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.     

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.     

Regulation of step frequency in transtibial amputee endurance athletes using a running-specific prosthesis

Running specific prostheses (RSP) are designed to replicate the spring-like behaviour of the human leg during running, by incorporating a real physical spring in the prosthesis. Leg stiffness is an important parameter in running as it is strongly related to step frequency and running economy. To be able to select a prosthesis that contributes to the required leg stiffness of the athlete, it needs to be known to what extent the behaviour of the prosthetic leg during running is dominated by the stiffness of the prosthesis or whether it can be regulated by adaptations of the residual joints. The aim of this study was to investigate whether and how athletes with an RSP could regulate leg stiffness during distance running at different step frequencies.Seven endurance runners with an unilateral transtibial amputation performed five running trials on a treadmill at a fixed speed, while different step frequencies were imposed (preferred step frequency (PSF) and −15%, −7.5%, +7.5% and +15% of PSF). Among others, step time, ground contact time, flight time, leg stiffness and joint kinetics were measured for both legs.In the intact leg, increasing step frequency was accompanied by a decrease in both contact and flight time, while in the prosthetic leg contact time remained constant and only flight time decreased. In accordance, leg stiffness increased in the intact leg, but not in the prosthetic leg. Although a substantial contribution of the residual leg to total leg stiffness was observed, this contribution did not change considerably with changing step frequency.Amputee athletes do not seem to be able to alter prosthetic leg stiffness to regulate step frequency during running. This invariant behaviour indicates that RSP stiffness has a large effect on total leg stiffness and therefore can have an important influence on running performance. Nevertheless, since prosthetic leg stiffness was considerably lower than stiffness of the RSP, compliance of the residual leg should not be ignored when selecting RSP stiffness.     

Walking and running at resonance

Humans and other animals can temporarily store mechanical energy in elastic oscillations, f(el), of body parts and in pendulum oscillations, f(p) = const sq.rt (g/L), of legs, length L, or other appendages, and thereby reduce the energy consumption of locomotion. However, energy saving only occurs if these oscillations are tuned to the leg propagation frequency f. It has long been known that f is tuned to the pendulum frequency of the free-swinging leg of walkers. During running the leg frequency increases to some new value f = f(r). We propose that in order to maintain resonance the animal, mass M, actively increases its leg pendulum frequency to the new value f(p,r) =const sq.rt (a(y)/L)=f(r), by giving its hips a vertical acceleration a(y)= F(y)/M. The pendulum frequency is increased if the impact force F(y) of the stance foot is larger than Mg, explaining the observation by Alexander and Bennet-Clark (1976) that F(v) becomes larger than Mg when animals start to run. Our model predictions of the running velocity U(r) as function of L, F(v), are in agreement with measurements of these quantities (Farley et al. 1993). The leg's longitudinal elastic oscillation frequency scales as f(el) = const sq.rt (k/M). Experiments by Ferris et al., (1998) show that runners adjust their leg's stiffness, k, when running on surfaces of different elasticity so that the total stiffness k remains constant. Our analysis of their data suggests that the longitudinal oscillations of the stance leg are indeed kept in tune with the running frequency. Therefore we conclude that humans, and by extension all animals, maintain resonance during running. Our model also predicts the Froude number of walking-running transitions, Fr = U(2)/gL approximately 0.5 in good agreement with measurements.     

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.     

The effect of speed on leg stiffness and joint kinetics in human running

The goals of this study were to examine the following hypotheses: (a) there is a difference between the theoretically calculated (McMahon and Cheng, 1990. Journal of Biomechanics 23, 65-78) and the kinematically measured length changes of the spring-mass model and (b) the leg spring stiffness, the ankle spring stiffness and the knee spring stiffness are influenced by running speed. Thirteen athletes took part in this study. Force was measured using a "Kistler" force plate (1000 Hz). Kinematic data were recorded using two high-speed (120 Hz) video cameras. Each athlete completed trials running at five different velocities (approx. 2.5, 3.5, 4.5, 5.5 and 6.5 m/s). Running velocity influences the leg spring stiffness, the effective vertical spring stiffness and the spring stiffness at the knee joint. The spring stiffness at the ankle joint showed no statistical difference (p < 0.05) for the five velocities. The theoretically calculated length change of the spring-mass model significantly (p < 0.05) overestimated the actual length change. For running velocities up to 6.5 m/s the leg spring stiffness is influenced mostly by changes in stiffness at the knee joint.     

