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EMG to force processing II: Estimation of parameters of the Hill muscle model for the human triceps surae by means of a calfergometer
c-w-929081
1107i9 II WMO Pergamon Prrsr Lid
EMG TO FORCE PROCESSING II: ESTIMATION OF PARAMETERS OF THE HILL MUSCLE MODEL FOR THE HUMAN TRICEPS SURAE BY MEANS OF A CALFERGOMETER* A. L. HOF and Jw. VAN
DEN BERG
Laboratory for Medical Physics. State University Groningen, Bloemsingel 10.9712 KZ Groningen, The Netherlands
Abslract- On a calfergometer isotonic contractions were performed by the calf musclesof human subjects. The measured torque M, was compared lo the torque M processed from rhc EMG and the joint angle b! means of an electronic processor based on the Hill muscle model. By means of these experiments the model parameters were determined for the torque-angle relation, the torque-angular velocity relation and the paralkl elastic component (PEC) Experiments were done (a) for the soleus only (4 subjects) and (bl for the combinarion of soleus and gastrocnemius (8 subjects). It turned out that: --the torque-angle parameters varied among subjects; ---the torque-angular velocity parameters for sokus only varied also among subjects: -for the combined calf musck group the interindividual differences in the torque-angular velocity parameters could be neglected ; -one of the parameters of the PEC was the same for all subjects, while the other was variable among the subjects. The obtained parameter values are discussed and compared with available literature data. Values for the r.ms. and the peak error in the isotonic phases of the contractIon are given and compared with values predicted from the stochastic properties of the EMG signal.
lNlRODUCTiON
In Part I of this series of papers (Hof and Van den Berg, 1981a) a method was described for the processing of the EMG to the muscle torque. It consists of an electrical analogue of the Hill muscle model ‘driven’by the rectified EMG. For a proper functioning it is necessary to know the parameters of the Hill model. The EMG gain factors (parameters) can be determined in quasi-static isometric contractions with a calfergometer, see a previous paper (Hof and Van den Berg, 1977). In the present paper we also used the calfergometer to determine a number of parameters of the Hill model (cf. Table 1 of Part I): -the parameters of the parallel elastic component (PEC), by measuring the passive torque around the ankle joint at complete relaxation of the muscles. -the torque-angle (cq. force-length) relation of the calf muscles, by quasi-static isometric contractions at various angles of the ankle joint. --the torque-velocity (Hill) relation. by isotonic contractions of the calf muscles. Isotonic contractions were used to separate the effects due to the series elastic component (SEC) and the contractile component (CC), as the stretch of the SEC is (virtually) constant during a (virtually) isotonic contraction. The parameters of the SEC and the time constants of the active state of the CC (apart from rIr see Part I) * Receired
in rerised.form
9 April
1981.
might be estimated from fast isometric contractions but the calfergometer is not well suited to these contractions: it is typically a low-inertia instrument for fast movements. As a consequence. its stiffness is too low for quick isometric contractions to lx real& isometric. These parameters will therefore be given only provisional values, in order to enable the model to work properly; their values are not critical for the present experiments. More definitive values will be determined in experiments with a more suitable instrument, a torque plate. 10 be described in Part III ~Hof and Van den Berg. 198 I b 1. To evaluate the functioning of the processor it IS necessary lo compare the calculated muscle torque with the measured torque. .4 problem with this is that an externally measured joint torque is the algebraic sum of all muscle torques from agonists and antagonists. whereas the processor gives in principle the torque from one muscle at a time. With the calf muscles - plantarflexors of the foot this problem ISnot too difficult fortunately. In most cases the antagonIsts can be kept inactlve (to be checked by EMG) and the contribution of the smaller plantarflexors can br neglected due to their relatively small cross-Kctlon and short lever arms. The plantarflecting torque ISthus mainly provided by M.M. soleus and gastrocnemlus. The parameters of the torque-angle and the torque-angular velocity relatwn have been detcrmined for two cases. (a) With the sub,iecl In sItrIng posirlon. \rlth strongly bent knees. onl! 111~solcus musck contra759
760
A. L. HOF and Jw. VAN DEN
butes to the torque (Hof and Van den Berg, 1977). This gives the opportunity tocompare the processoroutput with the torque of this muscle alone (4 subjects). (b) By adding the weighted EMGs of soleus and gastrocnemius and using the sum as the processor input these two musclesare taken together (8 subjects). This approach is similar to the one we used in quasistatic contractions (Hof and Van den Berg, 1977). The parameter values to be obtained are somewhat of an average of those of soleus and gastrocnemius. This approach is less idea1 in principle but it gives more practical results, because now the processor can be used to determine the total calf muscle torque in standing or walking subjects. METHODS
The essential parts of the calfergometer are shown in Fig. 1. The foot of the subject is fixed onto footplate A, which is connected by the rods B to the bar D. The whole pedal is suspended on heavy ball bearings in the main frame E. The axis of rotation C can be made coaxial with the ankle axis (identified by the medial and lateral malleoli) by adjusting the footplate A and the heel rest M. The angle is measured with potentiometer P. The range of the pedal rotation is restricted to 90’-105’. The load consists ofa set of roll springs (Tensator) F which can be attached by hooks to bar D. They give an almost constant torque (for stretching the torque is slightly larger than for shortening). A nominal torque of 56 Nm was used in most cases. The torque was measured by the strain gauges S on bar D, very close to the axis of rotation. For static torques the accuracy was better than + 3”,. The ergometer was constructed to have low inertia, but at the fastest contractions the inertia cannot be neglected. The moment of inertia of the parts above the axis of rotation is 0.13 kg. m2, the corresponding torque is measured by the strain gauges. The moment of inertia of the parts below the axis ofrotation, the footplate (0.04 kg. m’) and the foot itself (0.02-0.04 kg m’). goes along with a torque which is not measured by the strain gauges. This torque amounts to ca 5 Nm (i.e. 10:; of the isotonic torque) at the highest acceler-
BERG
ations encountered during the onset of the isotonic part of a contraction. Apart from introducing a measuring error, inertia implies that the contraction cannot be completely ,isotonic. In a first approximation we shall neglect this, but we come back upon it in the Discussion. Bar D can also be tixed at various positions for recording isometric contractions. The light construction of the pedal resulted in a low moment of inertia but it has the disadvantage of a rather low stiffness, ca 1200 Nm,‘rad. This must be accounted for when interpreting isometric contractions. The shod foot of the subject is fixed rigidly to the footplate by means of the heel rest M and an adjustable hoop H over the instep. The hoop is made of thin stainless steel and fitted with a leather pad L. When the subject is sitting, his knees are bent at ca 60; and are pressed down with a knee pad (cf. Lippold, Naylor and Treadwell, 1952). When standing the subject maintains his equilibrium by holding a fixed bar in front of him. To the moving bar D also a dashpot G can be attached, which damps the action of the springs in the backward movement (negative velocity). This was appreciated by most subjects as this part of the movement is difficult to control. But since in this case the return movement is no longer isotonic, the dashpot is disconnected in the contractions used for studying the behaviour of the muscle at negative velocities. The dashpot does not influence the torque in the forward part oi the movement. Recording The gastrocnemius electrodes were positioned one on the medial and one on the lateral head of the muscle. The soleus electrode pair (surface area 0.5 cmz) was placed 2.5 cm apart on the edge at the medial side. where the muscle protrudes from under the gastrocnemius (Hof and Van den Berg, 1977. see Fig. 2b). The angle of the ankle was recorded with the fixed potentiometer on the ergometer. The EMGs and the angular position were fed into the processor. The unprocessed signals were also recorded on a fourchannel tape recorder (HewlettPackard model 3960), together with the measured torque M,. Subjects The subjects were six men and two women. data of whom are listed in Table I. With the subjects 5-8 she torque-angle and torque-angular velocity parameters were determined only for the combination of soleus and gastrocnemius. For subject 3. who was of slight build, an isotonic load of 40Nm was used, because she could not move the pedal over its whole range with heavier loads. For all other subjects the isotonic torque was 56 Nm. Procedure
Fig. 1. The
calfergometer (schematic). See Methods
For all subjects the soleus gain factor (I, was determined in the sitting position with an isometric contraction at a joint angle of the foot 4, = 90’. The subject increased the torque slowly from zero IO maximum and back. By plotting the mean rectified soleus EMG against the measured torque on a X-Y recorder, g, could be determined as described previously (Hoi and Van den Berg, 1977). With the subjects 1-4 this procedure was repeated with 4 set at anglesfrom 75”-110”,atintervals0f5’.Thisgaveaseries0f points for the torque-angle relation from which I#J;and 4; could be determined. In our model the torque-angle relation /(#,)isnot a function ofthejoint angle 4 but of thecontractile component ‘length’ 4, the difference being the stretch of the series elastic component 4,(M). This differena must also be accounted for in the parameters: thus 4, = I$; + #,(M) and f#~a= 4; + +,(M), see part I, equation 8. Alter this a senes of isotomc contractions was made, with speeds varying from very slow to as fast as possible, to obtain the parameters of the torque-angular velocity relation. They were monitored by displaying the signals on a Mingograffast inkwriter.
