JOURNALOF
APPLIED
Vol. 39, No. 1, July
PHYSIOLOGY
1985.
Printed in U.S.A.
Force platforms
as ergometers
GIOVANNI A. CAVAGNA Istituto di Fisiologia Umana, Uniuersitci di Milan0 and Centro di Studio per la Fisiologia de1 Lavoro Muscolare del CNR,
CAVAGNA, GIOVANNI A. Force platforms as ergometers. J. Appl. 1975.-Walking and running on the Physiol. 39( 1) : 174-l 79. level involves external mechanical work, even when speed averaged over a complete stride remains constant. This work must be performed by the muscles to accelerate and/or raise the center of mass of the body during parts of the stride, replacing energy which is lost as the body slows and/or falls during other parts of the stride. External work can be measured with fair approximation by means of a force plate, which records the horizontal and vertical components of the resultant force applied by the body to the ground over a complete stride. The horizontal force and the vertical force minus the body weight are integrated electronically to determine the instantaneous velocity in each plane. These velocities are squared and multiplied by one-half the mass to yield the instantaneous kinetic energy. The change in potential energy is calculated by integrating vertical velocity as a function of time to yield vertical displacement and multiplying this by body weight. The total mechanical energy as a function of time is obtained by adding the instantaneous kinetic and potential energies. The positive external mechanical work is obtained by adding the increments in total mechanical energy over an integral number of strides.
biomechanics; external and
locomotion; internal work
walking
.> running;
mechanical
Milano,
Italy
lowered, requiring external positive work during the increase in potential energy and the speed of progression oscillates from below to above the average value requiring external positive work during the increase in kinetic energy of the centre of mass of the body. The muscles are active and exerting a force during the phases of the step of decreasing mechanical energy to retard and control the movement; in these phases they are being stretched and the muscular force is doing negative work. This braking action requires the expenditure of chemical energy by the muscles and is equivalent to applying the brakes on an automobile. Thus the chemical energy expended in applying the brakes and the mechanical energy which is not stored in and recovered from elastic elements appears as heat in the muscle and is lost. This mechanical energy is then replaced by contraction of the muscles as they do positive work to raise and reaccelerate the center of mass of the body. If complete storage and recovery of energy in elastic elements (contracted muscles and tendons) were possible, then the decrease in potential and/or kinetic energy during one phase of the step could be utilized in another phase of the step, and no additional energy from the muscles would be needed except that necessary to maintain tension (a slack muscle is unable to store any appreciable amount of elastic energy). Although there is some recovery of energy in elastic elements (1, 2, 5, 13), there is presently no way to measure it directly.
work;
FORCEPLATFORMASANERGOMETERTOMEASUREMECHANICALWORR WORK IN WALKING AND RUNNING. Muscles transform chemical energy into mechanical work during exercise. The rate of energy utilization is routinely determined by measuring oxygen consumption and lactic acid production. However, the rate at which muscles perform mechanical work is seldom measured, and then for specialized types of exercise, such as pedaling a bicycle ergometer. The mechanical work performed during such common exercises as walking and running on the level has only been measured a few times (3-5, 9, 10, 11) and then with an enormous effort. This paper describes a convenient means of measuring the external mechanical work performed in walking, running, and jumping from the forces applied by the body to the ground. It is frequently argued that walking and running at a constant speed on the level involves only a very small amount of external mechanical work to overcome air resistance. This argument is based on the fact that the mechanical energy possessed by the body (both potent .ial a nd kinetic) is the same at the beginning and the end of each step. On the other hand, it is commonly concluded that external mechanical work is done when a change in the total mechanical energy of the body is observed after one or more steps, for example, when a hill is climbed (increasing potential energy) or a sprinter accelerates (increasing kinetic energy). These situations differ from level walking and running at a constant speed only in the duration of time between performance of positive and negative work. For instance within a step cycle of “level” running at a “constant speed,” the center of gravity of the body is raised and
As long ago as 1885 Marey and Demeny (12) used a force platform to measure the vertical component of the force exerted by the feet against the ground during standing vertical jumps. In the 1930’s Fenn (11) utilized a similar type of platform to measure forward and backward components of force applied by the body to the ground during running. From these force measurements he calculated the mechanical work necessary to account for the velocity changes of the center of mass of a running man during each stride. In 1939 Elftman (8) used a force platform to measure the force exerted by one foot against the ground during walking. Although he did not measure the actual velocity and the displacements of the center of gravity, he showed that this was possible cc. . . by proper evaluation of integration constants.” Cavagna and colleagues (3, 5) utilized force platforms to measure the external mechanical work done in level running and walking at different speeds. The procedure was laborious since these authors were limited by I) the small dimensions of the platforms (35 x 35 cm; in walking only the phase of the step in which a single foot was on the ground could be studied, and in running it was difficult to step on the platform without altering the length of the stride); 2) the elaboration of the force-time tracing from the platform, made entirely by graphical computation (this was necessarily discontinuous, and the shape irregularities of the tracing and the oscillations due to the natural frequency of the platform at the highest speeds decreased the accuracy); 3) the method was SO slow that only a small number of determinations of mechanical work could be obtained.
MECHANICAL
174
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FORCE
PLATFORMS
175
AS ERGOMETERS
These difficulties have been overcome I) by using a large platform (4 x 0.5 m) sensitive to the forward and the vertical components of the force exerted by the foot and 2) by electronically integrating the force measurements to yield a direct readout of velocity.
WALK VF
(m/set)
FF (kg)
0.5r const.
CALCULATIONS FORCE
The tential
INVOLVED
IN UTILIZING
T O DETERMINE
EXTERNAL
MEASUREMENT WORK
Vv
= Fd, + Ffsf + F,sl = w, + Wf + WI
/F/./s/COSQ
Fv = weight
(P)
+ frictional
(or lateral)
forces
+
m ‘(I~
forces
+ m .af
= m,af
(2)
(2’)
(3) (3')
directly during exercise by zeroing the platform while the subject stands on it quietly immediately before the experiment. Thus it is possible to calculate the acceleration of the center of gravity from direct measurements of the force exerted in the vertical and forward directions. It is much easier to work from instantaneous velocity than from instantaneous force in calculating the mechanical work. This procedure abolishes the interference caused by vibrations of the plate and provides a much cleaner starting point. In addition the velocity tracings are much simpler and their interpretation much easier. Since the velocity is the integral of acceleration it can be obtained directly by electronically integrating the signal measured with the force plate (Fig. 1) P)dt
jFfdt
=
t
‘d
\
0 100 200
F, - P can be measured
-
-2
i
0
1
2 set
1. Tracings recorded when a subject was walking at 5.5 km/h (left) and running at 11.9 km/h over the force platform. Ff and F, are, respectively, the horizontal and vertical components of the resultant force impressed by the feet on the platform; whereas F, oscillates around the zero (positive values = acceleration forward of subject, i.e., backward push on platform), F,, oscillates around a force value (59 kg) equal to the subject’s body weight, P (corresponding to 1 g = 9.8 m/s2 on the right-hand scale). Simultaneously the horizontal force and the vertical force minus the body weight (F, - P) are integrated electronically to determine the velocity tracings Vf and V,; these give the forward and vertical components of the velocity of the center of gravity of the whole body plus an integration constant to be determined later according to the procedure indicated in Fig. 2. Electronic integrators are operated by subject crossing photocells at the platform level (2.7 m apart when walking and 3.5 m apart when running, Fig. 5); photocells are placed in such a way that the subject crosses them when he does not contact the ground before or after the platform. FIG.
