A simple method for measuring stiffness during running

The spring-mass model, representing a runner as a point mass supported by a single linear leg spring, has been a widely used concept in studies on running and bouncing mechanics. However, the measurement of leg and vertical stiffness has previously required force platforms and high-speed kinematic measurement systems that are costly and difficult to handle in field conditions. We propose a new "sine-wave" method for measuring stiffness during running. Based on the modeling of the force-time curve by a sine function,this method allows leg and vertical stiffness to be estimated from just a few simple mechanical parameters: body mass, forward velocity, leg length, flight time, and contact time. We compared this method to force-platform-derived stiffness measurements for treadmill dynamometer and overground running conditions, at velocities ranging from 3.33 m.s-1 to maximal running velocity in both recreational and highly trained runners. Stiffness values calculated with the proposed method ranged from 0.67 % to 6.93 % less than the force platform method, and thus were judged to be acceptable. Furthermore, significant linear regressions (p < 0.01) close to the identity line were obtained between force platform and sine-wave model values of stiffness. Given the limits inherent in the use of the spring-mass model, it was concluded that this sine-wave method allows leg and stiffness estimates in running on the basis of a few mechanical parameters, and could be useful in further field measurements.     

Resonant hopping of a robot controlled by an artificial neural oscillator

The bouncing gaits of terrestrial animals (hopping, running, trotting) can be modeled as a hybrid dynamic system, with spring-mass dynamics during stance and ballistic motion during the aerial phase. We used a simple hopping robot controlled by an artificial neural oscillator to test the ability of the neural oscillator to adaptively drive this hybrid dynamic system. The robot had a single joint, actuated by an artificial pneumatic muscle in series with a tendon spring. We examined how the oscillator-robot system responded to variation in two neural control parameters: descending neural drive and neuromuscular gain. We also tested the ability of the oscillator-robot system to adapt to variations in mechanical properties by changing the series and parallel spring stiffnesses. Across a 100-fold variation in both supraspinal gain and muscle gain, hopping frequency changed by less than 10%. The neural oscillator consistently drove the system at the resonant half-period for the stance phase, and adapted to a new resonant half-period when the muscle series and parallel stiffnesses were altered. Passive cycling of elastic energy in the tendon accounted for 70-79% of the mechanical work done during each hop cycle. Our results demonstrate that hopping dynamics were largely determined by the intrinsic properties of the mechanical system, not the specific choice of neural oscillator parameters. The findings provide the first evidence that an artificial neural oscillator will drive a hybrid dynamic system at partial resonance.     

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.     

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.     

Gender differences in active musculoskeletal stiffness. Part II. Quantification of leg stiffness during functional hopping tasks.

Leg stiffness was compared between age-matched males and females during hopping at preferred and controlled frequencies. Stiffness was defined as the linear regression slope between the vertical center of mass (COM) displacement and ground-reaction forces recorded from a force plate during the stance phase of the hopping task. Results demonstrate that subjects modulated the vertical displacement of the COM during ground contact in relation to the square of hopping frequency. This supports the accuracy of the spring-mass oscillator as a representative model of hopping. It also maintained peak vertical ground-reaction load at approximately three times body weight. Leg stiffness values in males (33.9+/-8.7 kN/m) were significantly (p<0.01) greater than in females (26.3+/-6.5 kN/m) at each of three hopping frequencies, 3.0, 2.5 Hz, and a preferred hopping rate. In the spring-mass oscillator model leg stiffness and body mass are related to the frequency of motion. Thus male subjects necessarily recruited greater leg stiffness to drive their heavier body mass at the same frequency as the lighter female subjects during the controlled frequency trials. However, in the preferred hopping condition the stiffness was not constrained by the task because frequency was self-selected. Nonetheless, both male and female subjects hopped at statistically similar preferred frequencies (2.34+/-0.22 Hz), therefore, the females continued to demonstrate less leg stiffness. Recognizing the active muscle stiffness contributes to biomechanical stability as well as leg stiffness, these results may provide insight into the gender bias in risk of musculoskeletal knee injury.     

