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Short communication

The natural frequency of the foot-surface cushion during the stance phaseof running

Wangdo Kim a,n, John Tan b, Antonio Veloso a, Veronica Vleck a, Arkady S. Voloshin c

a Biomechanics Laboratory, Faculty of Human Kinetics, Technical University of Lisbon, Estrada da Costa, 1495-688 Cruz Quebrada, Portugalb Physical Education and Sports Science, National Institute of Education, Singapore 639798, Singaporec Department of Mechanical Engineering and Mechanics, Lehigh University Bethlehem, PA 18015-3085, USA

a r t i c l e i n f o

Article history:

Accepted 27 October 2010

Keywords:

Heel strike

Leg-compliance

Leg-stiffness

Extended Kalman filter

Inverse dynamics

a b s t r a c t

Researchers have reported on the stiffness of running in holistic terms, i.e. for the structures that are

undergoing deformation as a whole rather than in terms of specific locations. This study aimed to estimate

both the natural frequency and the viscous damping coefficient of the human foot-surface cushion, during

the period between the heel strike and the mid-stance phase of running, using a purposely developed one

degree-of-freedom inverted pendulum state space model of the leg. The model, which was validated via a

comparison of measured and estimated ground reaction forces, incorporated a novel use of linearized and

extended Kalman filter estimators. Investigation of the effect of variation of the natural frequency and/or

the damping of the cushioning mechanism during running, using the said model, revealed the natural

frequency of running on said foot-surface cushion, during the stance phase, to lie between 5 and 11 Hz.

The ‘‘extended Kalman filter (EKF)’’ approach, that was used here for the first time to directly apply

measured ground forces, may be widely applicable to the identification process of combined estimation of

both unknown physiological state and mechanical characteristics of the environment in an inverse

dynamic model.

& 2010 Elsevier Ltd. All rights reserved.

1. Introduction

The strategy of minimization of energy loss as a means ofimproving sporting performance is one that has received substan-tial attention from researchers and practitioners (Nigg, 1997;Stefanyshyn and Nigg, 1998). Various mathematical models havebeen used to predict the forces, acceleration or system parametersthat are related to such an impact activity (Bullimore and Burn,2007; Farley and Gonzalez, 1996; Cavagna et al., 1997). Forexample, a simple spring–mass system was introduced to modelthe basic mechanics of running and it works remarkably well(Blickhan, 1989; McMahon and Cheng, 1990). Ozguven and Berme(1988) later developed a two-degree-of-freedom dynamic model,as a means of predicting gymnasts’ landing forces on a hard surface.They took into account the interaction between gravity, impactvelocity, body mass and effective vertical stiffness. The extent towhich running mechanics could influence the downward velocityof the center of mass during a period of contact, and as aconsequence, vertical stiffness was also being investigated byMcMahon and Greene (1979). Nigg and Segesser (1992) on theother hand, explored the role of the sports shoe. These authors

showed that for the manufacturers’ claims that the sole can be usedas a means of ‘‘returning energy’’ to be supported, energy returnmust be substantial enough to make a difference, and be returnedboth at the right time and at the right frequency. Nigg et al. (2000)later indicated that the natural frequency of the sole of an unloadedshoe (i.e. a shoe without a subject in it) is typically between 100 and200 Hz, but that of a loaded shoe sole was as yet unreported. Theyestimated the natural frequency of the latter to be 20–30 Hz.

The relationship between stiffness (vertical, leg and jointparameters) and running surfaces was studied using spring–mass system models (Ferris and Farley, 1997; Ferris et al., 1998,1999; Kerdok et al., 2002). The influence of different types ofrunning surface on the amplitude of the shock waves that areinitiated during foot impact, and subsequently propagated throughthe human locomotory system, were evaluated, using an accel-erometer externally attached to the runner’s tibial tuberosity, byKim and Voloshin (1992). The authors reported that running ongrass results in nearly 25% higher shock wave amplitude thanrunning on asphalt does. They followed up the above-mentionedresults in a new study (Kim et al., 1994) with a three-degree-of-freedom spring–damper–mass model to describe both the dissipa-tion and attenuation of the resultant shock wave. The model reliedon quasilinearization of shank velocity output as proposed byBellman et al. (1976). For system identification, the velocity datawas obtained through numerical integration of the acceleration

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Journal of Biomechanics

0021-9290/$ - see front matter & 2010 Elsevier Ltd. All rights reserved.

doi:10.1016/j.jbiomech.2010.10.041

n Corresponding author. Tel.: +351 214148127.

E-mail address: wangdokim@fmh.utl.pt (W. Kim).

