Compliant leg behaviour explains basic dynamics of walking and running

The basic mechanics of human locomotion are associated with vaulting over stiff legs in walking and rebounding on compliant legs in running. However, while rebounding legs well explain the stance dynamics of running, stiff legs cannot reproduce that of walking. With a simple bipedal spring-mass model, we show that not stiff but compliant legs are essential to obtain the basic walking mechanics; incorporating the double support as an essential part of the walking motion, the model reproduces the characteristic stance dynamics that result in the observed small vertical oscillation of the body and the observed out-of-phase changes in forward kinetic and gravitational potential energies. Exploring the parameter space of this model, we further show that it not only combines the basic dynamics of walking and running in one mechanical system, but also reveals these gaits to be just two out of the many solutions to legged locomotion offered by compliant leg behaviour and accessed by energy or speed.     

Extended Evolutionary Fast Learn-to-Walk Approach for Four-Legged Robots

Robot locomotion is an active research area. In this paper we focus on the locomotion of quadruped robots. An effective walking gait of quadruped robots is mainly concerned with two key aspects, namely speed and stability. The large search space of potential parameter settings for leg joints means that hand tuning is not feasible in general. As a result walking parameters are typically determined using machine learning techniques. A major shortcoming of using machine learning techniques is the significant wear and tear of robots since many parameter combinations need to be evaluated before an optimal solution is found.This paper proposes a direct walking gait learning approach, which is specifically designed to reduce wear and tear of robot motors, joints and other hardware. In essence we provide an effective learning mechanism that leads to a solution in a faster convergence time than previous algorithms. The results demonstrate that the new learning algorithm obtains a faster convergence to the best solutions in a short run. This approach is significant in obtaining faster walking gaits which will be useful for a wide range of applications where speed and stability are important. Future work will extend our methods so that the faster convergence algorithm can be applied to a two legged humanoid and lead to less wear and tear whilst still developing a fast and stable gait.     

Stable walking with asymmetric legs

Asymmetric leg function is often an undesired side-effect in artificial legged systems and may reflect functional deficits or variations in the mechanical construction. It can also be found in legged locomotion in humans and animals such as after an accident or in specific gait patterns. So far, it is not clear to what extent differences in the leg function of contralateral limbs can be tolerated during walking or running. Here, we address this issue using a bipedal spring-mass model for simulating walking with compliant legs. With the help of the model, we show that considerable differences between contralateral legs can be tolerated and may even provide advantages to the robustness of the system dynamics. A better understanding of the mechanisms and potential benefits of asymmetric leg operation may help to guide the development of artificial limbs or the design novel therapeutic concepts and rehabilitation strategies.     

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.

     

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.     

Ground reaction forces and center of mass mechanics of bipedal capuchin monkeys: Implications for the evolution of human bipedalism

Tufted capuchin monkeys are known to use both quadrupedalism and bipedalism in their natural environments. Although previous studies have investigated limb kinematics and metabolic costs, their ground reaction forces (GRFs) and center of mass (CoM) mechanics during two and four‐legged locomotion are unknown. Here, we determine the hind limb GRFs and CoM energy, work, and power during bipedalism and quadrupedalism over a range of speeds and gaits to investigate the effect of differential limb number on locomotor performance. Our results indicate that capuchin monkeys use a “grounded run” during bipedalism (0.83–1.43 ms^?1) and primarily ambling and galloping gaits during quadrupedalism (0.91–6.0 ms^?1). CoM energy recoveries are quite low during bipedalism (2–17%), and in general higher during quadrupedalism (4–72%). Consistent with this, hind limb vertical GRFs as well as CoM work, power, and collisional losses are higher in bipedalism than quadrupedalism. The positive CoM work is 2.04 ± 0.40 Jkg^?1 m^?1 (bipedalism) and 0.70 ± 0.29 Jkg^?1 m^?1 (quadrupedalism), which is within the range of published values for two and four‐legged terrestrial animals. The results of this study confirm that facultative bipedalism in capuchins and other nonhuman primates need not be restricted to a pendulum‐like walking gait, but rather can include running, albeit without an aerial phase. Based on these results and similar studies of other facultative bipeds, we suggest that important transitions in the evolution of hominin locomotor performance were the emergences of an obligate, pendulum‐like walking gait and a bouncy running gait that included a whole‐body aerial phase. Am J Phys Anthropol, 2013. © 2012 Wiley Periodicals, Inc.     

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.     

