Ability of the planar spring-mass model to predict mechanical parameters in running humans

The planar spring-mass model is a simple mathematical model of bouncing gaits, such as running, trotting and hopping. Although this model has been widely used in the study of locomotion, its accuracy in predicting locomotor mechanics has not been systematically quantified. We determined the percent error of the model in predicting 10 locomotor parameters in running humans by comparing the model predictions to experimental data from humans running in normal gravity and simulated reduced gravity. We tested the hypotheses that the model would overestimate horizontal impulse and the change in mechanical energy of the centre of mass (COM) during stance. The model provided good predictions of stance time, vertical impulse, contact length, duty factor, relative stride length and relative peak force. All predictions of these parameters were within 20% of measured values and at least 90% of predictions of each parameter were within 10% of measured values (median absolute errors: <7%). This suggests that the model incorporates all features of running humans that have a significant influence upon these six parameters. As simulated gravity level decreased, the magnitude of the errors in predicting each of these parameters either decreased or stayed constant, indicating that this is a good model of running in simulated reduced gravity. As hypothesised, horizontal impulse and change in mechanical energy of the COM during stance were overestimated (median absolute errors: 43.6% and 26.2%, respectively). Aerial time and peak vertical COM displacement during stance were also systematically overestimated (median absolute errors: 17.7% and 22.9%, respectively). Care should be taken to ensure that the model is used only to investigate parameters which it can predict accurately. It would be useful to extend this analysis to other species and gaits.     

Simultaneous recording of longitudinal displacements of both feet during human walking

1. Recordings of longitudinal displacements of both feet have been performed by linking each foot to a length-voltage transducer by means of threads. The movement transmitted to the transducer was reduced by winding the thread around a sixteen strand pulley block. 2. The aspect of the displacement curves allows a direct analysis of the walk. Some typical curves of normal and pathological gaits are presented. Graphic measurements done on several right and left cycles of a 6 m walk episode in 50 adult subjects provided data to determine the mean and the standard deviation of spatial and temporal parameters of the walk. 3. An advantage of this method is to permit the measurement of the parameters of several successive cycles of both sides and so to detect changes in the length, the duration and the velocity of successive cycles of the same foot and of alternated cycles of both feet. This being important to characterize pathological gaits.     

The assessment of posture control in the elderly using the displacement of the center of pressure after forward platform translation

We investigated the change of the center of pressure (COP) after forward platform translations in healthy subjects. These studies were performed on 26 normal young subjects and 20 healthy elderly subjects, who had a normal neurologic examination. Subjects stood barefoot on a three dimensional force plate on the platform, with feet parallel. The duration of the forward platform translations was 0.15 s, and the displacements were 3.75, 7.5, 10, 15, 20, and 30 mm. Six trials were carried out at random. The COP data were recorded for 35 s during standing, and were analyzed for 5 s after translation. With the platform translation displacements from 3.75 to 15 mm, displacement of the COP showed a tendency to increase in all subjects. Whereas with the stimuli between 20 and 30 mm, the results were more varied. The elderly group showed significantly (p<0.05) larger sway than the young group. These results indicate that the individual ability of posture control may be assessed by means of measuring the sway of the center of gravity after platform translation. Electromyography was carried out simultaneously, it showed that elderly people contrary to young subjects used proximal biceps femoris and distal foot muscles at an early stage of the platform translation (p<0.05), suggesting lack of ankle stability with aging.     

Biomechanical analysis of primate bipedal walking by computer simulation

A computer simulation technique was applied to make clear the mechanical characteristics of primate bipedal walking. A primate body and the walking mechanism were modeled mathematically with a set of dynamic equations. Using a digital computer, the following were calculated from these equations by substituting measured displacements and morphological data of each segment of the primate: the acceleration, joint angle, center of gravity, foot force, joint moment, muscular force, transmitted force at the joint, electric activity of the muscle, generated power by the leg and energy expenditure in walking.The model was evaluated by comparing some of the calculated results with the experimental results such as foot force and electromyographic data, and improved in order to obtain the agreement between them.The level bipedal walking of man, chimpanzee and Japanese monkey and several types of synthesized walking were analyzed from the viewpoint of biomechanics.It is concluded that the bipedal walking of chimpanzee is nearer to that of man than to that of the Japanese monkey because of its propulsive mechanism, but it requires large muscular force for supporting the body weight.     

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.     

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.     

Mechanics of running under simulated low gravity.

