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.     

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.     

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 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.     

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.     

Bipedal walking and running with spring-like biarticular muscles

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

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.     

Vertical stiffness during one-legged hopping with and without using a running-specific prosthesis

Although athletes with unilateral below-the-knee amputations (BKAs) generally use their affected leg, including their prosthesis, as their take-off leg for the long jump, little is known about the spring-like leg behavior and stiffness regulation of the affected leg. The purpose of this study was to investigate vertical stiffness during one-legged hopping in an elite-level long jump athlete with a unilateral BKA. We used the spring-mass model to calculate vertical stiffness, which equals the ratio of maximum vertical ground reaction force to maximum center of mass displacement, while the athlete with a BKA hopped on one leg at a range of frequencies. Then, we compared the vertical stiffness of this athlete to seven non-amputee elite-level long-jumpers. We found that from 1.8 to 3.4 Hz, the vertical stiffness of the unaffected leg for an athlete with a BKA increases with faster hopping frequencies, but the vertical stiffness of the affected leg remains nearly constant across frequencies. The athlete with a BKA attained the desired hopping frequencies at 2.2 and 2.6 Hz, but was unable to match the lowest (1.8 Hz) and two highest frequencies (3.0 and 3.4 Hz) using his affected leg. We also found that at 2.5 Hz, unaffected leg vertical stiffness was 15% greater than affected leg vertical stiffness, and the vertical stiffness of non-amputee long-jumpers was 32% greater than the affected leg vertical stiffness of an athlete with a BKA. The results of the present study suggest that the vertical stiffness regulation strategy of an athlete with a unilateral BKA is not the same in the unaffected versus affected legs, and compared to non-amputees.     

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.     

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.     

Dynamics of the long jump

A mechanical model is proposed which quantitatively describes the dynamics of the centre of gravity (c.g.) during the take-off phase of the long jump. The model entails a minimal but necessary number of components: a linear leg spring with the ability of lengthening to describe the active peak of the force time curve and a distal mass coupled with nonlinear visco-elastic elements to describe the passive peak. The influence of the positions and velocities of the supported body and the jumper's leg as well as of systemic parameters such as leg stiffness and mass distribution on the jumping distance were investigated. Techniques for optimum operation are identified: (1) There is a minimum stiffness for optimum performance. Further increase of the stiffness does not lead to longer jumps. (2) For any given stiffness there is always an optimum angle of attack. (3) The same distance can be achieved by different techniques. (4) The losses due to deceleration of the supporting leg do not result in reduced jumping distance as this deceleration results in a higher vertical momentum. (5) Thus, increasing the touch-down velocity of the jumper's supporting leg increases jumping distance.     

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.     

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.     

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     

Mechanical power in running: a comparison of different approaches

The purposes of the present study were: (1) to compare four different methods of calculating mechanical power in running on the basis of comparable data over a wide range of running velocity; (2) to examine the linearity of the relation between mechanical power as calculated with the four methods and running velocity. Eight runners participated in the investigation (height: 1.82 +/- 0.03 m, body mass: 81.05 +/- 4.69 kg). A Kistler force platform registered all components of the ground reaction force (1000 Hz) during one foot ground contact, which was additionally video taped using two high-speed video cameras running at 120 Hz. Four different methods were used to calculate mechanical power. Two methods determined the mechanical power due to the work done on the athletes' center of mass and two were calculated from the motion of the athletes' segments. The four different methods provided different relations between mechanical power and running velocity. The calculations on the basis of kinematic data cannot be recommended to determine efficiency of movement. The methods based on ground reaction force measurements revealed significant linear relations (r = 0.90, r2 = 0.84) between running velocity and mechanical power.     

Does a crouched leg posture enhance running stability and robustness?

Humans and birds both walk and run bipedally on compliant legs. However, differences in leg architecture may result in species-specific leg control strategies as indicated by the observed gait patterns. In this work, control strategies for stable running are derived based on a conceptual model and compared with experimental data on running humans and pheasants (Phasianus colchicus). From a model perspective, running with compliant legs can be represented by the planar spring mass model and stabilized by applying swing leg control. Here, linear adaptations of the three leg parameters, leg angle, leg length and leg stiffness during late swing phase are assumed. Experimentally observed kinematic control parameters (leg rotation and leg length change) of human and avian running are compared, and interpreted within the context of this model, with specific focus on stability and robustness characteristics. The results suggest differences in stability characteristics and applied control strategies of human and avian running, which may relate to differences in leg posture (straight leg posture in humans, and crouched leg posture in birds). It has been suggested that crouched leg postures may improve stability. However, as the system of control strategies is overdetermined, our model findings suggest that a crouched leg posture does not necessarily enhance running stability. The model also predicts different leg stiffness adaptation rates for human and avian running, and suggests that a crouched avian leg posture, which is capable of both leg shortening and lengthening, allows for stable running without adjusting leg stiffness. In contrast, in straight-legged human running, the preparation of the ground contact seems to be more critical, requiring leg stiffness adjustment to remain stable. Finally, analysis of a simple robustness measure, the normalized maximum drop, suggests that the crouched leg posture may provide greater robustness to changes in terrain height.     

