Group 1 animals (Figure 1, Table 1) walk with a duty factor around 0.8 and phase between 40% and 50%. This group consists of 11 species of reptile, 3 amphibians and 3 mammals. Members of this group might be linked by being relatively slow (e.g. hippo), slow-muscled (‘cold-blooded’ – consider tortoise muscle properties [Woledge, 1968]) or very small (mouse). If so, their high duty factors might be considered a strategy to limit the muscle activation cost due to instantaneous power demands (see [Usherwood, 2016]) (an argument related to, but distinct from, that proposed in (Jayes and Alexander, 1980) for tortoises). The issues of differential scaling between work and power has been covered elsewhere, and related to the scaling of: jumping performance (Bennet-Clark, 1977), gait selection and dynamic similarity (Alexander and Jayes, 1983), posture (Usherwood, 2013) and ontogeny of bipedal gaits (Hubel and Usherwood, 2015). It is sufficient here to note that it is not surprising that slow, slow-muscled and small walking quadrupeds operate with high duty factors; one aim of this paper is to account for why phases of 40–50% are adopted in this group.

Group 2 covers the majority of self-selected walking observed in upright, medium to large mammals: duty factors between around 0.6 and 0.75, and phases at or somewhat below 25%. A range of exceptions exists, notably some dogs [Hildebrand, 1968; Cartmill et al., 2002], which may adopt phases close to 0.5 (trotting) at higher speeds (lower duty factors). It appears likely that trotting mechanics can dominate at duty factors close to 0.5, equivalent to grounded running (Biknevicius and Reilly, 2006), and we do not attempt to include an account for this within the models of walking presented here.

Primates appear to adopt a discretely different limb phasing at the duty factors of Group 2, close to 75% ([Cartmill et al., 2002; Hildebrand, 1967] sometimes termed a ‘diagonal sequence’). Some other quadrupeds (opossum, [Schmitt and Lemelin, 2002]) occupy a similar region, potentially due to their grasping forelimbs and/or to cope with the hazards of walking along branches with a risk of failure (Cartmill et al., 2002).

Data published for 2-toed sloths locomoting with a suspended, under-branch quadrupedal gait (Nyakatura et al., 2010) cover a large range of duty factors. The relationship between duty factor and phase appears to contrast with that found across non-primate quadrupeds (Group 1 vs. Group 2): sloths adopt phases close to 25% at high duty factors (low speed), and 50% at lower duty factors (higher speeds).

There appears to be no record of a steady gait occurring at this phase in nature.

A numerical model was developed to calculate the mechanical work requirements of all the limbs on the center of mass as a function of duty factor and phase for symmetrical quadrupedal gaits. Mass-normalized vertical, horizontal and lateral forces for each limb were modeled with sinusoidal waveforms (see Figure 2). Vertically, a half-sine force profile ${\displaystyle F_{z,\mathrm{l}\mathrm{i}\mathrm{m}\mathrm{b}}}$ was used through time t over the stance period T_stance:

Figure 2

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where amplitude A_z is sufficient to oppose gravity g for a given duty factor DF ($\text{DF}=\raisebox{1ex}{$T_{\text{stance}}$}\!\left/ \!\raisebox{-1ex}{$T_{\text{stride}}$}\right.$, where T_stride is the period of a complete stride cycle):

Fore-aft forces were modeled with a full sine wave with bias from a half sine-wave:

where the amplitude ${\displaystyle A_x}$, as a proportion of the vertical amplitude, would relate closely to stance angle. The maximum leg angle from vertical ${\displaystyle \mathrm{\Phi }}$ can be approximated by considering an instant a quarter of the way through stance, assuming the combination of ${\displaystyle F_{z,\mathrm{l}\mathrm{i}\mathrm{m}\mathrm{b}}}$ and ${\displaystyle F_{x,\mathrm{l}\mathrm{i}\mathrm{m}\mathrm{b}}}$ to result in pure compression forces along the leg, and assuming the leg to sweep over the foot at a constant angular rate:

${\displaystyle \mathrm{\Phi }\approx 2{\tan }^{-1}\left(\frac{A_{\mathrm{z}}\sqrt{2}}{A_{\mathrm{x}}}\right)}$A single value of ${\displaystyle A_z/A_x}$ (so a single stance angle) is used; we take no account of any variation in stance angle or stance length with duty factor or speed.

Fore-aft force profiles were supplemented with a bias (shown in the ${\displaystyle A_{x,\mathrm{b}\mathrm{i}\mathrm{a}\mathrm{s}}}$ term of Equation 3) in which the hindlimbs provide a net forward impulse, counteracted by an equivalent backward impulse from the forelimbs. Finally, small medial forces ${\displaystyle F_{y,\mathrm{l}\mathrm{i}\mathrm{m}\mathrm{b}}}$ were introduced, applied equally and medially by each leg. Both of these latter additions follow the half-sine waveform of the vertical force, but with much lower amplitudes (10% and 5% of vertical respectively). At this stage, these values merely serve to demonstrate tendencies; while their influences can easily be modeled, we do not explore this aspect of the parameter space quantitatively here as we have limited data on how they might vary with size, species, speed, duty factor etc.

