Bipedal walking and running with spring-like biarticular muscles

Compliant elements in the leg musculoskeletal system appear to be important not only for running but also for walking in human locomotion as shown in the energetics and kinematics studies of spring-mass model. While the spring-mass model assumes a whole leg as a linear spring, it is still not clear how the compliant elements of muscle-tendon systems behave in a human-like segmented leg structure. This study presents a minimalistic model of compliant leg structure that exploits dynamics of biarticular tension springs. In the proposed bipedal model, each leg consists of three leg segments with passive knee and ankle joints that are constrained by four linear tension springs. We found that biarticular arrangements of the springs that correspond to rectus femoris, biceps femoris and gastrocnemius in human legs provide self-stabilizing characteristics for both walking and running gaits. Through the experiments in simulation and a real-world robotic platform, we show how behavioral characteristics of the proposed model agree with basic patterns of human locomotion including joint kinematics and ground reaction force, which could not be explained in the previous models.     

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.     

Dynamics and stability of legged locomotion in the horizontal plane: a test case using insects

 Motivated by experimental studies of insects, we propose a model for legged locomotion in the horizontal plane. A three-degree-of freedom, energetically conservative, rigid-body model with a pair of compliant virtual legs in intermittent contact with the ground allows us to study how dynamics depends on parameters such as mass, moment of inertia, leg stiffness, and length. We find periodic gaits, and show that mechanics alone can confer asymptotic stability of relative heading and body angular velocity. We discuss the relevance of our idealized models to experiments and simulations on insect running, showing that their gait and force characteristics match observations reasonably well. We perform parameter studies and suggest that our model is relevant to the understanding of locomotion dynamics across species. Received: 17 April 2001 / Accepted in revised form: 20 November 2001     

Tuataras and salamanders show that walking and running mechanics are ancient features of tetrapod locomotion

The lumbering locomotor behaviours of tuataras and salamanders are the best examples of quadrupedal locomotion of early terrestrial vertebrates. We show they use the same walking (out-of-phase) and running (in-phase) patterns of external mechanical energy fluctuations of the centre-of-mass known in fast moving (cursorial) animals. Thus, walking and running centre-of-mass mechanics have been a feature of tetrapods since quadrupedal locomotion emerged over 400 million years ago. When walking, these sprawling animals save external mechanical energy with the same pendular effectiveness observed in cursorial animals. However, unlike cursorial animals (that change footfall patterns and mechanics with speed), tuataras and salamanders use only diagonal couplet gaits and indifferently change from walking to running mechanics with no significant change in total mechanical energy. Thus, the change from walking to running is not related to speed and the advantage of walking versus running is unclear. Furthermore, lumbering mechanics in primitive tetrapods is reflected in having total mechanical energy driven by potential energy (rather than kinetic energy as in cursorial animals) and relative centre-of-mass displacements an order of magnitude greater than cursorial animals. Thus, large vertical displacements associated with lumbering locomotion in primitive tetrapods may preclude their ability to increase speed.     

Leg stiffness increases with speed to modulate gait frequency and propulsion energy

Bipedal walking models with compliant legs have been employed to represent the ground reaction forces (GRFs) observed in human subjects. Quantification of the leg stiffness at varying gait speeds, therefore, would improve our understanding of the contributions of spring-like leg behavior to gait dynamics. In this study, we tuned a model of bipedal walking with damped compliant legs to match human GRFs at different gait speeds. Eight subjects walked at four different gait speeds, ranging from their self-selected speed to their maximum speed, in a random order. To examine the correlation between leg stiffness and the oscillatory behavior of the center of mass (CoM) during the single support phase, the damped natural frequency of the single compliant leg was compared with the duration of the single support phase. We observed that leg stiffness increased with speed and that the damping ratio was low and increased slightly with speed. The duration of the single support phase correlated well with the oscillation period of the damped complaint walking model, suggesting that CoM oscillations during single support may take advantage of resonance characteristics of the spring-like leg. The theoretical leg stiffness that maximizes the elastic energy stored in the compliant leg at the end of the single support phase is approximated by the empirical leg stiffness used to match model GRFs to human GRFs. This result implies that the CoM momentum change during the double support phase requires maximum forward propulsion and that an increase in leg stiffness with speed would beneficially increase the propulsion energy. Our results suggest that humans emulate, and may benefit from, spring-like leg mechanics.     