EMG
761
IO force processing II
c j
--
__$_
_c-.-_-_ I
IL .--. --
/_. _..: T’ t
-.A
._‘
47-
:
r
763
EMG IO force processing II Table 1. Subject data and division of experiments.
Subject
Sex
Age (yr)
Body mass (kg)
Bad) height (m) 1.73 1.87 1.60 1.77 1.72 1.92 1.78 1.73
I
r
28
64
2 3 4 5 6
m r m m m
28 21 39 26 30
1
m
23
8
m
30
71 47 82 70 91 65 63
The other experiments. with all subjects, were performed in standing position. First the gaslrocnemius gain factor gP was determined at #J = 90’ asdescribed previously (Hofand Van den Berg, 1977). With gI and go set at their respective values the procedure as described above - isometric contractions at various foot angles and isotonic contractions-was repeated in the standing posture. Finally the passive torque due to the parallel elastic component was recorded as a function of 4, for all subjects in the standing position and for subjects l-4 in the sitting position. To this end the springs were disconnected and the ergometer arm was moved slowly by the experimenter while the subject had to keep his muscles relaxed. This was checked with the EMG. Parameter
esfimarion
As the unprocessed signals were recorded on tape, they could be played back through the processor as often and with as many parameter settings as needed. To facilitate a comparison between the measured torque .%I, and the processor torque M they were displayed next to one another on the Mingograf inkwriter together with their difference (M - M,; see Fig. 2). Systematic errors in the processing thus stood OUIclearly. With the help of this signal the setting of the gain factors (g, and g.), the parameters of rhe torque-angle relation (0, and 4, - #1) and the PEC parameter M, could be checked and, if necessary, be adjusted. For the other parameters approximate values could be found in this way. As a more quantitative measure of the error the integrated squared error (ISE) J(M - M,l’ dr was used. It was computed with the analogue electronic circuit intended for computing the work W. For the Isotonic contractions the integrator was reset by the zero-crossings of the angular velociry 4. In this way separate values of the ISE for both isotonic phases in the contracrion, that with positive and that with negative velocity, were obtained. The isotonic contractions were divided in three groups according 10 their peak positive velocity: slow (S) with Cpi 0.5 rad,s. moderate (M) with 0.5 5 @ < 1.5 radisandfast (F)with cp 1 1.5 radis. The number of contractions has been given in the Tables 4 and 5. Optimal values for the parameters of the torque-angular velocity equation could be found by plotting the total ISE as a function of the parameter value and looking for the minimum (Fig. 3k In determining b (and n) the phases with positive velocity of the moderate and fast groups were considered, for c the phases of negative velocity of the same groups were taken together, provided that the dashpot had not been used in the experiments. In the other case an equivalent group of contractions without dashpor was used. Represenrarion of the errors
Data on the in two forms:
processing
errors have been given in the lables
(a) The root mean square error A.t! (rms.! the ISE according IO its definition:
from
was obtained
Soleus + gastrocnemius
Soleus only
X X X X X X X X
X X X X
AM (r.m.s.) = J-F
(1) .r in which T is the total duration of the isotonic phases with positive or negative velocity in a series of conrractions. (b) The peak error has been given as the negative and positive maxima of AM = M - M, during the isotonic phases of a series of contractions. The reading-off accuracy was about f 2 Nm. RESULTS
EMG gainfactors.
series elastic component.
acrice stare
The EMG gain factors gs and g4 ranged from 1.4 to 6.5, and from 0.60 to 1.8, respectively. As they change for each electrode placing they were not listed in the Tables 2 and 3. As could be expected the values of the SEC para-
ISEl
/
‘5
SOO-
:J/I// 0
,w,,___
-_---
-._
j2fy&____‘_______
0
:
KKp/.___4__;___ I
1
,
,
’
1
I
‘I:
1
>
:
0
1
Fig. 3. Theintegrated squared error (1SE)integrated over the whole set of contractions as a function of the parameter b, with n = b/10. Subject 4, soleus only. 0: Medium speed, A: fast. For the fast speed it has been indicated how the ‘bars’ in Figs 6 and 7 are constructed: the symbol in the middle denotes the minimum of the ISE curwz,the vertical dashes the points at which the ISE is lw’/, and iSOo/,respectiwly of the minimum.
764
A. L. HOF and Jw. VAN DEN BERG
Table 2. Parameters of the soleus (sitting posture). For 6. n and c the optimum values for each individual have been given. Torgue-angle relation Subject
($gJ
‘jd;g:’
Torque- velocity relation b (rad/s)
I
159
50
1.5
2 3 4
154 146 I57
40 40 32
1.2 2.0 1.2
PEC
n
C
0.15 0.12 0.20 0.12
8 0.5 0
k, 4.0 1.0 1.0 3.0
Table 3. Parameters of the combined calf muscle group (standing position). Parameter values for b, n and c are averaged for the whole group of subjects. Torque-angle relation
Subject
(&I
‘idig:’
Torque-velocity relation
n
b (Nm/rad)
PEC C a,
2 3 4 5
159 169 146 169 174
40 40 40 40 40
1.2 1.2 1.2 1.2 1.2
0.12 0.12 0.12 0.12 0.12
0 0 0 0 0
6.0 2.0 2.0 3.8 1.5
6 7 8
159 169 164
40 40
1.2 1.2
0.12 0.12
0 0
::: 1.0
1
meters /I and K and those of the active state time constants f2 and ?3 (TVbeing preset at 25 ms, see Part I) had little influence on the error in the isotonic phases of the contraction, on which we concentrated. The most obvious effect was that the error decreased for lower values of /I and K (i.e. lower stiffness of the SEC). In our opinion this effect was related to the low-pass filtering properties of the muscle model, and not to a better fit of the SEC parameters (see Discussion). By inspection of the error during the isometric con-
traction phases (before, between, and after the two isotonic phases), approximate values for the active state and the SEC parameters could be found, bearing in mind that the former shape the descending part ofan isometric contraction while the rising part is influenccd by the SEC parameters (together with the parameters b and n of the torque-angular velocity relation). The values thus estimated were TV= 30 ms, ~~ = 6Oms, /I = lOrad-’ and K = 5OONm/rad (subjects 1, 3 and 6) or ZOONm/rad (other subjects).
MlP
INIlII
td
50-
(4
(Nn
o60
Fig. 4. Parallel elastic component: passive torque as a function of joint angk 4. Some recordings of dimerent amplitudes, subject 6 (thin lines, ach recording with dimerent symbols). Output of proafJor PEC circuit with M, = 3.5 Nm and 4, = 8” (thick line) (a) Linear scale (b) Logarithmic scale for M,, to show exponential course for & < 90”.
(b)
,
700
so0
9o”co
lo force processing II
EMG Parallel
elastic componenf
Figure 4 gives some recordings of the passive torque as a function of ankle rotation for one subject. For 4 -Z 90’/M,(4) increases exponentially with decreasing 4. in accordance with the PEC characteristics of the model (Part I, equation (9)). In order to show this, M,(4) is plotted logarithmically in Fig. 4(b). The curves show a considerable hysteresis, which is dependent on the previous stretch. As a consequence one finds that the value of the exponential parameter 4p should in fact be different for increasing or decreasing 4. For the subjects in this study ranges were found for 4, of 8.3- 12.2 and 6.0-8.4 degrees respectively. We see that the interindividual differences are about equally large as the difference in the same subject due to the hysteresis, which, for reasons of simplicity, has not been modelled in the processor. It is therefore reasonable to adopt a single average value of 8” for dp for all subjects. The proportionality factor M, was more variable between subjects, it has been given in Tables 2 and 3. The PEC curve from the processor has been included in Figs. 4a and 4b. It is seen that, in spite of the hysteresis effects not being modelled, it is accurate within some Nm in the practical range of 75” < d, < 1 lo”. Torque-angle
parameters
parameters of the torque-angle relation were obtained from measurements of the EMG gain factor as a function of the angle of the ankle (Fig. 5). The breakpoints of the measuredf(&) relation at/(d) = 0 andf(&) = 1 gave &‘,and 4; respectively, g, is equal to the level of the horizontal part at small 4. The values of 4, were obtained from &‘,by adding the stretch of the SEC : I$, = t#~‘,+ t#~,( M ), which depended of course on the values adopted for /? and K and on the torque at The
765
which the measurement was done: 4, was 30’ for subjects 1 and 6,26’for subject 3 and 39’ for the others. In subjects l-4 these values of 9, and 9, - 4z for the soleus gave satisfactory results in the isotonic contractions. For the combination of gastrocnemius and soleus the determination of 4, and 4, - 4r from isometric measurements was somewhat less precise. In several cases these parameters had to be changed somewhat lo obtain a good fit for slow isotonic contractions. Torque-angular
velocity
parameters
At positive velocity the torque-angular velocity (Hill) relation (Part 1, equation 5) is determined by the parameters b and n, of which the latter is onlj important at the higher speeds. The part at negative velocity is determined by the parameter c. It turned out to be impracticable to find b and n by varying them both and minimizing the ISE. Therefore they were kept in a fixed proportion on the basis of the following. From the Hill relation it follows that the intrinsic velocity Cp,.i.e. the maximal unloaded speed of the muscle, is equal to b/n. All subjects were asked to plantarllex their unloaded foot as fast as possible. This gave peak velocities of 8-12 rad!s with an average ofca 10 rad/s and without large systematic differences between subjects. The intrinsic velocity of the muscle analogue has been kept at this value by always taking )I = b/g,, with Cp, = 10 rad/s. In fact tt could be varied quite widely around this value without noticeable effects, the often cited literature value t1 = 0.25 was certainly too high. however. The parameters b and c were determined by minimizing the ISE for the isotonic contractions at positive and negative velocity. respectively. Figure 3 gives the ISE at positive speed as a function of b for subject 4. Contractions of moderate and fast positive
trir f
lp; 3 g,+ --_ _ - - _ __ J\
‘\
\
Fig. 5. /(o) relation. The EMG gain factor has been plotted as a function of the angle $ for subject 4, soleus only. These ratios have been obtained by drawing straight lines through X-Y plots of the mean recrified EMG vs measured torque. Fluctuations in the EMG cause some uncertainty. the estimated range of which is indicated by the vertical bars. From this figure can be concluded. ti, = 2.8. 6; = 1I P @>= X6
766
A.