Journal of Biomechanics 44 (2011) 774–779

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measurement. They found that changes in model stiffness para-meters do not significantly affect estimated ground reaction forces(GRF), while increases in model damping parameter led todecreases in GRF. As predicted by the said model (Kim et al.,1994), the damping seems to be the dominant factor controlling theeffect of landing impact on the human musculoskeletal system.

It should be noted that estimation of running stiffness can beclassified as an inverse problem (with the local perturbation beingthe known or prescribed variable and the equations that describethe process being the unknown (Kim et al., 1993, 1994; Kim andVoloshin, 1995). Alternatively, the external environment could bethe unknown (Kim et al., 2009)). The set of equations that need tobe solved constitute ‘‘metamodels’’ that are affected by informationthat is inferred from observations that are usually affected by the‘‘noisy data’’. Thus, they are normally more difficult to solve thanclassic direct boundary value problems. As the inverse model doesnot necessarily possess smooth mathematical properties, thesolutions that are obtained may be subject to an unacceptablelevel of error. Any means of decreasing the error of initial estimatescould, therefore, prove to be of real value.

The Kalman filter (KF) is an algorithm that estimates the state ofa system from noisy measurements using least-squares. It shouldallow the researcher to obtain both the optimal system stateestimate and a measure of how certain this state estimate is inrelation to the true state (Mohinder and Angus, 2001; Habibi,2008). In its extended (EKF) form (Habibi, 2008, see Appendix A fordetails [De Groote et al., 2008]), it can be used to ‘‘linearize’’ non-linear data about the current mean and covariance. Using an EKF,therefore, could prove to be a novel way of improving the quality ofestimation of running stiffness.

In view of the above, our main aim was to develop, and thengauge the validity of, an extended, non-linear EKF model for thefoot-surface cushion during the stance phase of running. Oursubsidiary aim was to use the EKF to explore the influence ofvariation in ground surface compliance and or the shock absorbingcapacity of the runner’s foot on damping. The last could allow forthe study of potential effect of landing impact on the humanmusculoskeletal system.

2. Methodology

In the model, stiffness (K) and damping (C) of the foot-surface cushion are

considered below the ground (as illustrated in Fig. 1, in which the inverted-

pendulum that represents the runner’s entire mass is hinged at the ground during

the stance phase, see Appendix B for details). For the second part of this study, five

young male subjects participated, from whom we first obtained a written informed

consent. Subject mass and trochanteric bursa-to-malleolus lengths were measured

using a Kistler force plate (Kistler Instruments, Switzerland) and anthropometric

measuring tape, respectively. Three passive markers were attached to the lateral

side of the skin above the trochanter, femoral condyle, and malleolus of each subject,

to allow for easy and accurate identification of bony landmarks (as illustrated in

Fig. 2).

Impact forces for the period between barefoot heel strike and toe-off period

were acquired for the subjects, using the Kistler force plate, when they ran over both

a compliant, polyurethane surface (as commonly used for running tracks in athletic

stadiums), and a non-compliant, ceramic, running surface (similar to that used for

house flooring). The length, width, and thickness of the polyurethane and ceramic

slabs that were used were 52 cm�46 cm�25 mm, and 46 cm�44 cm�12 mm,

respectively. In each case, the slab was placed directly on top of the Kistler force

plate and held tightly to it by adhesives. The force plate was then located about 10 m

from the starting point of a straight running path of approximately 25 m. The Peak

Motus System (Peak Performance Technologies, CO, USA) was also utilized to

capture kinematic motion, in the sagittal plane, for each subject as he ran barefoot, at

a self-selected running pace, over said path. We have assumed that the values for

relevant leg parameters remained constant as speed fluctuated from slow to

moderate, in accordance with the work of Brughelli and Cronin (2008) and

McMahon and Cheng (1990).

Only the force plate data that corresponded to pivoting of the leg were extracted.

Both the ground reaction force data and the kinematic data obtained from the Peak

Motus System associated to the subject’s mass and leg length were used as the

observation data for the inverse problem. The damping of the foot-surface cushion (C)

(a function of the softening effect of both the fatty pad under the calcaneus and the

surface on which the foot impact was made (Valiant, 1990)), the resonant frequency

and the ground reaction forces for running on polyurethane and running on the

ceramic plate, were estimated, using our EKF model. In this way, we were able to

explore the effect of variation of both the damping factor x and natural frequencyo of

the human foot-surface cushion.