Spring-mass running: simple approximate solution and application to gait stability

The planar spring-mass model is frequently used to describe bouncing gaits (running, hopping, trotting, galloping) in animal and human locomotion and robotics. Although this model represents a rather simple mechanical system, an analytical solution predicting the center of mass trajectory during stance remains open. We derive an approximate solution in elementary functions assuming a small angular sweep and a small spring compression during stance. The predictive power and quality of this solution is investigated for model parameters relevant to human locomotion. The analysis shows that (i), for spring compressions of up to 20% (angle of attack > or = 60 degree, angular sweep < or = 60 degree) the approximate solution describes the stance dynamics of the center of mass within a 1% tolerance of spring compression and 0.6 degree tolerance of angular motion compared to numerical calculations, and (ii), despite its relative simplicity, the approximate solution accurately predicts stable locomotion well extending into the physiologically reasonable parameter domain. (iii) Furthermore, in a particular case, an explicit parametric dependency required for gait stability can be revealed extending an earlier, empirically found relationship. It is suggested that this approximation of the planar spring-mass dynamics may serve as an analytical tool for application in robotics and further research on legged locomotion.     

Stance leg control: variation of leg parameters supports stable hopping

The spring-loaded inverted pendulum describes the planar center-of-mass dynamics of legged locomotion. This model features linear springs with constant parameters as legs. In biological systems, however, spring-like properties of limbs can change over time. Therefore, in this study, it is asked how variation of spring parameters during ground contact would affect the dynamics of the spring-mass model. Neglecting damping initially, it is found that decreasing stiffness and increasing rest length of the leg during a stance phase are required for orbitally stable hopping. With damping, stable hopping is found for a larger region of rest-length rates and stiffness rates. Here, also increasing stiffness and decreasing rest length can result in stable hopping. Within the predicted range of leg parameter variations for stable hopping, there is no need for precise parameter tuning. Since hopping gaits form a subset of the running gaits (with vanishing horizontal velocity), these results may help to improve leg design in robots and prostheses.     

Positive force feedback in bouncing gaits?

During bouncing gaits (running, hopping, trotting), passive compliant structures (e.g. tendons, ligaments) store and release part of the stride energy. Here, active muscles must provide the required force to withstand the developing tendon strain and to compensate for the inevitable energy losses. This requires an appropriate control of muscle activation. In this study, for hopping, the potential involvement of afferent information from muscle receptors (muscle spindles, Golgi tendon organs) is investigated using a two-segment leg model with one extensor muscle. It is found that: (i) positive feedbacks of muscle-fibre length and muscle force can result in periodic bouncing; (ii) positive force feedback (F+) stabilizes bouncing patterns within a large range of stride energies (maximum hopping height of 16.3 cm, almost twofold higher than the length feedback); and (iii) when employing this reflex scheme, for moderate hopping heights (up to 8.8 cm), an overall elastic leg behaviour is predicted (hopping frequency of 1.4-3 Hz, leg stiffness of 9-27 kN m(-1)). Furthermore, F+ could stabilize running. It is suggested that, during the stance phase of bouncing tasks, the reflex-generated motor control based on feedbacks might be an efficient and reliable alternative to central motor commands.     

Changes in running mechanics and spring-mass behavior induced by a mountain ultra-marathon race

Changes in running mechanics and spring-mass behavior due to fatigue induced by a mountain ultra-marathon race (MUM, 166km, total positive and negative elevation of 9500m) were studied in 18 ultra-marathon runners. Mechanical measurements were undertaken pre- and 3h post-MUM at 12km h(-1) on a 7m long pressure walkway: contact (t(c)), aerial (t(a)) times, step frequency (f), and running velocity (v) were sampled and averaged over 5-8 steps. From these variables, spring-mass parameters of peak vertical ground reaction force (F(max)), vertical downward displacement of the center of mass (Δz), leg length change (ΔL), vertical (k(vert)) and leg (k(leg)) stiffness were computed. After the MUM, there was a significant increase in f (5.9±5.5%; P<0.001) associated with reduced t(a) (-18.5±17.4%; P<0.001) with no change in t(c), and a significant decrease in both Δz and F(max) (-11.6±10.5 and -6.3±7.3%, respectively; P<0.001). k(vert) increased by 5.6±11.7% (P=0.053), and k(leg) remained unchanged. These results show that 3h post-MUM, subjects ran with a reduced vertical oscillation of their spring-mass system. This is consistent with (i) previous studies concerning muscular structure/function impairment in running and (ii) the hypothesis that these changes in the running pattern could be associated with lower overall impact (especially during the braking phase) supported by the locomotor system at each step, potentially leading to reduced pain during running.     

