An analytical estimation of the energy cost for legged locomotion

Legged locomotion requires the determination of a number of parameters such as stride period, stride length, order of leg movements, leg trajectory, etc. How are these parameters determined? It has been reported that the locomotor patterns of many legged animals exhibit common characteristics, which suggests that there exists a basic strategy for legged locomotion. In this study we derive an equation to estimate the cost of transport for legged locomotion and examine a criterion of the minimization of the transport cost as a candidate of the strategy. The obtained optimal locomotor pattern that minimizes the cost suitably represents many characteristics of the pattern observed in legged animals. This suggests that the locomotor pattern of legged animals is well optimized with regard to the energetic cost. The result also suggests that the existence of specific gait patterns and the phase transition between them could be the result due to optimization; they are induced by the change in the distribution of ground reaction forces for each leg during locomotion.     

A collisional model of the energetic cost of support work qualitatively explains leg sequencing in walking and galloping, pseudo-elastic leg behavior in running and the walk-to-run transition

Terrestrial legged locomotion requires repeated support forces to redirect the body's vertical velocity component from down to up. We assume that the redirection is accomplished by impulsive leg forces that cause small-angle glancing collisions of a point-mass model of the animal. We estimate the energetic costs of these collisions by assuming a metabolic cost proportional to positive muscle work involved in generating the impulses. The cost of bipedal running estimated from this collisional model becomes less than that of walking at a Froude number (v2/gl) of about 0.7. Two strategies to reduce locomotion costs associated with the motion redirection are: (1) having legs simulate purely elastic springs, as is observed in human running; and (2) sequencing the leg forces during the redirection phase; examples of this sequencing are the ba-da-dump pattern of a horse gallop and having push-off followed by heel-strike in human walking.     

Posture, gait and the ecological relevance of locomotor costs and energy-saving mechanisms in tetrapods

A reanalysis of locomotor data from functional, energetic, mechanical and ecological perspectives reveals that limb posture has major effects on limb biomechanics, energy-saving mechanisms and the costs of locomotion. Regressions of data coded by posture (crouched vs. erect) reveal nonlinear patterns in metabolic cost, limb muscle mass, effective mechanical advantage, and stride characteristics. In small crouched animals energy savings from spring and pendular mechanisms are inconsequential and thus the metabolic cost of locomotion is driven by muscle activation costs. Stride frequency appears to be the principal functional parameter related to the decreasing cost of locomotion in crouched animals. By contrast, the shift to erect limb postures invoked a series of correlated effects on the metabolic cost of locomotion: effective mechanical advantage increases, relative muscle masses decrease, metapodial limb segments elongate dramatically (as limbs shift from digitigrade to unguligrade designs) and biological springs increase in size and effectiveness. Each of these factors leads to decreases in the metabolic cost of locomotion in erect forms resulting from real and increasing contributions of pendular savings and spring savings. Comparisons of the relative costs and ecological relevance of different gaits reveal that running is cheaper than walking in smaller animals up to the size of dogs but running is more expensive than walking in horses. Animals do not necessarily use their cheapest gaits for their predominant locomotor activity. Therefore, locomotor costs are driven more by ecological relevance than by the need to optimize locomotor economy.     

Whole Body Mechanics of Stealthy Walking in Cats

The metabolic cost associated with locomotion represents a significant part of an animal''s metabolic energy budget. Therefore understanding the ways in which animals manage the energy required for locomotion by controlling muscular effort is critical to understanding limb design and the evolution of locomotor behavior. The assumption that energetic economy is the most important target of natural selection underlies many analyses of steady animal locomotion, leading to the prediction that animals will choose gaits and postures that maximize energetic efficiency. Many quadrupedal animals, particularly those that specialize in long distance steady locomotion, do in fact reduce the muscular contribution required for walking by adopting pendulum-like center of mass movements that facilitate exchange between kinetic energy (KE) and potential energy (PE) [1]–[4]. However, animals that are not specialized for long distance steady locomotion may face a more complex set of requirements, some of which may conflict with the efficient exchange of mechanical energy. For example, the “stealthy” walking style of cats may demand slow movements performed with the center of mass close to the ground. Force plate and video data show that domestic cats (Felis catus, Linnaeus, 1758) have lower mechanical energy recovery than mammals specialized for distance. A strong negative correlation was found between mechanical energy recovery and diagonality in the footfalls and there was also a negative correlation between limb compression and diagonality of footfalls such that more crouched postures tended to have greater diagonality. These data show a previously unrecognized mechanical relationship in which crouched postures are associated with changes in footfall pattern which are in turn related to reduced mechanical energy recovery. Low energy recovery was not associated with decreased vertical oscillations of the center of mass as theoretically predicted, but rather with posture and footfall pattern on the phase relationship between potential and kinetic energy. An important implication of these results is the possibility of a tradeoff between stealthy walking and economy of locomotion. This potential tradeoff highlights the complex and conflicting pressures that may govern the locomotor choices that animals make.     

