to the emergence of distinct gaits with unique footfall patterns, a phenomenon we've generalized through dimensional analysis in this study. Consequently, each gait manifests distinct preferred speed ranges and specific transition speeds. This work offers a comprehensive method to solve the gait generation problem for a quadruped, including pronking, two types of bounding, four variations of half-bounding, and two forms of galloping, and it also elucidates the mechanical rationale behind the necessity of gait transitions, providing high-level insight into the diversity and underlying mechanics of quadrupedal locomotion.