Kinematics of lower limbs during walking are emulated by springy walking model with a compliantly connected,off-centered curvy foot

The dynamics of the center of mass (CoM) during walking and running at various gait conditions are well described by the mechanics of a simple passive spring loaded inverted pendulum (SLIP). Due to its simplicity, however, the current form of the SLIP model is limited at providing any further information about multi-segmental lower limbs that generate oscillatory CoM behaviors and their corresponding ground reaction forces. Considering that the dynamics of the CoM are simply achieved by mass-spring mechanics, we wondered whether any of the multi-joint motions could be demonstrated by simple mechanics. In this study, we expand a SLIP model of human locomotion with an off-centered curvy foot connected to the leg by a springy segment that emulates the asymmetric kinematics and kinetics of the ankle joint. The passive dynamics of the proposed expansion of the SLIP model demonstrated the empirical data of ground reaction forces, center of mass trajectories, ankle joint kinematics and corresponding ankle joint torque at various gait speeds. From the mechanically simulated trajectories of the ankle joint and CoM, the motion of lower-limb segments, such as thigh and shank angles, could be estimated from inverse kinematics. The estimation of lower limb kinematics showed a qualitative match with empirical data of walking at various speeds. The representability of passive compliant mechanics for the kinetics of the CoM and ankle joint and lower limb joint kinematics implies that the coordination of multi-joint lower limbs during gait can be understood with a mechanical framework.     

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).     

Modeling locomotion of a soft-bodied arthropod using inverse dynamics

Most bio-inspired robots have been based on animals with jointed, stiff skeletons. There is now an increasing interest in mimicking the robust performance of animals in natural environments by incorporating compliant materials into the locomotory system. However, the mechanics of moving, highly conformable structures are particularly difficult to predict. This paper proposes a planar, extensible-link model for the soft-bodied tobacco hornworm caterpillar, Manduca sexta, to provide insight for biologists and engineers studying locomotion by highly deformable animals and caterpillar-like robots. Using inverse dynamics to process experimentally acquired point-tracking data, ground reaction forces and internal forces were determined for a crawling caterpillar. Computed ground reaction forces were compared to experimental data to validate the model. The results show that a system of linked extendable joints can faithfully describe the general form and magnitude of the contact forces produced by a crawling caterpillar. Furthermore, the model can be used to compute internal forces that cannot be measured experimentally. It is predicted that between different body segments in stance phase the body is mostly kept in tension and that compression only occurs during the swing phase when the prolegs release their grip. This finding supports a recently proposed mechanism for locomotion by soft animals in which the substrate transfers compressive forces from one part of the body to another (the environmental skeleton) thereby minimizing the need for hydrostatic stiffening. The model also provides a new means to characterize and test control strategies used in caterpillar crawling and soft robot locomotion.     

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.     

Predictive Simulation Generates Human Adaptations during Loaded and Inclined Walking

Predictive simulation is a powerful approach for analyzing human locomotion. Unlike techniques that track experimental data, predictive simulations synthesize gaits by minimizing a high-level objective such as metabolic energy expenditure while satisfying task requirements like achieving a target velocity. The fidelity of predictive gait simulations has only been systematically evaluated for locomotion data on flat ground. In this study, we construct a predictive simulation framework based on energy minimization and use it to generate normal walking, along with walking with a range of carried loads and up a range of inclines. The simulation is muscle-driven and includes controllers based on muscle force and stretch reflexes and contact state of the legs. We demonstrate how human-like locomotor strategies emerge from adapting the model to a range of environmental changes. Our simulation dynamics not only show good agreement with experimental data for normal walking on flat ground (92% of joint angle trajectories and 78% of joint torque trajectories lie within 1 standard deviation of experimental data), but also reproduce many of the salient changes in joint angles, joint moments, muscle coordination, and metabolic energy expenditure observed in experimental studies of loaded and inclined walking.     

