Analysis of Some Dynamic Tests II
Introduction
This study is based on test data provided by KuLTu. The data were obtained from dynamic and isometric tests with HUR Leg Extension/Curl Research Line machine and software. The test group
“Kalpa” encompassed 27 male subjects. Of the dynamic (extension) tests we had to drop one test for left leg and two tests for right leg because of spurious data. The dynamic tests consisted of 13 kicks at increasing load from 2 to 8 bar. The Force-Velocity relationship is studied with reference to the Hill modell.
Analysis using A- and B-parameters
For dynamic tests the torque and angular data are collected from which the software calculates maximum torque (MTQ), maximun angular velocity (VEL), and power at maximum torque for each kick (POW). Naturally we expect the velocity to decrease with increasing load. In the following graph we have plotted the MTQ-VEL data for 25 tests (right leg).
300
71 MTqi
644.8
83.3 AVel
i
0 100 200 300 400 500 600 700
50 100 150 200 250 300
Fig. 1. Maximum torque (Nm) vs maximum angular velocity (deg/s) for 25 tests (right leg, extension).
The graph suggests we may try to fit the individual MTQ-VEL data to linear relationship
(1) MTQ=A $ VEL+B
The A-B parameters can be calculated for each test and then plotted in an A-B graph:
0.262182
0.85 LAj rLAj
390.2
218.5 LB
j
200 250 300 350 400
1 0.8 0.6 0.4
Fig. 2. The A-B parameters for the 25 tests of fig. 1 are plotted in an A-B diagram. Also the best linear fit curve is
indicated.
As can been seen the A-B parameters correlate fairly well with each other. In fact, their Pearson correlation coefficient is - 0.80. The linear best fit in fig. 2 is given by
(2) A= −0.00238 $ B+0.258 2-8 bar data
The relation (1) further suggests that the ratio - B/A could be related to maximum velocity at small load. Indeed, if take the maximum velocity for each test and plot it with - B/A on the same graph we obtain the following picture.
846.410256
365.4
mv maxvj
LBj LAj
24
0 j
0 5 10 15 20 25
300 400 500 600 700 800 900
Fig. 3. Maximum velocity for each test (maxv) plotted with the quotients - B/A for the tests.
From the figure one can see that there is an obvious covariance (0.87). The best linear fit predicts the maximum velocity from - B/A to within 10 - 15 %.
644.8
365.4 maxvj rvelj
582.580219 367.855468 rvel
j
350 400 450 500 550 600
300 400 500 600 700
Fig. 4. Maximum velocity of each test (point) and the prediction based on - B/A (line).
The linear prediction (regression) is given in this case by
(3) MAXVEL=0.554 $
−B A
+113.4
(velocity in deg/s). Similarily we may expect the maximum power (MPOW) of each test to be correlated with
(4) P= − B2 4 $ A $ o
180
(in Watt). This is demonstrated by the following picture (Pearson’s correlation = 0.932).
1.214975 10. 3
635.909713 Pj mpwj
24
0 j
0 5 10 15 20 25
600 800 1000 1200 1400
Fig. 5. Plotting of maximum power (mpw) and P calculated from equ (4).
Both from experimental and theoretical points of view there are no reasons to expect a linear Force-Velocity curve for the whole range of loads. The situation is schematically represented in fig.
6.
M Nm
deg/s B1
B2
0
Fig. 6. If the schematic Force-Velocity curve is like this (concave), then the B-parameter should always
be less then the isometric maximum (M ).
We can use our experimental data to investigate whether we have a situation like the one depicted in fig. 6. For this we make two selections of the data: (1) kicks for loads 5 - 8 bar and (2) kicks for loads 2 - 5 bar, and then calculate the corresponding A - B parameters.
432.634318
190.662178 MRj Bj
24
0 j
0 5 10 15 20 25
100 200 300 400 500
Fig. 7. Isometric maximum/right (MR) and the B-parameter for the test selection 1 (5 - 8 bar).
As we can see from fig. 7 the B-parameter tends to be lower than the isometric maximum
(extension) for the 25 subjects with a few exceptions. For the averages we have: mean(B) = 303.9 Nm, and mean(MR) = 331.4 Nm. Also to be noted is that the power estimates calculated according to equ (4) are bad when based on the 5 - 8 bar data (corr(P, mpw) = 0.65).
0.114129
1.042522 Aj rAj
432.634318
190.662178 B
j
150 200 250 300 350 400 450 1.2
1 0.8 0.6 0.4 0.2 0
Fig. 8. The A - B parameters for selection 1 (5 - 8 bar).