Symmetrical gaits of primates

Walking and symmetrical running gaits of 26 genera of primates are analyzed using numerical and graphical methods described previously. The raw data are 1701 feet of 16 mm motion picture film mostly exposed at 64 frames per second. Adult monkeys and apes usually use the walking trot or diagonal-sequence walks. Individual monkeys occasionally use lateral-sequence walks resembling those that are usual for human infants. Human children moving on hands and feet use gaits ranging from the walking pace through the lateral-sequence walks to the walking trot. An infant macaque studied from age 17 hours to 96 days first walked with a lateral-sequence, diagonal-couplets gait and then gradually shifted to the diagonal-sequence, diagonal-couplets gait of the adult. Few non-primates use the diagonal-sequence walks which are typical of primates. Typical support sequences are figured. Relative placement of feet and consequent slight asymmetry are described.     

The effects of natural substrate discontinuities on the quadrupedal gait kinematics of free‐ranging Saimiri sciureus

Wild primates encounter complex matrices of substrates that differ in size, orientation, height, and compliance, and often move on multiple, discontinuous substrates within a single bout of locomotion. Our current understanding of primate gait is limited by artificial laboratory settings in which primate quadrupedal gait has primarily been studied. This study analyzes wild Saimiri sciureus (common squirrel monkey) gait on discontinuous substrates to capture the realistic effects of the complex arboreal habitat on walking kinematics. We collected high‐speed video footage at Tiputini Biodiversity Station, Ecuador between August and October 2017. Overall, the squirrel monkeys used more asymmetrical walking gaits than symmetrical gaits, and specifically asymmetrical lateral sequence walking gaits when moving across discontinuous substrates. When individuals used symmetrical gaits, they used diagonal sequence gaits more than lateral sequence gaits. In addition, individuals were more likely to change their footfall sequence during strides on discontinuous substrates. Squirrel monkeys increased the time lag between touchdowns both of ipsilaterally paired limbs (pair lag) and of the paired forelimbs (forelimb lag) when walking across discontinuous substrates compared to continuous substrates. Results indicate that gait flexibility and the ability to alter footfall patterns during quadrupedal walking may be critical for primates to safely move in their complex arboreal habitats. Notably, wild squirrel monkey quadrupedalism is diverse and flexible with high proportions of asymmetrical walking. Studying kinematics in the wild is critical for understanding the complexity of primate quadrupedalism.     

Behaviour-based modelling of hexapod locomotion: linking biology and technical application

Walking in insects and most six-legged robots requires simultaneous control of up to 18 joints. Moreover, the number of joints that are mechanically coupled via body and ground varies from one moment to the next, and external conditions such as friction, compliance and slope of the substrate are often unpredictable. Thus, walking behaviour requires adaptive, context-dependent control of many degrees of freedom. As a consequence, modelling legged locomotion addresses many aspects of any motor behaviour in general. Based on results from behavioural experiments on arthropods, we describe a kinematic model of hexapod walking: the distributed artificial neural network controller walknet. Conceptually, the model addresses three basic problems in legged locomotion. (I) First, coordination of several legs requires coupling between the step cycles of adjacent legs, optimising synergistic propulsion, but ensuring stability through flexible adjustment to external disturbances. A set of behaviourally derived leg coordination rules can account for decentralised generation of different gaits, and allows stable walking of the insect model as well as of a number of legged robots. (II) Second, a wide range of different leg movements must be possible, e.g. to search for foothold, grasp for objects or groom the body surface. We present a simple neural network controller that can simulate targeted swing trajectories, obstacle avoidance reflexes and cyclic searching-movements. (III) Third, control of mechanically coupled joints of the legs in stance is achieved by exploiting the physical interactions between body, legs and substrate. A local positive displacement feedback, acting on individual leg joints, transforms passive displacement of a joint into active movement, generating synergistic assistance reflexes in all mechanically coupled joints.     

How locomotion sub-functions can control walking at different speeds?

Inspired from template models explaining biological locomotory systems and Raibert׳s pioneering legged robots, locomotion can be realized by basic sub-functions: elastic axial leg function, leg swinging and balancing. Combinations of these three can generate different gaits with diverse properties. In this paper we investigate how locomotion sub-functions contribute to stabilize walking at different speeds. Based on this trilogy, we introduce a conceptual model to quantify human locomotion sub-functions in walking. This model can produce stable walking and also predict human locomotion sub-function control during swing phase of walking. Analyzing experimental data based on this modeling shows different control strategies which are employed to increase speed from slow to moderate and moderate to fast gaits.     