The integration constants must be known in order to calculate absolute velocity. The value of the integration constants in Eq. 4 and 4’ is zero when the subject begins to move on the platform from the resting condition (V = 0). In this case the determination of the instantaneous velocity is easy and both absolute velocity and external mechanical work can be recorded as a function of time simultaneously with the performance of the exercise (1, 2, 6, 13). However, when the subject arrives on the platform with a velocity above zero, the values of the constants in Eq. 4 and 4’ are unknown and the absolute velocity cannot be measured simultaneously with the performance of exercise. The absolute forward velocity is obtained by I) measuring the average forward velocity of the subject moving across the platform by means of two photocells: these also turn on and off the integrators while the subject is over the platform without contacting the ground before or after it (Fig. 24); 2) measuring the area under the recorded velocity tracing during the time interval while the integrators are on (either by planimetry or by utilizing a computer); and 3) dividing the area by the time required to travel the distance between the photocells to position the average velocity forward in the tracing (Fig. 2B). Instantaneous forward velocity can then be calculated. In this procedure it is assumed that the average speed of the trunk between photocells is equal to the average speed of the center of gravity: this is reasonable since the displacements of the center of gravity within the body and the “tilting” of the trunk (5, 11) are small in comparison with the distance between the sights. A similar procedure is followed to determine the absolute value of the vertical component of the velocity (Fig. 2, C and D); in this case however the area A below the vertical velocity tracing is measured only for an interval of time, nr, corresponding to an integral number of steps. The ratio A/w gives an average vertical velocity equal to zero on the assumption that during an integral number of cycles, the upward displacement of the center of gravity the
F,, - P = m,a,
J(Fo
E
const
F,(kg)
where m is the mass of the body, a is the acceleration of the center of gravity of the body and the frictional forces are u) air resistance and 6) the force which opposes the displacement of the center of gravity within the body during a nonelastic deformation of the body itself. The frictional forces oppose muscular force during the acceleration of the body whereas they cooperate with it in decelerating the body. For the purpose of the present calculations the forces of friction even in man sprinting at high speed (2) can be neglected as discussed later. Therefore Eq. 2 and 2’ can be simplified to
Ff
1 const
(1)
direction
Ff = frictional
r
-1
where s is the displacement of the center of mass of the body, F is the resultant of all external forces exerted against the body by the ground and the air, and Q is the angle between the two vectors F and s; F. , s,,, Ff , etc. are projections of F and s in the vertical, forward, and lateral directions; W, , Wf , and WI are the work done, respectively, by the vertical, forward, and lateral components of the force. During level walking and running sf is obviously much greater than s0 and s1 , and F, is much greater than Ff and FI due to the acceleration of gravity. As a result of the greater displacement and force, respectively, Wf and W, are much greater than WI (= 100 times in walking (4)). Thus WI can be neglected. The work involved in rotating the body about its center of mass (7) was also neglected. To determine W,,t one can then start by measuring forces in the vertical (Fo) and forward (Ff) directions over a complete stride. The force exerted by the feet on the platform in the vertical direction is
and in the forward
(m/set)
po-
RUN
-05 50 0 -50 1 2
-50 t
external work required to change the kinetic and/or energy of the center of mass of the body is defined as
W ezt = F ,s =
const
t
-051 50 0
OF
0.5
mV, + const
= mV$ + const
(4) (4')
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176
Al
G. A. CAVAGNA
Forward
velocity
tracing
Cl
:
Vertical
velocity
f
tracing
time
:
integrators
on
ON
0.5
0
B)
Integrate are on are on
>
4i
1
set
0
0)
during time integrators and divide by thetime they in order to locate VF on tracing:
I-12 0
0.5
set
From
the
absolute
kinetic
the
potential
and
the
are
easily
values
energies
0.5
energy:
total
of
VF,V,
E,,
EP
S,
= weight
Sv
and
EKF :$mV,f,
set
FIG. 2. Procedure mine the absolute in the forward (A tical (C and D)
1
Integrate during time of one or more complete strides (oblique hatching ) and divide by this time to Locate v,,which is zero
1
El the
OFF
0.5
set
velocity
constant
tracings,
recorded
during the exercise, (Fig. 1). From the absolute value of velocity given in B and D the corresponding kinetic energy can be calculated ; vertical velocity tracing given in D is integrated further to yield the vertical oscillations of the center of
1
Vertical displacement downwards ( vertical hatching ) and upwards (horizontal hatching IS obtained by integration of Vv tracing:
+
followed to detervalue of the velocity and B) and the verdirections from the
gravity )
energy
sV (E) changes
and
then
(Fig.
3).
