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A simple method for measuring power, force, velocity properties,

and mechanical effectiveness in sprint running

P. Samozino1, G. Rabita2, S. Dorel3, J. Slawinski4,N.Peyrot

5, E. Saez de Villarreal6, J.-B. Morin7

1Laboratory of Exercise Physiology (EA4338), University Savoie Mont Blanc, Le Bourget du Lac, France, 2Research Department,

Laboratory Sport, Expertise and Performance, French Institute of Sport (INSEP), Paris, France, 3Laboratory “Motricité, Interactions,

Performance” (EA 4334), University of Nantes, Nantes, France, 4CeSERM – EA 2931, UFRSTAPS, Université de Paris Ouest

Nanterre la Défense, Paris, France, 5Laboratory IRISSE (EA4075), University of La Réunion, Le Tampon, La Réunion, France,

6Laboratory of Human Performance, Department of Sports, University Pablo de Olavide, Seville, Spain, 7Laboratory of Human

Motricity, Education Sport and Health (EA6312), University of Nice Sophia Antipolis, Nice, France

Corresponding author: Pierre Samozino, PhD, Laboratoire de Physiologie de l’Exercice, Université de Savoie Mont-Blanc, UFR

CISM – Technolac, 73376 Le Bourget du Lac, France. Tel: +33 4 79 75 81 77, Fax: +33 4 79 75 81 48, E-mail:

pierre.samozino@univ-savoie.fr

Accepted for publication 7 April 2015

This study aimed to validate a simple ﬁeld method for

determining force– and power–velocity relationships and

mechanical effectiveness of force application during

sprint running. The proposed method, based on an

inverse dynamic approach applied to the body center of

mass, estimates the step-averaged ground reaction forces

in runner’s sagittal plane of motion during overground

sprint acceleration from only anthropometric and spatio-

temporal data. Force– and power–velocity relationships,

the associated variables, and mechanical effectiveness

were determined (a) on nine sprinters using both the

proposed method and force plate measurements and (b)

on six other sprinters using the proposed method during

several consecutive trials to assess the inter-trial reliabil-

ity. The low bias (<5%) and narrow limits of agreement

between both methods for maximal horizontal force

(638 ±84 N), velocity (10.5 ±0.74 m/s), and power output

(1680 ±280 W); for the slope of the force–velocity rela-

tionships; and for the mechanical effectiveness of force

application showed high concurrent validity of the pro-

posed method. The low standard errors of measurements

between trials (<5%) highlighted the high reliability of

the method. These ﬁndings support the validity of the

proposed simple method, convenient for ﬁeld use, to

determine power, force, velocity properties, and mechani-

cal effectiveness in sprint running.

Sprint running is a key factor of performance in many

sport activities, not only to reach the highest top velocity

but also, and most importantly, to cover a given distance

in the shortest time possible, be it in track-and-ﬁeld

events or in team sports. This ability implies large

forward acceleration, which has been related to the

capacity to produce and apply high amounts of power

output in the horizontal direction onto the ground, i.e.,

high amounts of horizontal external force at various

velocities over sprint acceleration (Jaskolska et al.,

1999b; Morin et al., 2011a, 2012; Rabita et al., 2015).

The overall mechanical capability to produce horizon-

tal external force during sprint running is well described

by the inverse linear force–velocity (F–v) and the para-

bolic power–velocity (P–v) relationships (Jaskolska

et al., 1999a, b; Morin et al., 2010, 2011a, 2012; Rabita

et al., 2015). Indeed, although the F–v relationships

obtained on isolated muscles or mono-articular move-

ments are described by a hyperbolic equation (Hill,

1938; Thorstensson et al., 1976), linear relationships

were consistently obtained for multijoint lower limb

movements such as pedaling, squat, leg press, or sprint

running movements (e.g., Yamauchi & Ishii, 2007; Dorel

et al., 2010; Morin et al., 2010; Bobbert, 2012;

Samozino et al., 2012; Rabita et al., 2015). These rela-

tionships characterize the external mechanical limits of

the entire neuromuscular system during speciﬁc

multijoint movements and are well summarized through

the theoretical maximal force (F0) and velocity (v0) this

system can develop, and the associated maximal power

output (Pmax). Moreover, the slope of the F–v relationship

determines the F–v mechanical proﬁle (SFV), i.e., the

individual ratio between force and velocity qualities.

These mechanical properties obtained from multijoint

F–v and P–v relationships are a complex integration of

different mechanisms involved in the total external force

produced during one (for acyclic movements) or several

consecutive (for cyclic movements) limb extensions.

They encompass individual muscle mechanical proper-

ties, morphological factors, neural mechanisms, and

segmental dynamics (Cormie et al., 2010a, b, 2011;

Bobbert, 2012). Furthermore, because sprint running is a

dynamic movement mainly requiring force production

in two dimensions (in contrast to squat or leg press

Scand J Med Sci Sports 2015: ••: ••–••

doi: 10.1111/sms.12490

© 2015 John Wiley & Sons A/S.

Published by John Wiley & Sons Ltd

1

exercises), the F–v and P–v relationships during running

propulsion also integrate the ability to apply the external

force effectively (i.e., horizontally in the antero-posterior

direction) onto the ground (Morin et al., 2011a, b, 2012;

Rabita et al., 2015). This concept of mechanical effec-

tiveness of force application has been previously pro-

posed in sprint pedaling considering tangential and

normal force components of the force applied onto the

pedals (Dorel et al., 2010). The technical ability of force

application (or mechanical effectiveness) during sprint

running has been quantiﬁed at each step by the ratio (RF)

of the net horizontal and resultant ground reaction forces

(GRF), and over the entire acceleration phase by the rate

(DRF) of linear decrease in RF as velocity increases. DRF,

which is independent from the amount of total force

applied (i.e., physical capabilities), describes the run-

ner’s ability to maintain a forward horizontal orientation

of the resultant GRF vector despite increasing speed

(Morin et al., 2011a, b; Morin et al., 2012; Rabita et al.,

2015; see Methods section). So, F–v and P–v relation-

ships provide a macroscopic and integrative view of the

P–F–v mechanical proﬁle of an athlete in the speciﬁc

sprint running task.

