The Veterinary Journal 198 (2013) e3–e8
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Gait as solution, but what is the problem? Exploring cost, economy and compromise in locomotion John E.A. Bertram ⇑ Faculty of Medicine, University of Calgary, 3330 Hospital Drive NW, Calgary, Alberta T2N 1N4, Canada
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Keywords: Locomotion Gait
a b s t r a c t Many studies have examined how legged mammals move, defining ‘what’ happens in locomotion. However, few ask ‘why’ those motions occur as they do. The energetic and functional constraints acting on an animal require that locomotion should be metabolically ‘cost effective’ and this in large part determines the strategies available to accomplish the task. Understanding the gaits utilised, within the spectrum of gaits possible, and determination of the value of specific relationships among speed, stride length, stride frequency and morphology, depends on identifying the fundamental costs involved and the effects of different movement strategies on those costs. It is argued here that a fundamental loss associated with moving on limbs (centre of mass momentum and energy loss) and two costs involved with controlling and replacing that loss (muscular work of the supporting limb during stance and muscular work of repositioning the limbs during swing) interact to determine the cost trade-offs involved and the optimisation strategies available for each species and speed. These optimisation strategies are what has been observed and characterised as gait. Ó 2013 Elsevier Ltd. All rights reserved.
Introduction Gait is a pattern of movement that ultimately allows the animal to move effectively across a substrate. A wide variety of movement patterns could result in locomotion, but a few distinctive patterns (and variations on each of these main themes) are almost universally used by quadrupeds. Historically from as early as Aristotle (ca. 350 BC) (Goiffon and Vincent, 1779; Marey, 1873; Hildebrand, 1965; Abourachid, 2003; Farquharson, 2007) and by tradition in equine practice, gait has been characterised largely on the basis of footfall sequence. This is a relatively easily observed and quantified aspect of animal locomotion, especially with modern analytical technology, such as high speed video, and newly emerging mathematical techniques, such as linear discriminant analysis (Robilliard et al., 2007). Although detailed descriptions of footfall patterns can be used to define gaits and to document differences among patterns used by different species in various circumstances (or even different individuals), such characterisation does not inform us of the functional purpose of each gait or provide information on why each exists as it does. The commonly observed, or ‘normal’, gaits are the ones that emerge from the range of possible gaits because they allow for metabolically cost-effective movement under the circumstances ⇑ Tel.: +1 403 2109857. E-mail address: [email protected] 1090-0233/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.tvjl.2013.09.025
in which the animal operates. For example, each of the preferred gaits of horses is selected and utilised to provide the least cost of transport over a certain range of speed, with the walk being used at slow speeds, the trot at intermediate speeds and the gallop at high speeds (Hoyt and Taylor, 1981). The operational conditions of locomotion and the appropriate movement strategies available are predominantly influenced by the speed of progression, although other factors from the physical properties of the substrate, such as stiffness (Wilson and Pardoe, 2001) and slope (Self et al., 2012), also have an influence. It is generally assumed that the gait pattern observed under normal circumstances is the only one available, potentially programmed by a neural control system originating in ancient coordination centres of the brain and spinal cord (Grillner, 1975; McCrae and Rybak, 2008). However, evidence from analysis of human gait selection (humans are a reasonably tractable species sometimes willing to adapt to useful experimental conditions that other species resist), indicate that the control of key gait parameters, such as the relationship between speed, stride frequency and stride length, are largely determined by the metabolic cost associated with each option (Bertram and Ruina, 2001). This results in the normal and spontaneous selection of quite unusual gait parameters under experimentally imposed circumstances, such as moving to a specified frequency or stride length. These gait parameters are selected in a manner that minimises the cost of locomotion under each set of specific conditions (Bertram, 2005; Snaterse et al., 2011; Fig. 1). The brain and spinal
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Fig. 1. Diagram showing how gait parameters, such as speed, step length and step frequency, are selected in human walking to minimise energetic expenditure under the circumstances the system operates within. The light blue contours represent the metabolic cost surface, where each contour has greater magnitude than the one inside it (minimum cost occurs at the central circle where the three linear relationships cross). For an individual walking at a determined speed (constant level on the y-axis), such as on a treadmill, minimum cost will occur using the step frequency where a horizontal line from the speed axis touches the innermost cost contour it can. The speed–frequency relationship for walking on a treadmill at various speeds (red line) will be a series of such horizontal contours. Step frequency (dark blue line) and step length (green line) cost minimisations will be determined in a similar manner for restricted frequency or step length conditions, except that the line determining each will intersect the cost surface in a different location (and therefore determines a different optimisation solution).
centres play a critical role in operating the locomotory apparatus, but this role should be considered in the same way as the critical role of the driver in operating a motor vehicle. The driver is capable of steering the vehicle anywhere, but a good driver remains on the roadway and operates according to the traffic regulations. To do otherwise results in less than optimum performance. When considering the best operation of the limbs in animal locomotion, a major challenge is to discover the ‘road map’ and ‘driving regulations’ that the central control centre must integrate with; these will be dependent on the physical interaction of the organism with its environment and the energetic consequences of that interaction. Understanding why footfall patterns appear as they do requires an understanding of the energetic consequences of the motions and comparison of these with available options. This may appear to be a difficult task, given that mammals are such complex systems, using complex biological actuators (muscles) organised in often redundant functional networks (thus providing multiple control options for any given movement and apparent ‘cost’). However, substantial insight into the purpose of gaits can be gained by identifying the main determinants of locomotion cost.