Computational Models to Synthesize Human Walking

The synthesis of human walking is of great interest in biomechanics and biomimetic engineering due to its predictive capabilities and potential applications in clinical biomechanics, rehabilitation engineering and biomimetic robotics. In this paper, the various methods that have been used to synthesize humanwalking are reviewed from an engineering viewpoint. This involves a wide spectrum of approaches, from simple passive walking theories to large-scale computational models integrating the nervous, muscular and skeletal systems. These methods are roughly categorized under four headings： models inspired by the concept of a CPG （Central Pattern Generator）, methods based on the principles of control engineering, predictive gait simulation using optimisation, and models inspired by passive walking theory. The shortcomings and advantages of these methods are examined, and future directions are discussed in the context of providing insights into the neural control objectives driving gait and improving the stability of the predicted gaits. Future advancements are likely to be motivated by improved understanding of neural control strategies and the subtle complexities of the musculoskeletal system during human locomotion. It is only a matter of time before predictive gait models become a practical and valuable tool in clinical diagnosis, rehabilitation engineering and robotics.     

An Experimental Study on the Gait Patterns and Kinematics of Chinese Mitten Crabs

Despite the many studies on eight-legged animals and the importance of their mechanics of terrestrial locomotion, the mechanical energy of crabs in voluntary locomotion on uneven, unpredictable terrain surfaces has received little attention thus far. In this paper, motion video images of Chinese mitten crab (Eriocheir sinensis Milne-Edwards) locomotion on five types of terrains were recorded using a high-speed three-dimensional (3D) recording video system. The typical variables of locomotion such as gait patterns, duty factor, mechanical energy of the mass center, mass-specific rate of the total mechanical power of the mass center, and percentage recovery, were analyzed. Results show that the Chinese mitten crab uses random gaits instead of the alternating tetrapod gait with the increasing terrain roughness. The duty factors of the rows of the leading legs are greater for all terrains than those of the rows of the trailing legs. On smooth terrain, the duty factors of the rows of the trailing legs are greater than that on rough terrains. Kinematic measurements and calculations reveal that similar to mammals, birds, and arthropods, the Chinese mitten crab uses two fundamental gaits to save mechanical energy: the inverted pendulum gait and the bouncing gait. The bouncing gait is the main pattern of mechanical energy conservation. The low probability of injury and energy expenditure due to adaptations to various terrains induce the Chinese mitten crab to modify the mass-specific rate of the total mechanical power of the mass center. The statistical results of percentage recovery also reveal that the Chinese mitten crab has lower energy recovery efficiency over rough terrains compared with smooth terrains.     

A model of bipedal locomotion on compliant legs.

Simple mathematical models capable of walking or running are used to compare the merits of bipedal gaits. Stride length, duty factor (the fraction of the stride, for which the foot is on the ground) and the pattern of force on the ground are varied, and the optimum gait is deemed to be the one that minimizes the positive work that the muscles must perform, per unit distance travelled. Even the simplest model, whose legs have neither mass nor elastic compliance, predicts the changes of duty factor and force pattern that people make as they increase their speed of walking. It predicts a sudden change to running at a critical speed, but this is much faster than the speed at which people make the change. When elastic compliance is incorporated in the model, unnaturally fast walking becomes uncompetitive. However, a slow run with very brief foot contact becomes the optimum gait at low speeds, at which people would walk, unless severe energy dissipation occurs in the compliance. A model whose legs have mass as well as elastic compliance predicts well the relationship between speed and stride length in human walking.     

Bird terrestrial locomotion as revealed by 3D kinematics

Most birds use at least two modes of locomotion: flying and walking (terrestrial locomotion). Whereas the wings and tail are used for flying, the legs are mainly used for walking. The role of other body segments remains, however, poorly understood. In this study, we examine the kinematics of the head, the trunk, and the legs during terrestrial locomotion in the quail (Coturnix coturnix). Despite the trunk representing about 70% of the total body mass, its function in locomotion has received little scientific interest to date. This prompted us to focus on its role in terrestrial locomotion. We used high-speed video fluoroscopic recordings of quails walking at voluntary speeds on a trackway. Dorso-ventral and lateral views of the motion of the skeletal elements were recorded successively and reconstructed in three dimensions using a novel method based on the temporal synchronisation of both views. An analysis of the trajectories of the body parts and their coordination showed that the trunk plays an important role during walking. Moreover, two sub-systems participate in the gait kinematics: (i) the integrated 3D motion of the trunk and thighs allows for the adjustment of the path of the centre of mass; (ii) the motion of distal limbs transforms the alternating forward motion of the feet into a continuous forward motion at the knee and thus assures propulsion. Finally, head bobbing appears qualitatively synchronised to the movements of the trunk. An important role for the thigh muscles in generating the 3D motion of the trunk is suggested by an analysis of the pelvic anatomy.     