the
potential
:+rnV:
energy:
E T O T = EKF+ E,,+ obtained
Ep
( Fig. 3 >
0
0.5
is equal to the downward displacement (i.e., that in level walking and running the height of the center of gravity on the average is constant). This assumption may not be true at the end of a single step, but it is certainly valid over a number of steps. Once the zero vertical velocity is determined then the instantaneous vertical velocity is known. From the instantaneous velocity in the forward and vertical directions the instantaneous kinetic energy Ekf = >s(rnVj2) and +AEk is the positive work Ekv = >s(m’V,2) can be calculated; required to accelerate the center of gravity of the body. The instantaneous potential energy Ep = Pas, can be obtained by integrating the vertical velocity to determine the vertical displacement of the center of mass (Fig. 2E)
= s, + const
SKdt
(4
Geometrically the upward displacement is represented by the upper area enclosed by the instantaneous velocity tracing, V, , and the zero (Fig. 20). The positive work against gravity is given by the increments of the potential energy: +AE, . The total mechanical energy is then obtained by adding the instantaneous potential energy Ep and the instantaneous kinetic energies EkV and EIcj
E tot =
E,
+
Ekv
+
Ekf
(6)
It is clear that potential energy can be converted into kinetic energy and vice versa if the increase in one is out of phase with the increase in the other, and indeed this happens during walking (4). On the other hand, during running, the increase and decrease in potential and kinetic energy are almost completely in phase, and there is little interconversion between kinetic and potential energy (5). Finally the positive external mechanical work is obtained by adding the increments in mechanical energy, +AEt,t , over an integral number of strides. In Fig. 3 the upward curve indicates how the kinetic energy Ekf of the center of mass of the body of a 78-kg subject oscillates during two steps of running at 15.5 km/h. This kinetic energy
set
1
curve was calculated from the forward velocity curve given in Fig. 2B. The E, curve below indicates the oscillation of the potential energy of the body and was calculated from the vertical displacement curve given in Fig. 2E. The curve just above it is the sum of the potential and the kinetic energy Ekv (calculated from curve in Fig. 20). In the interval in which the sum Ep + Ekv is constant the subject is off the ground and the kinetic energy EkV is transformed into potential energy and vice versa. At the top and bottom points the two curves coincide since at these points the vertical velocity and then EkV is nil. Whereas Ekf is given in absolute units (the average kinetic energy of the subject being about 171 Cal), the potential energy Ep is not: in each trial the starting point for the computation of Ep was taken as zero. The two curves below indicate the total mechanical energy Etot = Ekf + Ep -/- EkV and, for comparison, the sum Ep -/- Ekf as usually measured in the past (Fig. 5 in (5)). Etot may differ from Ep + Ekf not only for the slope of the curve, giving the rate of work production, but also for the amplitude of the oscillation of the curve which represents the external mechanical work done; the difference, negligible in running, becomes appreciable in walking. The positive external mechanical work done is taken as the increments of the curve Etot during the interval of time, of one or more complete strides, in which the V, tracing was integrated to locate the average vertical velocity (Fig. 20). The same is done for the curve Ekf to determine the work necessary to sustain the kinetic energy changes Wf , and for the curve Ep + EkV to determine the work against gravity, WV . All the calculations reported in Fig. 2 as well as the curves in Fig. 3 were made directly by a computer. As mentioned above if the subject starts moving on the platform the constant of integration in Eq. 4 and 4’ is zero and it is possible to integrate the vertical velocity directly to give displacement (and potential energy). Simultaneously other operational amplifiers can be used to square velocity to obtain a direct readout of kinetic energy and to sum instantaneous potential and kinetic energy to give total instantaneous energy. Thus mechanical work can be recorded simultaneously with the exercise (1, 2, 6, 13). The limit of this method is that any drift of the first integrator (velocity)
is squared by the second (displacement).
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FORCE PLATFORMS
AS ERGOMETERS
177 apparent kinetic energy change ( - AEh), calculated above, will be smaller than the negative work done, -AEk
,E
d(rn v2> > >a(rn d2) + +$ (m vfr2> which is inequality During negative
7. work
-JFdt i.e., assuming that velocity v to zero
= -J the
deceleration
-mu
0.
1.00
0.50
SEC
FIG. 3. Mechanical energy of the centre of mass of a 78 kg subject running at 15.5 kn/h. E kf is the kinetic energy calculated from the forward velocity curve given in Fig. 2B. EP is the potential energy calculated from the sU curve in Fig. 2E. Ekv is the kinetic energy calculated from the curve in Fig. 211. Etot = Ekj j-- EkV -j-- Ep. All the calculations reported in Fig. 2 as well as the curves above were made directly by a computer.
ASSUMPTIONS
INVOLVED
T O DETERMINE
WORK
FROM
>
+
AE’k + &,,,,,
= positive
w to reduce
+ mvf,
the
W)
-m v’, the momentum actually lost, lost because of a) the braking action platform, -m v, and 6) the force of the work as a kinetic energy change,
- %(m
vr2)
+
$5 h
vfr2)
(14)
that the initial kinetic energy, which is a The signs - indicate positive quantity, is subtracted from the final kinetic energy (equal to zero) in order to get the kinetic energy change. Fortunately the error done by assuming AE, = AE’k + Wlosses is not great. This was checked by measuring the work actually to accelerate forward and by done (AE’k + VVl osses) by a sprinter comparing it with the same work measured as AE, (2). The positive and the negative work actually done by the muscles at each step was determined by multiplying the average horizontal force, IQ , exerted on the platform during the push or during the brake, by the actual displacement forward of the trunk Sf during these intervals determined by means of photocells. Since the displacements of the center of gravity within the trunk (11) were small in comparison with the distance between sights (3 m), the displacement of the trunk Sf could not differ appreciably from the displacement of the center of gravity. The mechanical work done at each step
IN CALCULATIONS
EXTERNAL
Ws tep
= Ff Sf
P)
FORCE
is positive during acceleration and negative during deceleration; the positive and the negative work were calculated from Eq. 15 for 19-20 steps taken by two subjects after the start. The total positive work is given by
The kinetic energy increase (+AE,& calculated disregarding the forces of friction, is greater than the total positive work actually done by the subject to accelerate himself (+AE’,& to overcome air resistance and to deform the body ( VVlosses), i.e. +AEk
v2)
a(rn
m a dt + J
work
+wot
actually done by muscles
(7)
During deceleration, when the forces of friction and muscular have the same direction (-F = - m a + forces of friction),
force the
and
the total
negative
=
z
+
Wstep
wtot =
x
-
Wstep
(16)
work -
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178
G. A. CAVAGNA
The algebraic
sum
agreement with those determined with motion picture analysis. + Wtot - Wtot = Wtot
(17)
was, according to inequalities 7 and 8, smaller than the work measured as AEk . The difference however was only 8% (2). If some skidding took place part of this difference would represent the work done against friction between the foot and the ground. Taking into account that in sprint running one attains the highest speed, i.e., the greatest air resistance, and that the forces deforming the body are the greatest met in locomotion, it is therefore possible to conclude that the error done with the procedure described above is tolerable. The vertical displacements of the body usually take place at a speed appreciably smaller than that attained in the forward direction; in addition the cross-sectional area of the body perpendicular to the direction of the movement is much smaller in the vertical than in the forward direction. Both these factors make air resistance to a vertical movement much smaller than that to a forward movement and therefore probably negligible. In addition the vertical component of the push acting along the vertebral column is probably less effective in deforming the body than an equal force acting in forward direction. On the other hand the work lost in the deformation of the body due to the forward component of the push was found to be smaller than that due to air resistance in sprint running (2). These considerations induce one to think that V, N V’, and therefore Ep N E’, In addition experiments in progress indicate that data of work done against gravity in running, obtained with the present method, are in good VERTICAL
FORWARD
4 I 9
7 1
SIDE ’ VIEW i
500 m m
at the same speed by Fenn (11)
: & I
I
APPARATUS
INVOLVED
MECHANICAL
WORK
IN MEASURING DURING
WALKING
EXTERNAL AND
RUNNING
The force platform used in these studies consists of a single row of eight (0.5 rn)” plates. Each plate has separate vertical and forward units and utilizes twelve spring assemblies (6 in the vertical and 6 in the forward) consisting of four springs and two strain gauges (Fig. 4). Half of the strain gauges in the assemblies are compressed and half are stretched when a force is applied in one direction. Each half is wired in series with those of the other platforms forming two opposite arms of a Wheatstone bridge as indicated in Fig. 4. The platform is inserted, with its surface at the level of the floor, 30 m from the beginning of a corridor 50 m long. Its natural frequency is 42 Hz in the forward and 30 Hz in the vertical direction. The maximal difference found between the electrical response of the eight platforms, when acting with a given force on the central portion of each one of them, is: 4y0 (with full scale 5 kg) or 4.7 7’ (full scale 80 kg) in the vertical direction; and 8-9 ‘$& (full scale 10 kg) or 7-lO$$, (full scale 50 kg) in the horizontal direction. The maximal difference found between the 32 corners of the 8 platforms was 11.3-14%. These differences can be reduced by shunting (with appropriate resistors) the strain gauges corresponding to the most sensitive points. The average response of the horizontal units to a force directed forward (deceleration) was usually found slightly greater than that to the same force directed backward (acceleration); the difference however was less than 5%. The platform was tested from 1 to 250 kg in the vertical and from 0.5 to 100 kg in the forward direction and found to give a linear response within an average error of 5 and 7 %, respectively. The output from each Wheatstone bridge (forward and vertical, Fig. 4) after amplification by a Philips PR 7510 preamplifier
TOP VIEW 4
“ERTlCAL I ‘-
FORWARD
1FORWARD
FORCE
69 ,$
P
15OVOC
*
b
150 voc
4
FIG. 4. Schema of vertical and forward units forming each of the eight plates constituting the whole platform. Forward unit is placed on top of the vertical one and fixed to it by the indicated bolts. Strain gauges indicated by D undergo a deformation opposite to that of strain gauges Z when a force is applied to the unit. All the strain gauges of the eight units sensitive to the force in a given direction (vertical or forward) are wired together in a Wheatstone bridge of 96 resistors as indicated at the bottom.
