Optimum walking techniques for quadrupeds and bipeds

A new theory is presented which describes quadrupedal as well as bipedal walking. It avoids errors which occurred in previous theories by evaluating separately the work done by each leg instead of deriving net work from mechanical energy fluctuations. It takes particular account of two parameters, the duty factor β (the fraction of the stride for which each foot is on the ground) and a parameter q which defines the time course of the force on each foot. It shows that for any given speed there is an optimum (β, q ) which minimizes the energy cost of locomotion. These (β, q ) are only a little different for bipeds and quadrupeds except near the critical speed at which the optimum moves abruptly from walking (high β) to running (low β). Walking men use (β, q ) close to the theoretical optima, but with slightly higher q. Walking dogs and sheep use q which are lower than the optimum values except at very low speeds. Some of the energy which would otherwise be required for locomotion may be saved by storage of elastic strain energy in tendons etc. This mechanism is more effective in running than in fast walking, which may be why men change from walking to running at lower speeds than the inelastic theory suggests.     

Mechanics of locomotion of dogs (Canis familiaris) and sheep (Ovis aries)

Records have been made of the forces exerted on the ground by dogs and a sheep, in walking, trotting, cantering and slow galloping. Film has been taken simultaneously. The difference between walking and trotting was much less marked for the sheep than for the dogs.
Step length and stride length increase as speed increases. They are expressed as functions of the Froude number.
The vertical component of the force exerted by a foot on the ground shows two main maxima in walking, except in the case of the fore feet of sheep. In this case and in other gaits there is only one main maximum. The vertical movements of the fore and hind quarters which occurred in examples of each gait have been calculated from the force records.
The force exerted by a foot on the ground changes direction in the course of a step so as to remain more or less in line with a point fixed relative to the animal, but dorsal to its back.
The force records show impact disturbances in the first 003 sec of contact of each foot with the ground.
The point of application of the force on the sole of a foot tends to move posteriorly as the force increases.
The results are discussed in relation to a theoretical account of the mechanics of locomotion on legs.     

Bipedal locomotion: effects of speed, size and limb posture in birds and humans

Seven species of ground-dwelling birds (body mass range: 0.045-90 kg) were filmed while walking and running on a treadmill. High-speed light films were also taken of humans to compare kinematic patterns of avian with human bipedalism. Consistent patterns of stride frequency, stride length, step length, duty factor and limb excursion were observed in all species, with most of the variation among species being due to differences in body size. In general, smaller bipeds have higher stride frequencies (α M ^−0.18), shorter stride lengths (α M ^0.38) and more limited ranges of speed within each gait than large bipeds. After normalizing for size (based on Froude number, after Alexander, 1977), remaining kinematic variation is largely due to interspecific differences in posture and relative limb segment lengths. For their size, smaller bipeds have greater step lengths, limb excursion angles and duty factors than large bipeds because of their more crouched posture and greater effective limb length. The most notable differences in limb kinematics between birds and humans occur at the walk-run transition and are maintained as running speed increases. Change of gait is smooth and difficult to discern in birds, but distinct in humans, involving abrupt decreases in step length and duty factor (time of contact) and a corresponding increase in limb swing time. These differences appear to reflect a spring-like run that is stiff in humans (favouring elastic energy recovery) but more compliant in birds (increasing time of ground contact). Differences between birds and humans in balance of the body's centre of mass not only affect femoral orientation and motion, but also affect pattern of limb excursion with speed.     

Optimum walking techniques for idealized animals

The vertical component of the force exerted by a foot on the ground, in the course of a step, may rise to a single maximum and decline again (as in human running) or may show two distinct maxima (as in human walking). A foot may remain on the ground for a large or small fraction of the duration of a stride. Mathematical models are used to investigate the effects of these differences of technique on the energy cost of locomotion. The optimum technique for a biped at a given speed is different from the optimum for a hypothetical many-legged animal. The optima for quadrupedal walking are likely to lie between these extremes.
The walking techniques adopted by men at different speeds are close to the optima indicated by the bipedal model. The two maxima of the force exerted by a foot are higher, and have a lower minimum between them, at higher speeds of walking. The techniques adopted by a sheep are close to the optima indicated by the many-legged model but dogs use techniques rather closer to the optima for bipeds.
The limitations of the models are discussed.     