L. HOF
and
Jw. VAN
DEN BERG
C
bfmd/sl .wbj.
2
I I
I
I
1
0 I
l-b
t-t&
1
0
0
+-
L
2.0
a5
* t:;
:I
1.2
0
I
-
Fig. 6. Parameters of the soleus.Survey of resultsof the parameter estimation for b and c. For the meaning of the bars see Fig. 3. 0: medium speed (0.5-1.5 rad/s), A: fast speed (> 1.5 rad/s), 0: negative speed. speed were considered separately. Figure 6 gives a survey for the four subjects in the experiments with soleus only. The error curves have been represented by bars on which the minimum and the points where the interpolated curve is at 120% and lSoo/, of the minimum are indicated (cf. Fig. 3 below). Even if in the subjects 2 and 4 the M and F minima do not coincide exactly, for all subjects suitable speed independent optimum values for b could be found, which are indicated in Fig. 6 and listed in Table 2. Three out of the four optimum values for c were zero : a boundary extreme as c cannot be negative. Figure 7 summarizes the results of the parameter estimations for b and c in the experiments with the combination of soleus and gastrocnemius. In a similar
Fig. 7. Parameters of the combined calf muscles. Parameter estimation for b and c. game meaning of symbols as in Fig. 6. The individual optima as well as the average valueshave been indicated.
way as with the soleus experiments it is possible to assign an individual optimum to each subject (these are indicated). From the figure it is clear, however, that the optimum values are so close together that it is possible to assign one single value to each parameter. which is valid for all subjects. These values are b = 1.2 rad/s and c = 0. It can be seen that c = 0 gives an ISE less than 150% of the mimimum in every subject: taking into account that the ISE is a quadratic measure of the error, a 50% increase in ISE corresponds to a 22% increase in r.m.s. error. The ISE does not exceed lSOo/;,of the minimum for fixed b either, with the exception of some cases: the subjects 2 (F), 6 and 7 (M). Processor
perfirniance
The overall performance of the processor. for isotonic contractions and at optimal parameter settings. can be assessed from Fig. 2. The high (up IO 400 Nm in torque measure) and strongly fluctuating rectified EMG U(r) is processed to a model torque which follows the measured torque of 56 Nm maximum quite satisfactory. The errors which remained at the optimum parameter setting have been given in Table 4 for the soleus. Both the r.m.s. error and the peak error are presented. The peak values suggest that the error had a systematic component in some cases. Table 5 gives the r.m.s. and peak errors for the contractions with the combination of soleus and gastrocnemius in the standing position. For the torqueangular velocity parameters the common optimum values b = 1.2 rad/s, n = 0.12 and c = 0 have been used in all subjects. The error is in general slightly larger when the muscles are combined. In part this is because the common optimum value b = 1.2 rad/s is not optimal for some subjects (viz. nos. 2 (F). 6 and 7 (M)). For these subjects the error at their individual optimum for b and II has been given as well (between brackets). The data of Tables 4 and 5 refer IO the isotonic phases of the contractions only.
9.5
N
6
5’ 6
Subject
I
: 4
-10: -8; -8;
-26;
+10 +12 +12
+I2
Peak error
12 14 4
16
N
1.9 5.4 5.0
5.6
r.m.s. error 13
+12 +I8 +I6 +12
-12; -4; -8; -4;
-9 9
N
Peak error
Moderate 0.5 c 4 < 1.5 rad/s
3.4 6.7
4.5 -6; -16;
-12; +I4 +16
+12
Fast $ > 1.5 rad/s r.m.s. error Peak error
2L 19 6
29
N
6.5 8.2 14.9
19.0
+34 +22 -&IO +22
-18; -12; -10; -20;
Negalive cp
2.8-6.7 3.6-4.8 2.8-6.7
3.9-6.7
Predicted stochastic r.m.s. error
6 4
4 6 4 3
2
4
2
3 4 5 6
1
8
IS.1 ~_
7.1)
7.5 6.H 6.9 14.1
13.2 5.5
N
Subject
I
r.m.s. error
range
.
Slow
0: -
-14:
-14; -10: -12; -12:
-14: -16:
+74
+lO
c4 +R +4 +1x
+20 +R
Peak error
N
I!)
7
9 4 R 6
13 7
-__-----
8.3 6.4 7.0 7.3 (4.5) 7.2 (2.8) 9.3
7.1 4.5
r.m.s. error -t16 +6 +R +1? +12 + 16 +I2) +I6 A-12) +24
-4; -8; --to; -6; -6; 0: (-2; -2; (-2; -4:
Peak error
Moderate
4 .__ -._-
5
14 14 8 9
8 14
N
IO0
15.8 6.8 (5.0) 8.9 6.3 5.9 10.5 (6.7) 5.0
r.m.s. error
Fast
~.
0:
-22; -8: I-R; -10: -6: -8. -1x: (-8; --I’.
6
23 IO 4 15
21 6
N
+20 4 ~_.-.~~._-._I._
+26 +4 +4) +I2 +12 +12 +10 +18) +8
Peak error
19.9
12.7
10.4 10.4 9.3 14.2
15.9 1.5
r.m.s. error
+lO +lO +14 +12 +I6 +20 +I2 +24 -__.
-26; -12; -12; -12; -16; -12: -16: -8:
Peak error
Negative
2.2-5.0 .-__.---
2.2-5.0
2.0-3.6 2.2-5.0 2.2-5.0 2.8-5.0
2.8-5.0 2.2-5.0
r.m.s. error
Predicted stochastic
Table 5. R.M.S. error and peak error (negative; positive)for the isotonic phases of the experiments with the combined calf muscle group (in Nm). Measured data are obtained at the average parameter setting for b, n and c (Table 3). For some subjects the error at their individual optimum b has been given in brackets. For predicted stochastic (r.m.s.)error see Discussion. Isotonic torque 40 Nm for subject 3, 56 Nm for the others. N = number of contractionsin a series. Spaed
5.2 6.0 8.2
r.m.s. error
0 -C4 -Z 0.5 rad/s
Slow
Table 4. R.m.s. error = J( I !T) f ,(M - M,)‘dr and peak error (negative ; posifive) for the isotonic phases of the soleusexperiments (in Nm). Measured data refer IO the individual optimum parameter setting of each subject. see Table 2. For predicted stochastic r.m.s. error see Discussion Isotonic torque: 40 Nm for subject 3 and 56 Nm for the others. With subject no. 3 no fast contractions could be done. N = number of contractions in a series,
1 5’ rJa s
$
2 0
768
A. L. HOF and Jw. VAN DEN BERG DISCUSSION
Parallel elastic component The properties of the passive muscle are reasonably modeled by the exponential characteristic of the PEC (Fig. 4). The hysteresis effects are considerable, especially at very low 4 (extreme dorsiflexion). Similar results have been found for many biological soft tissues (Crisp, 1972, Fung, 1972). At extreme plantarflexion the passive torque becomes negative due to the ligaments and the PEC of the dorsiflecting muscles (Gottlieb and Agarwal, 1978). Both effects are not important in the range from 75” to 1lo”, to which the ankle movements are usually confined according to our observations. In the calfergometer experiments 4 is between 90” and 105”. In this range the passive torque is small compared to the active torque and the hysteresis effects are negligible. According to the experiments the exponential factor d,can be considered constant at 8’. but it is necessary to measure the proportionality factor M, for each new subject. Torque-angle parameters The parameters of the torque-angle relation, 4, and 4‘ - & show considerable differences between subjects, for soleus only (Table 2) as well as for both muscles (Table 3). Moreover the model performance was found to be rather sensitive lo the correctness of these parameters. It will therefore be necessary to measure them for every new subject. It seems plausible that the recorded torque-angle relation for the soleus, cf. Fig. 5. reflects the force-length relationship of this muscle. It has been found that the lever arm of the ankle is quite constant over the range of its rotation (Grieve. Cavanagh and Pheasant, 1978). A change of the electrode position with respect lo the muscle fibres might give some contribution to the observed angular dependency of the torque-EMG ratio but the reproducibility of the found f(4,) relations goes against this. Moreover the values found for 4, - & are in reasonable agreement with those to be expected. Soleus (as well as gastrocnemius) is a pennate muscle, with short fibres compared to its origo-insertion length and a long tendon. In a dissection of one cadaver leg of average size (tibia length 40 cm) we measured an ankle lever arm of 5 cm and a soleus fibre length of 3.5 cm. The muscle had been fixed in the anatomical position. i.e. with the foot fully plantartlexed, 4 z 120’, and the fibres thus strongly shortened. The fibre length at C#J= 90’ was estimated from these data as (length at I#J= 120”) + (lever arm x 120’-90’ in radians) = 3.5 + (5 x 0.5) = 6 cm. The force-length relation of a single muscle fibre goes from 0 at 0.6 I, to maximum at I,. This results in til - ti2 = (0.4 x 6)/5 = 0.5 rad = 30’; a value comparable to those found (Table 2). The soleus has its optimum length I,, at an angle of the foot of QI; (cf. Fig. 5). From the data of Table 2 it can be found that this angle is between 75’ and 86”. i.e.