Therefore, determining individual F–v and P–v rela-

tionships and mechanical effectiveness during sprint

propulsion is of great interest for coaches, sport practi-

tioners, or physiotherapists. Indeed, sprint performance

is highly correlated to Pmax, be it quantiﬁed during sprint

running or other movements such as vertical jump or

sprint cycling (e.g., Cronin & Sleivert, 2005; Morin

et al., 2012; Rabita et al., 2015). In addition to this power

output capability, the F–v mechanical proﬁle (character-

ized by the slope of the F–v relationship) has recently

been shown to inﬂuence maximal jumping perfor-

mances, independently from the effect of Pmax, with the

existence of an individual optimal F–v proﬁle character-

izing the best balance, for a given subject, between force

and velocity qualities to maximize performances

(Samozino et al., 2012, 2014). These results suggest that

the F–v mechanical proﬁle in sprint running, which

shows high inter-individual differences (Jaskolska et al.,

1999b; Morin et al., 2010), can also be interesting to

consider and adjust by individualized training loads and

exercises. Finally, recent studies showed that sprint per-

formances (6-s sprints, 100-m events, repeated sprints)

are related more to the effectiveness of force application

to the ground than to the total force developed by lower

limbs (Morin et al., 2011a, 2012; Rabita et al., 2015). So,

quantifying individually the mechanical effectiveness

can help distinguish the physical and technical origins of

inter- or intra-individual differences in both P–F–v

mechanical proﬁles and sprint performances, which can

be useful to more appropriately orient the training

process toward the speciﬁc mechanical qualities to

develop.

To date, such evaluations require to measure horizon-

tal antero-posterior and vertical GRF components and

forward horizontal velocity during an entire sprint accel-

eration (∼30–60 m). Sprint F–v and P–v relationships

have hitherto been obtained using speciﬁc instrumented

treadmills on which subjects accelerate the belt them-

selves by the action of their lower limbs, while their

waist is tethered backward to a ﬁxed point (e.g.,

Jaskolska et al., 1999a, b; Chelly & Denis, 2001; Morin

et al., 2010). Despite the high accuracy of these methods,

their main limitation is the treadmill condition that does

not exactly reproduce natural overground sprint running

movement due to waist attachment, a belt narrower than

a typical track lane, the impossibility to use starting

block, and the need to set a default torque to, only partly,

compensate for the friction of the treadmill belt bed

(Morin et al., 2010; Morin & Seve, 2011). Because, to

date, no 30- to 60-m long force plate systems exist,

GRFs over an entire sprint acceleration phase were very

recently determined in elite sprinters using data from

several sprints measured on a 6.60-m long force plate

system, which allowed, for the ﬁrst time, to provide

the data to entirely characterize the mechanics of

overground sprint acceleration (Rabita et al., 2015).

Sport practitioners do not have easy access to such rare

and expensive devices, and often do not have the tech-

nical expertise to process the raw force data measured. In

the best cases, this forces athletes to report to a labora-

tory, which explains that, although very accurate and

potentially useful for training purposes, this kind of

evaluation is almost never performed. In addition, sport

scientists investigating sprint mechanics and perfor-

mance usually assess, at best, only very few steps of a

sprint because of this technical limitation (e.g., Lockie

et al., 2013; Kawamori et al., 2014). A simple method for

determining F–v and P–v relationships and force appli-

cation effectiveness during sprint running in overground

realistic conditions could therefore be very interesting to

generalize such evaluations for training or scientiﬁc

purposes.

Mechanics and energetics of sprint running have been

approached by various kinds of mathematical models

that aimed at describing sprint performance from the

balance between the mechanical energy demand of

sprint running acceleration and the energy release capac-

ity of the aerobic and anaerobic metabolism (e.g.,

Furusawa et al., 1927; van Ingen Schenau et al., 1991;

Arsac & Locatelli, 2002; Helene & Yamashita, 2010).

Based on the mechanical analyses used in these models,

an inverse dynamic approach applied to the runner body

center of mass (CM) could give valid estimation of GRF

during sprint running acceleration from simple kine-

matic data, as recently proposed by di Prampero et al.

(2015), but never compared with force plate measure-

ments. This could then be used to obtain the aforemen-

tioned sprint mechanical properties without force

platform system in typical ﬁeld conditions of practice.

The aim of this study was to propose and validate a

simple ﬁeld method based on an inverse dynamic

Samozino et al.

2

approach applied to the runner body CM during sprint

running acceleration for (a) estimating the GRFs in the

sagittal plane of motion from only anthropometric and

spatiotemporal data and (b) determining the associated

F–v and P–v relationships and effectiveness of force

application. The concurrent validity of this method

was tested by comparison to reference force plate

measurements.

Biomechanical model used in the proposed method

This section, devoted to present the biomechanical

model on which the proposed simple method is based, is

an analysis of kinematics and kinetics of the runner’s

body CM during sprint acceleration using a macroscopic

inverse dynamic approach aiming to be the simplest

possible (Furusawa et al., 1927; Helene & Yamashita,

2010). All variables presented in this section are

modeled over time, without considering intra-step

changes, and thus correspond to step-averaged values

(contact plus aerial times).