Gait Transitions in Human Infants: Coping with Extremes of Treadmill Speed

Spinal pattern generators in quadrupedal animals can coordinate different forms of locomotion, like trotting or galloping, by altering coordination between the limbs (interlimb coordination). In the human system, infants have been used to study the subcortical control of gait, since the cerebral cortex and corticospinal tract are immature early in life. Like other animals, human infants can modify interlimb coordination to jump or step. Do human infants possess functional neuronal circuitry necessary to modify coordination within a limb (intralimb coordination) in order to generate distinct forms of alternating bipedal gait, such as walking and running? We monitored twenty-eight infants (7–12 months) stepping on a treadmill at speeds ranging between 0.06–2.36 m/s, and seventeen adults (22–47 years) walking or running at speeds spanning the walk-to-run transition. Six of the adults were tested with body weight support to mimic the conditions of infant stepping. We found that infants could accommodate a wide range of speeds by altering stride length and frequency, similar to adults. Moreover, as the treadmill speed increased, we observed periods of flight during which neither foot was in ground contact in infants and in adults. However, while adults modified other aspects of intralimb coordination and the mechanics of progression to transition to a running gait, infants did not make comparable changes. The lack of evidence for distinct walking and running patterns in infants suggests that the expression of different functional, alternating gait patterns in humans may require neuromuscular maturation and a period of learning post-independent walking.     

Pedestrian locomotion energetics and gait characteristics of a diving bird, the great cormorant, Phalacrocorax carbo

Great cormorants Phalacrocorax carbo are foot propelled diving birds that seem poorly suited to locomotion on land. They have relatively short legs, which are presumably adapted for the generation of high forces during the power stroke of aquatic locomotion, and walk with a pronounced "clumsy waddle". We hypothesise (1) that the speed, independent minimum cost of locomotion (C min, ml O2 m(-1)) will be high for cormorants during treadmill exercise, and (2) that cormorants will have a relatively limited speed range in comparison to more cursorial birds. We measured the rate of oxygen consumption (V02) of cormorants during pedestrian locomotion on a treadmill, and filmed them to determine duty factor (the fraction of stride period that the foot is in contact with the ground), foot contact time (tc), stride frequency (f), swing phase duration and stride length. C min was 2.1-fold higher than that predicted by their body mass and phylogenetic position, but was not significantly different from the C min of runners (Galliformes and Struthioniformes). The extrapolated gamma-intercept of the relationship between V02 and speed was 1.9-fold higher than that predicted by allometry. Again, cormorants were not significantly different from runners. Contrary to our hypothesis, we therefore conclude that cormorants do not have high pedestrian transport costs. Cormorants were observed to use a grounded gait with two double support phases at all speeds measured, and showed an apparent gait transition between 0.17 and 0.25 m s(-1). This transition occurs at a Froude number between 0.016 and 0.037, which is lower than the value of approximately 0.5 observed for many other species. However, despite the use of a limited speed range, and a gait transition at relatively low speed, we conclude that the pedestrian locomotion of these foot propelled diving birds is otherwise generally similar to that of cursorial birds at comparable relative velocities.     

Speed dependent phase shifts and gait changes in cockroaches running on substrates of different slipperiness

Background

Many legged animals change gaits when increasing speed. In insects, only one gait change has been documented so far, from slow walking to fast running, which is characterised by an alternating tripod. Studies on some fast-running insects suggested a further gait change at higher running speeds. Apart from speed, insect gaits and leg co-ordination have been shown to be influenced by substrate properties, but the detailed effects of speed and substrate on gait changes are still unclear. Here we investigate high-speed locomotion and gait changes of the cockroach Nauphoeta cinerea, on two substrates of different slipperiness.