From the limb force profiles and timings relating to limb phasing, fluctuations in center of mass velocity $\overrightarrow{V_{\text{CoM}}}$ were derived by integrating the sum of (mass normalized) limb forces through time (and then mean subtracting to give fluctuations about zero). Assuming a zero mean velocity vertically and laterally, and a positive mean forward velocity over a stride cycle, center of mass velocities were calculated. Note that the value of mean velocity, as long as sufficient to keep instantaneous fore-aft velocities always forward, has no bearing on the model results. The power demanded from each limb is given by:

and the positive components of P_limb – for all legs – provide the mechanical work ‘cost’. Note that the instantaneous limb power is the sum of all the components of the right hand side: a positive power due to a vertical force and velocity at the same time as a negative power due to a horizontal force and velocity would cancel; a simple vaulting or pendulum action would be calculated as passive and demand no limb power.

Our approach assumes that the impulses from each limb – and the timing of these forces – act independently from the limb phasing; and that center of mass motions are due to the action of these limb forces. We assume that ‘limb work’ is costly – that negative work being performed by one limb cannot power the positive work being performed by another limb at the same time (as would be the assumption if only center of mass or ‘external’ work was being considered). In other words, simultaneous positive and negative power of equal magnitudes from different limbs would not change the mechanical energy of the centre of mass (no ‘external work’ would be performed) but would require mechanical power from a limb, and so impose a cost according to the current model. Further, all positive work is considered costly, with no capacity for recovery of negative work through elasticity. While a variety of cost functions may be important (peak power, muscle force etc.) – as discussed in (Usherwood, 2016) – we assume here that it is the sum of mechanical limb work – or a direct correlate thereof (which may include peak instantaneous power) – that is relevant to the animal.

While this method has much in common with the pioneering analytical work of Alexander from the 1980’s (Jayes and Alexander, 1980; Alexander and Jayes, 1980; Alexander, 1989), advances in computer capability now allow additional parameter spaces – including mediolateral and fore-aft force fluctuations – to be explored numerically without demanding detailed mathematical derivations.

Our approach to studying work-minimizing strategies does not directly address the motivations behind the selection of stance angles, duty factors, fore-aft or lateral impulses, or details of force profiles more subtle than the sine-wave approximations use here. Clearly, many of these aspects may be related to work minimization provided the constraints of animal form; however, details may also relate to other geometric constraints - such as maximum allowable pitch or roll deviations as proposed by Jayes and Alexander for tortoises (Jayes and Alexander, 1980). Further, it should be emphasized that the approach taken here does not include any interplay between limb phasing and impulses or timing of forces through the stance; it is therefore incapable of approaching some aspects of passive vaulting or falling mechanics generally considered important in walking at above moderate speeds. Our approach therefore has limitations: many aspects of both energetics and stability (if indeed these two aspects should be separated) remain to be explored (e.g. [Wilshin et al., 2017]).

An ideal rolling wheel can support the weight of a vehicle moving over level ground without performing mechanical work because the forces opposing weight are orientated perpendicular to the velocity of the mass. The same is not true when driving up hill: some component of body weight is in the direction of velocity, and an engine is required to deliver power. The value of this principle, that the orientation between the force (or impulse) and velocity accounts for the demand for mechanical power (or work), can be couched in terms of collision mechanics. This reduction is becoming to be appreciated in the study of human and animal mechanics (e.g. Kuo, 2002; Ruina et al., 2005; Lee et al., 2011; see Bertram, 2016 for an overview). We use the principle here to provide tractable, intuitive accounts for the outputs of the numerical models described above. We link the outputs of the numerical, sine-based models to the principles of collision mechanics by highlighting the angles between center of mass velocities and limb forces at key instants, or the net effect of limb forces in the form of impulses. This approach to explaining the principles underlying differences in mechanical work demand supports the case that the mechanisms found are reasonably general and not merely a mathematical peculiarity of the specific sine-based assumptions of the model.

When approaching work calculations from a collisional perspective, the angle between the limb force (or impulse) and center of mass velocity is critical. It should be highlighted that, with multiple limbs applying forces to the center of mass at the same time, the relationship between limb forces and changes in center of mass velocity is not constant, and may not be intuitive. The geometry provides equal insight in terms of both the mechanical work ‘lost’ or ‘demanded’ – all gaits considered here balance the negative and positive works over a cycle. In the discussion, we generally focus on the mechanisms influencing the loss of energy; however, the process of returning this energy presumably dominates the physiological cost.

