Leg stiffness measures depend on computational method

Journal of Biomechanics 47 (2014) 115–121

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Journal of Biomechanics journal homepage: www.elsevier.com/locate/jbiomech www.JBiomech.com

Leg stiffness measures depend on computational method Kim Hébert-Losier a,n, Anders Eriksson a,b a b

Swedish Winter Sports Research Centre, Department of Health Sciences, Mid Sweden University, Kunskapens väg 8, Hus D, 83125 Östersund, Sweden KTH Mechanics, Royal Institute of Technology, Osquars backe 18, 10044 Stockholm, Sweden

art ic l e i nf o

a b s t r a c t

Article history: Accepted 30 September 2013

Leg stiffness is often computed from ground reaction force (GRF) registrations of vertical hops to estimate the force-resisting capacity of the lower-extremity during ground contact, with leg stiffness values incorporated in a spring–mass model to describe human motion. Individual biomechanical characteristics, including leg stiffness, were investigated in 40 healthy males. Our aim is to report and discuss the use of 13 different computational methods for evaluating leg stiffness from a double-legged repetitive hopping task, using only GRF registrations. Four approximations for the velocity integration constant were combined with three mathematical expressions, giving 12 methods for computing stiffness using double integrations. One frequency-based method that considered ground contact times was also trialled. The 13 methods thus defined were used to compute stiffness in four extreme cases, which were the stiffest, and most compliant, consistent and variable subjects. All methods provided different stiffness measures for a given individual, but the between-method variations in stiffness were consistent across the four atypical subjects. The frequency-based method apparently overestimated the actual stiffness values, whereas double integrations’ measures were more consistent. In double integrations, the choice of the integration constant and mathematical expression considerably affected stiffness values, as variations during hopping were more or less emphasized. Stating a zero centre of mass position at takeoff gave more consistent results, and taking a weighted-average of the force or displacement curve was more forgiving to variations in performance. In any case, stiffness values should always be accompanied by a detailed description of their evaluation methods, as our results demonstrated that computational methods affect calculated stiffness. & 2013 Elsevier Ltd. All rights reserved.

Keywords: Biomechanics Locomotion Lower-extremity Methodology Spring–mass model

1. Introduction The human body is often modelled as a spring that is able to store and release elastic energy through its muscle–tendon unit structures. This mechanical simplification of the human body acting as a spring is regularly applied to cyclic or rebounding type motions observed during locomotion (e.g., walking, running and hopping) that depend on stretch-shortening cycle muscle actions. During these natural type of muscle actions, the pre-activated muscles are first stretched (eccentric action) before they are shortened (concentric action) with the eccentric–concentric combination enhancing the final (concentric action) performance (Komi, 2000). When using the simple spring–mass model to describe human motion, the mechanical concept of stiffness is a key parameter of data analysis (Blum et al., 2009) and related to the ratio between a force and its corresponding displacement (Fig. 1). Stiffness is thereby the inverse of flexibility or compliance.

n

Corresponding author. Tel.: þ 46 63 14 56 32; fax: þ46 63 16 57 40. E-mail address: [email protected] (K. Hébert-Losier).

0021-9290/$ - see front matter & 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.jbiomech.2013.09.027

In its most macroscopic form, vertical stiffness is suggested to represent the overall body stiffness and defined as the ratio between the ground reaction force (GRF) and vertical displacement of the centre of mass (CoM). Leg stiffness, on the other hand, characterizes the stiffness of the lower-extremity and is described as the ratio between the GRF and leg length deformation (Cavagna, 1975; McMahon and Cheng, 1990). During locomotion, vertical stiffness is always greater than leg stiffness because changes in leg length surpass the vertical displacements of the CoM (Blickhan, 1989; Brughelli and Cronin, 2008; McMahon and Cheng, 1990). Although vertical and leg stiffness are not synonymous per se, the two are equivalent when leg length deformations are estimated from vertical jumps or hops (McMahon and Cheng, 1990; Serpell et al., 2012). Applied to a hopping task, the force in Fig. 1 is the external vertical (upwards) force from the ground support, whereas the displacement is the vertical (downwards) movement of the CoM during the ground contact. There has been a rapid increase in the number of applied research studies documenting stiffness values for the lowerextremity (Hobara et al., 2012; Jacobs et al., 1996; Lloyd et al., 2012; Maquirriain, 2012; Moritz and Farley, 2006; Pruyn et al., 2012), with researchers suggesting that sufficient levels of stiffness

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from a double-legged repetitive hopping task, using only GRF registrations. A total of 13 different stiffness evaluation methods were tested and compared on a subset of data acquired from a larger cohort of healthy males. The implications and differences associated to the choice of the computational methods, integration constants, mathematical expressions and spring–mass model assumptions used to define stiffness are topics included in the discussion.

