Breaking Symmetries Leads to Diverse Quadrupedal Gaits

TL;DR

The paper addresses how symmetries govern quadrupedal gaits and how breaking these symmetries generates diverse locomotion patterns. It introduces a group-theoretic framework for classifying gaits as periodic orbits of a hybrid quadrupedal system, distinguishing temporal, spatial, and morphological symmetries, and uses a simplified floating-base/SLIP model to illustrate symmetry-breaking bifurcations. Key contributions include formal definitions of gait symmetries, demonstration of interconnections among pronking, bounding, half-bounding, and galloping via energy and COM-position parameters, and a dimensionless analysis that highlights robust bifurcation pathways. The results provide theoretical guidance for gait selection and lay a foundation for efficient, symmetry-aware controllers in quadruped robotics.

Abstract

Symmetry manifests itself in legged locomotion in a variety of ways. No matter where a legged system begins to move periodically, the torso and limbs coordinate with each other's movements in a similar manner. Also, in many gaits observed in nature, the legs on both sides of the torso move in exactly the same way, sometimes they are just half a period out of phase. Furthermore, when some animals move forward and backward, their movements are strikingly similar as if the time had been reversed. This work aims to generalize these phenomena and propose formal definitions of symmetries in legged locomotion using group theory terminology. Symmetries in some common quadrupedal gaits such as pronking, bounding, half-bounding, and galloping have been discussed. Moreover, a spring-mass model has been used to demonstrate how breaking symmetries can alter gaits in a legged system. Studying the symmetries may provide insight into which gaits may be suitable for a particular robotic design, or may enable roboticists to design more agile and efficient robot controllers by using certain gaits.

Figures (6)

• Figure 1: (A) A1 quadrupedal robot from Unitree Robotics with bilateral symmetry and (B) a simplistic spring-mass model.
• Figure 2: This figure illustrates three prevalent types of symmetries of a quadrupedal robot: (A) Time-Reversal Symmetry (a specific instance of Temporal Symmetry) denoted as $G_{\psi}$, wherein reversing all velocities yields a backward-moving gait. (B) Spatial Symmetry denoted as $G_\xi$, which corresponds to 2-dimensional Euclidean isometries within $SE(2)$ that involve translations and rotations of the robot on the moving plane. (C) Morphological Symmetry denoted as $G_{\sigma}$, representing permutations of leg motions that do not alter the periodic movements of the robots.
• Figure 3: Gait branches of pronking forward (PF), bounding with gathered suspension (BG), and bounding with extended suspension (BE) on the Poincare section with the torso's horizontal velocity $\dot{q}_x$ as the horizontal axis and pitching velocity $\dot{q}_{\text{pitch}}$ as the vertical axis. Examples of each gait (a-c) are shown as successive keyframes at the touch-down, lift-off moments, and the apex at the bottom. Black feet are used to highlight the legs in stance.
• Figure 4: The figure displays bounding gait branches for asymmetrical models where $l_{b,H} \neq 0$. Two specific cases, $l_{b,H}=0.40~[l_o]$ and $l_{b,H}=0.60~[l_o]$, are highlighted, with their corresponding solution branches illustrated in orange and red, respectively. Solid lines represent bounding gaits with gathered suspensions (BG), while dashed lines depict bounding gaits with extended suspensions (BE). In contrast to the bounding gaits of the symmetrical model shown in Fig. \ref{['fig:PronkBound']}, our numerical analysis indicates that as the COM shifts closer to the rear, BG manifests at intermediate speeds (d). In contrast, when the COM is nearer to the front at $l_{b,H} = 0.60~[l_o]$, BE remains the exclusive bounding gait.
• Figure 5: This figure illustrates the gait branches of half-bounding as identified from the proposed model. Since symmetry breaking can occur in either the front or hind leg pair, during the numerical search, a total of four such gait patterns were discovered from the two bounding gaits (BG and BE): (f) and (g) demonstrate half-bounding gaits with gathered suspensions; (h) and (i) illustrate half-bounding gaits with extended suspensions.

Theorems & Definitions (6)

• Definition 1: Gait
• Definition 2: Symmetry
• Definition 3: Temporal Symmetry
• Definition 4: Time-Reversal Symmetry
• Definition 5: Spatial Symmetry
• Definition 6: Morphological Symmetry