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.

翻译：对称性以多种方式表现在腿部运动中。无论腿部系统从何处开始周期性运动，躯干和四肢都会以相似的方式协调彼此的运动。此外，在自然界观察到的许多步态中，躯干两侧的腿部以完全相同的方式运动，有时只是相位相差半个周期。更重要的是，当某些动物向前和向后运动时，它们的动作惊人地相似，仿佛时间发生了逆转。本研究旨在概括这些现象，并利用群论术语提出腿部运动中对称性的正式定义。讨论了一些常见四足步态（如跳跃步、跳跃步、半跳跃步和疾驰步）中的对称性。此外，使用弹簧-质量模型展示了打破对称性如何改变腿部系统的步态。研究对称性可能有助于理解哪些步态适用于特定的机器人设计，或使机器人专家能够通过使用某些步态设计出更敏捷、更高效的机器人控制器。