During a running maximal acceleration, horizontal

velocity (vH)–time (t) curve has long been shown to

systematically follow a mono-exponential function for

recreational to highly trained sprinters (e.g., Furusawa

et al., 1927; Chelly & Denis, 2001; di Prampero et al.,

2005; Morin et al., 2006):

vt v e

HH

t

()

=⋅−

()

−

max 1τ[1]

where vHmax is the maximal velocity reached at the end

of the acceleration and τis the acceleration time con-

stant. The horizontal position (xH) and acceleration

(aH) of the body CM as a function of time during the

acceleration phase can be expressed after integration

and derivation of vH(t) over time, respectively, as

follows:

x t v t dt v e dt

t

HH H

()

=

()

=⋅−

⎛

⎝

⎜⎞

⎠

⎟

∫∫ −

max 1τ[2]

xt v t e v

t

()

=⋅+

⎛

⎝

⎜⎞

⎠

⎟−⋅

−

HH

max max

.ττ

τ[3]

at dv t

dt

ve

dt

t

H

H

Hmax

()

=

()

=

⋅−

⎛

⎝

⎜⎞

⎠

⎟

−

1τ

[4]

at ve

t

H

()

=⎛

⎝

⎜⎞

⎠

⎟⋅−

max

ττ[5]

Applying the fundamental laws of dynamics in the

horizontal direction, the net horizontal antero-posterior

GRF (FH) applied to the body CM can be modeled over

time as:

Ft ma t F t

H H aero

()

=⋅

()

+

()

[6]

where mis the runner’s body mass (in kg) and Faero(t)is

the aerodynamic drag to overcome during sprint running

that is proportional to the square of the velocity of air

relative to the runner:

Ftkvtv

aero

()

=⋅

()

−

()

Hw

2[7]

where vwis the wind velocity (if any) and kis the run-

ner’s aerodynamic friction coefﬁcient, which can be esti-

mated as proposed by Arsac and Locatelli (2002) from

values of air density (ρ, in kg/m3), frontal area of the

runner (Af,inm

2), and drag coefﬁcient (Cd =0.9; van

Ingen Schenau et al., 1991):

kAfCd=⋅⋅⋅05.ρ[8]

with

ρρ=⋅ ⋅ +°

0760

273

273

Pb

T[9]

Af h m=⋅⋅

()

⋅0 2025 0 266

0 725 0 425

..

.. [10]

where ρ0=1.293 kg/m is the ρat 760 Torr and 273 °K,

Pb is the barometric pressure (in Torr), T° is the air

temperature (in °C), and his the runner’s stature

(in m). The mean net horizontal antero-posterior

power output applied to the body CM (PHin W) can

then be modeled at each instant as the product of FH

and vH.

In the vertical direction, the runner’s body CM during

the acceleration phase of a sprint goes up from the start-

ing crouched position (be it with or without using start-

ing blocks) to the standing running position, and then

does not change from one complete step to another.

Because the initial upward movement of the CM is

overall smoothed through a relative long time/distance

(∼30–40 m; Cavagna et al., 1971; Slawinski et al.,

2010), we can consider that it does not require any large

vertical acceleration, and so that the mean net vertical

acceleration of the CM over each step is quasi-null

throughout the sprint acceleration phase. Consequently,

applying the fundamental laws of dynamics in the verti-

cal direction, the mean net vertical GRF (FV) applied to

the body CM over each complete step can be modeled

over time as equal to body weight (di Prampero et al.,

2015):

Ft mg

V

()

=⋅ [11]

where gis the gravitational acceleration (9.81 m/s2).

Morin et al. (2011a) proposed that the mechanical

effectiveness of force application during running could

be quantiﬁed over each support phase or step by the

ratio (RF in %, eqn. [12]) of FHto the corresponding

total resultant GRF (FRes, in N), and over the entire

Simple method to compute sprint mechanics

3

acceleration phase by the slope of the linear decrease in

RF when velocity increases (DRF, in %.s/m):

RF F

F

F

FF

=⋅= +⋅

H

Res

H

HV

100 100

22 [12]

To date, RF values have been computed from the

second step (i.e., the ﬁrst complete step) to the step at

maximal speed (Morin et al., 2011a, b, 2012; Rabita

et al., 2015). Because the starting block phase (push-off

and following aerial time) lasts between 0.5 and 0.6 s

(Slawinski et al., 2010; Rabita et al., 2015) and so occurs

at an averaged time of ∼0.3 s, RF and DRF can be rea-

sonably computed from FHand FVvalues modeled for

t>0.3 s.

The above-described biomechanical model makes

possible to estimate GRFs in the sagittal plane of motion

during one single sprint running acceleration from

simple inputs: anthropometric (body mass and stature)

and spatiotemporal (split times or instantaneous running

velocity) data. This model can then be used as a simple

method to determine the F–v and P–v relationships and

the associated variables, as well as the mechanical effec-

tiveness of force application parameters (see Methods

section for details about the practical methodology asso-

ciated to this computation method). Two different

experimental protocols were conducted (a) to test the

concurrent validity of the proposed computation method

by comparison to force plate measurements and (b) to

test its inter-trial reliability.

Methods

First protocol: concurrent validity compared with force

plate measurements

Subjects and protocol

Nine elite or sub-elite sprinters (age: 23.9 ±3.4 years; body mass:

76.4 ±7.1 kg; height: 1.82 ±0.69 m) gave their written informed

consent to participate in this study, which was approved by the

local ethical committee and in agreement with the Declaration

of Helsinki. Their personal 100-m ofﬁcial best times were

10.37 ±0.27 s (range: 9.95–10.63 s). After a standardized 45-min

warm-up, subjects performed seven maximal sprints in an indoor

stadium (2 ×10, 2 ×15, 20, 30 and 40 m with 4-min rest between

each trial) in order to collect GRF data over an entire 40-m

distance. From these sprints, antero-posterior and vertical GRF

components; F–v, P–v, and RF–v relationships; and associated

variables (F0,v0,Pmax,SFV,DRF) were obtained from both force

plate measurements and above-described computation method.