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.     

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.     

Runners adjust leg stiffness for their first step on a new running surface.

Human runners adjust the stiffness of their stance leg to accommodate surface stiffness during steady state running. This adjustment allows runners to maintain similar center of mass movement (e.g., ground contact time and stride frequency) regardless of surface stiffness. When runners encounter abrupt transitions in the running surface, they must either make a rapid adjustment or allow the change in the surface stiffness to disrupt their running mechanics. Our goal was to determine how quickly runners adjust leg stiffness when they encounter an abrupt but expected change in surface stiffness that they have encountered previously. Six human subjects ran at 3 m s(-1) on a rubber track with two types of rubber surfaces: a compliant "soft" surface (ksurf = 21.3 kN m(-1) and a non-compliant "hard" surface (ksurf = 533 kN m(-1). We found that runners completely adjusted leg stiffness for their first step on the new surface after the transition. For example, runners decreased leg stiffness by 29% between the last step on the soft surface and the first step on the hard surface (from 10.7 kN m(-1) to 7.6 kN m(-1), respectively). As a result, the vertical displacement of the center of mass during stance ( approximately 7 cm) did not change at the transition despite a reduction in surface compression from 6 cm to less than 0.25 cm. By rapidly adjusting leg stiffness, each runner made a smooth transition between surfaces so that the path of the center of mass was unaffected by the change in surface stiffness.     

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.     

Running in the real world: adjusting leg stiffness for different surfaces.

A running animal coordinates the actions of many muscles, tendons, and ligaments in its leg so that the overall leg behaves like a single mechanical spring during ground contact. Experimental observations have revealed that an animal''s leg stiffness is independent of both speed and gravity level, suggesting that it is dictated by inherent musculoskeletal properties. However, if leg stiffness was invariant, the biomechanics of running (e.g. peak ground reaction force and ground contact time) would change when an animal encountered different surfaces in the natural world. We found that human runners adjust their leg stiffness to accommodate changes in surface stiffness, allowing them to maintain similar running mechanics on different surfaces. These results provide important insight into mechanics and control of animal locomotion and suggest that incorporating an adjustable leg stiffness in the design of hopping and running robots is important if they are to match the agility and speed of animals on varied terrain.     

The spring-mass model for running and hopping

A simple spring-mass model consisting of a massless spring attached to a point mass describes the interdependency of mechanical parameters characterizing running and hopping of humans as a function of speed. The bouncing mechanism itself results in a confinement of the free parameter space where solutions can be found. In particular, bouncing frequency and vertical displacement are closely related. Only a few parameters, such as the vector of the specific landing velocity and the specific leg length, are sufficient to determine the point of operation of the system. There are more physiological constraints than independent parameters. As constraints limit the parameter space where hopping is possible, they must be tuned to each other in order to allow for hopping at all. Within the range of physiologically possible hopping frequencies, a human hopper selects a frequency where the largest amount of energy can be delivered and still be stored elastically. During running and hopping animals use flat angles of the landing velocity resulting in maximum contact length. In this situation ground reaction force is proportional to specific contact time and total displacement is proportional to the square of the step duration. Contact time and hopping frequency are not simply determined by the natural frequency of the spring-mass system, but are influenced largely by the vector of the landing velocity. Differences in the aerial phase or in the angle of the landing velocity result in the different kinematic and dynamic patterns observed during running and hopping. Despite these differences, the model predicts the mass specific energy fluctuations of the center of mass per distance to be similar for runners and hoppers and similar to empirical data obtained for animals of various size.     

A movement criterion for running

The adjustment of the leg during running was addressed using a spring-mass model with a fixed landing angle of attack. The objective was to obtain periodic movement patterns. Spring-like running was monitored by a one-dimensional stride-to-stride mapping of the apex height to identify mechanically stable fixed points. We found that for certain angles of attack, the system becomes self-stabilized if the leg stiffness was properly adjusted and a minimum running speed was exceeded. At a given speed, running techniques fulfilling a stable movement pattern are characterized by an almost constant maximum leg force. With increasing speed, the leg adjustment becomes less critical. The techniques predicted for stable running are in agreement with experimental studies. Mechanically self-stabilized running requires a spring-like leg operation, a minimum running speed and a proper adjustment of leg stiffness and angle of attack. These conditions can be considered as a movement criterion for running.     

