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.     

A comparison and update of direct kinematic-kinetic models of leg stiffness in human running

Direct kinematic-kinetic modelling currently represents the “Gold-standard” in leg stiffness quantification during three-dimensional (3D) motion capture experiments. However, the medial-lateral components of ground reaction force and leg length have been neglected in current leg stiffness formulations. It is unknown if accounting for all 3D would alter healthy biologic estimates of leg stiffness, compared to present direct modelling methods. This study compared running leg stiffness derived from a new method (multiplanar method) which includes all three Cartesian axes, against current methods which either only include the vertical axis (line method) or only the plane of progression (uniplanar method). Twenty healthy female runners performed shod overground running at 5.0 m/s. Three-dimensional motion capture and synchronised in-ground force plates were used to track the change in length of the leg vector (hip joint centre to centre of pressure) and resultant projected ground reaction force. Leg stiffness was expressed as dimensionless units, as a percentage of an individual’s bodyweight divided by standing leg length (BW/LL). Leg stiffness using the line method was larger than the uniplanar method by 15.6%BW/LL (P < .001), and multiplanar method by 24.2%BW/LL (P < .001). Leg stiffness from the uniplanar method was larger than the multiplanar method by 8.5%BW/LL (6.5 kN/m) (P < .001). The inclusion of medial-lateral components significantly increased leg deformation magnitude, accounting for the reduction in leg stiffness estimate with the multiplanar method. Given that limb movements typically occur in 3D, the new multiplanar method provides the most complete accounting of all force and length components in leg stiffness calculation.     

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.     

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.     

Resonance-based oscillations could describe human gait mechanics under various loading conditions

The oscillatory behavior of the center of mass (CoM) and the corresponding ground reaction force (GRF) of human gait for various gait speeds can be accurately described in terms of resonance using a spring–mass bipedal model. Resonance is a mechanical phenomenon that reflects the maximum responsiveness and energetic efficiency of a system. To use resonance to describe human gait, we need to investigate whether resonant mechanics is a common property under multiple walking conditions. Body mass and leg stiffness are determinants of resonance; thus, in this study, we investigated the following questions: (1) whether the estimated leg stiffness increased with inertia, (2) whether a resonance-based CoM oscillation could be sustained during a change in the stiffness, and (3) whether these relationships were consistently observed for different walking speeds. Seven healthy young subjects participated in over-ground walking trials at three different gait speeds with and without a 25-kg backpack. We measured the GRFs and the joint kinematics using three force platforms and a motion capture system. The leg stiffness was incorporated using a stiffness parameter in a compliant bipedal model that best fitted the empirical GRF data. The results showed that the leg stiffness increased with the load such that the resonance-based oscillatory behavior of the CoM was maintained for a given gait speed. The results imply that the resonance-based oscillation of the CoM is a consistent gait property and that resonant mechanics may be useful for modeling human gait.     

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.     

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.     

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)).     

Ground reaction force estimation using an insole-type pressure mat and joint kinematics during walking

Kinetic analysis of walking requires joint kinematics and ground reaction force (GRF) measurement, which are typically obtained from a force plate. GRF is difficult to measure in certain cases such as slope walking, stair climbing, and track running. Nevertheless, estimating GRF continues to be of great interest for simulating human walking. The purpose of the study was to develop reaction force models placed on the sole of the foot to estimate full GRF when only joint kinematics are provided (Type-I), and to estimate ground contact shear forces when both joint kinematics and foot pressure are provided (Type-II and Type-II-val). The GRF estimation models were attached to a commercial full body skeletal model using the AnyBody Modeling System, which has an inverse dynamics-based optimization solver. The anterior–posterior shear force and medial–lateral shear force could be estimated with approximate accuracies of 6% BW and 2% BW in all three methods, respectively. Vertical force could be estimated in the Type-I model with an accuracy of 13.75% BW. The accuracy of the force estimation was the highest during the mid-single-stance period with an average RMS for errors of 3.10% BW, 1.48% BW, and 7.48% BW for anterior–posterior force, medial–lateral force, and vertical force, respectively. The proposed GRF estimation models could predict full and partial GRF with high accuracy. The design of the contact elements of the proposed model should make it applicable to various activities where installation of a force measurement system is difficult, including track running and treadmill walking.     

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

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

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.     

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.     

Estimates of muscle function in human gait depend on how foot-ground contact is modelled

Computational analyses of leg-muscle function in human locomotion commonly assume that contact between the foot and the ground occurs at discrete points on the sole of the foot. Kinematic constraints acting at these contact points restrict the motion of the foot and, therefore, alter model calculations of muscle function. The aim of this study was to evaluate how predictions of muscle function obtained from musculoskeletal models are influenced by the model used to simulate ground contact. Both single- and multiple-point contact models were evaluated. Muscle function during walking and running was determined by quantifying the contributions of individual muscles to the vertical, fore-aft and mediolateral components of the ground reaction force (GRF). The results showed that two factors – the number of foot-ground contact points assumed in the model and the type of kinematic constraint enforced at each point – affect the model predictions of muscle coordination. Whereas single- and multiple-point contact models produced similar predictions of muscle function in the sagittal plane, inconsistent results were obtained in the mediolateral direction. Kinematic constraints applied in the sagittal plane altered the model predictions of muscle contributions to the vertical and fore-aft GRFs, while constraints applied in the frontal plane altered the calculations of muscle contributions to the mediolateral GRF. The results illustrate the sensitivity of calculations of muscle coordination to the model used to simulate foot-ground contact.     