Force plate method: materials and data processing

The experimental protocol used here to determine F–v and P–v

relationships from force plate measurements has recently been

proposed and detailed by Rabita et al. (2015). Brieﬂy, during seven

sprints, vertical and antero-posterior GRF components were mea-

sured by a 6.60-m long force platform system (natural fre-

quency ≥500 Hz). This system consisted of six individual force

plates (1.2 ×0.6 m) connected in series, time synchronized, and

covered with a tartan mat leveled with the stadium track. Each

force plate was equipped with piezoelectric sensors (KI 9067;

Kistler, Winterthur, Switzerland). The force signals were digitized

at a 1000-Hz sampling rate.

The protocol was designed in order to virtually reconstruct for

each athlete the GRF signal of an entire single 40-m sprint by

setting differently for each sprint the position of the starting blocks

relatively to the 6.60-m long force platform system. The starting

blocks were placed over the ﬁrst platform for the 10-m sprints and

were placed remotely for the other trials (15–40 m) so that 17

different steps (18 foot contacts) from the block to the 40-m mark

could be measured (cf. Fig. 1 in Rabita et al., 2015). Force plat-

form signal was low-pass ﬁltered (200-Hz cutoff, third-order zero-

phase Butterworth) and instantaneous data of vertical (FV,inN)

and horizontal antero-posterior (FH, in N) GRF components as

well as the resultant (FRes) were averaged for each step consisting

of a contact and an aerial phase (determined using 10-N threshold

on FVsignal).

Over each measurement areas, the instantaneous horizontal

velocity of the CM (vH, in m/s) was computed as the ﬁrst integra-

tion over time of the antero-posterior horizontal acceleration (aH,

in m/s2) obtained at each instant dividing FHby body mass, with

initial vHvalues as integration constants obtained as follows. For

the 10-m sprints, the initial vHwas set to 0 as the starting blocks

were placed over the force plate area. For the other sprints, the

initial vHvalues were measured by high-speed video using a 300

frames per second digital camera (Exilim EX-F1, Casio, Tokyo,

Japan) placed perpendicularly to the sagittal plane of motion of the

athletes in a ﬁxed position focusing on the entrance of the force

plate area (for more details, see Rabita et al., 2015). The instanta-

neous power output in the horizontal direction (PH) was computed

as the product at each instant of FHand vH. Instantaneous data of

vHand PHwere averaged over each step. For the following analy-

ses, the data of the seven sprints were pooled to reconstruct a

complete 40-m dataset for each subject. For each step, RF was

obtained from averaged step values of FHand FRes using eqn. [12].

Individual DRF values were determined as the slope of the linear

RF–vrelationships.

Proposed computation method: material and speciﬁc

data processing

Sprint times were measured with a pair of photocells (Microgate,

Bolzano, Italy) located at the ﬁnish line of the seven sprints. For

Fig. 1. Correlation between force values measured at each step

using the force plate method and values computed by the pro-

posed method. Horizontal and vertical components and resultant

ground reaction force values are represented for all subjects. The

identity line is represented by the continuous black line.

Samozino et al.

4

each sprint, the timer was triggered when subjects’ right thumb left

the ground. To remove all possible bias due to this kind of trig-

gering procedure on variables obtained with this computation

method, the time between the beginning of the force production on

the starting blocks (which represents the actual start of the sprint)

and the trigger of the timer was determined and added to photocell

times using sagittal high-speed video recording of sprint starts

(Exilim EX-F1, Casio, Tokyo, Japan) synchronized with force

plate data.

For the two 10- and 15-m trials, only the best times were

considered at each distance for data analysis. For each subject, the

ﬁve split times at 10, 15, 20, 30, and 40 m were then used to

determine vHmax and τusing eqn. [3] and least-square regression

method. From these two parameters, vH(t) and aH(t) were modeled

over time using eqns. [1] and [5], respectively. From aH(t), FH(t)

was modeled over time using eqn. [6] and estimation of Faero(t)

from eqns. [7] to [10]. For individual computations of k, both the

subject’s body mass and stature were measured before tests, Pb

was 760 Torr, and T° was 20 °C. No wind was present during this

indoor testing session and the very limited effect of air humidity

on ρwas not considered. PHcan then be modeled at each instant as

the product of FHand vH.FV(t) and RF(t) were obtained using

eqns. [11] and [12], respectively, and individual DRF values were

determined as described in the theoretical background section. All

these variables were computed every 0.1 s (from eqns. [1], [6],

[11], and [12]) over each individual acceleration phase.

Common data analysis for both methods

Individual F–v and P–v relationships were determined for both

methods from step averaged (for force plate method) and modeled

(for proposed method) FH,PH, and vHvalues using least-square

linear and second-order polynomial regressions, respectively

(Jaskolska et al., 1999b; Morin et al., 2010, 2012; Rabita et al.,

2015). F–v relationships were extrapolated to obtain F0and v0as

the intercepts of the F–v curve with the force and velocity axis,

respectively. SFV value was determined for each subject as the

slope of the F–v relationship, and Pmax was determined as the apex

of the P–v relationship using the ﬁrst mathematical derivation of

the associated quadratic equation. Pmax values were also computed

as previously proposed and validated (Vandewalle et al., 1987;

Samozino et al., 2012, 2014) as follows:

PFv

max =⋅

00

4[13]

Statistical analysis

All data are presented as mean ±standard deviation (SD). For each

subject, the standard errors of estimate (SEE) of FH,FV,FRes, and

RF were computed between values obtained from force plate data

at each step and values estimated from model computations for

corresponding vHvalues:

SEE FF

N

=−

()

−

∑Force Plate Model

steps

2

2[14]

The correlation between force values (FH,FV,FRes) obtained by

both methods was analyzed. F–v relationships, power output capa-

bilities, and mechanical effectiveness obtained with both methods

were compared through F0,v0,SFV,Pmax, and DRF values using bias

and limits of agreements (Bland & Altman, 1986). To complete

this quantiﬁcation of inter-method differences, absolute bias was

also calculated for each subject as follows: absolute bias =|(com-

putation method–force plate method)/force plate method|.100. For

all statistical analyses, a P-value of 0.05 was accepted as the level

of signiﬁcance.