The bounce of the body in hopping,running and trotting: different machines with the same motor

The bouncing mechanism of human running is characterized by a shorter duration of the brake after ‘landing’ compared with a longer duration of the push before ‘takeoff’. This landing–takeoff asymmetry has been thought to be a consequence of the force–velocity relation of the muscle, resulting in a greater force exerted during stretching after landing and a lower force developed during shortening before takeoff. However, the asymmetric lever system of the human foot during stance may also be the cause. Here, we measure the landing–takeoff asymmetry in bouncing steps of running, hopping and trotting animals using diverse lever systems. We find that the duration of the push exceeds that of the brake in all the animals, indicating that the different lever systems comply with the basic property of muscle to resist stretching with a force greater than that developed during shortening. In addition, results show both the landing–takeoff asymmetry and the mass-specific vertical stiffness to be greater in small animals than in large animals. We suggest that the landing–takeoff asymmetry is an index of a lack of elasticity, which increases with increasing the role of muscle relative to that of tendon within muscle–tendon units.     

Human hopping on damped surfaces: strategies for adjusting leg mechanics

Fast-moving legged animals bounce along the ground with spring-like legs and agilely traverse variable terrain. Previous research has shown that hopping and running humans maintain the same bouncing movement of the body's centre of mass on a range of elastic surfaces by adjusting their spring-like legs to exactly offset changes in surface stiffness. This study investigated human hopping on damped surfaces that dissipated up to 72% of the hopper's mechanical energy. On these surfaces, the legs did not act like pure springs. Leg muscles performed up to 24-fold more net work to replace the energy lost by the damped surface. However, considering the leg and surface together, the combination appeared to behave like a constant stiffness spring on all damped surfaces. By conserving the mechanics of the leg-surface combination regardless of surface damping, hoppers also conserved centre-of-mass motions. Thus, the normal bouncing movements of the centre of mass in hopping are not always a direct result of spring-like leg behaviour. Conserving the trajectory of the centre of mass by maintaining spring-like mechanics of the leg-surface combination may be an important control strategy for fast-legged locomotion on variable terrain.     

The Effect of Body Pitching on Leg-Spring Behavior in Quadruped Running

<正> In this paper we investigated how the running speed would affect the dynamics of body pitching, and whether body inertiais important for animals. Passive trotting of spring-mass model and passive bounding of spring-beam model were studied atdifferent speeds for different sets of body parameters respectively. Furthermore, different body inertias were used in bounding.We found that running speed exerts effect on leg performance by means of centrifugal force. The centrifugal force can be understoodas an enhancement to the natural frequency of the spring-mass system. The disadvantage of body pitching may beoffset by the great increase in centrifugal force at high speed. The results also reveal that body mass distribution might not be themain reason for the difference in maximal running speeds of different animals.     

Consequences of forward translation of the point of force application for the mechanics of running

In running humans, the point of force application between the foot and the ground moves forwards during the stance phase. Our aim was to determine the mechanical consequences of this 'point of force translation' (POFT). We modified the planar spring-mass model of locomotion to incorporate POFT, and then compared spring-mass simulations with and without POFT. We found that, if leg stiffness is adjusted appropriately, it is possible to maintain very similar values of peak vertical ground reaction force (GRF), stance time, contact length and vertical centre of mass displacement, whether or not POFT occurs. The leg stiffness required to achieve this increased as the distance of POFT increased. Peak horizontal GRF and mechanical work per step were lower when POFT occurred. The results indicate that the lack of POFT in the traditional spring-mass model should not prevent it from providing good predictions of peak vertical GRF, stance time, contact length and vertical centre of mass displacement in running humans, if an appropriate spring stiffness is used. However, the model can be expected to overestimate peak horizontal GRF and mechanical work per step. When POFT occurs, the spring stiffness in the traditional spring-mass model is not equivalent to leg stiffness. Therefore, caution should be exercised when using spring stiffness to understand how the musculoskeletal system adapts to different running conditions. This can explain the contradictory results in the literature regarding the effect of running speed on leg stiffness.     