Energetic consequences of gait As much as we are all familiar with the fact that locomotion using legs involves substantial energetic cost, it is not as apparent why this must be so. Flying and swimming require much less metabolic investment per distance travelled (Tucker, 1970). The high cost of terrestrial locomotion is most commonly attributed to the negative work expended during the stride cycle when moving on a solid surface. It is possible to observe and verify that negative work occurs in legged locomotion, where the kinetic energy of the organism’s moving mass is expended as work on the muscles of the animal’s limbs instead of the muscles moving the mass of
the animal by producing positive work. Negative work, however, turns out to be just one of many mechanisms through which mechanical energy can be lost from the system (where the organism is the ‘system’ of interest). The process of energy loss actually depends on a far more fundamental, but largely neglected, feature of legged locomotion: collision-based loss caused by the interaction of the moving mass and its solid support. When the term ‘collision’ is mentioned, it is common to think of violent crashes between objects such as automobiles. However, in dynamics, a collision is defined simply as a discontinuity of the velocity (magnitude or direction) of a travelling mass. Any such discontinuity requires the application of force and will involve the transfer or loss of energy. Within the precision of dynamics, the term ‘discontinuity’ also has a specific meaning, where it is limited to an instantaneous event. In the real world, instantaneous events require infinite forces, so they do not exist. However, although such events occur more slowly in reality than in theory, there are some aspects of collision theory that do translate to energetic consequences in the real world of locomotion. As a simple example that most of us can relate to, consider a playground swing. A common playground swing is an example of an active pendulum. Much of the swinging motion is maintained through the spontaneous (and passive) exchange between potential energy (at the highest point of the swing) and translational kinetic energy (at the lowest point of the swing). Children learn early that they are able to contort their bodies in registry with the swinging motion to add momentum (and energy) to the swing; this is usually called ‘pumping’ (Wirkus et al., 1998). However, the height of the swing can be increased in this way only so far. As the swing moves beyond horizontal, a small jolt is felt on the down swing, following which the height of the next swing is much reduced (indicating that energy has been extracted from the system). The jolt that is felt indicates a small collision caused by the mass of the individual falling within the arc of the swing and being forced to change direction (a discontinuity in its path) as the supporting cables of the swing become taut. The energy loss occurs because the radial component of the falling centre of mass (CoM) velocity is directed along the supporting cables and is lost to the collision interaction (Fig. 2). A common acrobatic circus act involves demonstrating that a full 360° swing revolving completely around the rotational axis can be accomplished. Understanding how this can be done in a circus, when it cannot be done in the playground, indicates the value of considering the phenomena responsible for energy loss, since it affects functional capability so directly. Success in the full rotation of the circus swing depends on a single adjustment of equipment (a change of ‘morphology’), replacing the flexible supporting cables of the playground swing with rigid struts that do not flex when the individual swings higher than the horizontal. With rigid struts, the CoM is prevented from falling vertically, so is forced to remain on the swing arc regardless of position in the rotation. Without a vertical fall, there is no jolt and therefore no collision loss, so the energy state of the system can increase continuously until enough has accumulated to allow the mass to move completely around the axis. In the normal playground swing, the higher the swing above horizontal, the greater the jolt and the greater the loss.
Deflecting the centre of mass in legged locomotion Gaits differ, but there are features that all the gaits have in common. Most obviously, they all involve moving the mass of the animal across the substrate. Cyclic vertical motions occur while the animal moves forward at speeds that vary systematically over the period of the stride. To a much lesser extent, lateral movements also occur. The interaction of the forward and vertical movement is
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Fig. 2. Collision loss involved in a playground swing. If an individual on a playground swing swings above the horizontal, the centre of mass (CoM) will fall vertically. When the supporting cables again become tight, the individual will experience a small jolt and the next swing will reach a much lower height. The small jolt extracts the radially oriented velocity, and consequently that component of the kinetic energy. Deflecting the CoM when a limb contacts the ground surface results in a similar ‘collision’, with losses determined by the geometry of the velocity and constraining force applied by the contact limb.
a key factor in determining the energetic cost of legged locomotion, so the remainder of this discussion will neglect the lateral motions. Although the mass of the animal continues travelling forward in locomotion, the cyclic changes in height of the CoM requires that the vertical component of its direction change through the stride cycle: from forward and upward to forward and downward, and from forward and downward to forward and upward. These changes of CoM height are natural aspects of all gaits, but they are key factors in determining the consequences of any gait strategy. In steady state locomotion (constant average horizontal speed), changes in the height of the CoM as the animal moves forward mean that there are two important portions in each stride: (1) a portion of low energetic cost; and (2) a portion of high energetic cost. The low cost portion occurs when the CoM direction changes from moving upward to moving downward. This direction change involves little energy cost to the animal because it occurs spontaneously as a result of the passive action of gravity. The high cost portion occurs when the CoM direction changes from moving downward to moving upward. This direction change must be mediated by the action of the limbs and involves high cost because it is during this phase that energy can be lost and/or work must be performed by the limb. The footfall patterns that are recognised as gaits are employed to allow for the downward to upward direction change of the CoM, while limiting the net work that is required to maintain consistent average forward speed. Different footfall strategies (gaits) will suit different circumstances of locomotion, largely determined by the speed of forward travel. Understanding why each gait is used requires understanding how energy is lost from the system and what strategies are available to minimise and replace that loss. If animals could travel across the substrate with virtually no loss, like a wheel rolling, it would not be necessary to add much energy to each step. It is easy to recognise that animals do not roll like a wheel, but it is not as easy to identify why legged locomotion involves energy loss. This is because animals have developed