Robust and efficient walking with spring-like legs

The development of bipedal walking robots is inspired by human walking. A way of implementing walking could be performed by mimicking human leg dynamics. A fundamental model, representing human leg dynamics during walking and running, is the bipedal spring-mass model which is the basis for this paper. The aim of this study is the identification of leg parameters leading to a compromise between robustness and energy efficiency in walking. It is found that, compared to asymmetric walking, symmetric walking with flatter angles of attack reveals such a compromise. With increasing leg stiffness, energy efficiency increases continuously. However, robustness is the maximum at moderate leg stiffness and decreases slightly with increasing stiffness. Hence, an adjustable leg compliance would be preferred, which is adaptable to the environment. If the ground is even, a high leg stiffness leads to energy efficient walking. However, if external perturbations are expected, e.g. when the robot walks on uneven terrain, the leg should be softer and the angle of attack flatter. In the case of underactuated robots with constant physical springs, the leg stiffness should be larger than k = 14 in order to use the most robust gait. Soft legs, however, lack in both robustness and efficiency.     

Ambulatory estimates of maximal aerobic power from foot -ground contact times and heart rates in running humans.

Seeking to develop a simple ambulatory test of maximal aerobic power (VO(2 max)), we hypothesized that the ratio of inverse foot-ground contact time (1/t(c)) to heart rate (HR) during steady-speed running would accurately predict VO(2 max). Given the direct relationship between 1/t(c) and mass-specific O(2) uptake during running, the ratio 1/t(c). HR should reflect mass-specific O(2) pulse and, in turn, aerobic power. We divided 36 volunteers into matched experimental and validation groups. VO(2 max) was determined by a treadmill test to volitional fatigue. Ambulatory monitors on the shoe and chest recorded foot-ground contact time (t(c)) and steady-state HR, respectively, at a series of submaximal running speeds. In the experimental group, aerobic fitness index (1/t(c). HR) was nearly constant across running speed and correlated with VO(2 max) (r = 0.90). The regression equation derived from data from the experimental group predicted VO(2 max) from the 1/t(c). HR values in the validation group within 8.3% and 4.7 ml O(2) x kg(-1) x min(-1) (r = 0.84) of measured values. We conclude that simultaneous measurements of foot-ground constant times and heart rates during level running at a freely chosen constant speed can provide accurate estimates of maximal aerobic power.     

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.     

The spring-mass model and the energy cost of treadmill running

During running, the behaviour of the support leg was studied by modelling the runner using an oscillating system composed of a spring (the leg) and of a mass (the body mass). This model was applied to eight middle-distance runners running on a level treadmill at a velocity corresponding to 90% of their maximal aerobic velocity [mean 5.10 (SD 0.33) m · s⁻¹]. Their energy cost of running (C _r ), was determined from the measurement of O₂ consumption. The work, the stiffness and the resonant frequency of both legs were computed from measurements performed with a kinematic arm. The C _r was significantly related to the stiffness (P < 0.05, r = −0.80) and the absolute difference between the resonant frequency and the step frequency (P < 0.05, r = 0.79) computed for the leg producing the highest positive work. Neither of these significant relationships were obtained when analysing data from the other leg probably because of the work asymmetry observed between legs. It was concluded that the spring-mass model is a good approach further to understand mechanisms underlying the interindividual differences in C _r . Accepted: 18 August 1997     

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.     

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.     

The influence of track compliance on running

A model of running is proposed in which the leg is represented as a rack-and-pinion element in series with a damped spring. The rack-and-pinion element emphasizes the role of descending commands, while the damped spring represents the dynamic properties of muscles and the position and the rate sensitivity of reflexes. This model is used to predict separately the effect of track compliance on step length and ground contact time. The predictions are compared with experiments in which athletes ran over tracks of controlled spring stiffness. A sharp spike in foot force up to 5 times body weight was found on hard surfaces, but this spike disappeared as the athletes ran on soft experimental tracks. Both ground contact time and step length increased on very compliant surfaces, leading to moderately reduced running speeds, but a range of track stiffness was discovered which actually enhances speed.