2. Methods 2.1. Subjects

Fig. 1. Mechanical model for a human body of mass m in contact with the ground during a hop, with a mechanical spring representing the lower-extremity between the center of mass (CoM) and ground. The acting forces are the gravitational force mg, and the ground reaction force Fg. The position of the CoM is described by a vertical coordinate p of arbitrary origin, and the stiffness of the spring by a given constant k.

are required to optimize the utilization of the stretch-shortening cycle (Belli and Bosco, 1992; Kubo et al., 1999) and minimize the risk of musculoskeletal injury or re-injury (Maquirriain, 2012; Watsford et al., 2010). More specifically, high leg stiffness has been associated to a heightened risk of bony injuries (Granata et al., 2002; Williams et al., 2004), whereas low leg stiffness gives an increased susceptibility to soft tissue injuries (Butler et al., 2003; McMahon et al., 2012). Although the appropriate value of stiffness for performance and injury prevention has not yet been coined and depends on individual characteristics – i.e., sex, age, ethnicity and training background – the consistent use of a standard and valid computational method for measuring stiffness in research is important to attain a greater scientific merit and promote inferential and inductive reasoning in applied sciences (Borenstein and Hedges, 2009). Without registrations of a vertical position coordinate during experimentation, double integrations and resonant frequencies are the bases for the two most common methods used to quantify leg stiffness from GRF registrations (Butler et al., 2003). In the first method, the vertical GRF-curve is integrated twice to determine the vertical displacement of the CoM during the ground contact period. An indeterminate integration constant is selected, with the different approaches for defining this integration constant being discussed further below. In the second method, the period of oscillation and body mass of the individual are used to calculate stiffness from the net-GRF, which subtracts the gravity force from the original GRF-curve. Near identical results from the double integration and resonant frequency methods have been reported (Granata et al., 2002), which are of equal validity when used in conjunction with the conceptualized leg spring and stretchshortening models (Blum et al., 2009). However, there is a paucity of papers that report the actual stiffness values calculated from both approaches, using either one or citing that similar values were obtained if both were attempted (Granata et al., 2002). It is therefore challenging to contrast the different methods or make robust inferences from their respective results. Furthermore, the conventional methods used to quantify leg stiffness in clinical sciences are not always motivated by or verified against mechanical arguments. The aim of this paper is to describe and contrast a range of computational methods used to quantify leg stiffness in humans

After providing written informed consent, 40 healthy males participated in a multi-phase research project that evaluated several individual biomechanical characteristics, with only those pertaining to leg stiffness measurements from a double-legged repetitive hopping task reported here. The cohort had a mean 7standard deviation (SD) for age, height and mass of 30.4 7 8.9 yr, 181.9 7 7.0 cm and 77.3 79.3 kg, respectively. The research protocol was pre-approved by the Regional Ethical Review Board and adhered to the Declaration of Helsinki. Individuals in good self-reported general health were included, and excluded when reporting a current or recent musculoskeletal injury, joint pathology or other medical condition that could limit performance of repeated double-legged hops. All subjects provided verbal and written informed consent prior to participation.

2.2. Experimental procedures Each subject was familiarized with the experimental protocol and tested in a single laboratory-based session. After recording height using a telescopic measuring rod (Secas, DE) and body mass on a calibrated force-plate (Kistlers, CH); each subject watched an instructional video that demonstrated the hopping task, performed a light-intensity 5-min warm-up on a cycling ergometer (Monark AB, SE), and practiced the experimental task under supervision and guidance from the examiner. During this specific pre-test familiarization period, the investigator provided corrective feedback designed to ensure that the hopping task was performed in an appropriate manner. The familiarization period was followed by 2 min of rest, after which data collection was performed.