Using a linear mass-spring model of the body and leg (T. A. McMahon and G. C. Cheng. J. Biomech. 23: 65-78, 1990), we present experimental observations of human running under simulated low gravity and an analysis of these experiments. The purpose of the study was to investigate how the spring properties of the leg are adjusted to different levels of gravity. We hypothesized that leg spring stiffness would not change under simulated low-gravity conditions. To simulate low gravity, a nearly constant vertical force was applied to human subjects via a bicycle seat. The force was obtained by stretching long steel springs via a hand-operated winch. Subjects ran on a motorized treadmill that had been modified to include a force platform under the tread. Four subjects ran at one speed (3.0 m/s) under conditions of normal gravity and six simulated fractions of normal gravity from 0.2 to 0.7 G. For comparison, subjects also ran under normal gravity at five speeds from 2.0 to 6.0 m/s. Two basic principles emerged from all comparisons: both the stiffness of the leg, considered as a linear spring, and the vertical excursion of the center of mass during the flight phase did not change with forward speed or gravity. With these results as inputs, the mathematical model is able to account correctly for many of the changes in dynamic parameters that do take place, including the increasing vertical stiffness with speed at normal gravity and the decreasing peak force observed under conditions simulating low gravity.     

A gravitational impulse model predicts collision impulse and mechanical work during a step-to-step transition

The simplest walking model, which assumes an instantaneous collision with negligible gravity effect, is limited in its representation of the collision mechanics of human gaits because the actual step-to-step transition occurs over a finite duration of time with finite impulsive ground reaction forces (GRFs) that have the same order of magnitude as the gravitational force. In this study, we propose a new collision model that includes the contribution of the gravitational impulse to the momentum change of the center of mass (COM) during a step-to-step transition. To validate the model, we measured the GRFs of six subjects' over-ground walking at five different gait speeds and calculated the collision impulses and mechanical work. The data showed a significant contribution of the gravitational impulse to the momentum change during collision. To compensate for the gravity, the magnitudes of collision impulse and COM work were estimated to be much greater than in previous predictions. Consistent with the model prediction, push-off propulsion fully compensated for the collision loss, implying the step-to-step transition occurred in an energetically optimal manner. The new model predicted a moderate change in the collision mechanics with gait speed, which seems to be physiologically achievable. The gravitational collision model enables us to better understand collision dynamics during a step-to-step transition.     

An estimation of center of gravity from force platform data

A general, dynamic relationship between the data obtained from a force platform, center of gravity of the body on the platform and the time rate of change of moment of momentum of the body about its center of gravity was derived from principles of dynamics for a system of particles. The derived equations are useful for processing and interpreting the force platform data. Displacement and path of center of gravity of human body during standing on one foot and level walking were estimated by using the derived equations. An estimation of the time rate of change of moment of momentum of the body was also obtained. A biomechanical interpretation of point of application of the resultant of ground reactions was presented.     

Modelling of the biomechanics of posture and balance

A technique for studying the relationship of posture to balance has been developed. To investigate this relationship quantitatively, the human body was treated as consisting of 11 rigid body segments, each with six degrees of freedom. A bilateral Selspot II/TRACK data acquisition system provides position and orientation kinematic data for estimation of the trajectories of the individual body segment centers of gravity. From these, the whole body center of gravity is estimated and compared to concurrent force plate center of force data. Center of gravity and center of force excursions agree where dynamics are not significant. The technique may be employed to study quiet stance, response to postural disturbances, or the initiation and coordination of complex movements such as gait.     

Independent metabolic costs of supporting body weight and accelerating body mass during walking.

The metabolic cost of walking is determined by many mechanical tasks, but the individual contribution of each task remains unclear. We hypothesized that the force generated to support body weight and the work performed to redirect and accelerate body mass each individually incur a significant metabolic cost during normal walking. To test our hypothesis, we measured changes in metabolic rate in response to combinations of simulated reduced gravity and added loading. We found that reducing body weight by simulating reduced gravity modestly decreased net metabolic rate. By calculating the metabolic cost per Newton of reduced body weight, we deduced that generating force to support body weight comprises approximately 28% of the metabolic cost of normal walking. Similar to previous loading studies, we found that adding both weight and mass increased net metabolic rate in more than direct proportion to load. However, when we added mass alone by using a combination of simulated reduced gravity and added load, net metabolic rate increased about one-half as much as when we added both weight and mass. By calculating the cost per kilogram of added mass, we deduced that the work performed on the center of mass comprises approximately 45% of the metabolic cost of normal walking. Our findings support the hypothesis that force and work each incur a significant metabolic cost. Specifically, the cost of performing work to redirect and accelerate the center of mass is almost twice as great as the cost of generating force to support body weight.     