at slight dorsiflexion. The 4, and 4, - @Jovalues given for the combination of both muscles (Table 3) represent a kind of weighted average of those for the soleus and gastrocnemius, the weighting factors depending on the relative contribution of each muscle lo the total torque. This might be the reason why determining these parameters is less precise than when only the soleus is involved. The gastrocnemius length depends not only on the angle of the ankle but also on the knee angle. Grieve er al. (1978) however, found that the effective lever arm of this muscle with respect to the knee is only l/3 of that with respect to the ankle. Thus a slight variation in knee flexion during standing, 10’ at most, does not give serious errors. Torque-angular
velocity parameters
For every subject an optimum value for the torqueangular velocity parameter b (at n = b/10) could be found. In the soleus experiments these optima were somewhat different for three of the four subjects. The individual optimal values have been used in the determination of the errors. For the combined calf muscles the individual optima were close together. Using a common value for b and n introduced little excess error (Table 5). Even in the subjects 2 (F). 6 and 7 (Ml, where the ISE at the common parameter value was more than lSoO/, of its minimum value, the total (r.m.s.) error was not excessively high when compared to those of the other subjects. It was concluded from this that b = 1.2 rad/s and n = 0.12 can be adopted as standard values for normal subjects, though it must be remembered that the value of H relies on the assumption that the intrinsic velocity 4, = 10 rad,s for ail subjects.
At the highest velocities inertial effects become apparent (see the’Calfergometer’section and Fig. 2 right). For example during the onset of the movement the torque of the calf muscles has a peak (of 13.5 Nm max.) above the nominal torque of the spring. As stated in the ‘Calfergometer’ section. part of this inertial torque (5 Nm max.) is not measured by the strain gauges. According to equation (5) of Part 1 this may have caused an at most 157; too high estimation for b. The other part of the peak inertial torque (8.5Nm max. I was taken into account when comparing the calculated torque M and the measured torque M,. However. the contraction was no longer exactly isotonic. The elTect of the torque not being constant during the whole movement is nil if the SEC parameters have their correct values. Because thrs is not certain (they had only provisional values in these experiments), b may have been estimated slightly too high or too low in the fastest contractions. We were not able to discriminate such systematiceffects from those due to the random EMG fluctuations (Figs. 6 and 7). Since the inertial torques are comparable in magnitude to the random (peak) errors. this could indeed hardly be expected (Tables 4 and 5 1. There are only a few references with which the values for b and n can be checked. Bigland and Lippold (1954) do not give explicit numbers. but the torque-velocity curves of their Fig. 5 can quite well be fitted with b = 1.2 rad/s over the range of velocities they covered. Wiikie (1950) gives values of h for the parameter
elbow flexors. Ifconverted
to relative length changes of
EMG IO force processing II
the biceps brachii (fibre length I4 cm) they range from 1.6 to 3.4 fibre lengths per second. For the calf muscles (fibre length 6 cm, lever arm 5 cm) these values range from 1.0 to 1.7 1,/s (corresponding to 1.2-2.0 radis), and are thus somewhat lower. This is consistent with the finding that the calf muscles consist mainly of slow muscle fibres, while the biceps contains a large amount of fast fibres (Schmalbruch. 1970). Literature values for n in human muscles are 0.20-0.48 (Wilkie, 1950, elbow flexors) and 0.29-1.3 (Binkhorst, Hoofd and Vissers, 1977, handgrip muscles). The value we found, n = 0.12, is clearly lower. For most subjects c = 0 was optimal for soleus only, and this value was also found as the common optimum for both muscles. This is the lowest possible value in the model. but in spite of this in several subjects the torque was on the average too high at negative speeds. suggesting that c had to be made still smaller. On the other hand values in the range 0.2-0.5 are suggested by the literature (Joyce and Rack, 1969; Flitney and Hirst, 1978). This contradiction was reflected in the recordings: in some contractions the measured torque was approximated best by choosing c = 0.2 or higher. in other ones the model torque was systematically too high. It seems that the latter condition goesalong with the appearance of large fluctuations in the mean rectified EMG U(r), indicated with arrows in Fig. 2. These synchronous discharges of the EMG are a well known effect (Bigland and Lippold. 1954; Lippold. et al.. 1957; Milner-Brown et al.. 1975). They will give rise to larger fluctuations in the model torque than estimated under the assumption that the EMG is a stationary noise (see below). Moreover. the active state block in the analogue will transform these plus and minus fluctuations into a systematically too high M, because of its inherent non-linear properties. The choice of c = 0 must therefore be seen as a compromise. Errors Even if the processor would work perfectly there would remain a stochastic error. This is because the EMG input signal is a stochastic signal: an amplitude modulated Gaussian noise with zero mean (Schwedyk. Balasubramanian and Scott. 1977). Part of the stochastic fluctuations will remain after the processing, setting a limit to the achievable performance. When a stationary Gaussian noise with equivalent statistical bandwidth B is rectified and subsequently filtered by a linear low-pass filter. the remaining fluctuations have a variation coefficient E = AM (r.m.s.)/M,with (Bendat and Piersol. 1971, Chapter 8): s= &
(r.m.s.)
From a measured power spectrum of the soleus surface EMG we calculated B = 29OHz. In (2) 7 is the elTective averaging time. which depends on the filtering properties of the processor after the rectifier. An analysis is complicated by the non-linearity of the processor (i.e. the muscle model).
769
As a result the filtering properties depend on the state variables M, 4 and 4. Refraining from a detailed analysis, we will give here two estimations for T,which can serve as a lower and an upper bound respectively. (a) T is determined by the properties of the active state converter. As an estimate we may adopt for this case : T =
7,
+
T2 +
53
(3)
which gives T = 25 + 30 + 60 = 115 ms, and, from (2). & = 12%. On the basis of the model properties this value of E can be expected to hold for negative velocities. (b) T is determined by the low-pass properties of the Hill model at positive velocities. The linearized Hill model (Calvert and Chapman, 1977 ; Bawa, Mannard and Stein, 1976) is a first order low-pass filter with time constant T,. The highest value for T,,,. valid at low positive velocity, is: m 7 For a first order filter it holds (Bendat and Piersol. 1971) that: T = 2 T,.
(51
Depending on the value of K used, this gives 7 between 400 and 7OOms and E between 5 and 7”$ Equation (4) explains why the random error decreases at lower values of fi and K (cf. section on EMG gain factors. etc.). Predicted values for the error in the contractions with soleus only, both for (a) and (b). are given in Table 4. It is seen that the measured r.m.s. error in several series of contractions is within the predicted range. In other series, often characterized by asymmetric peak values, some systematic error was present. A possible source of the high. systematically positive error at negative velocities (subjects I and 4) has already been discussed. It is apparen t. nevertheless. that in these experiments the accuracy of the processor in a number of cases is not limited so much by the exactness of the model and its parameters. as by the inevitable statistical fluctuations caused by the EMG signal. In the experiments with the combination of both calf muscles the EMGs of soleus and gastrocnemius are added after rectification. If both signals are uncorrelated (which they supposedly are) and if they are equally strong (which they are not always, of course I the sum signal is twice the component signals, while the r.m.s. error in the sum signal is only ,i2 times as large as the error due to one of the components, This gives, instead of (2):
In the prediction of the error for the experiments with soleus and gastrocnemius combined. Table 5. equation (6) has been used. Although in one suhiect (no. 2) the theoretical minimum is approximated. in
710
A. L. HOF and Jw. VA?; DES BERG
Subjects the measured r.m.s. error is about 2-3 times the theoretical value, while the peak values indicate systematic errors. From inspection of the recordings it appeared that subjects with large errors, like no 8, showed irregularities in the EMG: in almost identical contractions the (rectified) EMG could be considerably different. Maybe in these cases the recorded EMG was less representative for the activity of the whole muscle mm
group. Acknowledgements - The calfergometer was designed by Dr. J. Lubbers and constructed by Mr. L. Martijn. The advice of Professor H. B. K. Boom was ofgreat value in the preparation of the manuscript.