2.3. Double-legged hopping task For the evaluation of leg stiffness; each subject hopped barefooted using both legs in the middle of the calibrated force-pate (Kistlers, CH) with hands placed on hips, feet shoulder width apart and eyes directed forward. Subjects were instructed to keep their knees straight and land in a similar position to that of take-off from the force-plate (i.e., ankles plantar-flexed). Since contact time instructions can influence performance, stiffness values and stiffness regulation during hopping (Arampatzis et al., 2001; Hobara et al., 2007; Voigt et al., 1998), subjects were instructed to minimize ground contact times during hops, which implied minimal secondary movements in other joints. The force-plate was zeroed prior to each trial and was used to collect GRF data at an 1000-Hz sampling frequency with the Kistler Measurement, Analysis and Reporting Software v.1.0.3 (S2P Ltd., SI). Each trial consisted of 33 successive hops performed at 2.2 Hz indicated audibly to subjects via the TempoPerfect© v.2.02a computerized metronome (NCH Software, AUS). Therefore, each hopping trial was meant to last approximately 15 s. If subjects failed to perform the hopping trial adequately – e.g., did not maintain the pace – the trial was disregarded and repeated after 2 min of rest. Only 5 subjects required a second attempt to achieve an acceptable performance.

2.4. Extraction of individual hops The evaluation of leg stiffness was based on recorded GRF data with a constant time interval between registrations, Δτ ¼ 0:001 (s). The total registration contained N force values in a vector F , with individual components F i , treated as discrete values of the force variation F ¼ FðτÞ, with τ as a time variable. The vector F was first split into individual hops by identifying subsets of the vector with positive GRF values, interpreted as a sufficiently long range of components i1 r i r i2 , where F i1 o ε, and all other F i 4 ε. These vectorial subsets were identified choosing a tolerance ε towards registration disturbances, set to ε ¼ 5 (N) in this work. Each of the positive GRF sequences of a hop, with a time span denoted T, was collected in a vector f containing components f i , with 0 r ir ns , and ns ¼ ðT=ΔτÞ ¼ i2  i1 . A time τ ¼ 0 was thereby associated to the first component of each vector f .

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2.5. Integration of forces to positions

117

GRF

The GRF values associated to one hop were transformed into vertical accelerations at each discrete time step, 0r ir ns , from the net vertical forces, as ai ¼

1 f g m i

ð1Þ

with m the mass of the subject, and g ¼ 9:82 m  s  2 the gravitational acceleration. The accelerations were subsequently integrated to velocities by central difference expressions. With a chosen value for v0 ¼ vð0Þ, to be discussed below, the integrations were performed as v1=2 ¼ v0 þ

1 Δτ a0 2

vi þ 1=2 ¼ vi  1=2 þ Δτ ai

ð2aÞ ðfor 1 r ir ns  1Þ

ð2bÞ

for 1r ir ns , velocities were then integrated into vertical displacements pi , as pi ¼ pi  1 þ Δτ vi  1=2

ð3Þ

The integration constant p0 ¼ pð0Þ ¼ 0 was defined by stating a zero vertical position at first ground contact. This choice ascertained that the CoM displacement became negative exactly when the GRF became positive, defining the start of the ground contact phase. The discrete positions pi were then collected in a vector p with components corresponding to the vector f . To choose a value for v0 , preliminary integrations were first performed to give provisional velocities v n and positions p n , using v0 ¼ 0: 2.6. Initial velocity integration constant A key issue in the integration of forces f to positions p is the integration constant v0 , which is the vertical velocity at touch-down and affects all calculated position values linearly. With limited theoretical support for any assumption on this constant, there are four reasonable criteria for setting the integration constant v0 , which are tested in this paper. Two of them consider only the data derived from one hop in isolation, denoted ‘F’ and ‘P’ below, whereas the two others consider the individual hops as part of a larger sequence, denoted ‘T’ and ‘V’ below. All four methods are thereby based on physical necessities and relationships – essentially mechanical energy principles – that are applied to a one-degree-of-freedom system under specified assumptions to approximate the more complex human system. The ‘F ’ criterion states that the most negative position and a vanishing velocity of the CoM occur at the same time as the maximal GRF (Ferris and Farley, 1997), giving the integration constant v0 ¼  vn ðτF Þ