Results

Analyses of leg co-ordination and body oscillations for straight and steady escape runs revealed that at high speeds, blaberid cockroaches changed from an alternating tripod to a rather metachronal gait, which to our knowledge, has not been described before for terrestrial arthropods. Despite low duty factors, this new gait is characterised by low vertical amplitudes of the centre of mass (COM), low vertical accelerations and presumably reduced total vertical peak forces. However, lateral amplitudes and accelerations were higher in the faster gait with reduced leg synchronisation than in the tripod gait with distinct leg synchronisation.

Conclusions

Temporally distributed leg force application as resulting from metachronal leg coordination at high running speeds may be particularly useful in animals with limited capabilities for elastic energy storage within the legs, as energy efficiency can be increased without the need for elasticity in the legs. It may also facilitate locomotion on slippery surfaces, which usually reduce leg force transmission to the ground. Moreover, increased temporal overlap of the stance phases of the legs likely improves locomotion control, which might result in a higher dynamic stability.

     

Influence of gait parameters on the loading of the lower limb

The ground reaction force which acts on the foot during normal walking consists of the sum of two components: the support of the weight of the body and the acceleration of the body. The relationships between the initial loading rate of the lower limb (ignoring the contribution of the heelstrike transient) and the general gait parameters — cadence, stride length, and velocity — have been examined. Plots of the resultant ground reaction force were used to calculate the loading rate of the limb. A sample of 13 normal male subjects, aged from 18 to 63 years, walked at five different self-selected speeds. Velocity showed the highest correlation with loading rate (r = 0.95) and stride length the lowest (r = 0.85). The relationship between cadence and loading rate was non-linear.     

Estimation of stride length in level walking using an inertial measurement unit attached to the foot: a validation of the zero velocity assumption during stance

In a variety of applications, inertial sensors are used to estimate spatial parameters by double integrating over time their coordinate acceleration components. In human movement applications, the drift inherent to the accelerometer signals is often reduced by exploiting the cyclical nature of gait and under the hypothesis that the velocity of the sensor is zero at some point in stance. In this study, the validity of the latter hypothesis was investigated by determining the minimum velocity of progression of selected points of the foot and shank during the stance phase of the gait cycle while walking at three different speeds on level ground. The errors affecting the accuracy of the stride length estimation resulting from assuming a zero velocity at the beginning of the integration interval were evaluated on twenty healthy subjects. Results showed that the minimum velocity of the selected points on the foot and shank increased as gait speed increased. Whereas the average minimum velocity of the foot locations was lower than 0.011 m/s, the velocity of the shank locations were up to 0.049 m/s corresponding to a percent error of the stride length equal to 3.3%. The preferable foot locations for an inertial sensor resulted to be the calcaneus and the lateral aspect of the rearfoot. In estimating the stride length, the hypothesis that the velocity of the sensor can be set to zero sometimes during stance is acceptable only if the sensor is attached to the foot.     

Spatio-temporal gait characteristics of the hind-limb cycles during voluntary bipedal and quadrupedal walking in bonobos (Pan paniscus)

Spatio-temporal gait characteristics (step and stride length, stride frequency, duty factor) were determined for the hind-limb cycles of nine bonobos (Pan paniscus) walking quadrupedally and bipedally at a range of speeds. The data were recalculated to dimensionless quantities according to the principle of dynamic similarity. Lower leg length was used as the reference length. Interindividual variability in speed modulation strategy of bonobos appears to be low. Compared to quadrupedal walking, bipedal bonobos use smaller steps to attain a given speed (differences increase with speed), resulting in shorter strides at a higher frequency. In the context of the ("hybrid") dynamic pattern approach to locomotion (Latach, 1998) we argue that, despite these absolute differences, intended walking speed is the basic control variable which elicits both quadrupedal and bipedal walking kinematics in a similar way. Differences in the initial status of the dynamic system may be responsible for the differences in step length between both gaits. Comparison with data deduced from the literature shows that the effects of walking speed on stride length and frequency are similar in bonobos, common chimpanzees, and humans. This suggests that (at least) within extant homininae, spatio-temporal gait characteristics are highly comparable, and this in spite of obvious differences in mass distribution and bipedal posture.     