sophisticated strategies to limit energy loss as much as possible, and these strategies camouflage many aspects of the processes involved. Recognising the factors involved, and the strategies available, to limit energy loss within the stride cycle provides substantial insight into why many of the motions observed occur as they do and why others that we might imagine do not. The contact of each limb is much like the contact of an individual spoke of a wheel that does not have a rim to roll on. The key loss occurs with each contact and is basically equivalent to the jolt felt in the playground swing that travels too high; it is derived from the collision-like deflection of the CoM as the limb strut applies force to the ground (along its axis), constrains the CoM path (to vault over the contact point) and shifts the CoM direction from forward and downward, prior to contact, to forward and upward. If the limb simply acted as a rigid strut, then locomotion would involve violent transitions and large amounts of energy would be lost at each contact. Maintaining steady forward speed requires the replacement of all lost energy, so the cost of locomotion in this case would be very high, involving both large energy losses and the cost of work required to repay the loss. Instead, animals have developed strategies to limit this loss and to allow them to travel for relatively low overall energetic investment. These strategies involve decreasing the loss involved in the high cost portion of the gait cycle, while increasing the travel accomplished in the low cost portion. There is a trade-off between these strategies and the active work that must be invested to implement them. An understanding of the movement patterns that we define as gaits requires identifying the factors involved in the optimisation solution and recognising the features and consequences of the trade-offs involved. Reducing loss in the high cost portion of the cycle: Walking and galloping Although they appear to be quite different gaits, walking and galloping are both strategies that primarily limit the energy loss
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associated with the transition in the CoM from moving forward but downward to forward and upward (the high cost portion of the stride cycle). Two aspects of this transition contribute to the energetic cost. The first is the collision-based energy loss that occurs when the limb contacts the substrate and deflects the animal’s mass. This loss could be substantial, so it is not surprising that strategies have been developed to limit the magnitude of loss. Some of these strategies are subtle and non-intuitive, even though they are effective. In walking, collision-based loss occurs when the leading limb in the transition process contacts the substrate and deflects the animal’s mass. This loss must be replaced in order to maintain constant average speed; this requires thrust from the trailing limb. However, the sequence of action of the limbs (deflection then thrust vs. a pre-emptive thrust then deflection), substantially alters the net energetic cost. This is because the geometry of the collision event differs between the two alternatives, even if everything else remains the same. This is most easily seen in the more straightforward geometry of bipedal walking (McGeer, 1990; Donelan et al., 2002; Collins et al., 2005; Ruina et al., 2005; Kuo, 2007; Fig. 3A), but the same process occurs in quadrupedal gaits (Birkemeier, 1998; Usherwood et al., 2007; Lee et al., 2011). The key factor to recognise in understanding this strategy is that only that portion of the CoM velocity vector aligned with the contact is involved in the collision; for the playground swing, this was the radial portion of the child’s total velocity, while in legged locomotion it is the trigonometric portion aligned with the axis of the limb. The collision loss associated with deflecting the CoM is largely dependent on the geometric relationship between the two dynamic features of the motion that interact: the velocity vector representing motion of the animal’s mass and the vector representing the net ground reaction force (which contributes to the impulse responsible for altering the direction of travel of the CoM, as described above). Since only the component of the two vectors that align can be involved in the collision interaction, keeping these vectors as close to perpendicular as possible limits collision loss (Lee et al., 2011). As a consequence, collision loss can also be reduced by making the deflection smaller. Interestingly, dividing a large deflection into a series of smaller sub-deflections, even when the sub-deflections sum to the magnitude of the large deflection, can substantially reduce overall loss. This is because the relative energy loss is a function of the square of the deflection angle (Ruina et al., 2005; Fig. 3B). Even with more contacts (collisions), the substantial reduction of loss in each contact reduces the overall loss. This is the strategy used in the three-beat canter and four-beat gallop (Bertram and Gutmann, 2009); the deflection in the canter requires three contacts (collisions), but each involves the cubed root of the energy of a single, large collision. In the end, if x = the proportion of energy lost in a single contact, using a sequenced footfall canter instead would require only 3(x)1/3 = 1/3x. With an extremely large number of limbs (n) the proportional loss declines to zero (1/n)x; equivalent to a wheel with a rim.
Increasing the low cost portion of the cycle: Running, trotting and hopping Bipedal running, kangaroo hopping and the trot used by horses and dogs all employ a series of discrete contacts with an intervening non-contact, or ‘flight’, phase. Since there is virtually no overlap between contacts as in the walk, there is no opportunity to decrease loss by applying a pre-emptive thrust prior to the next contact. However, there are strategies available that allow these gaits to be energetically cost effective (explaining why they so commonly appear).
Fig. 3. (A) A simple biped model indicates the value of considering collision energy loss. The transition from one support limb to the next requires a contact from the lead limb and a push-off thrust from the trailing limb. Centre of mass (CoM) velocity vectors are shown, where V and V+ are just before and just after transition between support limbs, respectively. If contact deflection occurs first, followed by push-off, CoM velocity will change as indicated by the grey vectors. The vector tips will follow line a as the lead limb acts to change the CoM direction (note a is parallel to the lead limb), then line b as the previous stance limb adds thrust (b parallels the effective previous stance limb, a direct line between the point of contact and the CoM). If, instead, the previous limb adds thrust before the next limb makes contact, the vector tips will follow paths d and e. Altering the sequence of contacts, and nothing else, changes the momentum and energy loss because it alters the relationship between the CoM velocity vector and the limbs at points in the cycle when substantial energy can be lost. (B) Dividing a single deflection (shown at left) into a series of sub-deflections results in an overall decrease in energy loss. Asymmetric gaits, such as the equine canter and gallop, appear to use this strategy to lower locomotion cost as higher speeds.