Estimates of muscle function in human gait depend on how foot-ground contact is modelled

Computational analyses of leg-muscle function in human locomotion commonly assume that contact between the foot and the ground occurs at discrete points on the sole of the foot. Kinematic constraints acting at these contact points restrict the motion of the foot and, therefore, alter model calculations of muscle function. The aim of this study was to evaluate how predictions of muscle function obtained from musculoskeletal models are influenced by the model used to simulate ground contact. Both single- and multiple-point contact models were evaluated. Muscle function during walking and running was determined by quantifying the contributions of individual muscles to the vertical, fore-aft and mediolateral components of the ground reaction force (GRF). The results showed that two factors--the number of foot-ground contact points assumed in the model and the type of kinematic constraint enforced at each point--affect the model predictions of muscle coordination. Whereas single- and multiple-point contact models produced similar predictions of muscle function in the sagittal plane, inconsistent results were obtained in the mediolateral direction. Kinematic constraints applied in the sagittal plane altered the model predictions of muscle contributions to the vertical and fore-aft GRFs, while constraints applied in the frontal plane altered the calculations of muscle contributions to the mediolateral GRF. The results illustrate the sensitivity of calculations of muscle coordination to the model used to simulate foot-ground contact.     

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.     

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.     

Joint stiffness of the ankle and the knee in running

The spring-mass model is a valid fundament to understand global dynamics of fast legged locomotion under gravity. The underlying concept of elasticity, implying leg stiffness as a crucial parameter, is also found on lower motor control levels, i.e. in muscle-reflex and muscle-tendon systems. Therefore, it seems reasonable that global leg stiffness emerges from local elasticity established by appropriate joint torques. A recently published model of an elastically operating, segmented leg predicts that proper adjustment of joint elasticities to the leg geometry and initial conditions of ground contact provides internal leg stability. Another recent study suggests that in turn the leg segmentation and the initial conditions may be a consequence of metabolic and bone stress constraints. In this study, the theoretical predictions were verified experimentally with respect to initial conditions and elastic joint characteristics in human running. Kinematics and kinetics were measured and the joint torques were estimated by inverse dynamics. Stiffnesses and elastic nonlinearities describing the resulting joint characteristics were extracted from parameter fits. Our results clearly support the theoretical predictions: the knee joint is always stiffer and more extended than the ankle joint. Moreover, the knee torque characteristic on the average shows the higher nonlinearity. According to literature, the leg geometry is a consequence of metabolic and material stress limitations. Adapted to this given geometry, the initial joint angle conditions in fast locomotion are a compromise between metabolic and control effort minimisation. Based on this adaptation, an appropriate joint stiffness ratio between ankle and knee passively safeguards the internal leg stability. The identified joint nonlinearities contribute to the linearisation of the leg spring.     

Effects of altered stride frequency and contact time on leg-spring behavior in human running

Many studies have demonstrated that contact time is a key factor affecting both the energetics and mechanics of running. The purpose of the present study was to further explore the relationships between contact time (t(c)), step frequency (f) and leg stiffness (k(leg)) in human running. Since f is a compound parameter, depending on both contact and aerial time, the specific goal of this study was to independently vary f and t(c) and to investigate their respective effects on spring-mass characteristics during running, seeking to determine if the changes in k(leg) observed when running at different f are mainly due to inherent changes in t(c). We compared three types of constant 3.33 m s(-1) running conditions in 10 male subjects: normal running at the subject's freely chosen f, running with decreased and increased f, and decreased and increased t(c) at the imposed freely chosen f. The data from the varied f trials showed that the variation of t(c) was strongly correlated to that of k(leg) (r(2)=0.90), and the variation of f was also significantly correlated to that of k(leg) (r(2)=0.47). Further, changes in t(c) obtained in various t(c) conditions were significantly correlated to changes in k(leg) (r(2)=0.96). These results confirm that leg stiffness was significantly influenced by step frequency variations during constant speed running, as earlier demonstrated, but our more novel finding is that compared to step frequency, the effect of contact time variations appears to be a stronger and more direct determinant of k(leg). Indeed, 90-96% of the variance in k(leg) can be explained by contact time, whether this latter parameter is directly controlled, or indirectly controlled through its close relationship with step frequency. In conclusion, from the comparison of two experimental procedures, i.e. imposing various step frequency conditions vs. asking subjects to intentionally vary contact time at their freely chosen step frequency, it appears that changes in leg stiffness are mainly related to changes in contact time, rather than to those in step frequency. Step frequency appears to be an indirect factor influencing leg stiffness, through its effect on contact time, which could be considered a major determinant of this spring-mass characteristic of human running.