Energy exchange between subject and belt during treadmill walking

Treadmill walking aims to simulate overground walking, but intra-stride belt speed variations of treadmills result in some interaction between treadmill and subject, possibly obstructing this aim. Especially in self-paced treadmill walking, in which the belt speed constantly adjusts to the subject, these interactions might affect the gait pattern significantly. The aim of this study was to quantify the energy exchange between subject and treadmill, during the fixed speed (FS) and self-paced (SP) modes of treadmill walking. Eighteen subjects walked on a dual-belt instrumented treadmill at both modes. The energy exchange was calculated as the integration of the product of the belt speed deviation and the fore-aft ground reaction force over the stride cycle. The total positive energy exchange was 0.44 J/stride and the negative exchange was 0.11 J/stride, which was both less than 1.6% of the performed work on the center of mass. Energy was mainly exchanged from subject to treadmill during both the braking and propulsive phase of gait. The two treadmill modes showed a similar pattern of energy exchange, with a slightly increased energy exchange during the braking phase of SP walking. It is concluded that treadmill walking is only mildly disturbed by subject-belt interactions when using instrumented treadmills with adequate belt control.     

Using body size to understand the structural design of animals: quadrupedal locomotion.

Many parameters of gait and performance, including stride frequency, stride length, maximum speed, and rate of O2 uptake are experimentally found to be power-law functions of body weight in running quadrupeds. All of these parameters are reasonably easy to measure except maximum speed, where the question arises whether one means top sprinting speed or top speed for sustained running. Moreover, differences in training and motivation make comparisons of top speed difficult. The problem is circumvented by comparing animals running at the transition between trotting and galloping, a physiologically similar speed. Theoretical models are proposed which preserve either geometric similarity, elastic similarity, or static stress similarity between animals of large and small body weights. The model postulating elastic similarity provides the best correlation with published data on body and bone proportions, body surface area, resting metabolic rate, and basal heart and lung frequencies. It also makes the most successful prediction of stride frequency, stride length, limb excursion angles, and the metabolic power required for running at the trot-gallop transition in quadrupeds ranging in size from mice to horses.     

Form in the context of function: Fundamentals of an energy effective striding walk,the role of the plantigrade foot and its expected size

What morphological and functional factors allow for the unique and characteristic upright striding walk of the hominin lineage? Predictive models of locomotion that arise from considering mechanisms of energy loss indicate that collision-like losses at the transition between stance limbs are important determinants of bipedal gait. Theoretical predictions argue that these collisional losses can be reduced by having “functional extra legs” which are physically the heel and the toe part of a single anatomical foot. The ideal spacing for these “functional legs” are up to a quarter of a stride length, depending on the model employed. We evaluate the foot in the context of the dynamics of a bipedal system and compare predictions of optimal foot size against empirical data from modern humans, the Laetoli footprint trackways, and chimpanzees walking bipedally. The dynamics-based modeling approach provides substantial insight into how, and why, walking works as it does, even though current models are too simple to make predictions at a level adequate to anticipate specific morphology except at the most general level.     

Gait transitions in simulated reduced gravity

Gravity has a strong effect on gait and the speed of gait transitions. A gait has been defined as a pattern of locomotion that changes discontinuously at the transition to another gait. On Earth, during gradual speed changes, humans exhibit a sudden discontinuous switch from walking to running at a specific speed. To study the effects of altered gravity on both the stance and swing legs, we developed a novel unloading exoskeleton that allows a person to step in simulated reduced gravity by tilting the body relative to the vertical. Using different simulation techniques, we confirmed that at lower gravity levels the transition speed is slower (in accordance with the previously reported Froude number ～0.5). Surprisingly, however, we found that at lower levels of simulated gravity the transition between walking and running was generally gradual, without any noticeable abrupt change in gait parameters. This was associated with a significant prolongation of the swing phase, whose duration became virtually equal to that of stance in the vicinity of the walk-run transition speed, and with a gradual shift from inverted-pendulum gait (walking) to bouncing gait (running).     