Second experimental protocol: inter-trial reliability

Six high-level sprinters (age: 22.5 ±3.9 years; body mass:

81.8 ±5.1 kg; height: 1.86 ±0.04 m) performed three maximal

50-m sprints using starting blocks with 10 min of rest between

each trial. Instantaneous vHwas measured at a sampling rate of

46.875 Hz with a radar system (Stalker ATS System, Radar Sales,

Minneapolis, Minnesota, USA) placed on a tripod 10 m behind the

subjects at a height of 1 m corresponding approximately to the

height of subjects’ CM (di Prampero et al., 2005; Morin et al.,

2012). F0,v0,SFV,Pmax, and DRF values were obtained with the

same data processing as presented before for the proposed com-

putation method, except that vHmax and τwere determined from

vH(t) using eqn. [1] and least-square regression method. Because

only the best of several trials is commonly used to be considered

during explosive performance testing, the inter-trial reliability of

each variable was quantiﬁed by the coefﬁcient of variation (CV in

%), the change in the mean, and the standard error of measurement

(SEM, expressed in percentage of mean values) between the two

best trials (Hopkins, 2000). These data were used to determine the

smallest worthwhile change (SWC) for intra-individual (SWCintra)

and for inter-individual (SWCinter) comparisons for each variable as

0.3 of the SEM (expressed in the variable unity) and 0.2 of the

between-subject SDs, respectively (Hopkins, 2004).

Results

For the validation protocol, the split times at 10, 15, 20,

30, and 40 m were 1.84 ±0.10, 2.49 ±0.11, 3.05 ±0.13,

4.08 ±0.18, and 5.10 ±0.25 s, respectively. For each

subject, the change in horizontal position with time

given by these split times was well ﬁtted by the expo-

nential model described by eqn. [3] (r2>0.999;

P<0.0001). The associated vHmax and τwere

10.05 ±0.66 m/s and 1.24 ±0.14 s, respectively. The

modeled values of FH,FV,FRes, and PHwell ﬁtted the

experimental values measured at each step using force

plates, which is shown by SEE of 39.8 ±13.3 N,

49.5 ±17.0 N, 52.8 ±16.5 N, and 234.4 ±69.9 W,

respectively. Force values measured by force plate

method at each step and force values computed for the

corresponding step using the proposed method were

highly correlated (P<0.001; Fig. 1). Figure 2 shows the

changes in step-averaged (force plate method) and

modeled (computation method) values of force, power

output, velocity, and RF over the acceleration phase

obtained for a typical subject (subject #5). The SEE of

RF over the acceleration phase was 3.72% ±0.76%.

For all subjects and considering both methods, F–v

relationships were well ﬁtted by linear regressions

(median r2=0.953 from 0.920 to 0.987 for the force

plate method and r2>0.999 for the proposed simple

method, P<0.001), P–v relationships were well ﬁtted

by second-order polynomial regressions (median

r2=0.886 from 0.857 to 0.961 for the force plate method

and r2=1.000 for the proposed simple method,

P<0.001), and RF–vrelationships were well ﬁtted by

linear regressions (median r2=0.965 from 0.955 to

Simple method to compute sprint mechanics

5

0.976 for the force plate method and r2>0.996 for the

proposed simple method, P<0.001). Typical examples

of these relationships obtained by both methods are

shown in Fig. 3 for subject #5. Values of F0,v0,Pmax (both

computed as the apex of the P–v relationship and using

eqn. [13]), SFV, and DRF are presented in Table 1, as well

as the associated bias, 95% limits of agreement, and

absolute bias between both methods. Table 2 presents

the inter-trial reliability of different mechanical variables

through CV, change in the mean and SEM between the

two best trials of the reliability protocol, as well as

values of SWCintra and SWCinter.

Discussion

Although running mechanics over the entire sprint accel-

eration phase have very recently been described for the

ﬁrst time in overground conditions using in-serie force

plates (Rabita et al., 2015), the present study showed

valid estimates of the main sprinting mechanical

variables from only basic anthropometric and spatiotem-

poral data (i.e., distance–time or speed–time measure-

ments). The proposed simple computation method

makes possible to determine force– and power–velocity

relationships and effectiveness of force application

during sprint running acceleration in real-practice con-

ditions. This new method shows very strong agreement

with the gold standard force plate measurements and a

Fig. 2. Changes over the acceleration phase in horizontal veloc-

ity (vH, black diamonds), horizontal antero-posterior (FH, black

ﬁlled circles) and vertical (FV, open triangles) force components,

horizontal antero-posterior power output (PH, open circles), and

ratio of force (RF, open diamonds) for a typical subject (subject

#5). Points represent averaged values over each step obtained

from force plate method (from ﬁve sprints) and lines represent

modeled values computed by the proposed simple method.

Fig. 3. Force–velocity (a), power–velocity (b), and RF–velocity

(c) relationships obtained by both methods for a typical subject

(subject #5). Open circles represent averaged values over each

step obtained from force plate method, dashed lines the associ-

ated regressions, and thin lines the modeled values computed by

the proposed simple method confounded with the associated

regressions (dotted lines).