Stability in legged locomotion

Stability is a key element in a gait synthesis. Static stability margins are widely adopted in crawlers, while no similar approach has been developed for dynamically stable systems. Utilizing an analytical approach, we developed a set of easy-to-calculate stability indices to describe instantaneous static and dynamic (In)stability for a certain group of walking systems. The analysis is based on a thorough analysis of the interaction between ground reaction forces and the walking system. The indices are applicable to walking systems regardless of the number of legs or mechanical/biological design. We show that static and dynamic stability are independent of each other. We suggest a possible categorization of gait modes based on stability. Stability characteristics are analyzed in a healthy and highly pathological human gait. Finally, we discuss the applicability of the proposed methods.     

The locomotion of the camel (Camelus dromedarius)

The feet and gaits of many camels Camelus dromedarius were studied and filmed in Mauritania, Africa. The camel has a digitigrade stance, large feet to support the animal in soft sand, and soles of flexible pads that step readily onto small stones where necessary. The walking stride is long and slow, with the body supported for much of each stride on the two right or two left legs. The pattern of supporting legs was significantly different in slow compared to fast walking camels, and in young compared to adult camels and compared to adults pulling water at the wells. There was no difference in pattern in one individual's walk, when it was either loaded or unloaded. The angles that the leg bones made with each other and with the horizon are depicted for the walk and the pace. The camel is the only animal which paces often and never trots. The pace is an unstable gait only suitable for flat terrain such as that in deserts. It may have evolved from the pace-like walk which is by far the dominant gait in this animal, which spends most of each day walking from plant to plant browsing or grazing. The pace is not used by all camelids, as one author has claimed. The pace and the gallop were only used by the camels at wells, when the animals were chased from the water by men.     

Footfall patterns and interlimb co-ordination in opossums (Family Didelphidae): evidence for the evolution of diagonal-sequence walking gaits in primates

Most primates typically use a diagonal-sequence footfall pattern during walking. This footfall pattern, which is unusual for mammals, is believed to have originated in ancestral primates in association with the use of grasping extremities for movement and foraging on thin, flexible branches. This theory was tested by comparing gait parameters between the grey short-tailed opossum Monodelphis domestica and the woolly opossum Caluromys philander , two didelphid marsupials that are strongly differentiated in grasping morphology of the extremities and in their reliance on foraging strategies involving thin branches. One hundred and thirty gait cycles were analysed quantitatively from videotapes of subjects moving quadrupedally on a runway and on poles of different diameters (7 and 28 mm). Duty factor (i.e. duration of the stance phase as a percentage of the stride period) for the forelimb and hindlimb, as well as diagonality (i.e. phase relationship between the forelimb and hindlimb cycles), were calculated for each of these symmetrical gait cycles. We found that the highly terrestrial Monodelphis , like most other non-primate mammals, relies primarily on lateral-sequence walking gaits on both runway and poles, and has relatively higher forelimb duty factors. Like primates, the highly arboreal Caluromys uses primarily diagonal-sequence walking gaits on the runway and pole, with relatively higher hindlimb duty factors and diagonality. The fact that the woolly opossum, a marsupial with primate-like feet that moves and forages mainly on thin branches, uses primarily diagonal-sequence gaits when walking supports the view that primate gaits evolved to meet the demands of locomotion on narrow supports. This also demonstrates the functional role of a grasping foot, in association with relatively higher hindlimb duty factors, protraction, and substrate reaction forces, in the production of such walking gaits.     

Biped gait stabilization via foot placement

It is shown that stable biped gaits can be achieved by discrete foot placement based on feedback of information available at the time of foot placement. The model, developed by Townsend (1981, J. Biomechanics 14, p. 727) to evaluate the coordinations of torso motions, subsumes most of the salient body members and motions. The modeling yielded a generalized inverted pendulum with a movable support point which physically defines lateral foot placement. The principal result is that stable gaits can be defined by foot placements which are a linear function of the system center of mass position and velocity at the time of foot placement (only). Gaits may be 'smooth' or may have impulsive corrections to adjust the character of the motions and foot placement. Several general algorithms and specific simulations are presented, and calculations for non-impulsive gaits and impulsive corrections are presented. The model predictions are compared with published data. The predictions are sufficiently close to the data such that the general algorithms appear to be validated. Of particular interest are the non-sinusoidal character of the motions and the relatively simple algorithms. Indeed, the simplicity of the algorithms suggests the practical possibility of legged mobile robots. Accordingly, further investigation seems warranted for determining the parametric variation and control of gait. Some attention is also given to continuous-feedback control such as would exist during double-leg support and in specialized tasks such as rope walking or skating. Subsequent investigation will consider superposition of single and double leg support, although clearly the discrete gaits pose the more restrictive stability problem.