A unified theory for the energy cost of legged locomotion

Small animals are remarkably efficient climbers but comparatively poor runners, a well-established phenomenon in locomotor energetics that drives size-related differences in locomotor ecology yet remains poorly understood. Here, I derive the energy cost of legged locomotion from two complementary components of muscle metabolism, Activation–Relaxation and Cross-bridge cycling. A mathematical model incorporating these costs explains observed patterns of locomotor cost both within and between species, across a broad range of animals (insects to ungulates), for a wide range of substrate slopes including level running and vertical climbing. This ARC model unifies work- and force-based models for locomotor cost and integrates whole-organism locomotor cost with cellular muscle physiology, creating a predictive framework for investigating evolutionary and ecological pressures shaping limb design and ranging behaviour.     

Convergent evolution and locomotion through complex terrain by insects, vertebrates and robots

Arthropods are the most successful members of the animal kingdom largely because of their ability to move efficiently through a range of environments. Their agility has not been lost on engineers seeking to design agile legged robots. However, one cannot simply copy mechanical and neural control systems from insects into robotic designs. Rather one has to select the properties that are critical for specific behaviors that the engineer wants to capture in a particular robot. Convergent evolution provides an important clue to the properties of legged locomotion that are critical for success. Arthropods and vertebrates evolved legged locomotion independently. Nevertheless, many neural control properties and mechanical schemes are remarkably similar. Here we describe three aspects of legged locomotion that are found in both insects and vertebrates and that provide enhancements to legged robots. They are leg specialization, body flexion and the development of a complex head structure. Although these properties are commonly seen in legged animals, most robotic vehicles have similar legs throughout, rigid bodies and rudimentary sensors on what would be considered the head region. We describe these convergent properties in the context of robots that we developed to capture the agility of insects in moving through complex terrain.     

Metabolic implications of a 'run now,pay later' strategy in lizards: an analysis of post-exercise oxygen consumption

Lizards and many other animals often engage in locomotor behaviors that are of such short duration that physiological steady-state conditions are not attained. It is sometimes difficult to estimate the energetic costs of this type of locomotor activity. This difficulty is addressed by considering as reflective of the metabolic cost of activity (C(act)) not only the oxygen consumed during the activity itself, but also the excess post-exercise oxygen consumption (EPOC) and any excess metabolites persisting at the end of EPOC. Data from both lizards and mammals demonstrate that EPOC is the major energetic cost when activity is short and intense. This paper evaluates the major metabolic components of EPOC in lizards. We then examine how behavioral variables associated with locomotion (duration, intensity, frequency) can influence EPOC and C(act). Short and intense activity is much more expensive by this measure than is steady-state locomotion. Evidence is provided that intermittent activity of short duration can be more economical relative to single bouts of the same activity. Metabolic savings appear greatest when the pause period between behaviors is short. In contrast, endurance is enhanced by short activity periods and longer pause periods, suggesting a tradeoff between endurance and EPOC-related metabolic costs.     

Locomotor development in the Svalbard rock ptarmigan (Lagopus muta hyperborea)

Juvenile animals often suffer from high levels of predation. Development of an effective and efficient locomotor system is therefore likely to be crucial towards ensuring their survival. However, our understanding of locomotor efficiency, at least in terms of energetic cost in young animals is poor. We performed this study as Svalbard rock ptarmigan, Lagopus muta hyperborea must rapidly develop the ability to locomote prior to the onset of their first winter, during which conditions are extreme. To aid survival, adult ptarmigan deposit large winter fat stores, whilst at the same time males exhibit a reduced metabolic cost of locomotion. Sub-adult males, however, are unable to fully acquire fat stores during their first winter and the maturity of their locomotor systems is unknown. Here, we investigate the energetics and kinematics of terrestrial locomotion in sub-adult male birds using flow-through respirometry and high-speed video recordings, respectively. We demonstrate that in terms of running speed and metabolic cost, sub-adult ptarmigan develop a mature functioning locomotor system prior to the onset of winter. This research indicates that achieving a mature locomotor system allows young males to emerge from the winter with the ability to compete for territories and mates during the breeding season.     