Locomotor mechanics of the slender loris (Loris tardigradus)

The quadrupedal walking gaits of most primates can be distinguished from those of most other mammals by the presence of diagonal-sequence (DS) footfall patterns and higher peak vertical forces on the hindlimbs compared to the forelimbs. The walking gait of the woolly opossum (Caluromys philander), a highly arboreal marsupial, is also characterized by diagonal-sequence footfalls and relatively low peak forelimb forces. Among primates, three species--Callithrix, Nycticebus, and Loris--have been reported to frequently use lateral-sequence (LS) gaits and experience relatively higher peak vertical forces on the forelimbs. These patterns among primates and other mammals suggest a strong association between footfall patterns and force distribution on the limbs. However, current data for lorises are limited and the frequency of DS vs. LS walking gaits in Loris is still ambiguous. To test the hypothesis that patterns of footfalls and force distribution on the limbs are functionally linked, kinematic and kinetic data were collected simultaneously for three adult slender lorises (Loris tardigradus) walking on a 1.25 cm horizontal pole. All subjects in this study consistently used diagonal-sequence walking gaits and always had higher peak vertical forces on their forelimbs relative to their hindlimbs. These results call into question the hypothesis that a functional link exists between the presence of diagonal-sequence walking gaits and relatively higher peak vertical forces on the hindlimbs. In addition, this study tested models that explain patterns of force distribution based on limb protraction angle or limb compliance. None of the Loris subjects examined showed kinematic patterns that would support current models proposing that weight distribution can be adjusted by actively shifting weight posteriorly or by changing limb stiffness. These data reveal the complexity of adaptations to arboreal locomotion in primates and indicate that diagonal-sequence walking gaits and relatively low forelimb forces could have evolved independently.     

Influence of an asymmetrical body weight distribution on the control of undisturbed upright stance

Postural asymmetry in humans is generally associated with different pathologies. However, its specific influence on undisturbed upright stance is poorly understood. To evaluate its separate effects on each support, the centre of pressure (CP) displacements were recorded through two force platforms. In a second step, the complex resultant centre of pressure trajectories (CP(Res)) were computed and decomposed into two elementary components: the horizontal displacements of the centre of gravity (CG(h)) and the difference in the plane of support between the vertical projection of CG(h) and CP(Res) (CP-CG(v)). These motions were then processed through a frequency analysis and modelled as fractional Brownian motion to gain some additional insight into their spatio-temporal organisation. Ten healthy adults were tested in three conditions consisting of various weight distributions. The quality of the mechanism involved in the control of the unloaded support CP motions appears to decrease as the asymmetry becomes more pronounced. To be precise, larger increases of the CP displacements are observed for the unloaded support compared to the loaded one. As a result, the CP(Res) motions are themselves augmented in the ML direction, inducing in turn larger CG(h) and CP-CG(v) motions. Postural asymmetry thus constitutes an important constraint on the control of upright undisturbed stance by generating changes in the control of both supports and by reducing the efficiency of the hip load/unload mechanisms. On the other hand, by inducing larger body sways, postural asymmetry necessitates higher energy expenditure and the setting of particular control mechanisms.     

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.     

Evolutionary implications of the unusual walking mechanics of the common marmoset (C. jacchus)

Several features that appear to differentiate the walking gaits of most primates from those of most other mammals (the prevalence of diagonal-sequence footfalls, high degrees of humeral protraction, and low forelimb vs. hindlimb peak vertical forces) are believed to have evolved in response to requirements of locomotion on thin arboreal supports by early primates that had developed clawless grasping hands and feet. This putative relationship between anatomy, behavior, and ecology is tested here by examining gait mechanics in the common marmoset (Callithrix jacchus), a primate that has sharp claws and reduced pedal grasping, and that spends much of its time clinging on large trunks. Kinematic and kinetic data were collected on three male Callithrix jacchus as they walked across a force platform attached to the ground or to raised horizontal poles. The vast majority of all walking gaits were lateral-sequence. For all steps, the humerus was retracted (<90 degrees relative to a horizontal axis) or held in a neutral (90 degrees ) position at forelimb touchdown. Peak vertical forces on the forelimb were always higher than those on the hindlimb. These three features of the walking gaits of C. jacchus separate it from any other primate studied (including other callitrichids). The walking gaits of C. jacchus are mechanically more similar to those of small, nonprimate mammals. The results of this study support previous models that suggest that the unusual suite of features that typify the walking gaits of most primates are adaptations to the requirements of locomotion on thin arboreal supports. These data, along with data from other primates and marsupials, suggest that primate postcranial and locomotor characteristics are part of a basal adaptation for walking on thin branches.     