Kinematics and motor activity during tethered walking and turning in the cockroach, <Emphasis Type="Italic">Blaberus discoidalis</Emphasis>

When insects turn from walking straight, their legs have to follow different motor patterns. In order to examine such pattern change precisely, we stimulated single antenna of an insect, thereby initiating its turning behavior, tethered over a lightly oiled glass plate. The resulting behavior included asymmetrical movements of prothoracic and mesothoracic legs. The mesothoracic leg on the inside of the turn (in the apparent direction of turning) extended the coxa-trochanter and femur-tibia joints during swing rather than during stance as in walking, while the outside mesothoracic leg kept a slow walking pattern. Electromyograms in mesothoracic legs revealed consistent changes in the motor neuron activity controlling extension of the coxa-trochanter and femur-tibia joints. In tethered walking, depressor trochanter activity consistently preceded slow extensor tibia activity. This pattern was reversed in the inside mesothoracic leg during turning. Also for turning, extensor and depressor motor neurons of the inside legs were activated in swing phase instead of stance. Turning was also examined in free ranging animals. Although more variable, some trials resembled the pattern generated by tethered animals. The distinct inter-joint and inter-leg coordination between tethered turning and walking, therefore, provides a good model to further study the neural control of changing locomotion patterns.     

Compliant bipedal model with the center of pressure excursion associated with oscillatory behavior of the center of mass reproduces the human gait dynamics

Although the compliant bipedal model could reproduce qualitative ground reaction force (GRF) of human walking, the model with a fixed pivot showed overestimations in stance leg rotation and the ratio of horizontal to vertical GRF. The human walking data showed a continuous forward progression of the center of pressure (CoP) during the stance phase and the suspension of the CoP near the forefoot before the onset of step transition. To better describe human gait dynamics with a minimal expense of model complexity, we proposed a compliant bipedal model with the accelerated pivot which associated the CoP excursion with the oscillatory behavior of the center of mass (CoM) with the existing simulation parameter and leg stiffness. Owing to the pivot acceleration defined to emulate human CoP profile, the arrival of the CoP at the limit of the stance foot over the single stance duration initiated the step-to-step transition. The proposed model showed an improved match of walking data. As the forward motion of CoM during single stance was partly accounted by forward pivot translation, the previously overestimated rotation of the stance leg was reduced and the corresponding horizontal GRF became closer to human data. The walking solutions of the model ranged over higher speed ranges (~1.7 m/s) than those of the fixed pivoted compliant bipedal model (~1.5 m/s) and exhibited other gait parameters, such as touchdown angle, step length and step frequency, comparable to the experimental observations. The good matches between the model and experimental GRF data imply that the continuous pivot acceleration associated with CoM oscillatory behavior could serve as a useful framework of bipedal model.     

Pygmy locomotion

The hypothesis that Pygmies may differ from Caucasians in some aspects of the mechanics of locomotion was tested. A total of 13 Pygmies and 7 Caucasians were asked to walk and run on a treadmill at 4–12 km · h^–1. Simultaneous metabolic measurements and three-dimensional motion analysis were performed allowing the energy expenditure and the mechanical external and internal work to be calculated. In Pygmies the metabolic energy cost was higher during walking at all speeds (P < 0.05), but tended to be lower during running (NS). The stride frequency and the internal mechanical work were higher for Pygmies at all walking (P < 0.05) and running (NS) speeds although the external mechanical work was similar. The total mechanical work for Pygmies was higher during walking (P < 0.05), but not during running and the efficiency of locomotion was similar in all subjects and speeds. The higher cost of walking in Pygmies is consistent with the allometric prediction for smaller subjects. The major determinants of the higher cost of walking was the difference in stride frequency (+9.45, SD 0.44% for Pygmies), which affected the mechanical internal work. This explains the observed higher total mechanical work of walking in Pygmies, even when the external component was the same. Most of the differences between Pygmies and Caucasians, observed during walking, tended to disappear when the speed was normalized as the Fronde number. However, this was not the case for running. Thus, whereas the tested hypothesis must be rejected for walking, the data from running, do indeed suggest that Pygmies may differ in some aspects of the mechanics of locomotion.     