Locomotor Performance Varies With Adult Phenotype in Ornamented/Non‐Ornamented Wolf Spiders

Locomotor performance constitutes a major component of whole‐animal performance and is involved in several fitness‐related behaviors. Locomotor capabilities may also correspond positively or negatively to sexually selected traits (e.g., male ornamentation and/or courtship displays). Negative correlations are predicted if secondary sexual traits take the form of morphological modifications that impose physical or energetic limitations. We tested this cost of secondary sexual traits by comparing locomotor performance in male Schizocosa wolf spiders that exhibit two distinct phenotypes. These phenotypes vary in the presence/absence of a morphological feature assumed to function as sexual ornamentation—foreleg brushes. Given the conspicuously large brushes of hair on the brush‐legged phenotype, we expected these males to suffer in locomotor performance. We tested this cost by comparing locomotor performance among male phenotypes (brush‐legged and non‐ornamented) and females at immature and adult life stages. We did not find strong support for costs of brushes on locomotion. First, brush‐legged males showed similar average speeds and endurance as both non‐ornamented males and females. Second, while brush‐legged males were slower at maximum speeds than non‐ornamented males as matures (but not as immatures), they were no slower than mature females. Further, we found no variation in endurance among phenotypes or life stages. Finally, brush size did not correspond to speed. Patterns of morphological variation in traits other than ornamentation may explain these patterns: when morphological variation in leg lengths was accounted for, differences in maximum speed among groups disappeared. We suggest that the faster speeds achieved by non‐ornamented males arise as a by‐product of selection on morphology and musculature potentially necessary for vigorous courtship.     

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.     

Energetic output of subadult polar bears (Ursus maritimus): resting, disturbance and locomotion.

1. Indirect calorimetry was used to determine metabolic rates in subadult polar bears at rest after human-controlled disturbance and at four rates of locomotion. 2. Disturbance factors that do not result in locomotion would only have a significant effect on energy expenditure if they occurred over an extended period of time. 3. Human disturbance resulting in locomotion would have a relatively high energetic cost to individual animals. 4. Polar bears may require a relatively high energetic output to initiate walking.     

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.     

Joint stiffness of the ankle and the knee in running

The spring-mass model is a valid fundament to understand global dynamics of fast legged locomotion under gravity. The underlying concept of elasticity, implying leg stiffness as a crucial parameter, is also found on lower motor control levels, i.e. in muscle-reflex and muscle-tendon systems. Therefore, it seems reasonable that global leg stiffness emerges from local elasticity established by appropriate joint torques. A recently published model of an elastically operating, segmented leg predicts that proper adjustment of joint elasticities to the leg geometry and initial conditions of ground contact provides internal leg stability. Another recent study suggests that in turn the leg segmentation and the initial conditions may be a consequence of metabolic and bone stress constraints. In this study, the theoretical predictions were verified experimentally with respect to initial conditions and elastic joint characteristics in human running. Kinematics and kinetics were measured and the joint torques were estimated by inverse dynamics. Stiffnesses and elastic nonlinearities describing the resulting joint characteristics were extracted from parameter fits. Our results clearly support the theoretical predictions: the knee joint is always stiffer and more extended than the ankle joint. Moreover, the knee torque characteristic on the average shows the higher nonlinearity. According to literature, the leg geometry is a consequence of metabolic and material stress limitations. Adapted to this given geometry, the initial joint angle conditions in fast locomotion are a compromise between metabolic and control effort minimisation. Based on this adaptation, an appropriate joint stiffness ratio between ankle and knee passively safeguards the internal leg stability. The identified joint nonlinearities contribute to the linearisation of the leg spring.     