Samozino et al.

6

high test–retest reliability, which supports its interest for

practitioners in numerous sports that involve sprint

running accelerations.

Modeled GRFs during sprint acceleration

The proposed method is based on a macroscopic biome-

chanical model using an inverse dynamic approach

applied to the runner body CM during sprint running

acceleration. This approach models the horizontal

antero-posterior and vertical GRF components applied

to the runner’s CM, and in turn the force developed

by the runner onto the ground in the sagittal plane,

during the entire acceleration phase of maximal intensity

sprint. The main simplifying assumptions admitted in

this model are those inherent to the application of fun-

damental laws of dynamics to the whole human body

considered as a system represented by its CM (e.g.,

Cavagna et al., 1971; van Ingen Schenau et al., 1991;

Samozino et al., 2008; Helene & Yamashita, 2010;

Samozino et al., 2010; Samozino et al., 2012; Rabita

et al., 2015), the estimation of the horizontal aerody-

namic drag from only stature, body mass and a ﬁxed

drag coefﬁcient (Arsac & Locatelli, 2002), and the

assumption of a quasi-null CM vertical acceleration over

the acceleration phase of the sprint. Note that our com-

putations lead to modeled values over complete steps,

i.e., contact plus aerial times, in contrast with previous

kinetic measurements during sprint running that aver-

aged mechanical variables over each support phase

(Morin et al., 2010, 2011a, 2012; Lockie et al., 2013;

Kawamori et al., 2014; Rabita et al., 2015). This induces

lower values of force or power output and a different

averaging period than support phase-averaged values:

step-averaged variables characterize more the mechanics

of the overall sprint running propulsion than speciﬁcally

the mechanical capabilities of lower limb neuromuscular

system during each contact phase. This does not affect

RF (and in turn DRF) values as it is a ratio between two

force components averaged over the same duration.

Despite the above-described simplifying assumptions,

present results showed that the modeled force values (FH,

FV,FRes) were very close to values measured by force

plates at each step with low SEE values of ∼30–50 N.

Furthermore, the values obtained by both methods were

highly correlated and closely distributed around the

identity line, even if the correlation was slightly lower

for FVthan for FHdenoting a better accuracy of estima-

tion in the horizontal than in the vertical direction. It is

worth noting that the SEE quantiﬁed here are mainly due

to the relatively high inter-step variability measured by

force plates rather than an inaccuracy of the proposed

method. Indeed, as shown in Fig. 2 for a typical subject,

force-modeled values well ﬁtted the force plate data with

a scattering of the latter due to a possible asymmetry

between right and left legs, an inter-step variability

inherent to the complexity of this multijoint free move-

ment characterized by a great muscle coordination com-

plexity, and the fact that force plate data were obtained

from ﬁve different sprints. The inter-step variability is

not detectable by the proposed method as the model

gives the average tendency of change in GRF compo-

nents with time. The inter-step variability in force plate

data is more obvious in power output values combining

the variability of both force and velocity values, which

lead to a relatively higher, but still very acceptable,

Table 1. Mean ±SD values of variables attesting the concurrent validity of the proposed method

Reference

method

Proposed

method

Bias 95% agreement

limits

Absolute

bias (%)

F0(N) 654 ±80 638 ±84 −15.9 ±25.7 (−66.3; 34.5) 3.74 ±2.69

v0(m/s) 10.20 ±0.36 10.51 ±0.74 0.32 ±0.52 (−0.7; 1.3) 4.77 ±3.26

Pmax (W)apex of the P–v relationship 1546 ±195 1661 ±277 115 ±107 (−94.7; 324.7) 8.04 ±5.01

Pmax (W)from F0and v0(eqn. [13]) 1669 ±253 1680 ±280 10.56 ±45.01 (−77.7; 98.8) 1.88 ±1.88

SFV (N/s/m) −64.06 ±6.30 −60.8 ±7.71 3.26 ±5.22 (−6.97; 13.49) 7.93 ±5.32

DRF (%/s/m) −6.80 ±0.28 −6.80 ±0.74 −0.002 ±0.58 (−1.139; 1.135) 6.04 ±5.70

SD, standard deviation.

Table 2. Mean ±SD of the main variables attesting the reliability of the proposed method

CV (%) Change in

the mean

Standard error of

measurement (%)

SWCintra SWCinter

F0(N) 2.93 ±2.00 −1.53 ±32.2 3.57 6.76 12.06

v0(m/s) 1.11 ±0.86 −0.171 ±0.776 1.40 0.166 0.268

Pmax (W)apex of the P–v relationship 1.90 ±1.40 −0.164 ±0.669 2.37 0.147 0.441

Pmax (W)from F0and v0(eqn. [13])

1.87 ±1.36

1.87 ±1.36 −0.167 ±0.66 2.33 0.144 0.446

SFV (N/s/m) 4.04 ±2.72 −0.20 ±4.18 4.94 0.888 0.963

DRF (%/s/m) 3.99 ±2.80 −0.110 ±0.45 4.86 0.096 0.080

SD, standard deviation; SWC, smallest worthwhile change.

Simple method to compute sprint mechanics

7

power output SEE (∼234 W). For all subjects, FVvalues

measured with force plates over the ﬁrst 20–30 m were

not particularly higher than body weight (as shown for a

typical subject in Fig. 2) and were very close to body

weight when averaged over the entire 40 m (grand aver-

aged difference between mean FVand body weight of

∼2.40%). This supports the assumption of a quasi-null

vertical acceleration of the CM over this phase due to a

very smoothed upward movement, and in turn supports

the validity of step averaged FVmodeled values as equal

to body weight. Collectively, these results support a very

good agreement in the model determination of GRF in

the sagittal plane of motion (horizontal, vertical, and

resultant) during sprint running acceleration.