FIG. 5. Diagram of the setup used to record the tracings given in Fig. 1: when the subject crosses the first photocell the integrators are operated and the velocity tracings begin (Vv and Vf in Fig. 1). Velocity tracings together with a ZOO-Hz signal are also recorded on tape for later treatment by a computer according to the procedure indicated in Fig. 2. For further details see text.
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FORCE
PLATFORMS
179
AS ERGOMETERS
goes a) directly to a Hewlett-Packard amplifier (350-1000B) and recorder (77 19) to yield force, 6) through a DC offset, to an integrator (which is turned on and off by the photocells) and then to the amplifier and recorder to yield velocity. The output of this amplifier is also recorded on tape (Hewlett-Packard 3960A tape recorder) for later treatment of the signal by a computer. The photocells also turn on and off a ZOO-Hz signal which is also recorded on the tape (Fig. 5). This provides the time base needed for calculating the integration constants as described earlier. In addition it is used to trigger the analog-to-digital converter. Before =P+ m a,> it integrating the vertical component of the force V earlier. is necessary to subtract the body weight as mentioned This is done by setting the platform’s output exactly to zero using the DC offset voltage while the subject stands absolutely still on the platform before the beginning of the exercise. This operation, and also the setting of the forward output exactly to zero with another DC offset voltage, must be done with great care immediately before each trial otherwise an appreciable drift of the output of the integrators takes place. In our setup the base line from the platform to the integrators has noise, mainly due to vibrations of the time the ground, attaining about & 100 g, and its drift during of a trial is usually less than 50 g.
(F
EXTERNAL,
INTERNAL,
The methods work necessary
AND
TOTAL
WORK
described above allow to sustain the displacements
us to determine of the center
only the of gravity,
i.e., the external work. In exercise, however, also internal work is done, and this cannot be measured by means of the force platform. The main source of internal work (and the only one directly measurable) is the kinetic energy changes of the limbs calculated from their velocity relative to the center of gravity. This can be measured by motion picture analysis as described for example by Fenn (10). The increments of the curve indicating the total kinetic energy of the limbs represent the internal work done. This curve should not be added to the curve of the mechanical energy level, E tot, given in Fig. 3 (as sometimes reported in the literature), because by this summation one implicitly assumes the transfer of some of the work done by the internal forces to an increase in the energy level of the center of mass, which is not possible by definition The absolute value of the increments of the curve Etot (giving the external positive work) must therefore be added to the absolute value of the increments of the curve of the kinetic energy of the limbs (giving the internal positive work) to obtain the total positive work done. The author thanks Prof. C. R. Taylor of Harvard University for his much helpful and constructive advice during the completion of this manuscript. The program to obtain the records of Fig. 3 was prepared by Dr. Carlo Cavagna of the Politecnico of M’ilan. The platform was projected by the Viterra firm, Wallisellen (Zurich, Switzerland). Received
for publication
25 October
1974.
REFERENCES 1. CAVAGNA, G. A., L. KOMAREK, G. CITTERIO, AND R. MARGARIA. Power output of the previously stretched muscle. Med. Sport 6: 159-167, 1971. 2. CAVAGNA, G. A., L. KOMAREK, AND S. MAZZOLENI. The mechanics of sprint running. J. Physiol., London 2 17 : 709-72 1, 197 1. 3. CAVAGNA, G. A., AND R. MARGARIA. Mechanics of walking. J. Ap~l. Physiol. 21 : 271-278, 1966. 4. CAVAGNA, G. A., F. P. SAIBENE, AND R. MARGARIA. External work in walking. J. A/@. Physiol. 18 : 1-9, 1963. 5. CAVAGNA, G. A., F. P. SAIBENE, AND R. MARGARIA. Mechanical work in running. J. AppZ. Physiol. 19 : 249-256, 1964. 6. CAVAGNA, G. A., A. ZAMBONI, T. FARAGGIANA, AND R. MARCARIA. Jumping on the moon: power output at different gravity values. Aerospace Med. 43 : 408-4 14, 1972.
7. ELFTMAN, H. The rotation of the body in walking. Arbeitsphysiologie 10: 477-484, 1939. 8. ELFTMAN, H. The force exerted by the ground in walking. Arbeitsphysiologie 10 : 485-49 1, 1939. 9. ELFTMAN, H. The work done by muscles in running. Am. J. Physiol. 129 : 672-684, 1940. 10. FENN, W. 0. Frictional and kinetic factors in the work of sprint running. Am. J. Physiol. 92 : 583-611, 1930. 11. FENN, W. 0. Work against gravity and work due to velocity changes in running. Am. J. Physiol. 93 : 433-462, 1930. 12. MAREY, J., AND G. DEMENY. Locomotion humaine, mecanisme du saut. Compt. Rend. Acad. Sci. 101 : 489-494, 1885. 13. THYS, H., T. FARAGGIANA, AND R. MARGARIA. Utilization of muscle elasticity in exercise. J. ApfZ. Physiol. 32 : 491-494, 1972.