Intersegmental transfer of sensory signals in the stick insect leg muscle control system

Intersegmental coordination during locomotion in legged animals arises from mechanical couplings and the exchange of neuronal information between legs. Here, the information flow from a single leg sense organ of the stick insect Cuniculina impigra onto motoneurons and interneurons of other legs was investigated. The femoral chordotonal organ (fCO) of the right middle leg, which measures posture and movement of the femur-tibia joint, was stimulated, and the responses of the tibial motoneuron pools of the other legs were recorded. In resting animals, fCO signals did not affect motoneuronal activity in neighboring legs. When the locomotor system was activated and antagonistic motoneurons were bursting in alternation, fCO stimuli facilitated transitions from flexor to extensor activity and vice versa in the contralateral leg. Following pharmacological treatment with picrotoxin, a blocker of GABA-ergic inhibition, the tibial motoneurons of all legs showed specific responses to signals from the middle leg fCO. For the contralateral middle leg we show that fCO signals encoding velocity and position of the tibia were processed by those identified local premotor nonspiking interneurons known to contribute to posture and movement control during standing and voluntary leg movements. Interneurons received both excitatory and inhibitory inputs, so that the response of some interneurons supported the motoneuronal output, while others opposed it. Our results demonstrate that sensory information from the fCO specifically affects the motoneuronal activity of other legs and that the layer of premotor nonspiking interneurons is a site of interaction between local proprioceptive sensory signals and proprioceptive signals from other legs.     

BigDog-inspired studies in the locomotion of goats and dogs

Collision-based expenditure of mechanical energy and the compliance and geometry of the leg are fundamental, interrelated considerations in the mechanical design of legged runners. This article provides a basic context and rationale for experiments designed to inform each of these key areas in Boston Dynamic's BigDog robot. Although these principles have been investigated throughout the past few decades within different academic disciplines, BigDog required that they be considered together and in concert with an impressive set of control algorithms that are not discussed here. Although collision reduction is an important strategy for reducing mechanical cost of transport in the slowest and fastest quadrupedal gaits, walking and galloping, BigDog employed an intermediate-speed trotting gait without collision reduction. Trotting, instead, uses a spring-loaded inverted pendulum mechanism with potential for storage and return of elastic strain energy in appropriately compliant structures. Rather than tuning BigDog's built-in leg springs according to a spring-mass model-based virtual leg-spring constant , a much stiffer distal leg spring together with actuation of the adjacent joint provided good trotting dynamics and avoided functional limitations that might have been imposed by too much compliance in real-world terrain. Adjusting the directional compliance of the legs by adopting a knee-forward, elbow-back geometry led to more robust trotting dynamics by reducing perturbations about the pitch axis of the robot's center of mass (CoM). BigDog is the most successful large-scale, all-terrain trotting machine built to date and it continues to stimulate our understanding of legged locomotion in comparative biomechanics as well as in robotics.     

Elastic energy savings and active energy cost in a simple model of running

The energetic economy of running benefits from tendon and other tissues that store and return elastic energy, thus saving muscles from costly mechanical work. The classic “Spring-mass” computational model successfully explains the forces, displacements and mechanical power of running, as the outcome of dynamical interactions between the body center of mass and a purely elastic spring for the leg. However, the Spring-mass model does not include active muscles and cannot explain the metabolic energy cost of running, whether on level ground or on a slope. Here we add explicit actuation and dissipation to the Spring-mass model, and show how they explain substantial active (and thus costly) work during human running, and much of the associated energetic cost. Dissipation is modeled as modest energy losses (5% of total mechanical energy for running at 3 m s^-1) from hysteresis and foot-ground collisions, that must be restored by active work each step. Even with substantial elastic energy return (59% of positive work, comparable to empirical observations), the active work could account for most of the metabolic cost of human running (about 68%, assuming human-like muscle efficiency). We also introduce a previously unappreciated energetic cost for rapid production of force, that helps explain the relatively smooth ground reaction forces of running, and why muscles might also actively perform negative work. With both work and rapid force costs, the model reproduces the energetics of human running at a range of speeds on level ground and on slopes. Although elastic return is key to energy savings, there are still losses that require restorative muscle work, which can cost substantial energy during running.     