The CoM during running and trotting has substantial horizontal velocity. Much of this is in jeopardy of being lost to large, jolting collisions if the contact dynamics are not managed properly. One simple strategy employed in these gaits is to maintain the contact limb(s) as upright as possible. If the limb made contact perpendicular to the horizontal path of the CoM, then no collision loss would occur (the CoM velocity and ground reaction force vectors would be perpendicular and could not interact with each other in a collision-like manner). However, this would limit the ability to apply the upward thrust that allows the ballistic flight phase of the gait; the flight phase is a very low cost portion of the stride cycle, so it is advantageous to emphasise this phase. One of the trade-offs in managing a cost effective trot is to decrease the contact loss, while increasing the low cost flight phase. Of course, these are mutually exclusive options, but the gait observed emerges as a compromise that optimises the net cost. Trots, runs and high speed hopping are also characterised by a distinctive bouncing motion. The supporting limb(s) deflect during contact and the CoM is at its lowest point at mid-contact, following which it accelerates upward into the non-contact phase of the cycle. The obvious bounce-like motion of these gaits, paralleling the bouncing of a rubber ball, has led to the expectation that these
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gaits depend on elastic recoil (the storage and return of elastic strain energy in limb tendons, ligaments and muscular components; Cavagna et al., 1977). However, it turns out that the bouncing motion is effective at managing collision costs and bounce-like running gaits are energetically cost-effective, even if all of the forces involved are actively generated and so have an active cost (Ruina et al., 2005; Srinivasan and Ruina, 2005). Passive elastic structures also contribute to reducing the cost of running and trotting (Alexander, 1984) and they do so without interfering with the collision loss saving. The motion required to limit collision loss during contact, smooth reversal of vertical motion, and that of passively deforming elastic tissues is the same (Ruina et al., 2005). The way animals trot and humans run is mechanically (and metabolically) cost effective, even without passive elasticity, but passive elasticity can be used to enhance the cost-effectiveness of the gait because the best leg deformation strategy for avoiding loss is coincident with the best leg deformation strategy for strain energy recovery. It has always been problematic to envision the adaptations required to produce the elegantly cursorial forms that exploit elastic recoil for effective locomotion. If the running gait were primarily dependent on effective elastic recovery, we would wonder how the elastic mechanisms could be developed prior to the origin of a gait that could exploit them. Instead, from understanding the energy changes within a running stride, it turns out that the bouncing gait is a cost-effective option, even without passive elasticity. Because these organisms already possessed connective tissues with properties that could augment these optimised motions, adaptive processes were then free to modify the passive components to work with bouncing gaits to provide a true ‘bounce’, as suited to the circumstances of each particular species.
Trade-offs that optimise gait Gaits and the manner that animals use their limbs for locomotion are best considered as an optimisation problem, where many features of the motion patterns may result in either positive or negative effects. The trade-off described above between strategies that decrease the energetic loss in the high cost portion of the gait cycle (a ‘positive’) being tied to a decrement of the low cost portion (a ‘negative’), is an example. The best strategy then, and the one used by animals while moving across a surface, is a compromise that results in the best overall performance. To understand and appreciate how gaits and the pattern of limb movements are determined, it is necessary to identify the factors that influence the optimisation process. A large number of factors will be involved, of course, but substantial insight can be gained by identifying the three most dominant factors; an important but neglected energy loss and two inter-dependent energetic costs associated with work performed by the limb. The discussion can become complex, even when restricted to the interplay between just three factors. It would be surprising if this were not the case, since locomotion is itself a complex dynamic process. The challenge is to recognise that negative effects will be eliminated as much as possible and so will become difficult to observe and identify. Such a decrease in negative effects comes at the expense of energetic cost (in the form of work performed) that is required to implement the available strategies. The gait and the details of its implementation will largely result from the best trade-off (balance) between reducing the negative effects, increasing the positive effects and generating the effort needed to accomplish these tasks. The basic trade-offs involved in legged locomotion can be summarised as (1) the collision-based energy loss caused by limb contact; and (2) the cost associated with leg work that can reduce that
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loss (depending on the strategy implemented), or at the very least must replace the loss in order to maintain motion. The work performed by the leg can be considered in terms of either stance leg work, altering the travel of the CoM while the limb is in contact with the substrate, or swing limb work, adjusting the position of limb when it is not in contact (to control step length and frequency). Although many details of these trade-offs/optimisations remain to be determined for specific species, it is valuable to have a conceptual understanding of the system. As discussed above, several strategies are available to reduce collision-based loss (e.g. altering the sequence of contact and thrust, as described above). For a gait such as the walk, collision of the CoM with the substrate (via contact of a stiff stance limb) could be eliminated by allowing the stance limb to bend and extend so the CoM travels on a path parallel to the substrate. However, such a strategy would require more work from the stance limb bending and extending under the load of the mass than would be saved by eliminating the collision-based loss (Srinivasan and Ruina, 2007; Srinivasan, 2011). Both the excessive loss of a jolting contact using a stiff, strut-like limb (and the work required to replace that loss) and the work required to eliminate that loss by bending and extending the limb (for a completely level motion of the CoM) are greater than a compromise that exists between these two extremes. The compromise optimises the combined decrease in loss with the cost of stance limb work to find a CoM path that varies in height in a manner that requires the least energetic investment. The jolting ‘collision’ is diminished to the point where it is difficult to detect, while the CoM vertical oscillations are relatively small, although not eliminated entirely. Collision-based losses could also be virtually eliminated by taking very small steps. If a step is small enough, the CoM falls very little between contacts, so very little loss is involved. However, to travel at a reasonable speed with such small steps would require substantial active swinging of the non-stance limb, in order to reposition it appropriately (Kuo, 2001). The numerous swing limb accelerations and decelerations would require greater energetic investment than is saved by totally eliminating the collision loss through small step length. Optimisation models indicate that a minimum cost strategy exists somewhere between the long, jolting steps that involve little swing limb cost, but large collision loss, and short, swift steps that decrease collision loss, but require substantial swing limb work. Conclusions The normal gaits we observe in animals largely represent the optimisation solution best suited to the circumstances the animal faces. The similarity of gaits between individuals of a given species indicates simply that each individual is ‘solving’ the same functional problem. Various biological and control factors may influence the details of these gaits, but it is useful to understand the functional ‘purpose’ of the gait in order to interpret how such factors affect locomotion and to anticipate the consequences of variation among animals and circumstances. Although we are still a long way from quantifiably interpreting (and predicting) the energetic consequences of specific movement strategies or the influence of any morphological variation, eventual success at these challenges will depend on understanding the fundamental compromises that define the range of effective movement strategies available to the organism. Conflict of interest statement The author of this paper has no financial or personal relationship with other people or organisations that could inappropriately influence or bias the content of the paper.