Joyce, G. C. and Rack, P. M. H. (1969) Isotonic lengthening and shortening movements ofcat soleus muscle. J. Physiol.
Land. 204.475-491. Lippold, 0. C. J.. Naylor, P. F. D. and Treadwell, E. E. E. (1952) A dynamometer for the human calf muscles. 1. sri. Instrum. 29, 365-366. Lippold 0. C. J., Redfcam, J. W. T., V&o. 1. (1957) The rhythmical activity of groups of motorunits in the voluntary contraction of muscle. J. Physiol., Land. 137,473487. Milncr-Brown, H. S., Stein, R. B., Lee, R. G. (1975)Synchronixation of human motorunits: possible roles ofexercise and supra spinal reflexes. EEG clin. Neurophysiol. 38,245-2X Schmalbntch, H. (1970)Dic quergcstrciftm Muskelfascm des Mmschcn. Erg&n. Anat. EnrwGesch. 43, Heft 1. Shwcdyk, E., Balasubramanian, R. and Scott, R. N. (1977) A nonstationary model for thcclcctromyogram. IEEE Truns. Biohfed. Engng 24,417-424. Wilkic. D. R. (1950) The relation between force and velocity in human muscle. J. Physiol. Land. 110, 249-280.
REFERENCES
Bawa. P., Mannard, A. and Stein, R. B. (1976) Predictions and experimental tests of a viscoclastic muscle model using elastic and inertial loads. Biol. Cybernet. 22, 139-147. Bcndat, J. S.. Piersol, A. G. (1971) Rondom Dara, Analysis ond Meosuremenr. Wiley-Intcrscima, New York. Bigland, B. and Lippold. 0. C. J. (1954) The relation between fora, velocity and integrated electrical activity in human muscles. J. Physiol. Land. 123, 214-224. Binkhorst, R. A., Hoofd. L. and Visscrs, A. C. A. (1977) Temperature and force-velocity relationship of human muscles. 1. appl. Physiol. 42,471-475. Calvert, T. W., Chapman, A. E. (1977) The relationship between the surface EMG and fora transients in muscle : Simulation and experimental studies. Proc. IEEE 65, 682-689. Crisp, J. D. C. (1972) Properties of tendon and skin. In Biomechanics. its Foundations and Objecriws, (Edited by Y. C. B. Fung et al.) pp. 14l- 180, hen tia-Hall, Englewood ClilTs,New Jersey. Flitney, F. W., Hirst, D. G. (1978) Cross-bridge detachment and sarcomere ‘give’ during stretch of active frog’s muscle. J. Physiol. Land. 276,449-465.
Fung, Y. C. B. (1972) Stress-strain-history relations of soft tissues in simple elongation. In Biomechunics. its Foundarions und Objectives, (Edited by Y. C. B. Fung et al.) pp. 181-208, Prentia-Hall, Engkwood ClitTs,New Jersey. Gottlieb, G. L. and Agarwal, G. C. (1978) Dcpmdcna of human ankle compliance on joint angle. J. Biomechanics 11, 177-181. Grieve, D. W., Cavanagh, P. R. and Pheasant, S. (1978) Prediction of gastrocnemius length from knee and ankle posture. In Biomechonics VI (Edited by E. Asmusscn and K. Jdrgcnsen) pp. 405-412, University Park Press, Baltimore. Hof. A. L. and Van den Berg, Jw. (1977) Linearity betweenthe weighted sum of the EMGs of the human triceps surac and the total torque. J. Biomechanics 10, 529-540. Hof, A. L. and Van den Berg, Jw. (1981a) EMG to fora processing I: An electrical analogue of the Hill muscle model. J. Biomechonics 14, 747-758. Hof, A. L. and Van den Berg, Jw. (1981b) EMG to fora processing III. Estimation of model parameters for the human triceps surae muscle and assessment of the accuracy by means of a torque plate. J. Biomechanics 14, 771-785.
NOMENCLATURE b B C
I(#) f(6,)
AM M, M, M, n
velocity parameter torque-velocity relation (5) (rad/s) equivalent statistical bandwidth of EMG signal (Hz) parameter torque-velocity relation at negative vclocities (6) torque-angle relation measured as a function of thC angle of the ankle torque-angle relation as function of CC-length 4, (form as used in the model) torque-EMG ratio gastrocnemius ditto for soleus linear elasticity parameter SEC (8) (Nm/rad) resting length of muscle (cm) total torque. as computed by the processor = .U, + M, (Nm) r.m.s. error in M = til’T){~M - M,Ydr (Nm) parameter of SEC = 1 Nm torque measured with the calfergometer (Nm) proportionality constant of PEC (9) (Nml parameter torque-velocity relation time (s) effective averaging time (s) rectified and smoothed EMG, gain set by g1 and gB tNm1 work done by the muscle = IM@dr (J) logarithmic elasticity parameter SEC (8) &ad- ‘ ) = AM/M_ coefficient of variation, in isotonic contraction time constant EMG smoothing filter = 25 ms plateau duration active state (ms) time constant exponential decay of active state (ms) time constant of linearized Hill muscle model (ms) angle of the ankle joint (deg) = d&dr (rad/s) angle corresponding to CC-length = 4 + 4. (deg) SEC stretch (rad) intrinsic velocity of muscle = (b/n)f(#,) &ad/s) exponential constant PEC (deg) parameters torque-angle relation (3) (degi parameters torque-angle relation before correction for the stretch of the SEC 4,: 4, = d’, + d,(M) (deg).
1107i9 II WMO Pergamon Prrsr Lid
EMG TO FORCE PROCESSING II: ESTIMATION OF PARAMETERS OF THE HILL MUSCLE MODEL FOR THE HUMAN TRICEPS SURAE BY MEANS OF A CALFERGOMETER* A. L. HOF and Jw. VAN
DEN BERG
Laboratory for Medical Physics. State University Groningen, Bloemsingel 10.9712 KZ Groningen, The Netherlands
Abslract- On a calfergometer isotonic contractions were performed by the calf musclesof human subjects. The measured torque M, was compared lo the torque M processed from rhc EMG and the joint angle b! means of an electronic processor based on the Hill muscle model. By means of these experiments the model parameters were determined for the torque-angle relation, the torque-angular velocity relation and the paralkl elastic component (PEC) Experiments were done (a) for the soleus only (4 subjects) and (bl for the combinarion of soleus and gastrocnemius (8 subjects). It turned out that: --the torque-angle parameters varied among subjects; ---the torque-angular velocity parameters for sokus only varied also among subjects: -for the combined calf musck group the interindividual differences in the torque-angular velocity parameters could be neglected ; -one of the parameters of the PEC was the same for all subjects, while the other was variable among the subjects. The obtained parameter values are discussed and compared with available literature data. Values for the r.ms. and the peak error in the isotonic phases of the contractIon are given and compared with values predicted from the stochastic properties of the EMG signal.
lNlRODUCTiON
In Part I of this series of papers (Hof and Van den Berg, 1981a) a method was described for the processing of the EMG to the muscle torque. It consists of an electrical analogue of the Hill muscle model ‘driven’by the rectified EMG. For a proper functioning it is necessary to know the parameters of the Hill model. The EMG gain factors (parameters) can be determined in quasi-static isometric contractions with a calfergometer, see a previous paper (Hof and Van den Berg, 1977). In the present paper we also used the calfergometer to determine a number of parameters of the Hill model (cf. Table 1 of Part I): -the parameters of the parallel elastic component (PEC), by measuring the passive torque around the ankle joint at complete relaxation of the muscles. -the torque-angle (cq. force-length) relation of the calf muscles, by quasi-static isometric contractions at various angles of the ankle joint. --the torque-velocity (Hill) relation. by isotonic contractions of the calf muscles. Isotonic contractions were used to separate the effects due to the series elastic component (SEC) and the contractile component (CC), as the stretch of the SEC is (virtually) constant during a (virtually) isotonic contraction. The parameters of the SEC and the time constants of the active state of the CC (apart from rIr see Part I) * Receired
in rerised.form
9 April
1981.