Intralimb compensation strategy depends on the nature of joint perturbation in human hopping

Due to the well-described spring-mass dynamics of bouncing gaits, human hopping is a tractable model for elucidating basic neuromuscular compensation principles. We tested whether subjects would employ a multi-joint or single-joint response to stabilize leg stiffness while wearing a spring-loaded ankle-foot orthosis (AFO) that applied localized resistive and assistive torques to the ankle. We analyzed kinematics and kinetics data from nine subjects hopping in place on one leg, at three frequencies (2.2, 2.4, and 2.8Hz) and three orthosis conditions (freely articulating AFO, AFO with plantarflexion resistance, and AFO with plantarflexion assistance). Leg stiffness was invariant across AFO conditions, however, compensation strategy depended upon the nature of the applied load. Biological ankle stiffness increased in response to a resistive load at twice the rate that it decreased with an assitive load. Ankle adjustments alone fully compensated for an assistive load with no net change in combined (biological plus applied) total ankle stiffness (p > or =0.133). In contrast, a resistive load resulted in a 7.4-9.0% increase in total ankle stiffness across frequencies and a concomitant 10-15% increase in knee joint stiffness at each frequency (p< or =0.037). The increased knee joint stiffness in response to resistive ankle load allowed subjects to maintain a more flexed knee at mid-stance, which attenuated the effect of the increased total ankle joint stiffness to preserve leg stiffness and whole limb biomechanical performance. Our findings suggest humans maintain invariant leg stiffness in bouncing gaits through different intralimb compensation strategies that are specific to the nature of the joint loading.     

Hopping frequency in humans: a test of how springs set stride frequency in bouncing gaits.

The storage and recovery of elastic energy in muscle-tendon springs is important in running, hopping, trotting, and galloping. We hypothesized that animals select the stride frequency at which they behave most like simple spring-mass systems. If higher or lower frequencies are used, they will not behave like simple spring-mass systems, and the storage and recovery of elastic energy will be reduced. We tested the hypothesis by having humans hop forward on a treadmill over a range of speeds and hop in place over a range of frequencies. The body was modeled as a simple spring-mass system, and the properties of the spring were measured by use of a force platform. Our subjects used nearly the same frequency (the "preferred frequency," 2.2 hops/s) when they hopped forward on a treadmill and when they hopped in place. At this frequency, the body behaved like a simple spring-mass system. Contrary to our predictions, it also behaved like a simple spring-mass system when the subjects hopped at higher frequencies, up to the maximum they could achieve. However, at the higher frequencies, the time available to apply force to the ground (the ground contact time) was shorter, perhaps resulting in a higher cost of generating muscular force. At frequencies below the preferred frequency, as predicted by the hypothesis, the body did not behave in a springlike manner, and it appeared likely that the storage and recovery of elastic energy was reduced. The combination of springlike behavior and a long ground contact time at the preferred frequency should minimize the cost of generating muscular force.     

Effective leg stiffness in running

Leg stiffness is a common parameter used to characterize leg function during bouncing gaits, like running and hopping. In the literature, different methods to approximate leg stiffness based on kinetic and kinematic parameters are described. A challenging point in estimating leg stiffness is the definition of leg compression during contact. In this paper four methods (methods A–D) based on ground reaction forces (GRF) and one method (method E) relying on temporal parameters are described. Leg stiffness calculated by these five methods is compared with running patterns, predicted by the spring mass model.The best and simplest approximation of leg stiffness is method E. It requires only easily accessible parameters (contact time, flight time, resting leg length, body mass and the leg's touch down angle). Method D is of similar quality but additionally requires the time-dependent progression of the GRF. The other three methods show clear differences from the model predictions by over- or underestimating leg stiffness, especially at slow speeds.Leg stiffness is derived from a conceptual model of legged locomotion and does not exist without this model. Therefore, it is important to prove which experimental method is suited best for approximating the stiffness in a specific task. This will help to interpret the predictions of the conceptual model in comparison with experimental data.     

Criteria for dynamic similarity in bouncing gaits

Animals of different sizes tend to move in a dynamically similar manner when travelling at speeds corresponding to equal values of a dimensionless parameter (DP) called the Froude number. Consequently, the Froude number has been widely used for defining equivalent speeds and predicting speeds of locomotion by extinct species and on other planets. However, experiments using simulated reduced gravity have demonstrated that equality of the Froude number does not guarantee dynamic similarity. This has cast doubt upon the usefulness of the Froude number in locomotion research. Here we use dimensional analysis of the planar spring-mass model, combined with Buckingham's Pi-Theorem, to demonstrate that four DPs must be equal for dynamic similarity in bouncing gaits such as trotting, hopping and bipedal running. This can be reduced to three DPs by applying the constraint of maintaining a constant average speed of locomotion. Sensitivity analysis indicates that all of these DPs are important for predicting dynamic similarity. We show that the reason humans do not run in a dynamically similar manner at equal Froude number in different levels of simulated reduced gravity is that dimensionless leg stiffness decreases as gravity increases. The reason that the Froude number can predict dynamic similarity in Earth gravity is that dimensionless leg stiffness and dimensionless vertical landing speed are both independent of size. In conclusion, although equal Froude number is not sufficient for dynamic similarity, it is a necessary condition. Therefore, to detect fundamental differences in locomotion, animals of different sizes should be compared at equal Froude number, so that they can be as close to dynamic similarity as possible. More generally, the concept of dynamic similarity provides a powerful framework within which similarities and differences in locomotion can be interpreted.     