ð4Þ

with τF defining the time instance of maximal GRF. The ‘P’ criterion states that the CoM position is zero at time T, which is at takeoff from the force plate (Bosco et al., 1983a, 1983b). This criterion gives the integration constant v0 ¼  pn ðTÞ=T

ð5Þ

The ‘T’ criterion considers the time span T F of a ballistic motion between the preceding and the current ground contacts, giving the touch-down velocity v0 ¼ 

1 g TF 2

ð6Þ

The ‘V ’ criterion assumes similarities between two successive hops. With a ballistic motion between two successive ground contacts, the touch-down velocity is equal to the take-off velocity, but of opposite sign. Assuming that two successive ground contacts are identical, the integration constant is v0 ¼ 

1 n v ðTÞ 2

ð7Þ

making touch-down and take-off velocities for the selected hop of equal magnitudes. After choosing a criterion for determining the integration constant, the selected value for v0 was added to all components of the vector v and a new position vector p was integrated according to Eqs. (2) and (3), still with p0 ¼ 0. The resulting vectors f and p were the basis for the evaluation of stiffness of each hop. 2.7. Stiffness evaluation methods Several methods are possible for evaluating leg stiffness from a repetitive hopping task, all describing stiffness as the ratio of ‘force by displacement’ and employing the fact that the time variations of position and GRF have similar shapes. Fig. 2 shows the measured force and calculated position variations for one typical hop, with differences in position coming fully from the choice of the velocity integration constant. At this stage, three different mathematical expressions for computing leg stiffness values from GRF registrations were considered.

Fig. 2. Time variations of the ground reaction force (GRF) and position of the centre of mass (CoM) during one single hop, with the velocity integration constant v0 introduced in different ways according to Eqs. (4)–(7), and denoted by ‘F ’, ‘P ’, ‘T ’, and ‘V ’, respectively. For comparison reasons, the negative of the CoM position calculated from these four equations is plotted and deflections are noted. Each curve is normalized to its maximum value. Ground contact time is defined by positive GRF. The simplest method to evaluate the stiffness of one hop is to create the ratio between the maximal GRF and the maximal negative position of the CoM, i.e., S1 ¼

maxðf Þ maxð  pÞ

ð8Þ

with v0 from Eq. (4), the two values maxðf Þ and maxð  pÞ will correspond to the same time instance, but this is not necessarily the case when using Eqs. (5)–(7). More elaborate methods consider that time variations of force and position are not congruent, as the deflections denote in Fig. 2. An average of the ratio between force and displacement, weighted by the position values, gives the stiffness S2 ¼ 

∑ðpi f i Þ ∑ðpi pi Þ

ð9Þ

in which the sums extend over the whole interval. A third possible method is to weigh the vector components by the force values instead, as S3 ¼ 

∑ðf i f i Þ ∑ðf i pi Þ

ð10Þ

Eqs. (8)–(10) define three different mathematical expressions for the evaluation of the leg stiffness from vectors f and p. Each expression can be used in conjunction with the four criteria defined for setting the initial velocity integration constant v0 , chosen from either of Eqs. (4) to (7). The resulting stiffness values are denoted as, e.g., SF1 , ST2 , where a subscript denotes the mathematical expression [Eqs. (8)–(10)], and a superscript the choice of the integration constant [Eqs. (4)–(7)]. Thus, twelve stiffness values were obtained from the same vector f , using double integrations and are compared below. A thirteenth stiffness value was computed based on a simple spring–mass system (Cavagna et al., 1988; Farley et al., 1991), where the compression of the leg is seen as half the oscillation of a spring. Isolating the time period T þ during which the net-GRF (f i  mg) is (upwards) positive, this period is one half of the oscillation, and the stiffness was evaluated from  Sπ ¼ m

4π Tþ

2 ð11Þ

2.8. Statistical treatment The expressions above describe 13 different methods for computing the leg stiffness for one single hop, using essentially only information from one isolated sequence of positive GRF registrations with or without information from the preceding flight phase. In this work, several hops from one double-legged trial were used to create the stiffness value for one subject. The entire GRF sequence was analysed, and all sequences of positive GRF were isolated. Each sequence was evaluated with one of the 13 specified methods, giving typically 34 stiffness values for one trial, where the first value resulted from a countermovement jump initiating the rhythmical sequence of hops (Fig. 3a). The set of stiffness values were then sorted in an ascending order, and the medial 22 from this ranked order were statistically analysed. Thereby, the outliers at both ends of the stiffness range were removed and a singular mean and standard