For the selection 1 (5 - 8 bar) the linear A - B regression curve is given by
(5)
A= −0.00324 $ B+0.504 corr(A, B) = −0.881 selection 1 (5 - 8 bar)
The result for the selection 2 (2 - 5 bar) is:
(6)
A= −0.00216 $ B+0.197 corr(A, B) = −0.833 selection 2 (2 - 5 bar)
For this selection we have B2 = mean(B) = 280.1 Nm. In neither case was there any significant correlation between the B-parameter and the mass of the subjects (corr(B1, M) = 0.40; corr(B2, M)
= -0.04), or between isometric maximum torque (extension) and the mass of the subjects (corr. = 0.23). The later may seem paradoxical, but is likely due to the small variation of mass and length in the group:
2.3 6.5
5.5 StDeviation
21.3 83
179.4 Mean
18 69
167 Min
26 100
189 Max
Age (year) Mass (kg)
Length (cm) n = 25
Above we have presented data for the right leg. For the left leg the data seem to be more spread (lower values for the the pertinent correlation coefficients), which may e.g. be due to fatigue (the right leg was tested first). This could be checked by testing with the left leg first. Anyway, comparing A- and B-parameters from different test is only meaningful if based on data from exercises in the same load range.
Hill modell
For the simple Hill modell [1] one has a relation of the form
(7) F= F0b−av b+v
between the maximal force F of a muscle and its velocity of contraction, v. Here a and b are parameters for the muscle, and F0 is maximal force at v = 0 (isometric).
1.25 10. 3
1 F v( )
100
0 v
0 20 40 60 80 100
0 500 1000
Hill modell
Velocity (cm/s)
Force (N)
Fig. 9. Typical Force-Velocity curve for the Hill modell.
The maximum contraction velocity (v0) for F = 0 is according to (7) given by
(8) v0= b a F0
Experiments with skeletal muscles show that
(9)
a F0 = b
v0 l 0.25
With this assumption one can calculate the maximum power P0, using P = Fv and (9), to be at
(10)
P0 =0.095 $ F0$ v0 where vmax P =0.31 $ v0
Fmax P =0.31 $ F0
and vmaxP and FmaxP indicate the velocity and force at maximum power. For our test data we can compare maximum power for each subject with the product of maximum torque (isometric test) and maximum veclocity in the test (which here really is not for zero load but for 2 bar load plus the weight of the leg).
1.145 10. 3
655 mpwj rmpj
2.604066 10. 3 1.447791 10. 3 mp
j
1400 1600 1800 2000 2200 2400 2600 2800 600
700 800 900 1000 1100 1200
Fig. 10. Measured maximum power (mpw) and the best linear fit (rmp) based on the product (mp) of
maximum torque (isometric) and maximum velocity (2 bar load).
In fig. 10 the product mp was calculated as
(11) mp=MTQ $ VEL $ o 180
(MTQ, maximum isometric torque; VEL, maximum velocity (deg/s) for dynamic extension). The correlation coefficient with actual measured maximal power mpw was 0.91, the linear best fit given by
(12) mpw=0.385 $ mp+133 (Watt)
This result is not directly comparable with the Hill modell because we could not use the maximum velocity for zero load (however the slope-coefficient in (12) is consistent with the prediction (10) that maximum power is attained at around one third of maximum isometric torque). This could e.g.
be indicated by the nonzero intercept in (12).
The Hill equation can also be written on the forms
(13)
v =v0 1− FF0 1+c $ FF
0
F=F0
1− vv0
1+c $ vv0
where c is the shape parameter ( related to a by c = F0/a ). In terms of the dimensionless variables the Hill curve is thus determined by a single parameter c. If we assume (13) to be valid then the shape parameter c is related to the force Fpmax at maximum power through
(14)
c= 1−2 $ Fp maxF
0
Fp maxF0
2
Fp max
F0 = 1
c 1+c −1
which can be calculated from the measurement data.
69.05 327.74
0.01 2.23
StDeviation
101.85 575.69
0.92 0.18
Min
214.38 1937.76
0.99 9.65
Max
73.68 985.85
0.97 3.04
Mean
144.99 872.73
0.98 2.12
25
110.66 712.22
0.98 0.18
24
99.61 1291.86
0.98 5.25
23
47.66 970.12
0.97 1.63
22
22.27 1044.91
0.97 3.68
21
101.85 1125.43
0.98 0.98
20
214.38 943.39
0.92 5.90
19
108.81 992.92
0.96 6.11
18
20.83 909.31
0.98 1.50
17
88.48 1053.56
0.98 2.79
16
47.12 592.37
0.97 1.02
15
147.12 980.06
0.98 1.88
14
73.39 646.51
0.99 1.20
13
112.54 693.67
0.99 1.68
12
58.69 575.69
0.99 0.70
11
159.71 631.67
0.98 4.56
10
89.70 724.44
0.98 1.71
9
130.85 639.91
0.98 1.49
8
16.42 1384.79
0.98 3.64
7
134.98 1011.01
0.97 4.33
6
0.93 791.85
0.98 0.81
5
4.60 1191.30
0.96 3.04
4
8.32 1937.76
0.98 6.11
3
133.67 1346.92
0.93 4.06
2
4.28 1581.94
0.99 9.65
1
intercept slope
correlation c - parameter
Test nr.
The above table summarizes the results (right leg). The first column gives the shape parameter calculated for each test. The next column gives the correlation between actual measured velocities (VEL) and the ones computed from Hill’s equation using the calculated shape parameter (VELh).