K C

y

m

L0

Fig. 1. The one-degree-of-freedom inverted pendulum model was used to predict

the dynamic forces during running based on the concept that runners hit the ground

with one leg at a time. The heel of the colliding leg is modeled as a pivot to push

forward. For simplicity, we assume the mass of the leg to be negligible and that the

entire mass of the runner is concentrated at the hip. Thus, the model consists of one

mass with a pivot joint, that is attached to the ground. K is the spring constant of the

foot/ground interface, C is the damping coefficient of the foot/ground interface, y0 is

the initial angle at the heel strike, which is measured from the y-axis, m is the mass of

the subject, L is the length of the leg–hip to ankle joint, and y is the direction of

deformation of the contact point at the foot/ground cushion.

Fig. 2. Positions of attached markers on a subject’s body. The line connects the hip to

the ankle joint, representing a rigid bar in the model in Fig. 1. It should be noted that

although Fig. 3 shows the subjects wearing shoes, this study only applies to the case

of barefoot running.

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3. Results

We verified our EKF model using the Monte Carlo numericalsimulation, before applying our experimental data to it. To do this,we used the values of 0.10 and 20 rad/s for x and o obtained fromthe results of Nigg et al. (2000). Our typical results were within 14%and 0.08% of the aforesaid values. The extent to which discrepancyoccurred between the predicted and actual values for position,

velocity, and ground reaction force, was of the order of 5%, 20%, and0.01%, respectively. The aforesaid numbers may be a measure ofKalman filter performance.

After model verification, this ‘‘extended Kalman filter (EKF)’’approach was directly applied to measured ground forces. It mayalso be observed from our data that the first force peak (A) for thecompliant surface (Fig. 3(c)) was approximately 95% of the ampli-tude of the second one (B) (980 vs 1040 N, respectively), as a result of

Fig. 3. State variables estimated by EKF for the case of running on a compliant (polyurethane) surface. The position and velocity were determined by the state Eq. (B6) in

supplementary material.

W. Kim et al. / Journal of Biomechanics 44 (2011) 774–779776

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force attenuation. However, on the non-compliant surface themaximum vertical impact force coincided with the first peak (C)(1350 N, Fig. 4(c)). During the period within which the runner is incontact with the ground, the contact surfaces suffer a degree ofdeformation as is illustrated in Figs. 3(d) and 4(d). The peak

deformation that was predicted by the model was greater for thecompliant surface than for the non-compliant surface (7.5 mm (D) vsless than 5 mm (E)). Velocity was also estimated to be higher on thenon-compliant surface (compare Figs. 3(e) and 4(e)). The typical GRFprofile for the non-compliant surface (Fig. 4(c)) exhibited a

Fig. 4. State variables estimated by EKF for the case of running on a non-compliant (ceramic) surface. The first peak is a passive force peak that is associated with the shock of

contact with hard surface. The second peak is due to active muscle force such as that exerted by the quadriceps muscle (Novacheck, 1998). On the hard surface, the initial

contact of the unshod foot with the ground produced a spike in foot force. Said spike was not observed on the compliant surface (McMahon and Greene, 1979; Ferris et al.,

1999).

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significant peak (po .01) at the point of heel strike (C), while the datafor compliant surface showed a relatively smoother transition(Fig. 3(c)) from the point of the heel strike to the mid-stance phaseof running. On average, K was found to reside in the region of199.1 kN/m for the non-compliant surface and 130.3 kN/m for thecompliant surface (Table 1). The damping factor, x resistance tovelocity of the vibrating system, was in the range of 0.17–0.23.

In order to explore the effect of the differing natural frequenciesof the two surfaces on the resulting GRF, we simulated data using

the mass–spring–damper model (Eq. (B1) in supplementary mate-rial), for several values of o, (rad/s) (Fig. 5). The model impliedthat increasing the damping factor will lead to a decrease inthe maximum amplitude of GRF. By doubling damping in thecushioning system, the peak GRF amplitude could, theoretically,be lessened by about 10% (Fig. 6). This agrees with the experimentaldata that we obtained for the peak values of A and C (Figs. 3 and 4).

4. Discussion

Previously obtained experimental data (Kim and Voloshin,1992) have indicated that acceleration measurements obtainedfor running on a non-compliant surface are higher than thoseobtained on a compliant surface. The results that we presented here(Fig. 5) indicated that although the amplitude of the GRF is not afunction of the natural frequency of the system, the loading rate ( _F )is. This result has potential clinical implications, given that highloading rates will equate to high bone strain rates (Burr et al., 2002;Forwood, 2001; Turner et al., 1995), and could increase the risk forthe onset of knee osteoarthritis and/or stress fracture(s). Qualita-tively, this is underlined by the relatively higher natural frequencyvalues (Table 1) that we observed for those of our subjects whowere Taekwondo practitioners, with a history of rigorous, barefoot,physical training.