Steps to Take to Enhance Gait Stability: The Effect of Stride Frequency,Stride Length,and Walking Speed on Local Dynamic Stability and Margins of Stability

The purpose of the current study was to investigate whether adaptations of stride length, stride frequency, and walking speed, independently influence local dynamic stability and the size of the medio-lateral and backward margins of stability during walking. Nine healthy subjects walked 25 trials on a treadmill at different combinations of stride frequency, stride length, and consequently at different walking speeds. Visual feedback about the required and the actual combination of stride frequency and stride length was given during the trials. Generalized Estimating Equations were used to investigate the independent contribution of stride length, stride frequency, and walking speed on the measures of gait stability. Increasing stride frequency was found to enhance medio-lateral margins of stability. Backward margins of stability became larger as stride length decreased or walking speed increased. For local dynamic stability no significant effects of stride frequency, stride length or walking speed were found. We conclude that adaptations in stride frequency, stride length and/or walking speed can result in an increase of the medio-lateral and backward margins of stability, while these adaptations do not seem to affect local dynamic stability. Gait training focusing on the observed stepping strategies to enhance margins of stability might be a useful contribution to programs aimed at fall prevention.     

Patterns of mechanical energy change in tetrapod gait: pendula, springs and work

Kinematic and center of mass (CoM) mechanical variables used to define terrestrial gaits are compared for various tetrapod species. Kinematic variables (limb phase, duty factor) provide important timing information regarding the neural control and limb coordination of various gaits. Whereas, mechanical variables (potential and kinetic energy relative phase, %Recovery, %Congruity) provide insight into the underlying mechanisms that minimize muscle work and the metabolic cost of locomotion, and also influence neural control strategies. Two basic mechanisms identified by Cavagna et al. (1977. Am J Physiol 233:R243-R261) are used broadly by various bipedal and quadrupedal species. During walking, animals exchange CoM potential energy (PE) with kinetic energy (KE) via an inverted pendulum mechanism to reduce muscle work. During the stance period of running (including trotting, hopping and galloping) gaits, animals convert PE and KE into elastic strain energy in spring elements of the limbs and trunk and regain this energy later during limb support. The bouncing motion of the body on the support limb(s) is well represented by a simple mass-spring system. Limb spring compliance allows the storage and return of elastic energy to reduce muscle work. These two distinct patterns of CoM mechanical energy exchange are fairly well correlated with kinematic distinctions of limb movement patterns associated with gait change. However, in some cases such correlations can be misleading. When running (or trotting) at low speeds many animals lack an aerial period and have limb duty factors that exceed 0.5. Rather than interpreting this as a change of gait, the underlying mechanics of the body's CoM motion indicate no fundamental change in limb movement pattern or CoM dynamics has occurred. Nevertheless, the idealized, distinctive patterns of CoM energy fluctuation predicted by an inverted pendulum for walking and a bouncing mass spring for running are often not clear cut, especially for less cursorial species. When the kinematic and mechanical patterns of a broader diversity of quadrupeds and bipeds are compared, more complex patterns emerge, indicating that some animals may combine walking and running mechanics at intermediate speeds or at very large size. These models also ignore energy costs that are likely associated with the opposing action of limbs that have overlapping support times during walking. A recent model of terrestrial gait (Ruina et al., 2005. J Theor Biol, in press) that treats limb contact with the ground in terms of collisional energy loss indicates that considerable CoM energy can be conserved simply by matching the path of CoM motion perpendicular to limb ground force. This model, coupled with the earlier ones of pendular exchange during walking and mass-spring elastic energy savings during running, provides compelling argument for the view that the legged locomotion of quadrupeds and other terrestrial animals has generally evolved to minimize muscle work during steady level movement.     

