Most quadrupedal mammals transition from a four-beat walk to a two-beat run (e.g. trot), but some transition to a four-beat run (e.g. amble). Recent analysis shows that a two-beat run minimizes work only for animals with a small pitch moment of inertia (MOI), though empirical MOI were not reported. It also remains unclear whether MOI affects gait energetics at slow speeds. Here I show that a particular normalization of the pitch moment of inertia (the Murphy number) has opposite effects on walking and running energetics. During walking, simultaneous fore and hindlimb contacts dampen pitching energy, favouring a four-beat gait that can distribute expensive transfer of support. However, the required pitching of a four-beat walk becomes more expensive as Murphy number increases. Using trajectory optimization of a simple model, I show that both the walking and slow running strategies used by dogs, horses, giraffes and elephants can be explained by work optimization under their specific Murphy numbers. Rotational dynamics have been largely ignored as a determining factor in quadrupedal locomotion, but appear to be a central factor in gait selection.

中文翻译：

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墨菲数：俯仰惯性矩如何决定四足步行和跑步的能量学

大多数四足动物从四步走转为两步走（例如小跑），但也有一些过渡到四步走（例如小跑）。最近的分析表明，两次节拍运行仅对具有较小俯仰惯性矩（MOI）的动物最大程度地减少了工作，尽管没有经验性MOI的报道。MOI是否会以慢速影响步态能量还不清楚。在这里，我证明了惯性俯仰力矩（墨菲数）的特定归一化对步行和跑步的能量学有相反的影响。在步行过程中，前肢和后肢同时接触会减弱俯仰能量，有利于四拍式步态，这种步态可以分配昂贵的支撑传递。但是，随着墨菲数的增加，四拍步行所需的俯仰变得更加昂贵。使用简单模型的轨迹优化 我表明，狗，马，长颈鹿和大象使用的步行和慢跑策略都可以通过在特定的墨菲数下进行工作优化来解释。作为四足运动的决定因素，旋转动力学已被很大程度上忽略，但似乎是步态选择的中心因素。