A comparative collision-based analysis of human gait

This study compares human walking and running, and places them within the context of other mammalian gaits. We use a collision-based approach to analyse the fundamental dynamics of the centre of mass (CoM) according to three angles derived from the instantaneous force and velocity vectors. These dimensionless angles permit comparisons across gait, species and size. The collision angle Φ, which is equivalent to the dimensionless mechanical cost of transport CoT_mech, is found to be three times greater during running than walking of humans. This threefold difference is consistent with previous studies of walking versus trotting of quadrupeds, albeit tends to be greater in the gaits of humans and hopping bipeds than in quadrupeds. Plotting the collision angle Φ together with the angles of the CoM force vector Θ and velocity vector Λ results in the functional grouping of bipedal and quadrupedal gaits according to their CoM dynamics—walking, galloping and ambling are distinguished as separate gaits that employ collision reduction, whereas trotting, running and hopping employ little collision reduction and represent more of a continuum that is influenced by dimensionless speed. Comparable with quadrupedal mammals, collision fraction (the ratio of actual to potential collision) is 0.51 during walking and 0.89 during running, indicating substantial collision reduction during walking, but not running, of humans.     

A new dimensionless number highlighted from mechanical energy exchange during running

This study aimed to highlight a new dimensionless number from mechanical energy transfer occurring at the centre of gravity (Cg) during running. We built two different-sized spring-mass models (SMM #1 and SMM #2). SMM #1 was built from the previously published data, and SMM #2 was built to be dynamically similar to SMM #1. The potential gravitational energy (E(P)), kinetic energy (E(K)), and potential elastic energy (E(E)) were taken into account to test our hypothesis. For both SMM #1 and SMM #2, N(Mo-Dela)=(E(P)+E(K))/E(E) reached the same mean value and was constant (4.1+/-0.7) between 30% and 70% of contact time. Values of N(Mo-Dela) obtained out of this time interval were due to the absence of E(E) at initial and final times of the simulation. This phenomenon does not occur during in vivo running because a leg muscle's pre-activation enables potential elastic energy storage prior to ground contact. Our findings also revealed that two different-sized spring-mass models bouncing with equal N(Mo-Dela) values moved in a dynamically similar fashion. N(Mo-Dela), which can be expressed by the combination of Strouhal and Froude numbers, could be of great interest in order to study animal and human locomotion under Earth's gravity or to induce dynamic similarity between different-sized individuals during bouncing gaits.     

Running backwards: soft landing–hard takeoff,a less efficient rebound

Human running at low and intermediate speeds is characterized by a greater average force exerted after ‘landing’, when muscle–tendon units are stretched (‘hard landing’), and a lower average force exerted before ‘takeoff’, when muscle–tendon units shorten (‘soft takeoff’). This landing–takeoff asymmetry is consistent with the force–velocity relation of the ‘motor’ (i.e. with the basic property of muscle to resist stretching with a force greater than that developed during shortening), but it may also be due to the ‘machine’ (e.g. to the asymmetric lever system of the foot operating during stance). Hard landing and soft takeoff—never the reverse—were found in running, hopping and trotting animals using diverse lever systems, suggesting that the different machines evolved to comply with the basic force–velocity relation of the motor. Here we measure the mechanical energy of the centre of mass of the body in backward running, an exercise where the normal coupling between motor and machine is voluntarily disrupted, in order to see the relevance of the motor–machine interplay in human running. We find that the landing–takeoff asymmetry is reversed. The resulting ‘soft landing’ and ‘hard takeoff’ are associated with a reduced efficiency of positive work production. We conclude that the landing–takeoff asymmetry found in running, hopping and trotting is the expression of a convenient interplay between motor and machine. More metabolic energy must be spent in the opposite case when muscle is forced to work against its basic property (i.e. when it must exert a greater force during shortening and a lower force during stretching).