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Fig. 3. (a) Ground reaction force (GRF) data during one double-legged hopping trial for one typical subject. The total number of stiffness measures (N¼ 34) include 33 hops performed at a 2.2-Hz frequency and 1 initial countermovement jump. (b) Stiffness values computed from the double-legged hopping trial shown in Fig. 3a, using the SP1 method. The stiffness values are presented in a chronological hopping order (Hop no.) and in a rank order (Rank no.). The final stiffness value for the subject is the average and standard deviation of the values associate to Rank no. 7 to 28, thereby disregarding the 6 minimum and 6 maximum outlying values from the ranked sequence.

deviation value was obtained, corresponding to one hopping trial (Fig. 3b) and representing 10 s of GRF data. 2.9. Comparison of methods To contrast the methods and their stiffness values, we targeted four atypical cases. These cases were identified using the mean and related SD stiffness values computed using the SP1 method. Hence, the subjects with the highest and lowest means were selected, as were those with the highest and lowest SDs, thereby discerning a stiff, compliant, consistent, and variable subject, respectively. This selection process permitted to examine how the 13 different computational methods handled extreme cases and computed stiffness, rather than basing study results and interpretations on statistical analyses of mean (group) values.

3. Results The stiffness measures resulting from the 13 different computational methods are documented in Table 1 for the four extreme cases defined above. There were no significant differences in the age (31.5 79.4 yr, P ¼0.8156), height (183.67 5.0 cm, P¼ 0.6250) and body mass (69.6 710.1 kg, P ¼0.1429) of the four selected subjects when compared to the entire cohort originally investigated (N ¼40). Note that, using the SP1 method, the mean and SD stiffness from the entire cohort were 32.69 77.42 kN m  1 and 1.61 70.75 kN m  1, respectively. 4. Discussion We have implemented and tested 13 different methods for evaluating leg stiffness from GRF registrations only. The methods

section provides some preliminary arguments for choosing any one method, with the selected computation influencing results, and this topic is discussed further here. The main distinction between methods, as defined in the literature, is whether the leg stiffness is evaluated from a doubly-integrated acceleration sequence or a frequency-based analysis (Butler et al., 2003). Our results showed frequencybased stiffness measures that were generally 10% to 30% higher than those calculated from doubly-integrated accelerations. Mathematically, this can be explained by the lack of fulfilment of a sineshaped force and displacement variation assumption; a projection of a variation on a particular functional form should usually result in an overestimation of stiffness values. Furthermore, frequencybased stiffness evaluations are more dependent on subject body mass and sensitive to incorrect body mass entry, which can have a considerable negative impact on the precision of the stiffness value computed. In light of these findings and mechanical arguments, researchers and clinicians alike should bear these distinctions between stiffness measures in mind when selecting or comparing results from doubly-integrated accelerations and frequency-based evaluation methods. In its most simplistic definition, the spring–mass model assumes that during rhythmical hopping, velocities of the CoM at touch-down and take-off are symmetrical and displacements of the CoM during ground contact are sinusoidal. Positions of the CoM at take-off and touch-down must therefore also be symmetrical (Blickhan, 1989). Consequently, at the middle of the ground contact phase of hops: velocities reach zero, vertical CoM descents are maximal, and the GRFs peak. However, as reviewed elsewhere

K. Hébert-Losier, A. Eriksson / Journal of Biomechanics 47 (2014) 115–121

119

Table 1 Mean 7 SD stiffness values computed from the double-legged hopping trial of four subjects, using 13 different computational methods. The four subjects were chosen from a larger cohort (N ¼ 40) based on their calculated mean stiffness and associated SD evaluated using SP1 . The first 12 methods are denoted by a subscript, referring to Eqs. (8)– (10) and a superscript defining an integration constant from Eqs. (4) to (7). The 13th method is frequency-based from Eq. (11) and is denoted using Sπ . Evaluation method