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APPLIED
Vol. 39, No. 1, July
PHYSIOLOGY
1985.
Printed in U.S.A.
Force platforms
as ergometers
GIOVANNI A. CAVAGNA Istituto di Fisiologia Umana, Uniuersitci di Milan0 and Centro di Studio per la Fisiologia de1 Lavoro Muscolare del CNR,
CAVAGNA, GIOVANNI A. Force platforms as ergometers. J. Appl. 1975.-Walking and running on the Physiol. 39( 1) : 174-l 79. level involves external mechanical work, even when speed averaged over a complete stride remains constant. This work must be performed by the muscles to accelerate and/or raise the center of mass of the body during parts of the stride, replacing energy which is lost as the body slows and/or falls during other parts of the stride. External work can be measured with fair approximation by means of a force plate, which records the horizontal and vertical components of the resultant force applied by the body to the ground over a complete stride. The horizontal force and the vertical force minus the body weight are integrated electronically to determine the instantaneous velocity in each plane. These velocities are squared and multiplied by one-half the mass to yield the instantaneous kinetic energy. The change in potential energy is calculated by integrating vertical velocity as a function of time to yield vertical displacement and multiplying this by body weight. The total mechanical energy as a function of time is obtained by adding the instantaneous kinetic and potential energies. The positive external mechanical work is obtained by adding the increments in total mechanical energy over an integral number of strides.
biomechanics; external and
locomotion; internal work
walking
.> running;
mechanical
Milano,
Italy
lowered, requiring external positive work during the increase in potential energy and the speed of progression oscillates from below to above the average value requiring external positive work during the increase in kinetic energy of the centre of mass of the body. The muscles are active and exerting a force during the phases of the step of decreasing mechanical energy to retard and control the movement; in these phases they are being stretched and the muscular force is doing negative work. This braking action requires the expenditure of chemical energy by the muscles and is equivalent to applying the brakes on an automobile. Thus the chemical energy expended in applying the brakes and the mechanical energy which is not stored in and recovered from elastic elements appears as heat in the muscle and is lost. This mechanical energy is then replaced by contraction of the muscles as they do positive work to raise and reaccelerate the center of mass of the body. If complete storage and recovery of energy in elastic elements (contracted muscles and tendons) were possible, then the decrease in potential and/or kinetic energy during one phase of the step could be utilized in another phase of the step, and no additional energy from the muscles would be needed except that necessary to maintain tension (a slack muscle is unable to store any appreciable amount of elastic energy). Although there is some recovery of energy in elastic elements (1, 2, 5, 13), there is presently no way to measure it directly.
work;
FORCEPLATFORMASANERGOMETERTOMEASUREMECHANICALWORR WORK IN WALKING AND RUNNING. Muscles transform chemical energy into mechanical work during exercise. The rate of energy utilization is routinely determined by measuring oxygen consumption and lactic acid production. However, the rate at which muscles perform mechanical work is seldom measured, and then for specialized types of exercise, such as pedaling a bicycle ergometer. The mechanical work performed during such common exercises as walking and running on the level has only been measured a few times (3-5, 9, 10, 11) and then with an enormous effort. This paper describes a convenient means of measuring the external mechanical work performed in walking, running, and jumping from the forces applied by the body to the ground. It is frequently argued that walking and running at a constant speed on the level involves only a very small amount of external mechanical work to overcome air resistance. This argument is based on the fact that the mechanical energy possessed by the body (both potent .ial a nd kinetic) is the same at the beginning and the end of each step. On the other hand, it is commonly concluded that external mechanical work is done when a change in the total mechanical energy of the body is observed after one or more steps, for example, when a hill is climbed (increasing potential energy) or a sprinter accelerates (increasing kinetic energy). These situations differ from level walking and running at a constant speed only in the duration of time between performance of positive and negative work. For instance within a step cycle of “level” running at a “constant speed,” the center of gravity of the body is raised and
As long ago as 1885 Marey and Demeny (12) used a force platform to measure the vertical component of the force exerted by the feet against the ground during standing vertical jumps. In the 1930’s Fenn (11) utilized a similar type of platform to measure forward and backward components of force applied by the body to the ground during running. From these force measurements he calculated the mechanical work necessary to account for the velocity changes of the center of mass of a running man during each stride. In 1939 Elftman (8) used a force platform to measure the force exerted by one foot against the ground during walking. Although he did not measure the actual velocity and the displacements of the center of gravity, he showed that this was possible cc. . . by proper evaluation of integration constants.” Cavagna and colleagues (3, 5) utilized force platforms to measure the external mechanical work done in level running and walking at different speeds. The procedure was laborious since these authors were limited by I) the small dimensions of the platforms (35 x 35 cm; in walking only the phase of the step in which a single foot was on the ground could be studied, and in running it was difficult to step on the platform without altering the length of the stride); 2) the elaboration of the force-time tracing from the platform, made entirely by graphical computation (this was necessarily discontinuous, and the shape irregularities of the tracing and the oscillations due to the natural frequency of the platform at the highest speeds decreased the accuracy); 3) the method was SO slow that only a small number of determinations of mechanical work could be obtained.
MECHANICAL
174
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FORCE
PLATFORMS
175
AS ERGOMETERS
These difficulties have been overcome I) by using a large platform (4 x 0.5 m) sensitive to the forward and the vertical components of the force exerted by the foot and 2) by electronically integrating the force measurements to yield a direct readout of velocity.
WALK VF
(m/set)
FF (kg)
0.5r const.
CALCULATIONS FORCE
The tential
INVOLVED
IN UTILIZING
T O DETERMINE
EXTERNAL
MEASUREMENT WORK
Vv
= Fd, + Ffsf + F,sl = w, + Wf + WI
/F/./s/COSQ
Fv = weight
(P)
+ frictional
(or lateral)
forces
+
m ‘(I~
forces
+ m .af
= m,af
(2)
(2’)
(3) (3')
directly during exercise by zeroing the platform while the subject stands on it quietly immediately before the experiment. Thus it is possible to calculate the acceleration of the center of gravity from direct measurements of the force exerted in the vertical and forward directions. It is much easier to work from instantaneous velocity than from instantaneous force in calculating the mechanical work. This procedure abolishes the interference caused by vibrations of the plate and provides a much cleaner starting point. In addition the velocity tracings are much simpler and their interpretation much easier. Since the velocity is the integral of acceleration it can be obtained directly by electronically integrating the signal measured with the force plate (Fig. 1) P)dt
jFfdt
=
t
‘d
\
0 100 200
F, - P can be measured
-
-2
i
0
1
2 set
1. Tracings recorded when a subject was walking at 5.5 km/h (left) and running at 11.9 km/h over the force platform. Ff and F, are, respectively, the horizontal and vertical components of the resultant force impressed by the feet on the platform; whereas F, oscillates around the zero (positive values = acceleration forward of subject, i.e., backward push on platform), F,, oscillates around a force value (59 kg) equal to the subject’s body weight, P (corresponding to 1 g = 9.8 m/s2 on the right-hand scale). Simultaneously the horizontal force and the vertical force minus the body weight (F, - P) are integrated electronically to determine the velocity tracings Vf and V,; these give the forward and vertical components of the velocity of the center of gravity of the whole body plus an integration constant to be determined later according to the procedure indicated in Fig. 2. Electronic integrators are operated by subject crossing photocells at the platform level (2.7 m apart when walking and 3.5 m apart when running, Fig. 5); photocells are placed in such a way that the subject crosses them when he does not contact the ground before or after the platform. FIG.