The simplest form of the model (Figure 2A) has vertical limb forces of a half sine-wave of sufficient amplitude to support body weight (increasing with decreasing duty factor) and horizontal forces of a sine-wave of a suitable amplitude – we choose a value of −0.2 the vertical amplitude, approximately equivalent to a stance sweep-angle of 30 degrees. All results are broadly insensitive to this assumption; systematic changes in stance angle with speed or duty factor are not included, but in any case are negligible after the normalization at each duty factor. The implications of duty factor and phase can be calculated in terms of limb work, and presented as a cost surface, normalized for each duty factor (blue minimum, red maximum for each duty factor). The surface shows work minimization at phases of 50% and 0/100% at duty factors above 0.75, and 25%/75% at duty factors below 0.75. The DF = 0.75 cut-off is not exact; it is slightly higher with increasing horizontal force amplitude – or stance angle.

At low duty factors (0.5 < DF < 0.75), the vertical forces produced by the legs relate closely to the vertical forces experienced by the center of mass (see Figure 3, which shows a 50% phasing). In this situation, center of mass velocity is forward and downward in the first half of stance (Figure 3C; Figure 4A), while it is met with an upward and decelerating force, resulting in negative work. The angle between the velocity and force vectors would be improved – made more nearly perpendicular – with a lower magnitude of vertical velocity. This would be achieved by distributing the vertical impulses evenly through time through using a 25% or 75% footfall phasing – benefits analogous to having more spokes on a rimless wheel. In contrast, at duty factors above 0.75, there is sufficient overlap for the forces produced by the limbs to combine such that sub-maximal vertical forces on the center of mass occur when an individual limb is maximally loaded. This results in a reversal in timing of upward and downward velocities compared with lower duty factors (Figure 3), at which point higher magnitudes of vertical velocity are favorable, as they increase the angle between center of mass velocity and limb force vectors (Figure 4A), reducing the limb power. This principle is analogous to that in bipedal walking with vaulting, stiff-limbed ‘inverted pendulum’ mechanics. The beneficial, higher vertical velocity magnitudes at the beginning of stance are achieved with synchronous footfalls – phases of 0% or 50% (pace or trot).

Figure 3

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Figure 4

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Suspended quadrupedal walking, as typical for sloths and used facultatively in a range of primates, provides a useful contrast. In this case, the sense of the horizontal forces is reversed (Figure 2B). In early stance each limb – acting predominantly in tension instead of compression – provides an accelerating force, and decelerates over the second half of stance (confirmed by force measurements in lemurs, [Granatosky et al., 2016]). With this difference, force-velocity angles are improved (Figure 4B) and work minimization is achieved with a reversed duty factor/phase relationship: high duty factor (slow) gaits benefit from low vertical velocity magnitudes and distributed foot timing; low duty factor (faster) gaits benefit from higher vertical velocity magnitudes and synchronous (trotting or pacing) foot contact phasing. This relationship appears to be supported by observations of locomoting sloths (Nyakatura et al., 2010).

The extension of the model to include net foreword impulses from the hind limbs and rear-ward impulses from the forelimbs (Figure 2C) is achieved with a fore-aft bias of an additional half sine-wave (positive for hindlimbs; negative for fore) of an amplitude 0.1 that of the vertical. The cost surface is shifted towards lower phases, with low-cost regions becoming <50% (100%) at high duty factor and <25% (75%) at low duty factors. The extent of this shifting increases with higher bias amplitudes.

The mechanism underlying this shift can again be described using the collisional framework (Figure 4C). In this case, the net impulse from each limb is treated as a single vector, orientated slightly forward for hindlimbs and backward in forelimbs. With this inclination the angles between velocities and impulses are most favorable – most near perpendicular – with a reduced interval between hind and forelimb contacts. This is the principle described by Ruina et al. (Ruina et al., 2005) to account for the ‘gathered’ gallop of horses, and is related to a previously speculated mechanism for walking dogs (Usherwood et al., 2007).

Up to this stage of model development, everything could be treated in a planar manner: there is no difference between left and right, and phases ± 50% are equivalent. The final addition to the model is small medial impulses (as generally reported, when measured ((e.g. dogs: [Griffin et al., 2004]) and considered of relevance) produced by a half sine-wave of amplitude 0.05 that of vertical (Figure 2D). The effect of this addition on the cost surface it to tip the low phases to be less costly; and make phases > 80% uneconomical. The cost at the troughs (blue regions) for each duty factor become slightly different; the dark blue line in Figure 2D denotes the global limb work minimum for each duty factor.