Locomotor energetics in primates: gait mechanics and their relationship to the energetics of vertical and horizontal locomotion

All primates regularly move within three-dimensional arboreal environments and must often climb, but little is known about the energetic costs of this critical activity. Limited previous work on the energetics of incline locomotion suggests that there may be differential selective pressures for large compared to small primates in choosing to exploit a complex arboreal environment. Necessary metabolic and gait data have never been collected to examine this possibility and biomechanical mechanisms that might explain size-based differences in the cost of arboreal movement. Energetics and kinematics were collected for five species of primate during climbing and horizontal locomotion. Subjects moved on a treadmill with a narrow vertical substrate and one with a narrow horizontal substrate at their maximum sustainable speed for 10–20 min while oxygen consumption was monitored. Data during climbing were compared to those during horizontal locomotion and across size. Results show that climbing energetic costs were similar to horizontal costs for small primates (<0.5 kg) but were nearly double for larger species. Spatio-temporal gait characteristics suggest that the relationship between the cost of locomotion and the rate of force production changes between the two locomotor modes. Thus, the main determinants of climbing costs are fundamentally different from those during horizontal locomotion. These new results combining spatiotemporal and energetic data confirm and expand on our previous argument (Hanna et al.: Science 320 (2008) 898) that similar costs of horizontal and vertical locomotion in small primates facilitated the successful occupation of a fine-branch arboreal milieu by the earliest primates.     

More than energy cost: multiple benefits of the long Achilles tendon in human walking and running

Elastic strain energy that is stored and released from long, distal tendons such as the Achilles during locomotion allows for muscle power amplification as well as for reduction of the locomotor energy cost: as distal tendons perform mechanical work during recoil, plantar flexor muscle fibres can work over smaller length ranges, at slower shortening speeds, and at lower activation levels. Scant evidence exists that long distal tendons evolved in humans (or were retained from our more distant Hominoidea ancestors) primarily to allow high muscle–tendon power outputs, and indeed we remain relatively powerless compared to many other species. Instead, the majority of evidence suggests that such tendons evolved to reduce total locomotor energy cost. However, numerous additional, often unrecognised, advantages of long tendons may speculatively be of greater evolutionary advantage, including the reduced limb inertia afforded by shorter and lighter muscles (reducing proximal muscle force requirement), reduced energy dissipation during the foot–ground collisions, capacity to store and reuse the muscle work done to dampen the vibrations triggered by foot–ground collisions, reduced muscle heat production (and thus core temperature), and attenuation of work-induced muscle damage. Cumulatively, these effects should reduce both neuromotor fatigue and sense of locomotor effort, allowing humans to choose to move at faster speeds for longer. As these benefits are greater at faster locomotor speeds, they are consistent with the hypothesis that running gaits used by our ancestors may have exerted substantial evolutionary pressure on Achilles tendon length. The long Achilles tendon may therefore be a singular adaptation that provided numerous physiological, biomechanical, and psychological benefits and thus influenced behaviour across multiple tasks, both including and additional to locomotion. While energy cost may be a variable of interest in locomotor studies, future research should consider the broader range of factors influencing our movement capacity, including our decision to move over given distances at specific speeds, in order to understand more fully the effects of Achilles tendon function as well as changes in this function in response to physical activity, inactivity, disuse and disease, on movement performance.     

Forward dynamic simulation of bipedal walking in the Japanese macaque: investigation of causal relationships among limb kinematics, speed, and energetics of bipedal locomotion in a nonhuman primate

Japanese macaques that have been trained for monkey performances exhibit a remarkable ability to walk bipedally. In this study, we dynamically reconstructed bipedal walking of the Japanese macaque to investigate causal relationships among limb kinematics, speed, and energetics, with a view to understanding the mechanisms underlying the evolution of human bipedalism. We constructed a two-dimensional macaque musculoskeletal model consisting of nine rigid links and eight principal muscles. To generate locomotion, we used a trajectory-tracking control law, the reference trajectories of which were obtained experimentally. Using this framework, we evaluated the effects of changes in cycle duration and gait kinematics on locomotor efficiency. The energetic cost of locomotion was estimated based on the calculation of mechanical energy generated by muscles. Our results demonstrated that the mass-specific metabolic cost of transport decreased as speed increased in bipedal walking of the Japanese macaque. Furthermore, the cost of transport in bipedal walking was reduced when vertical displacement of the hip joint was virtually modified in the simulation to be more humanlike. Human vertical fluctuations in the body's center of mass actually contributed to energy savings via an inverted pendulum mechanism.