The integration constants must be known in order to calculate absolute velocity. The value of the integration constants in Eq. 4 and 4’ is zero when the subject begins to move on the platform from the resting condition (V = 0). In this case the determination of the instantaneous velocity is easy and both absolute velocity and external mechanical work can be recorded as a function of time simultaneously with the performance of the exercise (1, 2, 6, 13). However, when the subject arrives on the platform with a velocity above zero, the values of the constants in Eq. 4 and 4’ are unknown and the absolute velocity cannot be measured simultaneously with the performance of exercise. The absolute forward velocity is obtained by I) measuring the average forward velocity of the subject moving across the platform by means of two photocells: these also turn on and off the integrators while the subject is over the platform without contacting the ground before or after it (Fig. 24); 2) measuring the area under the recorded velocity tracing during the time interval while the integrators are on (either by planimetry or by utilizing a computer); and 3) dividing the area by the time required to travel the distance between the photocells to position the average velocity forward in the tracing (Fig. 2B). Instantaneous forward velocity can then be calculated. In this procedure it is assumed that the average speed of the trunk between photocells is equal to the average speed of the center of gravity: this is reasonable since the displacements of the center of gravity within the body and the “tilting” of the trunk (5, 11) are small in comparison with the distance between the sights. A similar procedure is followed to determine the absolute value of the vertical component of the velocity (Fig. 2, C and D); in this case however the area A below the vertical velocity tracing is measured only for an interval of time, nr, corresponding to an integral number of steps. The ratio A/w gives an average vertical velocity equal to zero on the assumption that during an integral number of cycles, the upward displacement of the center of gravity the
F,, - P = m,a,
J(Fo
E
const
F,(kg)
where m is the mass of the body, a is the acceleration of the center of gravity of the body and the frictional forces are u) air resistance and 6) the force which opposes the displacement of the center of gravity within the body during a nonelastic deformation of the body itself. The frictional forces oppose muscular force during the acceleration of the body whereas they cooperate with it in decelerating the body. For the purpose of the present calculations the forces of friction even in man sprinting at high speed (2) can be neglected as discussed later. Therefore Eq. 2 and 2’ can be simplified to
Ff
1 const
(1)
direction
Ff = frictional
r
-1
where s is the displacement of the center of mass of the body, F is the resultant of all external forces exerted against the body by the ground and the air, and Q is the angle between the two vectors F and s; F. , s,,, Ff , etc. are projections of F and s in the vertical, forward, and lateral directions; W, , Wf , and WI are the work done, respectively, by the vertical, forward, and lateral components of the force. During level walking and running sf is obviously much greater than s0 and s1 , and F, is much greater than Ff and FI due to the acceleration of gravity. As a result of the greater displacement and force, respectively, Wf and W, are much greater than WI (= 100 times in walking (4)). Thus WI can be neglected. The work involved in rotating the body about its center of mass (7) was also neglected. To determine W,,t one can then start by measuring forces in the vertical (Fo) and forward (Ff) directions over a complete stride. The force exerted by the feet on the platform in the vertical direction is
and in the forward
(m/set)
po-
RUN
-05 50 0 -50 1 2
-50 t
external work required to change the kinetic and/or energy of the center of mass of the body is defined as
W ezt = F ,s =
const
t
-051 50 0
OF
0.5
mV, + const
= mV$ + const
(4) (4')
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176
Al
G. A. CAVAGNA
Forward
velocity
tracing
Cl
:
Vertical
velocity
f
tracing
time
:
integrators
on
ON
0.5
0
B)
Integrate are on are on
>
4i
1
set
0
0)
during time integrators and divide by thetime they in order to locate VF on tracing:
I-12 0
0.5
set
From
the
absolute
kinetic
the
potential
and
the
are
easily
values
energies
0.5
energy:
total
of
VF,V,
E,,
EP
S,
= weight
Sv
and
EKF :$mV,f,
set
FIG. 2. Procedure mine the absolute in the forward (A tical (C and D)
1
Integrate during time of one or more complete strides (oblique hatching ) and divide by this time to Locate v,,which is zero
1
El the
OFF
0.5
set
velocity
constant
tracings,
recorded
during the exercise, (Fig. 1). From the absolute value of velocity given in B and D the corresponding kinetic energy can be calculated ; vertical velocity tracing given in D is integrated further to yield the vertical oscillations of the center of
1
Vertical displacement downwards ( vertical hatching ) and upwards (horizontal hatching IS obtained by integration of Vv tracing:
+
followed to detervalue of the velocity and B) and the verdirections from the
gravity )
energy
sV (E) changes
and
then
(Fig.
3).