A bipedal compliant walking model generates periodic gait cycles with realistic swing dynamics

A simple spring mechanics model can capture the dynamics of the center of mass (CoM) during human walking, which is coordinated by multiple joints. This simple spring model, however, only describes the CoM during the stance phase, and the mechanics involved in the bipedality of the human gait are limited. In this study, a bipedal spring walking model was proposed to demonstrate the dynamics of bipedal walking, including swing dynamics followed by the step-to-step transition. The model consists of two springs with different stiffnesses and rest lengths representing the stance leg and swing leg. One end of each spring has a foot mass, and the other end is attached to the body mass. To induce a forward swing that matches the gait phase, a torsional hip joint spring was introduced at each leg. To reflect the active knee flexion for foot clearance, the rest length of the swing leg was set shorter than that of the stance leg, generating a discrete elastic restoring force. The number of model parameters was reduced by introducing dependencies among stiffness parameters. The proposed model generates periodic gaits with dynamics-driven step-to-step transitions and realistic swing dynamics. While preserving the mimicry of the CoM and ground reaction force (GRF) data at various gait speeds, the proposed model emulated the kinematics of the swing leg. This result implies that the dynamics of human walking generated by the actuations of multiple body segments is describable by a simple spring mechanics.     

Similarity in multilegged locomotion: Bouncing like a monopode

Despite impressive variation in leg number, length, position and type of skeleton, similarities of legged, pedestrian locomotion exist in energetics, gait, stride frequency and ground-reaction force. Analysis of data available in the literature showed that a bouncing, spring-mass, monopode model can approximate the energetics and dynamics of trotting, running, and hopping in animals as diverse as cockroaches, quail and kangaroos. From an animal's mechanical-energy fluctuation and ground-reaction force, we calculated the compression of a virtual monopode's leg and its stiffness. Comparison of dimensionless parameters revealed that locomotor dynamics depend on gait and leg number and not on body mass. Relative stiffness per leg was similar for all animals and appears to be a very conservative quantity in the design of legged locomotor systems. Differences in the general dynamics of gait are based largely on the number of legs acting simultaneously to determine the total stiffness of the system. Four- and six-legged trotters had a greater whole body stiffness than two-legged runners operating their systems at about the same relative speed. The greater whole body stiffness in trotters resulted in a smaller compression of the virtual leg and a higher natural frequency and stride frequency.     

Contributions of muscles to mediolateral ground reaction force over a range of walking speeds

Impaired control of mediolateral body motion during walking is an important health concern. Developing treatments to improve mediolateral control is challenging, partly because the mechanisms by which muscles modulate mediolateral ground reaction force (and thereby modulate mediolateral acceleration of the body mass center) during unimpaired walking are poorly understood. To investigate this, we examined mediolateral ground reaction forces in eight unimpaired subjects walking at four speeds and determined the contributions of muscles, gravity, and velocity-related forces to the mediolateral ground reaction force by analyzing muscle-driven simulations of these subjects. During early stance (0-6% gait cycle), peak ground reaction force on the leading foot was directed laterally and increased significantly (p<0.05) with walking speed. During early single support (14-30% gait cycle), peak ground reaction force on the stance foot was directed medially and increased significantly (p<0.01) with speed. Muscles accounted for more than 92% of the mediolateral ground reaction force over all walking speeds, whereas gravity and velocity-related forces made relatively small contributions. Muscles coordinate mediolateral acceleration via an interplay between the medial ground reaction force contributed by the abductors and the lateral ground reaction forces contributed by the knee extensors, plantarflexors, and adductors. Our findings show how muscles that contribute to forward progression and body-weight support also modulate mediolateral acceleration of the body mass center while weight is transferred from one leg to another during double support.