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 Wider Pelvis Does Not Increase Locomotor Cost in Humans,with Implications for the Evolution of Childbirth

The shape of the human female pelvis is thought to reflect an evolutionary trade-off between two competing demands: a pelvis wide enough to permit the birth of large-brained infants, and narrow enough for efficient bipedal locomotion. This trade-off, known as the obstetrical dilemma, is invoked to explain the relative difficulty of human childbirth and differences in locomotor performance between men and women. The basis for the obstetrical dilemma is a standard static biomechanical model that predicts wider pelves in females increase the metabolic cost of locomotion by decreasing the effective mechanical advantage of the hip abductor muscles for pelvic stabilization during the single-leg support phase of walking and running, requiring these muscles to produce more force. Here we experimentally test this model against a more accurate dynamic model of hip abductor mechanics in men and women. The results show that pelvic width does not predict hip abductor mechanics or locomotor cost in either women or men, and that women and men are equally efficient at both walking and running. Since a wider birth canal does not increase a woman’s locomotor cost, and because selection for successful birthing must be strong, other factors affecting maternal pelvic and fetal size should be investigated in order to help explain the prevalence of birth complications caused by a neonate too large to fit through the birth canal.     

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.     

Mechanisms of stick insect locomotion in a gap-crossing paradigm

Locomotion of stick insects climbing over gaps of more than twice their step length has proved to be a useful paradigm to investigate how locomotor behaviour is adapted to external conditions. In this study, swing amplitudes and extreme positions of single steps from gap-crossing sequences have been analysed and compared to corresponding parameters of undisturbed walking. We show that adaptations of the basic mechanisms concern movements of single legs as well as the coordination between the legs. Slowing down of stance velocity, searching movements of legs in protraction and the generation of short steps are crucial prerequisites in the gap-crossing task. The rules of leg coordination described for stick insect walking seem to be modified, and load on the supporting legs is assumed to have a major effect on coordination especially in slow walking. Stepping into the gap with a front leg and antennal contact with the far edge of the gap provide information, as both events influence the following leg movements, whereas antennal non-contact seems not to contain information. Integration of these results into the model of the walking controller can improve our understanding of insect locomotion in highly irregular environments.Abbreviations AEP anterior extreme position - fAEP fictive anterior extreme position - PEP posterior extreme position - TOT treading-on-tarsus     

Energetics of walking and running: insights from simulated reduced-gravity experiments.

On Earth, a person uses about one-half as much energy to walk a mile as to run a mile. On another planet with lower gravity, would walking still be more economical than running? When people carry weights while they walk or run, energetic cost increases in proportion to the added load. It would seem to follow that if gravity were reduced, energetic cost would decrease in proportion to body weight in both gaits. However, we find that under simulated reduced gravity, the rate of energy consumption decreases in proportion to body weight during running but not during walking. When gravity is reduced by 75%, the rate of energy consumption is reduced by 72% during running but only by 33% during walking. Because reducing gravity decreases the energetic cost much more for running than for walking, walking is not the cheapest way to travel a mile at low levels of gravity. These results suggest that the link between the mechanics of locomotion and energetic cost is fundamentally different for walking and for running.     

Mechanical models for insect locomotion: dynamics and stability in the horizontal plane I. Theory

We study the dynamics and stability of legged locomotion in the horizontal plane. Motivated by experimental studies of insects, we develop two- and three-degree-of freedom rigid body models with pairs of ‘virtual’ elastic legs in intermittent contact with the ground. We focus on conservative compliant-legged models, but we also consider prescribed forces, prescribed leg displacements, and combined strategies. The resulting mechanical systems exhibit periodic gaits whose stability characteristics are due to intermittent foot contact, and are largely determined by geometrical criteria. Most strikingly, we show that mechanics alone can confer asymptotic stability in heading and body orientation. In a companion paper, we apply our results to rapidly running cockroaches. Received: 6 September 1999 / Accepted in revised form: 8 May 2000