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.     

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.     

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.     

Leg stiffness can be maintained during reactive hopping despite modified acceleration conditions

AimThe aim of the present study was to evaluate reactive hops under systematically modified acceleration conditions. It was hypothesized that a high preactivity of the leg extensors and phase-specific adjustments of the leg muscle activation would compensate the alterations caused by the various acceleration levels in order to maintain a high leg stiffness, thus enabling the jumper to perform truly reactive jumps with short ground contact times despite the unaccustomed acceleration conditions.MethodsGround reaction forces (GRF), kinematic and electromyographic data of 20 healthy subjects were recorded during reactive hopping in a special sledge jump system for seven different acceleration levels: three acceleration levels with lower than normal gravity (0.7g, 0.8g, 0.9g), one with gravitational acceleration (1g) and three with higher acceleration (1.1g, 1.2g, 1.3g).ResultsThe increase of the acceleration from 0.7g to 1.3g had no significant effect on the preactivity of the leg extensors, the leg stiffness and the rate of force development. However, it resulted in increased peak GRF (+15%), longer ground contact time (+10%) and increased angular excursion at the ankle and knee joints (+3°).DiscussionThroughout a wide acceleration range, the subjects were able to maintain a high leg stiffness and perform reactive hops by keeping the preactivity constantly high and adjusting the muscle activity in the later phases. In consequence, it can be concluded that the neuromuscular system can cope with different acceleration levels, at least in the acceleration range used in this study.     

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.     

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.     

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)).     

Age-related differences in the neural regulation of stretch-shortening cycle activities in male youths during maximal and sub-maximal hopping

The aim of the current study was to investigate potential age-related differences in neural regulation strategies during maximal and sub-maximal hopping. Thirty-two boys from three different age groups (9-, 12- and 15-years), completed trials of both maximal and sub maximal hopping, and based on contact and flight times, measures of reactive strength index (RSI = jump height/contact time) and leg stiffness (peak ground reaction force/peak displacement of centre of mass) were collected respectively. During all trials, surface electromyograms (EMG) were recorded from four different muscle sites of the dominant lower limb, during 100 ms pre-ground contact, and then four subsequent stretch reflex phases: background muscle activity (0-30 ms), short-latency stretch reflex (31-60 ms), intermediate15 latency stretch reflex 61-90 ms and long-latency stretch reflex (91-120 ms). Reactive strength index and leg stiffness were measured during the hopping trials. During maximal hopping, both 12- and 15-year olds produced significantly greater RSI (P < 0.02) than 9-year olds, with 15-year olds utilising significantly greater soleus muscle activity during the 100 ms prior to ground contact than the younger age groups (P < 0.01). During sub-maximal hopping, 15-year olds produced significantly greater absolute leg stiffness than both 12- and 9-year olds (P < 0.01), with 9-year olds producing significantly less soleus muscle activity during the 31-60 ms time phase. For all age groups, sub-maximal hopping was associated with significantly greater background muscle activity and short-latency stretch reflex activity in the soleus and vastus lateralis, when compared to maximal hopping (P < 0.001). Results suggest that as children mature, they become more reliant on supra-spinal feed forward input and short latency stretch reflexes to regulate greater levels of leg stiffness and RSI when hopping.     

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.     

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.     

Influence of leg stiffness and its effect on myodynamic jumping performance.

The purposes of this study are: a) to examine the possibility of influencing the leg stiffness through instructions given to the subjects and b) to determine the effect of the leg stiffness on the mechanical power and take-off velocity during the drop jumps. A total of 15 athletes performed a series of drop jumps from heights of 20, 40 and 60 cm. The instructions given to the subjects were a) "jump as high as you can" and b) "jump high a little faster than your previous jump". The jumps were performed at each height until the athlete could not achieve a shorter ground contact time. The ground reaction forces were measured using a "Kistler" force plate (1000 Hz). The athletes body positions were recorded using a high speed (250 Hz) video camera. EMG was used to measure muscle activity in five leg muscles. The data was divided into 5 groups where group 1 was made up of the longest ground contact times of each athlete and group 5 the shortest. The leg and ankle stiffness values were higher when the contact times were shorter. This means that by influencing contact time through verbal instructions it is possible to control leg stiffness. Maximum center of mass take-off velocity the can be achieved with different levels of leg stiffness. The mechanical power acting on the human body during the positive phase of the drop jumps had the highest values in group 3. This means that there is an optimum stiffness value for the lower extremities to maximize mechanical power.     