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The Veterinary Journal journal homepage: www.elsevier.com/locate/tvjl
Gait as solution, but what is the problem? Exploring cost, economy and compromise in locomotion John E.A. Bertram ⇑ Faculty of Medicine, University of Calgary, 3330 Hospital Drive NW, Calgary, Alberta T2N 1N4, Canada
a r t i c l e
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Keywords: Locomotion Gait
a b s t r a c t Many studies have examined how legged mammals move, defining ‘what’ happens in locomotion. However, few ask ‘why’ those motions occur as they do. The energetic and functional constraints acting on an animal require that locomotion should be metabolically ‘cost effective’ and this in large part determines the strategies available to accomplish the task. Understanding the gaits utilised, within the spectrum of gaits possible, and determination of the value of specific relationships among speed, stride length, stride frequency and morphology, depends on identifying the fundamental costs involved and the effects of different movement strategies on those costs. It is argued here that a fundamental loss associated with moving on limbs (centre of mass momentum and energy loss) and two costs involved with controlling and replacing that loss (muscular work of the supporting limb during stance and muscular work of repositioning the limbs during swing) interact to determine the cost trade-offs involved and the optimisation strategies available for each species and speed. These optimisation strategies are what has been observed and characterised as gait. Ó 2013 Elsevier Ltd. All rights reserved.
Introduction Gait is a pattern of movement that ultimately allows the animal to move effectively across a substrate. A wide variety of movement patterns could result in locomotion, but a few distinctive patterns (and variations on each of these main themes) are almost universally used by quadrupeds. Historically from as early as Aristotle (ca. 350 BC) (Goiffon and Vincent, 1779; Marey, 1873; Hildebrand, 1965; Abourachid, 2003; Farquharson, 2007) and by tradition in equine practice, gait has been characterised largely on the basis of footfall sequence. This is a relatively easily observed and quantified aspect of animal locomotion, especially with modern analytical technology, such as high speed video, and newly emerging mathematical techniques, such as linear discriminant analysis (Robilliard et al., 2007). Although detailed descriptions of footfall patterns can be used to define gaits and to document differences among patterns used by different species in various circumstances (or even different individuals), such characterisation does not inform us of the functional purpose of each gait or provide information on why each exists as it does. The commonly observed, or ‘normal’, gaits are the ones that emerge from the range of possible gaits because they allow for metabolically cost-effective movement under the circumstances ⇑ Tel.: +1 403 2109857. E-mail address: [email protected] 1090-0233/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.tvjl.2013.09.025
in which the animal operates. For example, each of the preferred gaits of horses is selected and utilised to provide the least cost of transport over a certain range of speed, with the walk being used at slow speeds, the trot at intermediate speeds and the gallop at high speeds (Hoyt and Taylor, 1981). The operational conditions of locomotion and the appropriate movement strategies available are predominantly influenced by the speed of progression, although other factors from the physical properties of the substrate, such as stiffness (Wilson and Pardoe, 2001) and slope (Self et al., 2012), also have an influence. It is generally assumed that the gait pattern observed under normal circumstances is the only one available, potentially programmed by a neural control system originating in ancient coordination centres of the brain and spinal cord (Grillner, 1975; McCrae and Rybak, 2008). However, evidence from analysis of human gait selection (humans are a reasonably tractable species sometimes willing to adapt to useful experimental conditions that other species resist), indicate that the control of key gait parameters, such as the relationship between speed, stride frequency and stride length, are largely determined by the metabolic cost associated with each option (Bertram and Ruina, 2001). This results in the normal and spontaneous selection of quite unusual gait parameters under experimentally imposed circumstances, such as moving to a specified frequency or stride length. These gait parameters are selected in a manner that minimises the cost of locomotion under each set of specific conditions (Bertram, 2005; Snaterse et al., 2011; Fig. 1). The brain and spinal
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Fig. 1. Diagram showing how gait parameters, such as speed, step length and step frequency, are selected in human walking to minimise energetic expenditure under the circumstances the system operates within. The light blue contours represent the metabolic cost surface, where each contour has greater magnitude than the one inside it (minimum cost occurs at the central circle where the three linear relationships cross). For an individual walking at a determined speed (constant level on the y-axis), such as on a treadmill, minimum cost will occur using the step frequency where a horizontal line from the speed axis touches the innermost cost contour it can. The speed–frequency relationship for walking on a treadmill at various speeds (red line) will be a series of such horizontal contours. Step frequency (dark blue line) and step length (green line) cost minimisations will be determined in a similar manner for restricted frequency or step length conditions, except that the line determining each will intersect the cost surface in a different location (and therefore determines a different optimisation solution).
centres play a critical role in operating the locomotory apparatus, but this role should be considered in the same way as the critical role of the driver in operating a motor vehicle. The driver is capable of steering the vehicle anywhere, but a good driver remains on the roadway and operates according to the traffic regulations. To do otherwise results in less than optimum performance. When considering the best operation of the limbs in animal locomotion, a major challenge is to discover the ‘road map’ and ‘driving regulations’ that the central control centre must integrate with; these will be dependent on the physical interaction of the organism with its environment and the energetic consequences of that interaction. Understanding why footfall patterns appear as they do requires an understanding of the energetic consequences of the motions and comparison of these with available options. This may appear to be a difficult task, given that mammals are such complex systems, using complex biological actuators (muscles) organised in often redundant functional networks (thus providing multiple control options for any given movement and apparent ‘cost’). However, substantial insight into the purpose of gaits can be gained by identifying the main determinants of locomotion cost.