might be estimated from fast isometric contractions but the calfergometer is not well suited to these contractions: it is typically a low-inertia instrument for fast movements. As a consequence. its stiffness is too low for quick isometric contractions to lx real& isometric. These parameters will therefore be given only provisional values, in order to enable the model to work properly; their values are not critical for the present experiments. More definitive values will be determined in experiments with a more suitable instrument, a torque plate. 10 be described in Part III ~Hof and Van den Berg. 198 I b 1. To evaluate the functioning of the processor it IS necessary lo compare the calculated muscle torque with the measured torque. .4 problem with this is that an externally measured joint torque is the algebraic sum of all muscle torques from agonists and antagonists. whereas the processor gives in principle the torque from one muscle at a time. With the calf muscles - plantarflexors of the foot this problem ISnot too difficult fortunately. In most cases the antagonIsts can be kept inactlve (to be checked by EMG) and the contribution of the smaller plantarflexors can br neglected due to their relatively small cross-Kctlon and short lever arms. The plantarflecting torque ISthus mainly provided by M.M. soleus and gastrocnemlus. The parameters of the torque-angle and the torque-angular velocity relatwn have been detcrmined for two cases. (a) With the sub,iecl In sItrIng posirlon. \rlth strongly bent knees. onl! 111~solcus musck contra759
760
A. L. HOF and Jw. VAN DEN
butes to the torque (Hof and Van den Berg, 1977). This gives the opportunity tocompare the processoroutput with the torque of this muscle alone (4 subjects). (b) By adding the weighted EMGs of soleus and gastrocnemius and using the sum as the processor input these two musclesare taken together (8 subjects). This approach is similar to the one we used in quasistatic contractions (Hof and Van den Berg, 1977). The parameter values to be obtained are somewhat of an average of those of soleus and gastrocnemius. This approach is less idea1 in principle but it gives more practical results, because now the processor can be used to determine the total calf muscle torque in standing or walking subjects. METHODS
The essential parts of the calfergometer are shown in Fig. 1. The foot of the subject is fixed onto footplate A, which is connected by the rods B to the bar D. The whole pedal is suspended on heavy ball bearings in the main frame E. The axis of rotation C can be made coaxial with the ankle axis (identified by the medial and lateral malleoli) by adjusting the footplate A and the heel rest M. The angle is measured with potentiometer P. The range of the pedal rotation is restricted to 90’-105’. The load consists ofa set of roll springs (Tensator) F which can be attached by hooks to bar D. They give an almost constant torque (for stretching the torque is slightly larger than for shortening). A nominal torque of 56 Nm was used in most cases. The torque was measured by the strain gauges S on bar D, very close to the axis of rotation. For static torques the accuracy was better than + 3”,. The ergometer was constructed to have low inertia, but at the fastest contractions the inertia cannot be neglected. The moment of inertia of the parts above the axis of rotation is 0.13 kg. m2, the corresponding torque is measured by the strain gauges. The moment of inertia of the parts below the axis ofrotation, the footplate (0.04 kg. m’) and the foot itself (0.02-0.04 kg m’). goes along with a torque which is not measured by the strain gauges. This torque amounts to ca 5 Nm (i.e. 10:; of the isotonic torque) at the highest acceler-
BERG
ations encountered during the onset of the isotonic part of a contraction. Apart from introducing a measuring error, inertia implies that the contraction cannot be completely ,isotonic. In a first approximation we shall neglect this, but we come back upon it in the Discussion. Bar D can also be tixed at various positions for recording isometric contractions. The light construction of the pedal resulted in a low moment of inertia but it has the disadvantage of a rather low stiffness, ca 1200 Nm,‘rad. This must be accounted for when interpreting isometric contractions. The shod foot of the subject is fixed rigidly to the footplate by means of the heel rest M and an adjustable hoop H over the instep. The hoop is made of thin stainless steel and fitted with a leather pad L. When the subject is sitting, his knees are bent at ca 60; and are pressed down with a knee pad (cf. Lippold, Naylor and Treadwell, 1952). When standing the subject maintains his equilibrium by holding a fixed bar in front of him. To the moving bar D also a dashpot G can be attached, which damps the action of the springs in the backward movement (negative velocity). This was appreciated by most subjects as this part of the movement is difficult to control. But since in this case the return movement is no longer isotonic, the dashpot is disconnected in the contractions used for studying the behaviour of the muscle at negative velocities. The dashpot does not influence the torque in the forward part oi the movement. Recording The gastrocnemius electrodes were positioned one on the medial and one on the lateral head of the muscle. The soleus electrode pair (surface area 0.5 cmz) was placed 2.5 cm apart on the edge at the medial side. where the muscle protrudes from under the gastrocnemius (Hof and Van den Berg, 1977. see Fig. 2b). The angle of the ankle was recorded with the fixed potentiometer on the ergometer. The EMGs and the angular position were fed into the processor. The unprocessed signals were also recorded on a fourchannel tape recorder (HewlettPackard model 3960), together with the measured torque M,. Subjects The subjects were six men and two women. data of whom are listed in Table I. With the subjects 5-8 she torque-angle and torque-angular velocity parameters were determined only for the combination of soleus and gastrocnemius. For subject 3. who was of slight build, an isotonic load of 40Nm was used, because she could not move the pedal over its whole range with heavier loads. For all other subjects the isotonic torque was 56 Nm. Procedure
Fig. 1. The
calfergometer (schematic). See Methods
For all subjects the soleus gain factor (I, was determined in the sitting position with an isometric contraction at a joint angle of the foot 4, = 90’. The subject increased the torque slowly from zero IO maximum and back. By plotting the mean rectified soleus EMG against the measured torque on a X-Y recorder, g, could be determined as described previously (Hoi and Van den Berg, 1977). With the subjects 1-4 this procedure was repeated with 4 set at anglesfrom 75”-110”,atintervals0f5’.Thisgaveaseries0f points for the torque-angle relation from which I#J;and 4; could be determined. In our model the torque-angle relation /(#,)isnot a function ofthejoint angle 4 but of thecontractile component ‘length’ 4, the difference being the stretch of the series elastic component 4,(M). This differena must also be accounted for in the parameters: thus 4, = I$; + #,(M) and f#~a= 4; + +,(M), see part I, equation 8. Alter this a senes of isotomc contractions was made, with speeds varying from very slow to as fast as possible, to obtain the parameters of the torque-angular velocity relation. They were monitored by displaying the signals on a Mingograffast inkwriter.
EMG
761
IO force processing II
c j
--
__$_
_c-.-_-_ I
IL .--. --
/_. _..: T’ t
-.A
._‘
47-
:
r
763
EMG IO force processing II Table 1. Subject data and division of experiments.
Subject
Sex
Age (yr)
Body mass (kg)
Bad) height (m) 1.73 1.87 1.60 1.77 1.72 1.92 1.78 1.73
I
r
28
64
2 3 4 5 6
m r m m m
28 21 39 26 30
1
m
23
8
m
30
71 47 82 70 91 65 63
The other experiments. with all subjects, were performed in standing position. First the gaslrocnemius gain factor gP was determined at #J = 90’ asdescribed previously (Hofand Van den Berg, 1977). With gI and go set at their respective values the procedure as described above - isometric contractions at various foot angles and isotonic contractions-was repeated in the standing posture. Finally the passive torque due to the parallel elastic component was recorded as a function of 4, for all subjects in the standing position and for subjects l-4 in the sitting position. To this end the springs were disconnected and the ergometer arm was moved slowly by the experimenter while the subject had to keep his muscles relaxed. This was checked with the EMG. Parameter
esfimarion
As the unprocessed signals were recorded on tape, they could be played back through the processor as often and with as many parameter settings as needed. To facilitate a comparison between the measured torque .%I, and the processor torque M they were displayed next to one another on the Mingograf inkwriter together with their difference (M - M,; see Fig. 2). Systematic errors in the processing thus stood OUIclearly. With the help of this signal the setting of the gain factors (g, and g.), the parameters of rhe torque-angle relation (0, and 4, - #1) and the PEC parameter M, could be checked and, if necessary, be adjusted. For the other parameters approximate values could be found in this way. As a more quantitative measure of the error the integrated squared error (ISE) J(M - M,l’ dr was used. It was computed with the analogue electronic circuit intended for computing the work W. For the Isotonic contractions the integrator was reset by the zero-crossings of the angular velociry 4. In this way separate values of the ISE for both isotonic phases in the contracrion, that with positive and that with negative velocity, were obtained. The isotonic contractions were divided in three groups according 10 their peak positive velocity: slow (S) with Cpi 0.5 rad,s. moderate (M) with 0.5 5 @ < 1.5 radisandfast (F)with cp 1 1.5 radis. The number of contractions has been given in the Tables 4 and 5. Optimal values for the parameters of the torque-angular velocity equation could be found by plotting the total ISE as a function of the parameter value and looking for the minimum (Fig. 3k In determining b (and n) the phases with positive velocity of the moderate and fast groups were considered, for c the phases of negative velocity of the same groups were taken together, provided that the dashpot had not been used in the experiments. In the other case an equivalent group of contractions without dashpor was used. Represenrarion of the errors
Data on the in two forms:
processing
errors have been given in the lables
(a) The root mean square error A.t! (rms.! the ISE according IO its definition:
from
was obtained
Soleus + gastrocnemius
Soleus only
X X X X X X X X
X X X X
AM (r.m.s.) = J-F
(1) .r in which T is the total duration of the isotonic phases with positive or negative velocity in a series of conrractions. (b) The peak error has been given as the negative and positive maxima of AM = M - M, during the isotonic phases of a series of contractions. The reading-off accuracy was about f 2 Nm. RESULTS
EMG gainfactors.
series elastic component.
acrice stare
The EMG gain factors gs and g4 ranged from 1.4 to 6.5, and from 0.60 to 1.8, respectively. As they change for each electrode placing they were not listed in the Tables 2 and 3. As could be expected the values of the SEC para-
ISEl
/
‘5
SOO-
:J/I// 0
,w,,___
-_---
-._
j2fy&____‘_______
0
:
KKp/.___4__;___ I
1
,
,
’
1
I
‘I:
1
>
:
0
1
Fig. 3. Theintegrated squared error (1SE)integrated over the whole set of contractions as a function of the parameter b, with n = b/10. Subject 4, soleus only. 0: Medium speed, A: fast. For the fast speed it has been indicated how the ‘bars’ in Figs 6 and 7 are constructed: the symbol in the middle denotes the minimum of the ISE curwz,the vertical dashes the points at which the ISE is lw’/, and iSOo/,respectiwly of the minimum.