Validity of force– and power–velocity

relationships determination

Both methods presented individual F–v relationships

strongly described by linear regression (r2>0.920) as

reported in our recent paper (Rabita et al., 2015), during

treadmill sprint running protocols (Jaskolska et al.,

1999b; Morin et al., 2010, 2012) or more generally

during multijoint lower limb movements such as pedal-

ing, squat, or leg press movements (e.g., Yamauchi &

Ishii, 2007; Dorel et al., 2010; Bobbert, 2012; Samozino

et al., 2012). The adjustment quality of the linear regres-

sions was logically better for the proposed method

(mean r2=1.00) based on modeled values than for the

reference method (mean r2=0.956) using experimental

force plate data that were inevitably associated to a

higher measurement and inter-step variability. The low

bias associated to narrow 95% agreement limits crossing

0 in the determination of F0,v0, and SFV showed that the

difference between both methods is very low (−2.43%,

3.14%, and 5.09% when expressed relatively to refer-

ence values, respectively) and can be attributed to mea-

surements variability. The very low absolute bias

(<5%), representing exactly the mean absolute error

value between both methods at each F–v variable deter-

mination, clearly supports the validity of the proposed

method to determine F–v relationship in sprint running.

Note that the absolute bias for SFV was slightly higher

than for F0and v0as it represents a regression slope

(often associated to higher variability than other vari-

ables) computed from F0and v0, thus including twice

variability.

Individual P–v relationships presented well-ﬁtted qua-

dratic regressions for both methods, as expected from

previous sprint running protocols (Jaskolska et al.,

1999b; Morin et al., 2010, 2012; Rabita et al., 2015).

However, for force plate method, the adjustment quality

of P–v regressions was lower (mean r2=0.894) than that

of F–v relationships, and lower than for the proposed

method (mean r2=1.00). This can be explained by the

above-discussed higher inter-step and inter-sprint vari-

ability measured by force plate and by the very few

number of steps in the ascending part of the P–v rela-

tionship inherent to maximal power production occur-

ring after only approximately ﬁve to six steps (Rabita

et al., 2015). The lower adjustment quality of the qua-

dratic model obtained from force plate data induced

noise in the determination of Pmax from the apex of P–v

relationships, which explains the higher, but still very

acceptable, bias and absolute bias (∼8%) observed here

between both methods for this variable compared with

bias observed for F0and v0. Consequently, differences in

Pmax are mainly due to the inter-step variability detected

by force plates, which was supported by the very similar

Pmax values obtained using the proposed method from the

apex of the P–v relationships and using eqn. [13] (∼1%),

while they were quite different for the reference method

(∼8%). From a purely mathematical point of view, if the

F–v relationship is perfectly linear, the apex of the P–v

relationships should be equal to Pmax given by eqn. [13]

(Vandewalle et al., 1987; Samozino et al., 2012). More-

over, when computed from eqn. [13], Pmax values

obtained by both methods were very close with very low

bias values (absolute bias <2%) and narrowed limits of

agreement. These ﬁndings support the validity of the

proposed method in determining F–v and P–v relation-

ships and their associated mechanical variables (Pmax,F0,

v0,SFV) in sprint running.

Validity of the effectiveness of force

application determination

The very good agreement between both methods in GRF

components resulted in modeled RF values similar to

those computed at each step from force plate measure-

ments, as shown by the low SEE. RF values measured

here from both methods (RF values between 0% and

60%) were in line with values previously reported from

dynamometric treadmill measurements (between 10%

and 40%; Morin et al., 2011a, b; Morin et al., 2012), but

through a larger range of values as treadmill running

made impossible to measure RF for the ﬁrst step (due to

the starting crouch position) and forced subjects to over-

come treadmill belt bed horizontal friction force at peak

running velocity, the latter being ∼20% lower than the

overground one (Morin & Seve, 2011). Individual

RF–velocity relationships determined by both methods

were well ﬁtted by inverse linear regressions (r2>0.95),

as originally shown on treadmill (Morin et al., 2011a,

2012) and more recently during overground sprint

running (Rabita et al., 2015). Moreover, present results

show that the rate of decrease of these regressions (DRF)

were very similar between both methods (absolute

bias ∼6%). This supports the accuracy of the proposed

method to determine the effectiveness of force applica-

tion throughout the acceleration phase of a sprint and,

in turn, its rate of decrease when running velocity

increases.

Samozino et al.

8

Validation protocol

The validation of the proposed method was performed

using a recent validated multiple sprint protocol (Rabita

et al., 2015) as the reference method. The input variables

of the present model are basic anthropometric (body

mass and stature) and spatiotemporal data: split times (as

used here) or running velocity measurements (as could

be obtained from radar guns, e.g., di Prampero et al.,

2005; Morin et al., 2006, or laser beams, e.g., Bezodis

et al., 2012) during one single sprint. Because the aim of

the present study was to validate the whole approach and

the proposed equations, using split times from several

sprints allowed us to compare exactly the same exercises

between the two methods. Using radar data would have

forced us to compare mechanical variables obtained

from one single sprint using the proposed method to data

obtained from ﬁve sprints with the reference method,

which could have added a bias that would have only been

associated to the validation protocol itself. However, the

proposed method was also tested in the present protocol

from data measured using a radar (Stalker ATS System,

Radar Sales, 46.875 Hz) during the best sprints of the 30

and 40-m trials. Results were very similar to those

obtained from split times, with slightly higher bias

values (absolute bias from 3% to 7%) due to the above-

discussed point, which supports the validity of the

proposed method from running velocity measurements.