Energetic consequences of gait As much as we are all familiar with the fact that locomotion using legs involves substantial energetic cost, it is not as apparent why this must be so. Flying and swimming require much less metabolic investment per distance travelled (Tucker, 1970). The high cost of terrestrial locomotion is most commonly attributed to the negative work expended during the stride cycle when moving on a solid surface. It is possible to observe and verify that negative work occurs in legged locomotion, where the kinetic energy of the organism’s moving mass is expended as work on the muscles of the animal’s limbs instead of the muscles moving the mass of
the animal by producing positive work. Negative work, however, turns out to be just one of many mechanisms through which mechanical energy can be lost from the system (where the organism is the ‘system’ of interest). The process of energy loss actually depends on a far more fundamental, but largely neglected, feature of legged locomotion: collision-based loss caused by the interaction of the moving mass and its solid support. When the term ‘collision’ is mentioned, it is common to think of violent crashes between objects such as automobiles. However, in dynamics, a collision is defined simply as a discontinuity of the velocity (magnitude or direction) of a travelling mass. Any such discontinuity requires the application of force and will involve the transfer or loss of energy. Within the precision of dynamics, the term ‘discontinuity’ also has a specific meaning, where it is limited to an instantaneous event. In the real world, instantaneous events require infinite forces, so they do not exist. However, although such events occur more slowly in reality than in theory, there are some aspects of collision theory that do translate to energetic consequences in the real world of locomotion. As a simple example that most of us can relate to, consider a playground swing. A common playground swing is an example of an active pendulum. Much of the swinging motion is maintained through the spontaneous (and passive) exchange between potential energy (at the highest point of the swing) and translational kinetic energy (at the lowest point of the swing). Children learn early that they are able to contort their bodies in registry with the swinging motion to add momentum (and energy) to the swing; this is usually called ‘pumping’ (Wirkus et al., 1998). However, the height of the swing can be increased in this way only so far. As the swing moves beyond horizontal, a small jolt is felt on the down swing, following which the height of the next swing is much reduced (indicating that energy has been extracted from the system). The jolt that is felt indicates a small collision caused by the mass of the individual falling within the arc of the swing and being forced to change direction (a discontinuity in its path) as the supporting cables of the swing become taut. The energy loss occurs because the radial component of the falling centre of mass (CoM) velocity is directed along the supporting cables and is lost to the collision interaction (Fig. 2). A common acrobatic circus act involves demonstrating that a full 360° swing revolving completely around the rotational axis can be accomplished. Understanding how this can be done in a circus, when it cannot be done in the playground, indicates the value of considering the phenomena responsible for energy loss, since it affects functional capability so directly. Success in the full rotation of the circus swing depends on a single adjustment of equipment (a change of ‘morphology’), replacing the flexible supporting cables of the playground swing with rigid struts that do not flex when the individual swings higher than the horizontal. With rigid struts, the CoM is prevented from falling vertically, so is forced to remain on the swing arc regardless of position in the rotation. Without a vertical fall, there is no jolt and therefore no collision loss, so the energy state of the system can increase continuously until enough has accumulated to allow the mass to move completely around the axis. In the normal playground swing, the higher the swing above horizontal, the greater the jolt and the greater the loss.
Deflecting the centre of mass in legged locomotion Gaits differ, but there are features that all the gaits have in common. Most obviously, they all involve moving the mass of the animal across the substrate. Cyclic vertical motions occur while the animal moves forward at speeds that vary systematically over the period of the stride. To a much lesser extent, lateral movements also occur. The interaction of the forward and vertical movement is
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Fig. 2. Collision loss involved in a playground swing. If an individual on a playground swing swings above the horizontal, the centre of mass (CoM) will fall vertically. When the supporting cables again become tight, the individual will experience a small jolt and the next swing will reach a much lower height. The small jolt extracts the radially oriented velocity, and consequently that component of the kinetic energy. Deflecting the CoM when a limb contacts the ground surface results in a similar ‘collision’, with losses determined by the geometry of the velocity and constraining force applied by the contact limb.
a key factor in determining the energetic cost of legged locomotion, so the remainder of this discussion will neglect the lateral motions. Although the mass of the animal continues travelling forward in locomotion, the cyclic changes in height of the CoM requires that the vertical component of its direction change through the stride cycle: from forward and upward to forward and downward, and from forward and downward to forward and upward. These changes of CoM height are natural aspects of all gaits, but they are key factors in determining the consequences of any gait strategy. In steady state locomotion (constant average horizontal speed), changes in the height of the CoM as the animal moves forward mean that there are two important portions in each stride: (1) a portion of low energetic cost; and (2) a portion of high energetic cost. The low cost portion occurs when the CoM direction changes from moving upward to moving downward. This direction change involves little energy cost to the animal because it occurs spontaneously as a result of the passive action of gravity. The high cost portion occurs when the CoM direction changes from moving downward to moving upward. This direction change must be mediated by the action of the limbs and involves high cost because it is during this phase that energy can be lost and/or work must be performed by the limb. The footfall patterns that are recognised as gaits are employed to allow for the downward to upward direction change of the CoM, while limiting the net work that is required to maintain consistent average forward speed. Different footfall strategies (gaits) will suit different circumstances of locomotion, largely determined by the speed of forward travel. Understanding why each gait is used requires understanding how energy is lost from the system and what strategies are available to minimise and replace that loss. If animals could travel across the substrate with virtually no loss, like a wheel rolling, it would not be necessary to add much energy to each step. It is easy to recognise that animals do not roll like a wheel, but it is not as easy to identify why legged locomotion involves energy loss. This is because animals have developed