764
A. L. HOF and Jw. VAN DEN BERG
Table 2. Parameters of the soleus (sitting posture). For 6. n and c the optimum values for each individual have been given. Torgue-angle relation Subject
($gJ
‘jd;g:’
Torque- velocity relation b (rad/s)
I
159
50
1.5
2 3 4
154 146 I57
40 40 32
1.2 2.0 1.2
PEC
n
C
0.15 0.12 0.20 0.12
8 0.5 0
k, 4.0 1.0 1.0 3.0
Table 3. Parameters of the combined calf muscle group (standing position). Parameter values for b, n and c are averaged for the whole group of subjects. Torque-angle relation
Subject
(&I
‘idig:’
Torque-velocity relation
n
b (Nm/rad)
PEC C a,
2 3 4 5
159 169 146 169 174
40 40 40 40 40
1.2 1.2 1.2 1.2 1.2
0.12 0.12 0.12 0.12 0.12
0 0 0 0 0
6.0 2.0 2.0 3.8 1.5
6 7 8
159 169 164
40 40
1.2 1.2
0.12 0.12
0 0
::: 1.0
1
meters /I and K and those of the active state time constants f2 and ?3 (TVbeing preset at 25 ms, see Part I) had little influence on the error in the isotonic phases of the contraction, on which we concentrated. The most obvious effect was that the error decreased for lower values of /I and K (i.e. lower stiffness of the SEC). In our opinion this effect was related to the low-pass filtering properties of the muscle model, and not to a better fit of the SEC parameters (see Discussion). By inspection of the error during the isometric con-
traction phases (before, between, and after the two isotonic phases), approximate values for the active state and the SEC parameters could be found, bearing in mind that the former shape the descending part ofan isometric contraction while the rising part is influenccd by the SEC parameters (together with the parameters b and n of the torque-angular velocity relation). The values thus estimated were TV= 30 ms, ~~ = 6Oms, /I = lOrad-’ and K = 5OONm/rad (subjects 1, 3 and 6) or ZOONm/rad (other subjects).
MlP
INIlII
td
50-
(4
(Nn
o60
Fig. 4. Parallel elastic component: passive torque as a function of joint angk 4. Some recordings of dimerent amplitudes, subject 6 (thin lines, ach recording with dimerent symbols). Output of proafJor PEC circuit with M, = 3.5 Nm and 4, = 8” (thick line) (a) Linear scale (b) Logarithmic scale for M,, to show exponential course for & < 90”.
(b)
,
700
so0
9o”co
lo force processing II
EMG Parallel
elastic componenf
Figure 4 gives some recordings of the passive torque as a function of ankle rotation for one subject. For 4 -Z 90’/M,(4) increases exponentially with decreasing 4. in accordance with the PEC characteristics of the model (Part I, equation (9)). In order to show this, M,(4) is plotted logarithmically in Fig. 4(b). The curves show a considerable hysteresis, which is dependent on the previous stretch. As a consequence one finds that the value of the exponential parameter 4p should in fact be different for increasing or decreasing 4. For the subjects in this study ranges were found for 4, of 8.3- 12.2 and 6.0-8.4 degrees respectively. We see that the interindividual differences are about equally large as the difference in the same subject due to the hysteresis, which, for reasons of simplicity, has not been modelled in the processor. It is therefore reasonable to adopt a single average value of 8” for dp for all subjects. The proportionality factor M, was more variable between subjects, it has been given in Tables 2 and 3. The PEC curve from the processor has been included in Figs. 4a and 4b. It is seen that, in spite of the hysteresis effects not being modelled, it is accurate within some Nm in the practical range of 75” < d, < 1 lo”. Torque-angle
parameters
parameters of the torque-angle relation were obtained from measurements of the EMG gain factor as a function of the angle of the ankle (Fig. 5). The breakpoints of the measuredf(&) relation at/(d) = 0 andf(&) = 1 gave &‘,and 4; respectively, g, is equal to the level of the horizontal part at small 4. The values of 4, were obtained from &‘,by adding the stretch of the SEC : I$, = t#~‘,+ t#~,( M ), which depended of course on the values adopted for /? and K and on the torque at The
765
which the measurement was done: 4, was 30’ for subjects 1 and 6,26’for subject 3 and 39’ for the others. In subjects l-4 these values of 9, and 9, - 4z for the soleus gave satisfactory results in the isotonic contractions. For the combination of gastrocnemius and soleus the determination of 4, and 4, - 4r from isometric measurements was somewhat less precise. In several cases these parameters had to be changed somewhat lo obtain a good fit for slow isotonic contractions. Torque-angular
velocity
parameters
At positive velocity the torque-angular velocity (Hill) relation (Part 1, equation 5) is determined by the parameters b and n, of which the latter is onlj important at the higher speeds. The part at negative velocity is determined by the parameter c. It turned out to be impracticable to find b and n by varying them both and minimizing the ISE. Therefore they were kept in a fixed proportion on the basis of the following. From the Hill relation it follows that the intrinsic velocity Cp,.i.e. the maximal unloaded speed of the muscle, is equal to b/n. All subjects were asked to plantarllex their unloaded foot as fast as possible. This gave peak velocities of 8-12 rad!s with an average ofca 10 rad/s and without large systematic differences between subjects. The intrinsic velocity of the muscle analogue has been kept at this value by always taking )I = b/g,, with Cp, = 10 rad/s. In fact tt could be varied quite widely around this value without noticeable effects, the often cited literature value t1 = 0.25 was certainly too high. however. The parameters b and c were determined by minimizing the ISE for the isotonic contractions at positive and negative velocity. respectively. Figure 3 gives the ISE at positive speed as a function of b for subject 4. Contractions of moderate and fast positive
trir f
lp; 3 g,+ --_ _ - - _ __ J\
‘\
\
Fig. 5. /(o) relation. The EMG gain factor has been plotted as a function of the angle $ for subject 4, soleus only. These ratios have been obtained by drawing straight lines through X-Y plots of the mean recrified EMG vs measured torque. Fluctuations in the EMG cause some uncertainty. the estimated range of which is indicated by the vertical bars. From this figure can be concluded. ti, = 2.8. 6; = 1I P @>= X6
766
A.
L. HOF
and
Jw. VAN
DEN BERG
C
bfmd/sl .wbj.
2
I I
I
I
1
0 I
l-b
t-t&
1
0
0
+-
L
2.0
a5
* t:;
:I
1.2
0
I
-
Fig. 6. Parameters of the soleus.Survey of resultsof the parameter estimation for b and c. For the meaning of the bars see Fig. 3. 0: medium speed (0.5-1.5 rad/s), A: fast speed (> 1.5 rad/s), 0: negative speed. speed were considered separately. Figure 6 gives a survey for the four subjects in the experiments with soleus only. The error curves have been represented by bars on which the minimum and the points where the interpolated curve is at 120% and lSoo/, of the minimum are indicated (cf. Fig. 3 below). Even if in the subjects 2 and 4 the M and F minima do not coincide exactly, for all subjects suitable speed independent optimum values for b could be found, which are indicated in Fig. 6 and listed in Table 2. Three out of the four optimum values for c were zero : a boundary extreme as c cannot be negative. Figure 7 summarizes the results of the parameter estimations for b and c in the experiments with the combination of soleus and gastrocnemius. In a similar
Fig. 7. Parameters of the combined calf muscles. Parameter estimation for b and c. game meaning of symbols as in Fig. 6. The individual optima as well as the average valueshave been indicated.
way as with the soleus experiments it is possible to assign an individual optimum to each subject (these are indicated). From the figure it is clear, however, that the optimum values are so close together that it is possible to assign one single value to each parameter. which is valid for all subjects. These values are b = 1.2 rad/s and c = 0. It can be seen that c = 0 gives an ISE less than 150% of the mimimum in every subject: taking into account that the ISE is a quadratic measure of the error, a 50% increase in ISE corresponds to a 22% increase in r.m.s. error. The ISE does not exceed lSOo/;,of the minimum for fixed b either, with the exception of some cases: the subjects 2 (F), 6 and 7 (M). Processor
perfirniance
The overall performance of the processor. for isotonic contractions and at optimal parameter settings. can be assessed from Fig. 2. The high (up IO 400 Nm in torque measure) and strongly fluctuating rectified EMG U(r) is processed to a model torque which follows the measured torque of 56 Nm maximum quite satisfactory. The errors which remained at the optimum parameter setting have been given in Table 4 for the soleus. Both the r.m.s. error and the peak error are presented. The peak values suggest that the error had a systematic component in some cases. Table 5 gives the r.m.s. and peak errors for the contractions with the combination of soleus and gastrocnemius in the standing position. For the torqueangular velocity parameters the common optimum values b = 1.2 rad/s, n = 0.12 and c = 0 have been used in all subjects. The error is in general slightly larger when the muscles are combined. In part this is because the common optimum value b = 1.2 rad/s is not optimal for some subjects (viz. nos. 2 (F). 6 and 7 (M)). For these subjects the error at their individual optimum for b and II has been given as well (between brackets). The data of Tables 4 and 5 refer IO the isotonic phases of the contractions only.