Stiffness characteristic of subjecta Stiffest

Compliant

Consistent

Variable

SP1

44.08 7 1.26

SP2

41.82 71.42

13.317 0.77

25.79 70.57

42.53 73.47

11.93 7 0.85

23.277 0.67

SP3

35.59 7 2.61

42.29 7 1.42

12.157 0.86

23.80 70.63

37.79 7 2.80

SF1

48.29 72.36

11.93 7 1.20

28.50 70.88

43.88 74.39

46.75 7 2.59

10.127 1.32

26.56 70.93

36.46 74.25

SF3

47.52 72.70

10.62 7 1.25

26.95 70.98

38.487 4.18

SV1

SF2

49.40 72.64

13.977 1.49

29.677 1.15

45.39 74.47

SV2

48.15 72.66

12.577 1.43

27.717 1.23

38.34 73.76

SV3

48.707 2.94

12.89 7 1.59

28.40 71.45

40.737 3.92

ST1 ST2 ST3

49.86 72.41

13.36 7 1.16

29.107 1.57

43.647 4.06

48.127 2.68

11.78 7 1.25

26.89 71.67

37.117 3.55

48.75 73.01

12.02 7 1.24

27.417 1.91

39.197 3.60

Sπ

49.26 72.36

16.93 7 1.12

29.02 70.93

61.86 7 5.66

All methods Range (min, max) Variabilityb

47.157 2.69 41.82, 49.86 16%

12.587 1.68 10.12, 16.93 40%

27.167 1.96 23.27, 29.67 22%

41.617 6.83 35.59, 61.86 42%

a b

Calculated using the SP1 evaluation method, where the: highest mean ¼ stiffest, lowest mean ¼ compliant, lowest SD ¼ consistent, and highest SD ¼variable.
 min  100%: Variability ¼ maxmax

(Cavagna, 2010) and verified by Fig. 2, the assumptions of symmetric parameters for take-off and touch-down are not entirely valid in humanoid bounding gait, which explains some of the discrepancies in the stiffness values computed here from the 13 different methods. The ‘V’ criterion for setting the velocity integration constant presumes symmetrical velocities at touch-down and take-off; ‘P’ assumes identical CoM positions at touch-down and take-off; whereas ‘F’ postulates a synchronized peak (downwards) CoM displacement and peak (positive) GRF. Due to asymmetry of hops, all these criteria are false to some extent, which directly impacts the stiffness values computed. In fact, the most reliable computational method for determining the leg stiffness of an individual likely depends on the individualized level of hopping symmetry or asymmetry with respect to touch-down and take-off velocities and CoM positions, as well as on bodily displacement responses to the GRF during ground contact. Hence, we speculate that the most appropriate computational method to assess stiffness is subjectdependent. Of course, this assumption needs to be verified in future studies attempting to validate stiffness computational methods for sports scientist, biomechanics, and clinicians. In any case, stiffness values should always be accompanied by a detailed account of their evaluation methods. That being said, the method that assumed identical centre of mass positions at touch-down and take-off and evaluated stiffness as the ratio between the maximal ground reaction force and displacement of the centre of mass during ground contact (i.e., SP1 ) was the most robust. This method showed the least variance in values (i.e., smallest SD) for all four cases presented here, with the exception of the individual purposefully selected based on high SD using SP1 . Given its robustness, we recommend that the SP1 method be used when performing an individual-based evaluation of stiffness from hopping trials becomes impractical, for instance, in large cohort studies. Likewise, given its apparent robustness, employing the SP1 method can be recommended for computing stiffness from the GRF registrations of hopping trials performed in similar, but not exact, conditions than those described in our