sophisticated strategies to limit energy loss as much as possible, and these strategies camouflage many aspects of the processes involved. Recognising the factors involved, and the strategies available, to limit energy loss within the stride cycle provides substantial insight into why many of the motions observed occur as they do and why others that we might imagine do not. The contact of each limb is much like the contact of an individual spoke of a wheel that does not have a rim to roll on. The key loss occurs with each contact and is basically equivalent to the jolt felt in the playground swing that travels too high; it is derived from the collision-like deflection of the CoM as the limb strut applies force to the ground (along its axis), constrains the CoM path (to vault over the contact point) and shifts the CoM direction from forward and downward, prior to contact, to forward and upward. If the limb simply acted as a rigid strut, then locomotion would involve violent transitions and large amounts of energy would be lost at each contact. Maintaining steady forward speed requires the replacement of all lost energy, so the cost of locomotion in this case would be very high, involving both large energy losses and the cost of work required to repay the loss. Instead, animals have developed strategies to limit this loss and to allow them to travel for relatively low overall energetic investment. These strategies involve decreasing the loss involved in the high cost portion of the gait cycle, while increasing the travel accomplished in the low cost portion. There is a trade-off between these strategies and the active work that must be invested to implement them. An understanding of the movement patterns that we define as gaits requires identifying the factors involved in the optimisation solution and recognising the features and consequences of the trade-offs involved. Reducing loss in the high cost portion of the cycle: Walking and galloping Although they appear to be quite different gaits, walking and galloping are both strategies that primarily limit the energy loss
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associated with the transition in the CoM from moving forward but downward to forward and upward (the high cost portion of the stride cycle). Two aspects of this transition contribute to the energetic cost. The first is the collision-based energy loss that occurs when the limb contacts the substrate and deflects the animal’s mass. This loss could be substantial, so it is not surprising that strategies have been developed to limit the magnitude of loss. Some of these strategies are subtle and non-intuitive, even though they are effective. In walking, collision-based loss occurs when the leading limb in the transition process contacts the substrate and deflects the animal’s mass. This loss must be replaced in order to maintain constant average speed; this requires thrust from the trailing limb. However, the sequence of action of the limbs (deflection then thrust vs. a pre-emptive thrust then deflection), substantially alters the net energetic cost. This is because the geometry of the collision event differs between the two alternatives, even if everything else remains the same. This is most easily seen in the more straightforward geometry of bipedal walking (McGeer, 1990; Donelan et al., 2002; Collins et al., 2005; Ruina et al., 2005; Kuo, 2007; Fig. 3A), but the same process occurs in quadrupedal gaits (Birkemeier, 1998; Usherwood et al., 2007; Lee et al., 2011). The key factor to recognise in understanding this strategy is that only that portion of the CoM velocity vector aligned with the contact is involved in the collision; for the playground swing, this was the radial portion of the child’s total velocity, while in legged locomotion it is the trigonometric portion aligned with the axis of the limb. The collision loss associated with deflecting the CoM is largely dependent on the geometric relationship between the two dynamic features of the motion that interact: the velocity vector representing motion of the animal’s mass and the vector representing the net ground reaction force (which contributes to the impulse responsible for altering the direction of travel of the CoM, as described above). Since only the component of the two vectors that align can be involved in the collision interaction, keeping these vectors as close to perpendicular as possible limits collision loss (Lee et al., 2011). As a consequence, collision loss can also be reduced by making the deflection smaller. Interestingly, dividing a large deflection into a series of smaller sub-deflections, even when the sub-deflections sum to the magnitude of the large deflection, can substantially reduce overall loss. This is because the relative energy loss is a function of the square of the deflection angle (Ruina et al., 2005; Fig. 3B). Even with more contacts (collisions), the substantial reduction of loss in each contact reduces the overall loss. This is the strategy used in the three-beat canter and four-beat gallop (Bertram and Gutmann, 2009); the deflection in the canter requires three contacts (collisions), but each involves the cubed root of the energy of a single, large collision. In the end, if x = the proportion of energy lost in a single contact, using a sequenced footfall canter instead would require only 3(x)1/3 = 1/3x. With an extremely large number of limbs (n) the proportional loss declines to zero (1/n)x; equivalent to a wheel with a rim.
Increasing the low cost portion of the cycle: Running, trotting and hopping Bipedal running, kangaroo hopping and the trot used by horses and dogs all employ a series of discrete contacts with an intervening non-contact, or ‘flight’, phase. Since there is virtually no overlap between contacts as in the walk, there is no opportunity to decrease loss by applying a pre-emptive thrust prior to the next contact. However, there are strategies available that allow these gaits to be energetically cost effective (explaining why they so commonly appear).
Fig. 3. (A) A simple biped model indicates the value of considering collision energy loss. The transition from one support limb to the next requires a contact from the lead limb and a push-off thrust from the trailing limb. Centre of mass (CoM) velocity vectors are shown, where V and V+ are just before and just after transition between support limbs, respectively. If contact deflection occurs first, followed by push-off, CoM velocity will change as indicated by the grey vectors. The vector tips will follow line a as the lead limb acts to change the CoM direction (note a is parallel to the lead limb), then line b as the previous stance limb adds thrust (b parallels the effective previous stance limb, a direct line between the point of contact and the CoM). If, instead, the previous limb adds thrust before the next limb makes contact, the vector tips will follow paths d and e. Altering the sequence of contacts, and nothing else, changes the momentum and energy loss because it alters the relationship between the CoM velocity vector and the limbs at points in the cycle when substantial energy can be lost. (B) Dividing a single deflection (shown at left) into a series of sub-deflections results in an overall decrease in energy loss. Asymmetric gaits, such as the equine canter and gallop, appear to use this strategy to lower locomotion cost as higher speeds.