Groucho running

An important determinant of the mechanics of running is the effective vertical stiffness of the body. This stiffness increases with running speed. At any one speed, the stiffness may be reduced in a controlled fashion by running with the knees bent more than usual. In a series of experiments, subjects ran in both normal and flexed postures on a treadmill. In other experiments, they ran down a runway and over a force platform. Results show that running with the knees bent reduces the effective vertical stiffness and diminishes the transmission of mechanical shock from the foot to the skull but requires an increase of as much as 50% in the rate of O2 consumption. A new dimensionless parameter (u omega 0/g) is introduced to distinguish between hard and soft running modes. Here, omega 0 is the natural frequency of a mass-spring system representing the body, g is gravity, and u is the vertical landing velocity. In normal running, this parameter is near unity, but in deep-flexed running, where the aerial phase of the stride cycle almost disappears, u omega 0/g approaches zero.     

A virtual model of the bench press exercise

The objective of this study was to design and validate a three degrees of freedom model in the sagittal plane for the bench press exercise. The mechanical model was based on rigid segments connected by revolute and prismatic pairs, which enabled a kinematic approach and global force estimation. The method requires only three simple measurements: (i) horizontal position of the hand (x₀); (ii) vertical displacement of the barbell (Z) and (iii) elbow angle (θ). Eight adult male throwers performed maximal concentric bench press exercises against different masses. The kinematic results showed that the vertical displacement of each segment and the global centre of mass followed the vertical displacement of the lifted mass. Consequently, the vertical velocity and acceleration of the combined centre of mass and the lifted mass were identical. Finally, for each lifted mass, there were no practical differences between forces calculated from the bench press model and those simultaneously measured with a force platform. The error was lower than 2.5%. The validity of the mechanical method was also highlighted by a standard error of the estimate (SEE) ranging from 2.0 to 6.6 N in absolute terms, a coefficient of variation (CV) ?0.8%, and a correlation between the two scores ?0.99 for all the lifts (p<0.001). The method described here, which is based on three simple parameters, allows accurate evaluation of the force developed by the upper limb muscles during bench press exercises in both field and laboratory conditions.     

Leg stiffness in human running: Comparison of estimates derived from previously published models to direct kinematic-kinetic measures

It is not presently clear whether mathematical models used to estimate leg stiffness during human running are valid. Therefore, leg stiffness during the braking phase of ground contact of running was calculated directly using synchronous kinematic (high-speed motion analysis) and kinetic (force platform) analysis, and compared to stiffness calculated using four previously published kinetic models. Nineteen well-trained male middle distance runners (age=21.1±4.1yr; VO(2max)=69.5±7.5mlO(2)kg(-1)min(-1)) completed a series of runs of increasing speed from 2.5 to 6.5ms(-1). Leg stiffness was calculated directly from kinetic-kinematic analysis using both vertical and horizontal forces to obtain the resultant force in the line of leg compression (Model 1). Values were also estimated using four previously published mathematical models where only force platform derived and anthropometric measures were required (Models 2-5; Morin et al., 2005, Morin et al., 2011, Blum et al., 2009, Farley et al., 1993, respectively). The greatest statistical similarity between leg stiffness values occurred with Models 1 and 2. The poorest similarity occurred when values from Model 4 were compared with Model 1. Analyses suggest that the poor correlation between Model 1 other models may have resulted from errors in the estimation in change in leg length during the braking phase. Previously published mathematical models did not provide accurate leg stiffness estimates, although Model 2, used by Morin et al. (2005), provided reasonable estimates that could be further improved by the removal of systematic error using a correction factor (K=1.0496K(Model2)).     

How well can spring-mass-like telescoping leg models fit multi-pedal sagittal-plane locomotion data?

Idealized mathematical models of animals, with point-mass bodies and spring-like legs, have been used by researchers to study various aspects of terrestrial legged locomotion. Here, we fit a bipedal spring-mass model to the ground reaction forces of human running, a horse trotting, and a cockroach running. We find that, in all three cases, while the model captures center-of-mass motions and vertical force variations well, horizontal forces are less well reproduced, primarily due to variations in net force vector directions that the model cannot accommodate. The fits result in different apparent leg stiffnesses in the three animals. Assuming a simple fixed leg-angle touch-down strategy, we find that the gaits of these models are stable in different speed-step length regimes that overlap with those used by humans and horses, but not with that used by cockroaches.