experiment. Because leg stiffness is influenced by change in surface (Ferris and Farley, 1997; Ferris et al., 1998), knee angles (Greene and McMahon, 1979), task frequency (Farley et al., 1991; Farley and González, 1996), and verbal instructions (Arampatzis et al., 2001; Hobara et al., 2007; Voigt et al., 1998); the SP1 method should be appropriate for computing stiffness when hopping single-legged, at higher frequencies than 2.2 Hz, or on soft surfaces, for example. For evaluating stiffness using double integrated accelerations, it is obvious that the choice of the integration constant for the touchdown velocity v0 created differences in the obtained stiffness values. In each group (subjects and integration constants), the stiffness values were lowest from S2, largest from S1, and moderate from S3 computations (i.e., S2 o S3 oS1). In general, the S2 and S3 methods provided rather similar stiffness values that are mathematically expected to be lower than those from S1. The S2 and S3 methods consider the real shape of both force and position curves and do not postulate that they are congruent or take an instantaneous peak value from each curve as done in S1. Accordingly, we assume that the lower stiffness values associated to the S2 and S3 computational methods are more valid and representative of the actual stiffness of individuals, given that these values consider all data points from the force and position curves. The antithesis is that S2 and S3 provided, in general, higher SD values than S1, therefore demonstrating lower robustness in stiffness computations. The exception here was again the individual purposefully selected to have a high SD using the SP1 method. In this case, S1 gave higher SD than S2 and S3, meaning that the latter two methods were able to somewhat reduce the effects from a strongly varying hopping performance. Therefore, in individuals who display varying levels of hopping performance, employing the S2 or S3 methods for stiffness computation is recommended because these methods emerged as more forgiving to variations. For practicality purposes, the lower computational work demanded for frequency-based stiffness determination may be perceived as advantageous. Although a reasonable justification, the assessor needs to remain aware of the potential to

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overestimate stiffness and of the importance of correctly identifying the positive net-GRF duration. However, several computational software and programs are available to researchers that permit the rapid calculation of stiffness through the double integration methods, with negligible differences in computational time existing between S1, S2 and S3, or selected integration constant. Given the perceived higher validity of values computed from double integrations of the GRF-curve with respect to the actual leg stiffness of individuals, we recommend that scientists employ these approaches rather than frequency-based ones and aspire towards a common methodological underpinning to stiffness evaluation from hopping tasks. The differences found in the stiffness values and models used here reflect some of the limitations of applying the spring–mass model to human motion. First and foremost, the simplification of the model requires that a series of assumptions be made. In this work, the effect of the selected set of assumptions was observed in the estimated position of the CoM over time (see Fig. 2) and computed leg stiffness value. Furthermore, the lower-extremities are actuated by many muscles that respond in a more complex manner than suggests the spring–mass model, with simulations and optimization of dynamic tasks providing insight on the complex interplay between muscles (Anderson and Pandy, 1993; Hamner et al., 2010; McGowan et al., 2013). Several anatomical structures (Latash and Zatsiorsky, 1993) and neural mechanisms are responsible for determining and regulating stiffness (Zuur et al., 2010), with stiffness varying throughout the time course of a hop (Dyhre-Poulsen et al., 1991). There is evidence suggesting that during hopping, humans adjust leg stiffness to accommodate for changes in experimental conditions primarily by modulating ankle joint stiffness (Farley et al., 1998; Farley and Morgenroth, 1999; Hobara et al., 2011). At 2.2 Hz, Hobara et al. (2011) showed a significant correlation between leg and ankle stiffness values, but not between leg and knee or hip joint stiffness measures. From these data, we can presume that the stiffness measured in our study during hopping was more strongly related to ankle stiffness, although also comprised knee and hip joint stiffness. However, without registrations of vertical positions and angular displacements, we are not able to verify the relative contribution of each joint to the leg stiffness values computed here. In fact, to validate the results from the 13 kinetic-based computational methods that we applied to repetitive hopping herein, a future study could seek to contrast kinetic to kinematic methods as did Coleman et al. (2012) for running to validate the mathematical models used to compute leg stiffness from GRF registrations.

5. Conclusions In light of the fact that the frequency-based method apparently provides results that overestimate the actual leg stiffness of individuals, we recommend that scientists use the doublyintegrated acceleration methods more consistently to evaluate stiffness from double-legged repetitive hopping tasks. With respect to double integrations, our results clearly demonstrate how the choice of the indeterminate integration constant and mathematical expression computing stiffness create differences in the resulting values. In the 12 variations of the double integration methods investigated here, stating that the CoM position is zero at take-off provided more consistent stiffness values that were robust to variations, as did taking a weighted average of the ratio between the force and displacement curves when hops were highly variable. In any case, standardization of the methods used to determine stiffness using GRF registrations only is highly desirable, and a stiffness value should also be accompanied by a detailed account of its computational method.

Conflict of interest statement The authors declare they have no conflict of interest. No external source of funding was necessary for the preparation of this paper.

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