The CoM during running and trotting has substantial horizontal velocity. Much of this is in jeopardy of being lost to large, jolting collisions if the contact dynamics are not managed properly. One simple strategy employed in these gaits is to maintain the contact limb(s) as upright as possible. If the limb made contact perpendicular to the horizontal path of the CoM, then no collision loss would occur (the CoM velocity and ground reaction force vectors would be perpendicular and could not interact with each other in a collision-like manner). However, this would limit the ability to apply the upward thrust that allows the ballistic flight phase of the gait; the flight phase is a very low cost portion of the stride cycle, so it is advantageous to emphasise this phase. One of the trade-offs in managing a cost effective trot is to decrease the contact loss, while increasing the low cost flight phase. Of course, these are mutually exclusive options, but the gait observed emerges as a compromise that optimises the net cost. Trots, runs and high speed hopping are also characterised by a distinctive bouncing motion. The supporting limb(s) deflect during contact and the CoM is at its lowest point at mid-contact, following which it accelerates upward into the non-contact phase of the cycle. The obvious bounce-like motion of these gaits, paralleling the bouncing of a rubber ball, has led to the expectation that these
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gaits depend on elastic recoil (the storage and return of elastic strain energy in limb tendons, ligaments and muscular components; Cavagna et al., 1977). However, it turns out that the bouncing motion is effective at managing collision costs and bounce-like running gaits are energetically cost-effective, even if all of the forces involved are actively generated and so have an active cost (Ruina et al., 2005; Srinivasan and Ruina, 2005). Passive elastic structures also contribute to reducing the cost of running and trotting (Alexander, 1984) and they do so without interfering with the collision loss saving. The motion required to limit collision loss during contact, smooth reversal of vertical motion, and that of passively deforming elastic tissues is the same (Ruina et al., 2005). The way animals trot and humans run is mechanically (and metabolically) cost effective, even without passive elasticity, but passive elasticity can be used to enhance the cost-effectiveness of the gait because the best leg deformation strategy for avoiding loss is coincident with the best leg deformation strategy for strain energy recovery. It has always been problematic to envision the adaptations required to produce the elegantly cursorial forms that exploit elastic recoil for effective locomotion. If the running gait were primarily dependent on effective elastic recovery, we would wonder how the elastic mechanisms could be developed prior to the origin of a gait that could exploit them. Instead, from understanding the energy changes within a running stride, it turns out that the bouncing gait is a cost-effective option, even without passive elasticity. Because these organisms already possessed connective tissues with properties that could augment these optimised motions, adaptive processes were then free to modify the passive components to work with bouncing gaits to provide a true ‘bounce’, as suited to the circumstances of each particular species.
Trade-offs that optimise gait Gaits and the manner that animals use their limbs for locomotion are best considered as an optimisation problem, where many features of the motion patterns may result in either positive or negative effects. The trade-off described above between strategies that decrease the energetic loss in the high cost portion of the gait cycle (a ‘positive’) being tied to a decrement of the low cost portion (a ‘negative’), is an example. The best strategy then, and the one used by animals while moving across a surface, is a compromise that results in the best overall performance. To understand and appreciate how gaits and the pattern of limb movements are determined, it is necessary to identify the factors that influence the optimisation process. A large number of factors will be involved, of course, but substantial insight can be gained by identifying the three most dominant factors; an important but neglected energy loss and two inter-dependent energetic costs associated with work performed by the limb. The discussion can become complex, even when restricted to the interplay between just three factors. It would be surprising if this were not the case, since locomotion is itself a complex dynamic process. The challenge is to recognise that negative effects will be eliminated as much as possible and so will become difficult to observe and identify. Such a decrease in negative effects comes at the expense of energetic cost (in the form of work performed) that is required to implement the available strategies. The gait and the details of its implementation will largely result from the best trade-off (balance) between reducing the negative effects, increasing the positive effects and generating the effort needed to accomplish these tasks. The basic trade-offs involved in legged locomotion can be summarised as (1) the collision-based energy loss caused by limb contact; and (2) the cost associated with leg work that can reduce that
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loss (depending on the strategy implemented), or at the very least must replace the loss in order to maintain motion. The work performed by the leg can be considered in terms of either stance leg work, altering the travel of the CoM while the limb is in contact with the substrate, or swing limb work, adjusting the position of limb when it is not in contact (to control step length and frequency). Although many details of these trade-offs/optimisations remain to be determined for specific species, it is valuable to have a conceptual understanding of the system. As discussed above, several strategies are available to reduce collision-based loss (e.g. altering the sequence of contact and thrust, as described above). For a gait such as the walk, collision of the CoM with the substrate (via contact of a stiff stance limb) could be eliminated by allowing the stance limb to bend and extend so the CoM travels on a path parallel to the substrate. However, such a strategy would require more work from the stance limb bending and extending under the load of the mass than would be saved by eliminating the collision-based loss (Srinivasan and Ruina, 2007; Srinivasan, 2011). Both the excessive loss of a jolting contact using a stiff, strut-like limb (and the work required to replace that loss) and the work required to eliminate that loss by bending and extending the limb (for a completely level motion of the CoM) are greater than a compromise that exists between these two extremes. The compromise optimises the combined decrease in loss with the cost of stance limb work to find a CoM path that varies in height in a manner that requires the least energetic investment. The jolting ‘collision’ is diminished to the point where it is difficult to detect, while the CoM vertical oscillations are relatively small, although not eliminated entirely. Collision-based losses could also be virtually eliminated by taking very small steps. If a step is small enough, the CoM falls very little between contacts, so very little loss is involved. However, to travel at a reasonable speed with such small steps would require substantial active swinging of the non-stance limb, in order to reposition it appropriately (Kuo, 2001). The numerous swing limb accelerations and decelerations would require greater energetic investment than is saved by totally eliminating the collision loss through small step length. Optimisation models indicate that a minimum cost strategy exists somewhere between the long, jolting steps that involve little swing limb cost, but large collision loss, and short, swift steps that decrease collision loss, but require substantial swing limb work. Conclusions The normal gaits we observe in animals largely represent the optimisation solution best suited to the circumstances the animal faces. The similarity of gaits between individuals of a given species indicates simply that each individual is ‘solving’ the same functional problem. Various biological and control factors may influence the details of these gaits, but it is useful to understand the functional ‘purpose’ of the gait in order to interpret how such factors affect locomotion and to anticipate the consequences of variation among animals and circumstances. Although we are still a long way from quantifiably interpreting (and predicting) the energetic consequences of specific movement strategies or the influence of any morphological variation, eventual success at these challenges will depend on understanding the fundamental compromises that define the range of effective movement strategies available to the organism. Conflict of interest statement The author of this paper has no financial or personal relationship with other people or organisations that could inappropriately influence or bias the content of the paper.
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