Comparison of whole-body vertical stiffness and leg stiffness during single-leg hopping in place in children and adults

In the hopping literature, whole-body vertical stiffness and leg stiffness are used interchangeably, due to most of the movement occurring in the vertical direction. However, there is some anterior/posterior movement of the center of mass and displacements of the foot during hopping in place in both children and adults. Further it is not understood if leg stiffness show a similar pattern as whole-body vertical stiffness when increasing hopping frequency. The purpose of this study was to test if whole-body vertical stiffness and leg stiffness are different during single-leg hopping in-place in children and adults, across a range of frequencies. Seventeen children aged 5–11 years and 16 young adults participated in this study. The subjects hopped at their preferred frequency as well as 20% below, 20% above and 40% above preferred frequency. Our results demonstrate that both whole-body vertical stiffness and leg stiffness increase when increasing hopping frequency for children and adults. However, whole-body vertical stiffness consistently overestimates leg stiffness due to a similar peak force but a greater leg length change compared to vertical COM displacement. This suggests a considerable horizontal COM movement from landing to mid-stance during hopping. Children aged 5–11 years old showed lower absolute values but higher normalized values of two stiffness measures than adults. This suggests somewhat adult-like stiffness control in children, but a reduced ability to manipulate the horizontal movement during single-leg hopping in place when compared to adults.     

Active leg stiffness and energy stored in the muscles during maximal counter movement jump in the aged

The purpose of this study was to examine the effects of age on active leg stiffness adjustment, electromyogram (EMG) activities and energy stored during eccentric and concentric phases in performing a maximal functional task involving stretch-shorten cycle. Ten young (24.3 ± 2 years) and 10 old (68.6 ± 5 years) healthy male subjects were filmed during maximal performance of counter movement jump (CMJ) and squat jump (SJ) on force plate. Integrated EMG (IEMG), ground reaction force (GRF), active leg stiffness, energy stored/returned and active work done by the muscles were compared between two groups on eccentric (ECC) and concentric (CON) phases of CMJ. The GRF, leg stiffness and energy stored in ECC and GRF, IEMG, energy returned and active work in CON were less in the elderly (p < 0.05). These results demonstrate that the neuromuscular function of adjusting active stiffness, storing elastic energy and optimizing the performance may decrease with age during CMJ.     

Knee stiffness is a major determinant of leg stiffness during maximal hopping

Understanding stiffness of the lower extremities during human movement may provide important information for developing more effective training methods during sports activities. It has been reported that leg stiffness during submaximal hopping depends primarily on ankle stiffness, but the way stiffness is regulated in maximal hopping is unknown. The goal of this study was to examine the hypothesis that knee stiffness is a major determinant of leg stiffness during the maximal hopping. Ten well-trained male athletes performed two-legged hopping in place with a maximal effort. We determined leg and joint stiffness of the hip, knee, and ankle from kinetic and kinematic data. Knee stiffness was significantly higher than ankle and hip stiffness. Further, the regression model revealed that only knee stiffness was significantly correlated with leg stiffness. The results of the present study suggest that the knee stiffness, rather than those of the ankle or hip, is the major determinant of leg stiffness during maximal hopping.     

Spring-like leg behaviour, musculoskeletal mechanics and control in maximum and submaximum height human hopping

The purpose of this study was to understand how humans regulate their 'leg stiffness' in hopping, and to determine whether this regulation is intended to minimize energy expenditure. 'Leg stiffness' is the slope of the relationship between ground reaction force and displacement of the centre of mass (CM). Variations in leg stiffness were achieved in six subjects by having them hop at maximum and submaximum heights at a frequency of 1.7 Hz. Kinematics, ground reaction forces and electromyograms were measured. Leg stiffness decreased with hopping height, from 350 N m(-1) kg(-1) at 26 cm to 150 N m(-1) kg(-1) at 14 cm. Subjects reduced hopping height primarily by reducing the amplitude of muscle activation. Experimental results were reproduced with a model of the musculoskeletal system comprising four body segments and nine Hill-type muscles, with muscle stimulation STIM(t) as only input. Correspondence between simulated hops and experimental hops was poor when STIM(t) was optimized to minimize mechanical energy expenditure, but good when an objective function was used that penalized jerk of CM motion, suggesting that hopping subjects are not minimizing energy expenditure. Instead, we speculated, subjects are using a simple control strategy that results in smooth movements and a decrease in leg stiffness with hopping height.     