the
potential
:+rnV:
energy:
E T O T = EKF+ E,,+ obtained
Ep
( Fig. 3 >
0
0.5
is equal to the downward displacement (i.e., that in level walking and running the height of the center of gravity on the average is constant). This assumption may not be true at the end of a single step, but it is certainly valid over a number of steps. Once the zero vertical velocity is determined then the instantaneous vertical velocity is known. From the instantaneous velocity in the forward and vertical directions the instantaneous kinetic energy Ekf = >s(rnVj2) and +AEk is the positive work Ekv = >s(m’V,2) can be calculated; required to accelerate the center of gravity of the body. The instantaneous potential energy Ep = Pas, can be obtained by integrating the vertical velocity to determine the vertical displacement of the center of mass (Fig. 2E)
= s, + const
SKdt
(4
Geometrically the upward displacement is represented by the upper area enclosed by the instantaneous velocity tracing, V, , and the zero (Fig. 20). The positive work against gravity is given by the increments of the potential energy: +AE, . The total mechanical energy is then obtained by adding the instantaneous potential energy Ep and the instantaneous kinetic energies EkV and EIcj
E tot =
E,
+
Ekv
+
Ekf
(6)
It is clear that potential energy can be converted into kinetic energy and vice versa if the increase in one is out of phase with the increase in the other, and indeed this happens during walking (4). On the other hand, during running, the increase and decrease in potential and kinetic energy are almost completely in phase, and there is little interconversion between kinetic and potential energy (5). Finally the positive external mechanical work is obtained by adding the increments in mechanical energy, +AEt,t , over an integral number of strides. In Fig. 3 the upward curve indicates how the kinetic energy Ekf of the center of mass of the body of a 78-kg subject oscillates during two steps of running at 15.5 km/h. This kinetic energy
set
1
curve was calculated from the forward velocity curve given in Fig. 2B. The E, curve below indicates the oscillation of the potential energy of the body and was calculated from the vertical displacement curve given in Fig. 2E. The curve just above it is the sum of the potential and the kinetic energy Ekv (calculated from curve in Fig. 20). In the interval in which the sum Ep + Ekv is constant the subject is off the ground and the kinetic energy EkV is transformed into potential energy and vice versa. At the top and bottom points the two curves coincide since at these points the vertical velocity and then EkV is nil. Whereas Ekf is given in absolute units (the average kinetic energy of the subject being about 171 Cal), the potential energy Ep is not: in each trial the starting point for the computation of Ep was taken as zero. The two curves below indicate the total mechanical energy Etot = Ekf + Ep -/- EkV and, for comparison, the sum Ep -/- Ekf as usually measured in the past (Fig. 5 in (5)). Etot may differ from Ep + Ekf not only for the slope of the curve, giving the rate of work production, but also for the amplitude of the oscillation of the curve which represents the external mechanical work done; the difference, negligible in running, becomes appreciable in walking. The positive external mechanical work done is taken as the increments of the curve Etot during the interval of time, of one or more complete strides, in which the V, tracing was integrated to locate the average vertical velocity (Fig. 20). The same is done for the curve Ekf to determine the work necessary to sustain the kinetic energy changes Wf , and for the curve Ep + EkV to determine the work against gravity, WV . All the calculations reported in Fig. 2 as well as the curves in Fig. 3 were made directly by a computer. As mentioned above if the subject starts moving on the platform the constant of integration in Eq. 4 and 4’ is zero and it is possible to integrate the vertical velocity directly to give displacement (and potential energy). Simultaneously other operational amplifiers can be used to square velocity to obtain a direct readout of kinetic energy and to sum instantaneous potential and kinetic energy to give total instantaneous energy. Thus mechanical work can be recorded simultaneously with the exercise (1, 2, 6, 13). The limit of this method is that any drift of the first integrator (velocity)
is squared by the second (displacement).
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FORCE PLATFORMS
AS ERGOMETERS
177 apparent kinetic energy change ( - AEh), calculated above, will be smaller than the negative work done, -AEk
,E
d(rn v2> > >a(rn d2) + +$ (m vfr2> which is inequality During negative
7. work
-JFdt i.e., assuming that velocity v to zero
= -J the
deceleration
-mu
0.
1.00
0.50
SEC
FIG. 3. Mechanical energy of the centre of mass of a 78 kg subject running at 15.5 kn/h. E kf is the kinetic energy calculated from the forward velocity curve given in Fig. 2B. EP is the potential energy calculated from the sU curve in Fig. 2E. Ekv is the kinetic energy calculated from the curve in Fig. 211. Etot = Ekj j-- EkV -j-- Ep. All the calculations reported in Fig. 2 as well as the curves above were made directly by a computer.
ASSUMPTIONS
INVOLVED
T O DETERMINE
WORK
FROM
>
+
AE’k + &,,,,,
= positive
w to reduce
+ mvf,
the
W)
-m v’, the momentum actually lost, lost because of a) the braking action platform, -m v, and 6) the force of the work as a kinetic energy change,
- %(m
vr2)
+
$5 h
vfr2)
(14)
that the initial kinetic energy, which is a The signs - indicate positive quantity, is subtracted from the final kinetic energy (equal to zero) in order to get the kinetic energy change. Fortunately the error done by assuming AE, = AE’k + Wlosses is not great. This was checked by measuring the work actually to accelerate forward and by done (AE’k + VVl osses) by a sprinter comparing it with the same work measured as AE, (2). The positive and the negative work actually done by the muscles at each step was determined by multiplying the average horizontal force, IQ , exerted on the platform during the push or during the brake, by the actual displacement forward of the trunk Sf during these intervals determined by means of photocells. Since the displacements of the center of gravity within the trunk (11) were small in comparison with the distance between sights (3 m), the displacement of the trunk Sf could not differ appreciably from the displacement of the center of gravity. The mechanical work done at each step
IN CALCULATIONS
EXTERNAL
Ws tep
= Ff Sf
P)
FORCE
is positive during acceleration and negative during deceleration; the positive and the negative work were calculated from Eq. 15 for 19-20 steps taken by two subjects after the start. The total positive work is given by
The kinetic energy increase (+AE,& calculated disregarding the forces of friction, is greater than the total positive work actually done by the subject to accelerate himself (+AE’,& to overcome air resistance and to deform the body ( VVlosses), i.e. +AEk
v2)
a(rn
m a dt + J
work
+wot
actually done by muscles
(7)
During deceleration, when the forces of friction and muscular have the same direction (-F = - m a + forces of friction),
force the
and
the total
negative
=
z
+
Wstep
wtot =
x
-
Wstep
(16)
work -
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178
G. A. CAVAGNA
The algebraic
sum
agreement with those determined with motion picture analysis. + Wtot - Wtot = Wtot
(17)
was, according to inequalities 7 and 8, smaller than the work measured as AEk . The difference however was only 8% (2). If some skidding took place part of this difference would represent the work done against friction between the foot and the ground. Taking into account that in sprint running one attains the highest speed, i.e., the greatest air resistance, and that the forces deforming the body are the greatest met in locomotion, it is therefore possible to conclude that the error done with the procedure described above is tolerable. The vertical displacements of the body usually take place at a speed appreciably smaller than that attained in the forward direction; in addition the cross-sectional area of the body perpendicular to the direction of the movement is much smaller in the vertical than in the forward direction. Both these factors make air resistance to a vertical movement much smaller than that to a forward movement and therefore probably negligible. In addition the vertical component of the push acting along the vertebral column is probably less effective in deforming the body than an equal force acting in forward direction. On the other hand the work lost in the deformation of the body due to the forward component of the push was found to be smaller than that due to air resistance in sprint running (2). These considerations induce one to think that V, N V’, and therefore Ep N E’, In addition experiments in progress indicate that data of work done against gravity in running, obtained with the present method, are in good VERTICAL
FORWARD
4 I 9
7 1
SIDE ’ VIEW i
500 m m
at the same speed by Fenn (11)
: & I
I
APPARATUS
INVOLVED
MECHANICAL
WORK
IN MEASURING DURING
WALKING
EXTERNAL AND
RUNNING
The force platform used in these studies consists of a single row of eight (0.5 rn)” plates. Each plate has separate vertical and forward units and utilizes twelve spring assemblies (6 in the vertical and 6 in the forward) consisting of four springs and two strain gauges (Fig. 4). Half of the strain gauges in the assemblies are compressed and half are stretched when a force is applied in one direction. Each half is wired in series with those of the other platforms forming two opposite arms of a Wheatstone bridge as indicated in Fig. 4. The platform is inserted, with its surface at the level of the floor, 30 m from the beginning of a corridor 50 m long. Its natural frequency is 42 Hz in the forward and 30 Hz in the vertical direction. The maximal difference found between the electrical response of the eight platforms, when acting with a given force on the central portion of each one of them, is: 4y0 (with full scale 5 kg) or 4.7 7’ (full scale 80 kg) in the vertical direction; and 8-9 ‘$& (full scale 10 kg) or 7-lO$$, (full scale 50 kg) in the horizontal direction. The maximal difference found between the 32 corners of the 8 platforms was 11.3-14%. These differences can be reduced by shunting (with appropriate resistors) the strain gauges corresponding to the most sensitive points. The average response of the horizontal units to a force directed forward (deceleration) was usually found slightly greater than that to the same force directed backward (acceleration); the difference however was less than 5%. The platform was tested from 1 to 250 kg in the vertical and from 0.5 to 100 kg in the forward direction and found to give a linear response within an average error of 5 and 7 %, respectively. The output from each Wheatstone bridge (forward and vertical, Fig. 4) after amplification by a Philips PR 7510 preamplifier
TOP VIEW 4
“ERTlCAL I ‘-
FORWARD
1FORWARD
FORCE
69 ,$
P
15OVOC
*
b
150 voc
4
FIG. 4. Schema of vertical and forward units forming each of the eight plates constituting the whole platform. Forward unit is placed on top of the vertical one and fixed to it by the indicated bolts. Strain gauges indicated by D undergo a deformation opposite to that of strain gauges Z when a force is applied to the unit. All the strain gauges of the eight units sensitive to the force in a given direction (vertical or forward) are wired together in a Wheatstone bridge of 96 resistors as indicated at the bottom.