The mechanism underlying this change can again be related to collision mechanics (Figure 4D), this time considering only the horizontal plane. Each hindlimb produces an impulse that accelerates the body both forward and medially; each forelimb provides backward and medial impulses. Figure 4D demonstrates the difference between low and high phases: with low phasing, each impulse acts more closely to perpendicular to the center of mass velocities, and so lower mechanical limb work is required. With the addition of medial impulses, the undesirability of a 95% phase becomes apparent. Just prior to front-right foot contact, the center of mass velocity is a near-maximal downward, forward and lateral, before being opposed by an impulse orientated upward, backward and medially: much of the kinetic energy of the body cannot be maintained with small-angle collisions. This phenomenon, and the importance generally of timing on the energetic consequences of impulses, can be experienced directly. A human can be asked to crawl with a pacing footfall pattern – this is usually achieved with ease. A phasing just off pacing, with the knee landing just before the hand, is also easily achieved with comfort (phase roughly 5%). However, the slight change the other way, with the right hand landing just prior to the right knee (phase around 95%) is physically very demanding, and very challenging to maintain for any duration.

Both the limb-force-driven model and the geometric description of mechanism provide energetic accounts for selection of limb phasing in many quadrupeds undertaking slow locomotion. Phases somewhat below 50% are predicted at ${\displaystyle DF\approx 0.8}$; somewhat below 25% at ${\displaystyle DF\approx 0.65}$, with the reverse relation predicted for suspended sloth-like quadrupedal gaits. This provides a simple work-minimizing account for commonalities in quadrupedal gaits (Figure 2C) across extreme phylogenetic distance (Group 1 – Figure 1 – includes species that last shared a common ancestor around 300 million years ago), considerable difference in scale (Group 1 covers from hippo to mouse – a 500,000-fold difference in mass) and contrast in muscle properties (very ‘slow’ (tortoise [Woledge, 1968]) to ‘fast’ (mouse, e.g. [Askew and Marsh, 1997])). Group 2 (Figure 1), consisting of both familiar and exotic medium to large mammals walking at lower duty factors, approaches the even footfall or slightly lower phases predicted from simple planar collisional principles (Ruina et al., 2005; Usherwood et al., 2007). The advantage in terms of work minimization of near-25% vs. near-75% phase only becomes apparent when medial impulses are included (Figure 2D; Figure 4D).

A purely energetic account is not found for the unusual phases used by many primates (and opossum, [Schmitt and Lemelin, 2002]); however, the cost landscape is such that the phases observed, while not meeting the absolute energetic minimum, may be in a region that achieves a nearly equivalent performance (see also [Sellers et al., 2013]). Some other account (perhaps associated with managing locomotion on unreliable substrates – [Cartmill et al., 2002]) is presumably required but beyond the scope of this study. Further support for the explanatory power of the model comes from cases where duty factors transitions over 0.75. The sloth data set has already been discussed. The case of a macaque through ontogeny ([Hildebrand, 1967]; a generally consistent theme appears to hold across a range of primates, [Hurov, 1982]) also appears to fit model predictions very nicely (Figure 5). When very young, the macaque used a very high duty factor, and phase just below trotting. As it matured, it adopted lower duty factors and both horse-like and primate-like phases, but avoided the previous phase. As adult, it adopted a purely adult primate-like phase. As discussed above, the models presented here cannot account for the adult primate phase, but can account for the avoidance of phases around 40% at low duty factors.

Figure 5

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The driver behind the high duty factors of ‘reptile’, slow, small or juvenile gaits, while potentially related to the avoidance of excessive muscle activation for power demands, is not covered here. However, given high duty factors, phasing around 0.4 to 0.5 is consistent with limb work minimization. This provides an alternative account for the observation of these near-trotting (i.e. 0.4 to 0.5) phases in human infants (Patrick et al., 2009; Patrick et al., 2012): where previously they have been attributed to some immaturity in neural development (Patrick et al., 2012), they may actually be adaptive from a work minimization perspective. This has clear parallels with the contrasts between toddler and adult walking and running gaits (Hubel and Usherwood, 2015): might it be that we would do better viewing ontogeny of gait – whether quadrupedal or bipedal – first from an adaptive energetic framework before invoking some constraining limitation in the rate of neural development?

The funders had no role in study design, data collection and interpretation, or the decision to submit the work for publication.

The support and contributions of many members of the Structure and Motion Lab, past and present, are greatly appreciated; in particular, Tatjana Hubel, Vivian Allen, Sandy Kawano, and Andrew Spence (see [Spence et al., 2013]) for sharing their videos of a range of species (see Supplementary ile 1).

1. Received: June 9, 2017
2. Accepted: August 28, 2017
3. Version of Record published: September 14, 2017 (version 1)

© 2017, Usherwood et al.

This article is distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use and redistribution provided that the original author and source are credited.