Altered control strategy between leading and trailing leg increases knee adduction moment in the elderly while descending stairs

The aim of the study was to examine the external knee adduction moments in a group of older and younger adults while descending stairs and thus the possibility of an increased risk of knee osteoarthritis due to altered knee joint loading in the elderly. Twenty-seven older and 16 younger adults descended a purpose-built staircase. A motion capture system and a force plate were used to determine the subjects' 3D kinematics and ground reaction forces (GRF) during locomotion. Calculation of the leg kinematics and kinetics was done by means of a rigid, three-segment, 3D leg model. In the initial portion of the support phase, older adults showed a more medio-posterior GRF vector relative to the ankle joint, leading to lower ankle joint moments (P<0.05). At the knee, the older adults demonstrated a more medio-posterior directed GRF vector, increasing in knee flexion and adduction in the second part of the single support phase (P<0.05). Further, GRF magnitude was lower in the initial and higher in the mid-portions of the support phase for the elderly (P<0.05). The results show that older adults descend stairs by using the trailing leg before the initiation of the double support phase more compared to the younger ones. The consequence of this altered control strategy while stepping down is a more medially directed GRF vector increasing the magnitude of external knee adduction moment in the elderly. The observed changes between leading and trailing leg in the elderly may cause a redistribution of the mechanical load at the tibiofemoral joint, affecting the initiation and progression of knee osteoarthritis in the elderly.     

Leg stiffness primarily depends on ankle stiffness during human hopping

When humans hop in place or run forward, they adjust leg stiffness to accommodate changes in stride frequency or surface stiffness. The goal of the present study was to determine the mechanisms by which humans adjust leg stiffness during hopping in place. Five subjects hopped in place at 2.2 Hz while we collected force platform and kinematic data. Each subject completed trials in which they hopped to whatever height they chose ("preferred height hopping") and trials in which they hopped as high as possible ("maximum height hopping"). Leg stiffness was approximately twice as great for maximum height hopping as for preferred height hopping. Ankle torsional stiffness was 1.9-times greater while knee torsional stiffness was 1.7-times greater in maximum height hopping than in preferred height hopping. We used a computer simulation to examine the sensitivity of leg stiffness to the observed changes in ankle and knee stiffness. Our model consisted of four segments (foot, shank, thigh, head-arms-trunk) interconnected by three torsional springs (ankle, knee, hip). In the model, increasing ankle stiffness by 1.9-fold, as observed in the subjects, caused leg stiffness to increase by 2.0-fold. Increasing knee stiffness by 1.7-fold had virtually no effect on leg stiffness. Thus, we conclude that the primary mechanism for leg stiffness adjustment is the adjustment of ankle stiffness.     

Linear center-of-mass dynamics emerge from non-linear leg-spring properties in human hopping

Given the almost linear relationship between ground-reaction force and leg length, bouncy gaits are commonly described using spring–mass models with constant leg-spring parameters. In biological systems, however, spring-like properties of limbs may change over time. Therefore, it was investigated how much variation of leg-spring parameters is present during vertical human hopping. In order to do so, rest-length and stiffness profiles were estimated from ground-reaction forces and center-of-mass dynamics measured in human hopping. Trials included five hopping frequencies ranging from 1.2 to 3.6 Hz. Results show that, even though stiffness and rest length vary during stance, for most frequencies the center-of-mass dynamics still resemble those of a linear spring–mass hopper. Rest-length and stiffness profiles differ for slow and fast hopping. Furthermore, at 1.2 Hz two distinct control schemes were observed.     

Comparison of lower limb kinetics,kinematics and muscle activation during drop jumping under shod and barefoot conditions

This study was to investigate the acute effects of wearing shoes on lower limb kinetics, kinematics and muscle activation during a drop jump. Eighteen healthy men performed a drop jump under barefoot and shod conditions. Vertical ground reaction force (GRF) was measured on a force plate during the contact phase of a drop jump, and GRF valuables were calculated for each condition. The angles of the knee and ankle joints, and the foot strike angle (the angle between the plantar surface of the foot and the ground during ground contact) as well as the electromyography of 7 muscles were measured. The shod condition showed a significant larger first peak GRF, longer time to first peak GRF from the initial ground contact and lower initial loading rate than the barefoot condition. The shod condition showed a significant larger ankle joint angle at initial ground contact, smaller knee joint angle between the second peak GRF and take-off as well as smaller foot strike angle at both initial ground contact and take-off than the barefoot condition. There were significant correlations between relative differences in ankle joint at the initial ground contact and relative differences in the initial loading rate. The muscle activity of all muscles during foot ground contact did not differ between two conditions; however, in the shod condition, muscle activation of 150 ms before foot ground contact was significantly higher in the rectus femoris, whereas it was lower in the biceps femoris and tibialis anterior muscles than the barefoot condition. These results indicate that wearing shoes alternates the GRF variables at initial ground contact, joint kinematics at the ground contact and muscle activation before foot ground contact during a drop jump, suggesting that the effects of wearing shoes on drop jump training differ from being barefoot.     

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.