FIG. 5. Diagram of the setup used to record the tracings given in Fig. 1: when the subject crosses the first photocell the integrators are operated and the velocity tracings begin (Vv and Vf in Fig. 1). Velocity tracings together with a ZOO-Hz signal are also recorded on tape for later treatment by a computer according to the procedure indicated in Fig. 2. For further details see text.
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FORCE
PLATFORMS
179
AS ERGOMETERS
goes a) directly to a Hewlett-Packard amplifier (350-1000B) and recorder (77 19) to yield force, 6) through a DC offset, to an integrator (which is turned on and off by the photocells) and then to the amplifier and recorder to yield velocity. The output of this amplifier is also recorded on tape (Hewlett-Packard 3960A tape recorder) for later treatment of the signal by a computer. The photocells also turn on and off a ZOO-Hz signal which is also recorded on the tape (Fig. 5). This provides the time base needed for calculating the integration constants as described earlier. In addition it is used to trigger the analog-to-digital converter. Before =P+ m a,> it integrating the vertical component of the force V earlier. is necessary to subtract the body weight as mentioned This is done by setting the platform’s output exactly to zero using the DC offset voltage while the subject stands absolutely still on the platform before the beginning of the exercise. This operation, and also the setting of the forward output exactly to zero with another DC offset voltage, must be done with great care immediately before each trial otherwise an appreciable drift of the output of the integrators takes place. In our setup the base line from the platform to the integrators has noise, mainly due to vibrations of the time the ground, attaining about & 100 g, and its drift during of a trial is usually less than 50 g.
(F
EXTERNAL,
INTERNAL,
The methods work necessary
AND
TOTAL
WORK
described above allow to sustain the displacements
us to determine of the center
only the of gravity,
i.e., the external work. In exercise, however, also internal work is done, and this cannot be measured by means of the force platform. The main source of internal work (and the only one directly measurable) is the kinetic energy changes of the limbs calculated from their velocity relative to the center of gravity. This can be measured by motion picture analysis as described for example by Fenn (10). The increments of the curve indicating the total kinetic energy of the limbs represent the internal work done. This curve should not be added to the curve of the mechanical energy level, E tot, given in Fig. 3 (as sometimes reported in the literature), because by this summation one implicitly assumes the transfer of some of the work done by the internal forces to an increase in the energy level of the center of mass, which is not possible by definition The absolute value of the increments of the curve Etot (giving the external positive work) must therefore be added to the absolute value of the increments of the curve of the kinetic energy of the limbs (giving the internal positive work) to obtain the total positive work done. The author thanks Prof. C. R. Taylor of Harvard University for his much helpful and constructive advice during the completion of this manuscript. The program to obtain the records of Fig. 3 was prepared by Dr. Carlo Cavagna of the Politecnico of M’ilan. The platform was projected by the Viterra firm, Wallisellen (Zurich, Switzerland). Received
for publication
25 October
1974.
REFERENCES 1. CAVAGNA, G. A., L. KOMAREK, G. CITTERIO, AND R. MARGARIA. Power output of the previously stretched muscle. Med. Sport 6: 159-167, 1971. 2. CAVAGNA, G. A., L. KOMAREK, AND S. MAZZOLENI. The mechanics of sprint running. J. Physiol., London 2 17 : 709-72 1, 197 1. 3. CAVAGNA, G. A., AND R. MARGARIA. Mechanics of walking. J. Ap~l. Physiol. 21 : 271-278, 1966. 4. CAVAGNA, G. A., F. P. SAIBENE, AND R. MARGARIA. External work in walking. J. A/@. Physiol. 18 : 1-9, 1963. 5. CAVAGNA, G. A., F. P. SAIBENE, AND R. MARGARIA. Mechanical work in running. J. AppZ. Physiol. 19 : 249-256, 1964. 6. CAVAGNA, G. A., A. ZAMBONI, T. FARAGGIANA, AND R. MARCARIA. Jumping on the moon: power output at different gravity values. Aerospace Med. 43 : 408-4 14, 1972.
7. ELFTMAN, H. The rotation of the body in walking. Arbeitsphysiologie 10: 477-484, 1939. 8. ELFTMAN, H. The force exerted by the ground in walking. Arbeitsphysiologie 10 : 485-49 1, 1939. 9. ELFTMAN, H. The work done by muscles in running. Am. J. Physiol. 129 : 672-684, 1940. 10. FENN, W. 0. Frictional and kinetic factors in the work of sprint running. Am. J. Physiol. 92 : 583-611, 1930. 11. FENN, W. 0. Work against gravity and work due to velocity changes in running. Am. J. Physiol. 93 : 433-462, 1930. 12. MAREY, J., AND G. DEMENY. Locomotion humaine, mecanisme du saut. Compt. Rend. Acad. Sci. 101 : 489-494, 1885. 13. THYS, H., T. FARAGGIANA, AND R. MARGARIA. Utilization of muscle elasticity in exercise. J. ApfZ. Physiol. 32 : 491-494, 1972.
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