Mechanical energy and effective foot mass during impact loading of walking and running

The human heel pad is considered an important structure for attenuation of the transient force caused by heel-strike. Although the mechanical properties of heel pads are relatively well understood, the mechanical energy (Etot) absorbed by the heel pad during the impact phase has never been documented directly because data on the effective foot mass (Meff) was previously unavailable during normal forward locomotion. In this study, we use the impulse-momentum method (IMM) for calculating Meff from moving subjects. Mass-spring-damper models were developed to evaluate errors and to examine the effects of pad property, upper body mass, and effective leg spring on Meff. We simultaneously collected ground reaction forces, pad deformation, and lower limb kinematics during impact phase of barefoot walking, running, and crouched walking. The latter was included to examine the effect of knee angle on Meff. The magnitude of Meff as a percentage of body mass (M(B)) varies with knee angle at impact and significantly differs among gaits: 6.3%M(B) in walking, 5.3%M(B) in running, and 3.7%M(B) in crouched walking. Our modeling results suggested that Meff is insensitive to heel pad resilience and effective leg stiffness. At the instant prior to heel strike, Etot ranges from 0.24 to 3.99 J. The combination of video and forceplate data used in this study allows analyses of Etot and Etot as a function of heel-strike kinematics during normal locomotion. Relationship between Meff and knee angle provides insights into how changes in posture moderate impact transients at different gaits.     

Symmetrical gaits and center of mass mechanics in small-bodied,primitive mammals

Widely accepted relationships between gaits (footfall patterns) and center of mass mechanics have been formulated from observations for cursorial mammals. However, sparse data on smaller or more generalized forms suggest a fundamentally different relationship. This study explores locomotor dynamics in one eutherian and five metatherian (marsupials) mammals—all small-bodied (<2 kg) with generalized body plans that utilize symmetrical gaits. Across our sample, trials conforming to vaulting mechanics occurred least frequently (<10% of all trials) while bouncing mechanics was obtained most commonly (60%); the remaining trials represented mixed mechanics. Contrary to the common situation in large mammals, there was no evidence for discrete gait switching within symmetrical gaits as speed increased. This was in part due to the common practice of grounded running. The adaptive advantage of different patterns of center-of-mass motion and their putative energy savings remain questionable in small-bodied mammals.     

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.     

Leveling the playing field: Evaluation of a portable instrument for quantifying balance performance

Balance is a complex, sensorimotor task requiring an individual to maintain the center of gravity within the base of support. Quantifying balance in a reliable and valid manner is essential to evaluating disease progression, aging complications, and injuries in clinical and research settings. Typically, researchers use force plates to track motion of the center of gravity during a variety of tasks. However, limiting factors such as cost, portability, and availability have hindered postural stability evaluation in these settings. This study compared the “gold standard” for assessing postural stability (i.e., the laboratory-grade force plate) to a more affordable and portable assessment tool (i.e., BTrackS balance plate) in healthy young adults. Correlations and Bland-Altman plots between the center of pressure outcome measures derived from these two instruments were produced. Based on the results of this study, the measures attained from the portable balance plate objectively quantified postural stability with high validity on both rigid and compliant surfaces, demonstrated by thirty-five out of thirty-eight observed postural stability metrics in both surface conditions with a correlation of 0.98 or greater. The low cost, portable system performed similarly to the lab-grade force plate indicating the potential for practitioners and researchers to use the BTrackS balance plate as an alternative to the more expensive force plate option for assessing postural stability, whether in the lab setting or in the field.     

Deoxymyoglobin studied by the conformational normal mode analysis. II. The conformational change upon oxygenation

The conformational change taking place in myoglobin concomitantly with the observed geometrical change at the heme-His(F8) linkage upon oxygenation is studied by normal mode analysis, which is based on the quadratic approximation of the conformational energy function. The heme-globin interaction energy increases for this change by 8.114 kcal/mol when both the heme group and the globin molecule are held rigid. When they are permitted flexibility, the interaction energy relaxes by 7.038 kcal/mol, and the difference (1.076 kcal/mol) is distributed as strain energy within the molecule. This increase is the work necessary for the heme group to move against the force exerted by the globin. If the force is assumed to be invariable during this move, the work is small, 0.276 kcal/mol, meaning that the force is strongly variable. Furthermore, this means that the heme group is located near the equilibrium point of the potential energy of the heme-globin interaction. The change in the localized strain energy stored in the force field at the linkage between the heme and the imidazole of HisF8 is estimated to be of the same order of magnitude as the distributed energy. The largest atomic displacements are observed at the region from the F helix to the beginning of the G helix, and secondary large displacements occur at several regions, i.e, the A helix, from the C helix to the CD corner, the E helix, and the C-terminal side of the H helix. All of these regions have strong dynamic interactions with the heme group, either directly or indirectly. Their secondary structures show complex deformations. In other parts, relatively rigid segments undergo rotational and/or bending changes in a way consistent with the large changes described above and close atomic packing within the molecule. The calculated conformational change is decomposed to vibrational normal modes of deoxymyoglobin. The magnitude of the conformational change, measured by the mass-weighted mean-square atomic displacement, is accounted for up to 92.0% by the 151 normal modes with frequencies lower than 40 cm-1. In descending order of contribution, the first six modes, each of which has a frequency lower than 12 cm-1, account for up to 57.4%. This means that the functionally important conformational change can well be expressed in terms of a relatively small number of collective low frequency normal modes.