Kim et al. (1994) showed stiffness (K) to be related to the ‘‘storingof elastic energy’’. In the present study, where the runners ranbarefoot, K represented the combined stiffness of both the fat padlayer at the bottom of the heel and the running surface. The higherthe stiffness of said foot–ground mechanism, the higher the value ofK. However, K is both proportional to and will be affected by changesin both the runners’ mass and or the natural frequency of the foot-surface cushion (i.e. K¼mo 2). The latter is dictated by theinteraction of the foot, the compliance of the surface on which itlands, and its surrounding structures. Several studies support thepremise, which is one of the most important results of this studypaper that leg-stiffness is not directly related to running mechanics,but, rather, to the running environment. Recent studies (Ferris andFarley, 1997; Ferris et al., 1998, 1999; Kerdok et al., 2002) haverevealed that leg-stiffness is adjusted to accommodate differentsurfaces whilst similar running mechanics are maintained.

The K values that we obtained in this study lie midway betweenthose presented by Nigg et al. (2000) (of 1000–10,000 kN/m, forunloaded, typical ‘marathon runner’ and ‘high jumping’ shoes) andCavagna et al. (1997) (of 32715 kN/m, for the whole body). Wemay explain this by virtue of the EKF model being able to moreclosely track accurate ground reaction forces. As a result, our GRFvalues were more heavily weighted than the segment kinematics.This may have important implications for the accuracy of para-meter estimation, when performing inverse dynamics, in the caseof motion data. The latter method is more heavily weightedtowards segmental kinematics.

Table 1Subject data and model estimations for the natural frequency and spring stiffness.

Mass (kg) Hip to ankle

length(m)

Fitness level Natural frequency (rad/sec) and

(stiffness (kN/m)), respectively,

on the ceramic surface,

(mean7SD)

Natural frequency (rad/sec) and (Stiffness

(kN/m)), respectively, on the compliant

(polyurethane) surface (mean7SD)

64.5 0.940 Competitive sprinter 60.5714.5 (236.3713.6) 50.7716.7 (165.9717.9)

55.0 0.915 Fairly sedentary 55.7713.8 (170.6710.4) 40.5710.9 (90.476.5)

65.0 0.920 Fairly sedentary 50.0710.9 (160.177.7) 44.7714.5 (130.0713.7)

52.0 0.860 Taekwondo practitioner 70.077.6 (254.173.0) 49.9723.5 (129.5728.7)

51.5 0.905 Taekwondo practitioner 58.2732.2 (174.3753.5) 51.373.7 (135.570.7)

Averaged 58.977.3 (199.1742.9) 47.474.7 (130.3726.9)

Fig. 5. In an attempt to determine the interdependency of mechanical parameters

during running, peak ground reaction forces over a range of natural frequencies

(o¼35, 52.5, 70, 105, and 140 rad/s) of the foot/ground interface were generated by

the model whilst the damping factor x was held constant as 0.1. The mass–spring–

damper model shown in Eq. (B1) in supplementary material was used for the

simulation.

Fig. 6. Model generated peak ground reaction forces over a range of the damping

factors (x¼0.05, 0.075, 0.1, 0.15 and 0.2) of the foot/ground interface, whilst natural

frequencyowas held constant at 70 rad/s for the parameter thought relevant to the

typical subject (Table 1). The mass–spring–damper model shown in Eq. (B1) in

supplementary material was used for the simulation.

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Our results allow us to propose that the EKF model developedcould be used to explore the effect of both different running surfacesand the ergonomic contribution of running footwear to performance.Moreover, we tracked GRF as an observable measurement, with lessweighting of kinematic data (in contrast to the traditional modelingapproach of not applying measured GRF to a model, but, rather, onlyusing them to validate the model). Inverse dynamics modeling doesrequire realistic values for mechanical parameters in order to obtainadequate values for both stiffness and damping for the contact modelbetween the body segment and the environment. Our novel EKFmodeling technique apparently generates realistic estimations andcould be helpful as one of the techniques tracking the issue of ‘‘noisy’’data that is common in sports biomechanics research.

Conflict of interest statement

None declared.

Acknowledgement

The authors should like to acknowledge the help of NanyangTechnological University students, especially, Alex Chung Kiat andYew Choon Kiat, for both their contribution to the processing ofexperimental data and their participation as subjects in the study.

Appendix A. Supplementary material

Supplementary data associated with this article can be found inthe online version at doi:10.1016/j.jbiomech.2010.10.041.

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