As partly noticed by Furusawa et al. (1927), the present

macroscopic biomechanical model shows that, whatever

the kind of locomotion, when displacement velocity

changes with time follow an exponential function

(as described by eqn. [1] and previously reported during

maximal sprint acceleration from recreational to

highly trained sprinters; di Prampero et al., 2005; Morin

& Seve, 2011; Morin et al., 2012), the relationship

between horizontal GRF component and velocity is

quasi-linear.

Reliability

The reliability of the proposed method was tested here in

a second protocol as the ﬁrst mentioned protocol already

required seven maximal sprints to compare the proposed

to the reference method. For all the mechanical vari-

ables, low CV and SEM values (<5%) associated to

change in the mean close to 0 showed the high Test-retest

reliability of the proposed method. The present level of

reliability is in accordance with reliability previously

reported during isoinertial all out tests (e.g., Hopkins

et al., 2001; Samozino et al., 2008). This high reliability

led to low SWC values for both intra- and inter-

individual comparisons for each variable and strongly

that this simple method is of great interest for sport

practitioners and clinicians to detect training or rehabili-

tation effects, or for scientists when using progressive

magnitude statistics.

Practical applications

In light of the above-discussed points, the present study

proposed an accurate, reliable, valid, and simple method

to determine F–v, P–v, and RF–v relationships during

overground sprint running from variables easily obtained

in ﬁeld conditions and with a precision similar to that

obtained with speciﬁc laboratory devices (force plates;

Rabita et al., 2015). This new ﬁeld method is based on

the previously validated simple method focusing on the

same kind of mechanical properties during jumping

movements and based on a similar biomechanical mac-

roscopic approach (Samozino et al., 2008, 2014). The

present proposed method only requires individual basic

anthropometric data (body mass and stature) and ~5 split

times or instantaneous velocity measurements obtained

during one single sprint acceleration until maximal

velocity. Note that split times or instantaneous velocity

have become more accessible with new technologies

such as position trackers, GPS, or accelerometer-based

systems. Anthropometric and split times or instanta-

neous velocity data can then be used as inputs in the

basic data processing to determine vHmax and τusing

least-square regression method from eqns. [1] or [2] and

then to compute different mechanical variables (details

in the Methods section “Proposed computation method:

material and speciﬁc data processing” and “Common

data analysis for both methods”). This will contribute

to generalize such evaluations for both scientiﬁc and

training purposes as it has the potential to be easily

reproduced by coaches, sport practitioners, or physio-

therapists in their daily practice. Such a testing session

could take only 15–20 min after a regular warm-up with

two or three sprints by athlete, only the best one being

analyzed using the method proposed here.

The variables obtained from F–v, P–v, and RF–v rela-

tionships give key information about force, velocity, and

power output capabilities, and about the effectiveness of

force application, which are of great interest to optimize

sprint running acceleration performance by comparing

P–F–v qualities of different athletes, orienting and indi-

vidualizing training loads exercises, and monitoring

training or rehabilitation in sports using sprint accelera-

tions (e.g., track-and-ﬁeld events, team sports). Indeed,

as previously mentioned, sprint performance has been

shown to be related to these mechanical variables,

notably Pmax,v0, and DRF (Morin et al., 2011a, 2012;

Rabita et al., 2015). Furthermore, these mechanical vari-

ables seem to be sensitive to training modalities (notably

through in-season variations, unpublished personal

data). For instance, using weighted sled towing improves

horizontal force production and force application effec-

tiveness in sprint running (Cronin et al., 2008;

Kawamori et al., 2014). Note that methods using force

plate system with several sprints or instrumented sprint

treadmill would give the same kind of information, but

are currently impossible to set in training practice for

Simple method to compute sprint mechanics

9

most sport practitioners. That said, they are still very

interesting for studying inter-step variability or for intra-

step analyses, contact and aerial times, step length/

frequency and force impulse, and rate of development

during sprint running, or for other analyses requiring

additional and synchronized laboratory measurements.

Perspectives

This study proposed an accurate, reliable, and valid

simple method to evaluate mechanical properties of

overground sprint running propulsion and validated it

through a very good agreement with gold standard force

plate measurements. The proposed method is based on a

macroscopic biomechanical model using an inverse

dynamic approach applied to the runner’s CM. This

method is convenient for ﬁeld use by sport practitioners

and clinicians as it requires only anthropometric (body

mass and stature) and spatiotemporal (split times or

instantaneous velocity) variables easy to obtain out of a

laboratory during sprint running acceleration. This

method could be further used to increase the understand-

ing in the mechanical determinants of sprint acceleration

performance in many sports, to study the adaptations of

the mechanical properties of lower limb neuromuscular

system in a variety of sprint running propulsion, and to

optimize sprint performance by individualizing and ori-

enting training or rehabilitation programs.

Key words: Acceleration, maximal performance, evalu-

ation, sprint mechanics, lower limb explosive capabilities,

ratio of force application.

Acknowledgements

We are grateful to Dr Pascal Edouard and Philippe Gimenez (Uni-

versity of Saint-Etienne, France), Pedro Jiménez-Reyes (UCAM,

Murcia, Spain), Matt Brughelli (AUT, Auckland, New Zealand),

Antoine Couturier, Gaël Guilhem, Caroline Giroux, Stevy Farcy,

and Virha Despotova (INSEP, France) for their valuable participa-

tion to this project and their enthusiastic and friendly collabora-

tion. We would like to thank Guy Ontanon, Dimitri Demonière,

Michel Gilot, Pierre Carraz, and the athletes of both the French

Institute of Sport (INSEP) and the Athletic Sport Aixois who

voluntarily gave their best performance for this protocol.

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