The Murphy number: how pitch moment of inertia dictates quadrupedal walking and running energetics

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.

Tennessee Walking Horse [12]. Given the few morphological differences between gaited and non-gaited breeds, 32 it seems less likely that natural populations are physically constrained from performing a four-beat run, and 33 more likely that they reject it (whether through behavioural, developmental or evolutionary programming; 34 e.g. [13]). 35 In a recent article, Usherwood resolved the paradox by considering the energy of pitching the body [14]. 36 Assuming ground-contact forces are axial to the leg, then foot contact in a four-beat gait induces pitching, 37 but a two-beat gait can avoid it. The question, then, is when do the energetics of pitching outweigh the 38 energetics of COM translation? When pitching energetics dominate, trotting should be optimal, and when 39 translation dominates, tölting should be. 40 Usherwood [14] showed that the ratio of translational to rotational kinetic energy is related to the distance. This dimensionless MOI (called hereafter the "Murphy number" for expediency and in honour 45 of its discoverer), is exactly the ratio of the change in translational to rotational kinetic energy imparted 46 to a free object by a generating impulse perpendicular to L (supplemental information). ForÎ < 1 more rotational energy is imparted than translational, and the opposite is true forÎ > 1 (figure 1).
cheaper. For very small Murphy numbers, the rotational term dominates, all the energy goes into pitching, 51 and a trot is cheaper. But whenÎ = 1, the cost of tölting and trotting are equal. In general, a four-beat run 52 is optimal whenÎ > 1, but whenÎ < 1, a two-beat run is optimal. 53 This insight might point to why some mammals deviate from a two-beat run at moderate speeds. of support is independent of MOI. Since the rotational speed is independent of MOI, pitching energy should 71 be proportional to MOI-not inversely proportional, as in running (figure 1). 72 We would therefore expect the Murphy number to have the opposite effect on the energetics of walking as 73 compared to running. At largeÎ, a two-beat walk should be favoured to avoid costly pitching at the expense 74 of larger COM collisions. At lowÎ, the optimal strategy should be to distribute contacts in a four-beat walk, 75 but switch to a pitch-free two-beat run at higher speeds-the common 4 → 2 pattern. 76 However, mammals that avoid two-beat running typically do not avoid four-beat walking; the walking An impulse P is generated at the hindlimbs and produces an equal change in center of mass velocity (blue arrow) and translational kinetic energy (E trans ) across all cases. (Upper row ) In four-beat running as Murphy number increases (left to right), the angular velocity (purple arrow) and rotational energy (E rot ) decrease. WhenÎ < 1, E rot > E trans and a two-beat gait should be favoured. (Bottom row ) In four-beat walking, the impulse from hindlimb transfer of support generates a reaction impulse at the forelimbs ( P R ) forÎ < 1, as in this condition the induced forelimb velocity change is downward (red arrow). Because of this, the angular velocity in a four-beat walk does not change as Murphy number increases. However, the rotational energy increases proportionally with MOI. A two-beat gait should be favoured for someÎ > 1 when E rot > E trans . ForÎ > 1, the impulse P causes a positive change in forelimb vertical velocity. However, if forces are not instantaneous, the forelimb can compensate by reducing its applied force, maintaining a constant pitch rate.
mization of a simple quadrupedal model with a work-based cost function. I also use published data to test 82 the hypothesis that Murphy number is a predictor of differences gait choice between quadrupedal mammals.  For a given parameter combination, the lowest-cost solutions were selected among all local minima dis-121 covered. The beat number was determined post hoc by looking at peak negative power during the stride.

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Defining a beat as peak negative power is consistent with the collisional gait perspective, which points to 123 mechanisms of energy loss and approximates them as impulsive events [7,27]. Setting normalized (negative) 124 power as where P tot is instantaneous net power from all actuators, the number of beats were the number of local 126 maxima in P Ntot (t) > 0.3 . If two maxima were less than 0.03 T apart, the greater maximum among them 127 was counted as a single beat. This method eliminated some noise while selecting only the largest events of 128 energy loss as a "beat". However, it is somewhat arbitrary and the shape of gait "zones" changes to some 129 extent depending on tolerances (see supplemental information for results using other tolerances).

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Limb contact for a given limb was defined as its GRF > 0.01 mg. Walking was defined as having duty   Figure 2a shows optimal gaits at parameter combinations ofÎ and U . Optimal gaits generally fall into 153 four large regions. At highÎ, four-beat runs and two-beat walks are optimal. At lowÎ, the reverse is true.

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While the cutoff between two-and four-beat runs is approximatelyÎ ∼ 1, as predicted by equation (2), the 155 transitionÎ for four-beat to two-beat walking increases from aboutÎ = 1 at the highest walking speeds to There is some debate whether the canter, also used by giraffes, is a distinct gait or merely a slow gallop. 4 As U ≈ 0.8 was the slowest running gait observed, and the authors did not report on walking gaits in that study, it is unclear if this is truly the transition speed. (e) At slow speeds and highÎ, a two-beat walk is optimal. (f ) As speed increases withÎ = 10, a four-beat "running" solution emerges, with single limb vaulting swapping between fore and hind limbs. (g) At higher speeds, the optimal gait is a hybrid between vaulting in hind and bouncing in front, reminiscent of the slow tölt [11]. (h) At still higher speeds, a typical fast tölt pattern emerges.
limit for a pace given by Hildebrand [12]. These observations of a four-beat gait are for an extremely slow 180 normalized speed (0.14 < U < 0.30), matching the region where the work-minimizing model predicts a 181 four-beat walk (figure 2a). Is there any evidence of giraffes using a two-beat walk at intermediate speeds?

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The walk of the giraffe has been described in two-beat terms, including "rack-like" [34] or as a pace [36].

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The solution is to perform a single-limb vault over forelimbs, then hindlimbs, and repeat this pattern. This 223 solution is feasible because the Murphy number is so extreme that the body barely pitches during single 224 stance, even though it is supported only at one end.

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As we increase Murphy number, we approach the limit where any pitching can be effectively ignored. In 226 this limit, we expect all gaits to be four-beat; it is analogous to a point mass biped with half the stride length.  [7]-, the net torque about the COM is appreciable and energetically costly. Furthermore, these non-pitching 254 gaits may be commonly used precisely because pitching would otherwise be extremely costly. Pitching may 255 be so important energetically, that the optimal solution is often to render it absent.

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Data Availability

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The dataset supporting this article has been uploaded as part of the supplementary material (