The “slope” (α) and “intercept” (β) describe the best linear fit curve
(15) VEL=a $ VELh+b
The high correlations above are not spectacular; we get the same kind of figures even if we compute correlations between data from different tests (different persons). This is not surprising if the the data complies with the Hill modell, because then the data should follow the same simple pattern.
Indeed, if we map all the data as in fig. 1 but use dimensionless variables
(16) f= TQ
TQiso max
V= VEL VEL0
(here TQisomax is the isometric maximum, and VEL0 the maximum velocity in the test with least load;
i.e. 2 bar) the we get the figure:
0.821168
0.173302 MTqi MRf i( )
1.021228
0.172392 AVel
i VEL<f i( )>
0
0 0.2 0.4 0.6 0.8 1
0 0.2 0.4 0.6 0.8 1
Fig. 11. The same plot as in fig. 1 but with dimensionless variables (16) instead ( f along the vertical, V along
the horizontal axis).
Fig. 11 emphasizes the common systematic trend in the data which also explains the high
intercorrelation between different tests. One interesting feature of the Hill modell is the symmetrical way it treats force and velocity as can be seen from equ (13) and from its geometrical shape if plotted using F/F0 and v/v0.
F/F0
V/V0 Max power
vpmax fpmax
Fig. 12. The symmetrical Hill curve. The parameter c describes the deviation from a straight line. The maximum power ( P = F V ) is attained at the midpoint of the curve. The corresponding fractional velocities and forces will according to the Hill curve be equal at this point.
According to the Hill equation the maximum power will be attained at the midpoint of the curve where Fpmax/F0 = vpmax/v0. Now we can compute Fpmax/F0 for each test from the data.
0.06 StDeviation
0.24 Min
0.48 Max
0.35 Mean
Fpmax/F0 n = 25
Interestingly, the mean value 0.35 is quite close to the value 0.31 predicted by (10) based on a shape parameter equal to 4. Thus, the maximum power is attained at about one third of the maximum force and the maximum velocity. From this one could estimate the maximum velocity v0
which may be difficult to measure directly. Also, it can be determined from the differential quotient of velocity and force at the point of maximum power according to
(17) v0=F0 dv dF p max
From the data we compute the average for vpmax to be 410 grad/s which would correpsond to an average value 410/0.35 = 1170 grad/s (20.4 rad/s) for v0. (If we take the moment radius of the quadriceps muscle to be 3.3 cm [1] then 20 rad/s corresponds to a muscle contraction velocity of 20 × 3.3 cm/s = 0.66 m/s.)
An inspection of the data also reveals that the tests are concentrated at the left side of the Hill curve;
i.e. the low velocity/heavy load end.
F/F0
V/V0 Test data concentrated
vpmax fpmax
here
Fig. 13. Test are concentrated at the low velocity/high load end.
We will also study the connection between the Hill modell and the ( A, B ) - parameters discussed in the first part of this paper. If the Hill modell describes the data then the ( A, B ) - parameters should be approximated by the parameters decribing the tangent line to the Hill curve touching it at a point in the middle of the measurement range. This leads to the following expression for the parameters
(18)
A = − F
0v
01 + c
1 + c
vv0
2B = F + F
0v
01 + c
1 + c
vv0
2v
where F and v are in the middle of the measurement range. Equ (18) predicts a linear relationship between A and B of the sort
(19)
A = − 1
v $ B + F v
where the slope and the intercept are thus given by
(20)
a = − 1 v b = F
v
If we compute the average of 1/VEL and TQ/VEL for the 6th kick (of 13 in all) of each test for the whole data we get mean(1/VEL) = 0.00298, mean(TQ/VEL) = 0.425, to be compared with values 0.00238 and 0.258 given by equ (2). (Interestingly, if we compute the same averages for the 3d kicks instead we get the values 0.00238 resp. 0.255 which match the regression coefficients (2) very well !) This cursory comparision seems to show that the Hill modell can explain a number of important features of the data.
Discussion
It seems that the Hill modell could be of help interpreting the data even if the data is not optimal for a comparision in this case. Now we used only the maximum velocity for each kick, but calculating the velocity at maximum power would be more useful (in this case these are not too far off from each other, though). This calculation could be added to the HUR software. In the Hill modell one can also incorporate the muscle length factor which means that the constant maximum isometric force F0 in (13) is to be replaced by one that varies with the length of the muscle (or the joint angle). The length factor can be studied experimentally by measuring the maximum isometric force for a range of joint angles between 90 and 180 degree. This factor must furthermore include the geometry of the muscle and the leg which relate the net muscular force with net muscular torque. These things, and the effect of inertia, will be also studied in a simulation in order to assess their importance for analyzing the data.
In the theory the shape parameter is a measure how fast the muscle is. The larger the shape parameter the more fast fibers in the muscle. This could perhaps be tested by also taking the time for a 60 meter sprinting of the test subjects (in case of fairly well trained subjects).
References:
1. Nigg, B M, Herzog W: Biomechanics of the Musculo-skeletal System, Wiley 1995.
2. Keener J, Sneyd J: Mathematical Physiology, Springer 1998.
Frank Borg (borgbros@netti.fi)