9.5
N
6
5’ 6
Subject
I
: 4
-10: -8; -8;
-26;
+10 +12 +12
+I2
Peak error
12 14 4
16
N
1.9 5.4 5.0
5.6
r.m.s. error 13
+12 +I8 +I6 +12
-12; -4; -8; -4;
-9 9
N
Peak error
Moderate 0.5 c 4 < 1.5 rad/s
3.4 6.7
4.5 -6; -16;
-12; +I4 +16
+12
Fast $ > 1.5 rad/s r.m.s. error Peak error
2L 19 6
29
N
6.5 8.2 14.9
19.0
+34 +22 -&IO +22
-18; -12; -10; -20;
Negalive cp
2.8-6.7 3.6-4.8 2.8-6.7
3.9-6.7
Predicted stochastic r.m.s. error
6 4
4 6 4 3
2
4
2
3 4 5 6
1
8
IS.1 ~_
7.1)
7.5 6.H 6.9 14.1
13.2 5.5
N
Subject
I
r.m.s. error
range
.
Slow
0: -
-14:
-14; -10: -12; -12:
-14: -16:
+74
+lO
c4 +R +4 +1x
+20 +R
Peak error
N
I!)
7
9 4 R 6
13 7
-__-----
8.3 6.4 7.0 7.3 (4.5) 7.2 (2.8) 9.3
7.1 4.5
r.m.s. error -t16 +6 +R +1? +12 + 16 +I2) +I6 A-12) +24
-4; -8; --to; -6; -6; 0: (-2; -2; (-2; -4:
Peak error
Moderate
4 .__ -._-
5
14 14 8 9
8 14
N
IO0
15.8 6.8 (5.0) 8.9 6.3 5.9 10.5 (6.7) 5.0
r.m.s. error
Fast
~.
0:
-22; -8: I-R; -10: -6: -8. -1x: (-8; --I’.
6
23 IO 4 15
21 6
N
+20 4 ~_.-.~~._-._I._
+26 +4 +4) +I2 +12 +12 +10 +18) +8
Peak error
19.9
12.7
10.4 10.4 9.3 14.2
15.9 1.5
r.m.s. error
+lO +lO +14 +12 +I6 +20 +I2 +24 -__.
-26; -12; -12; -12; -16; -12: -16: -8:
Peak error
Negative
2.2-5.0 .-__.---
2.2-5.0
2.0-3.6 2.2-5.0 2.2-5.0 2.8-5.0
2.8-5.0 2.2-5.0
r.m.s. error
Predicted stochastic
Table 5. R.M.S. error and peak error (negative; positive)for the isotonic phases of the experiments with the combined calf muscle group (in Nm). Measured data are obtained at the average parameter setting for b, n and c (Table 3). For some subjects the error at their individual optimum b has been given in brackets. For predicted stochastic (r.m.s.)error see Discussion. Isotonic torque 40 Nm for subject 3, 56 Nm for the others. N = number of contractionsin a series. Spaed
5.2 6.0 8.2
r.m.s. error
0 -C4 -Z 0.5 rad/s
Slow
Table 4. R.m.s. error = J( I !T) f ,(M - M,)‘dr and peak error (negative ; posifive) for the isotonic phases of the soleusexperiments (in Nm). Measured data refer IO the individual optimum parameter setting of each subject. see Table 2. For predicted stochastic r.m.s. error see Discussion Isotonic torque: 40 Nm for subject 3 and 56 Nm for the others. With subject no. 3 no fast contractions could be done. N = number of contractions in a series,
1 5’ rJa s
$
2 0
768
A. L. HOF and Jw. VAN DEN BERG DISCUSSION
Parallel elastic component The properties of the passive muscle are reasonably modeled by the exponential characteristic of the PEC (Fig. 4). The hysteresis effects are considerable, especially at very low 4 (extreme dorsiflexion). Similar results have been found for many biological soft tissues (Crisp, 1972, Fung, 1972). At extreme plantarflexion the passive torque becomes negative due to the ligaments and the PEC of the dorsiflecting muscles (Gottlieb and Agarwal, 1978). Both effects are not important in the range from 75” to 1lo”, to which the ankle movements are usually confined according to our observations. In the calfergometer experiments 4 is between 90” and 105”. In this range the passive torque is small compared to the active torque and the hysteresis effects are negligible. According to the experiments the exponential factor d,can be considered constant at 8’. but it is necessary to measure the proportionality factor M, for each new subject. Torque-angle parameters The parameters of the torque-angle relation, 4, and 4‘ - & show considerable differences between subjects, for soleus only (Table 2) as well as for both muscles (Table 3). Moreover the model performance was found to be rather sensitive lo the correctness of these parameters. It will therefore be necessary to measure them for every new subject. It seems plausible that the recorded torque-angle relation for the soleus, cf. Fig. 5. reflects the force-length relationship of this muscle. It has been found that the lever arm of the ankle is quite constant over the range of its rotation (Grieve. Cavanagh and Pheasant, 1978). A change of the electrode position with respect lo the muscle fibres might give some contribution to the observed angular dependency of the torque-EMG ratio but the reproducibility of the found f(4,) relations goes against this. Moreover the values found for 4, - & are in reasonable agreement with those to be expected. Soleus (as well as gastrocnemius) is a pennate muscle, with short fibres compared to its origo-insertion length and a long tendon. In a dissection of one cadaver leg of average size (tibia length 40 cm) we measured an ankle lever arm of 5 cm and a soleus fibre length of 3.5 cm. The muscle had been fixed in the anatomical position. i.e. with the foot fully plantartlexed, 4 z 120’, and the fibres thus strongly shortened. The fibre length at C#J= 90’ was estimated from these data as (length at I#J= 120”) + (lever arm x 120’-90’ in radians) = 3.5 + (5 x 0.5) = 6 cm. The force-length relation of a single muscle fibre goes from 0 at 0.6 I, to maximum at I,. This results in til - ti2 = (0.4 x 6)/5 = 0.5 rad = 30’; a value comparable to those found (Table 2). The soleus has its optimum length I,, at an angle of the foot of QI; (cf. Fig. 5). From the data of Table 2 it can be found that this angle is between 75’ and 86”. i.e.
at slight dorsiflexion. The 4, and 4, - @Jovalues given for the combination of both muscles (Table 3) represent a kind of weighted average of those for the soleus and gastrocnemius, the weighting factors depending on the relative contribution of each muscle lo the total torque. This might be the reason why determining these parameters is less precise than when only the soleus is involved. The gastrocnemius length depends not only on the angle of the ankle but also on the knee angle. Grieve er al. (1978) however, found that the effective lever arm of this muscle with respect to the knee is only l/3 of that with respect to the ankle. Thus a slight variation in knee flexion during standing, 10’ at most, does not give serious errors. Torque-angular
velocity parameters
For every subject an optimum value for the torqueangular velocity parameter b (at n = b/10) could be found. In the soleus experiments these optima were somewhat different for three of the four subjects. The individual optimal values have been used in the determination of the errors. For the combined calf muscles the individual optima were close together. Using a common value for b and n introduced little excess error (Table 5). Even in the subjects 2 (F). 6 and 7 (Ml, where the ISE at the common parameter value was more than lSoO/, of its minimum value, the total (r.m.s.) error was not excessively high when compared to those of the other subjects. It was concluded from this that b = 1.2 rad/s and n = 0.12 can be adopted as standard values for normal subjects, though it must be remembered that the value of H relies on the assumption that the intrinsic velocity 4, = 10 rad,s for ail subjects.
At the highest velocities inertial effects become apparent (see the’Calfergometer’section and Fig. 2 right). For example during the onset of the movement the torque of the calf muscles has a peak (of 13.5 Nm max.) above the nominal torque of the spring. As stated in the ‘Calfergometer’ section. part of this inertial torque (5 Nm max.) is not measured by the strain gauges. According to equation (5) of Part 1 this may have caused an at most 157; too high estimation for b. The other part of the peak inertial torque (8.5Nm max. I was taken into account when comparing the calculated torque M and the measured torque M,. However. the contraction was no longer exactly isotonic. The elTect of the torque not being constant during the whole movement is nil if the SEC parameters have their correct values. Because thrs is not certain (they had only provisional values in these experiments), b may have been estimated slightly too high or too low in the fastest contractions. We were not able to discriminate such systematiceffects from those due to the random EMG fluctuations (Figs. 6 and 7). Since the inertial torques are comparable in magnitude to the random (peak) errors. this could indeed hardly be expected (Tables 4 and 5 1. There are only a few references with which the values for b and n can be checked. Bigland and Lippold (1954) do not give explicit numbers. but the torque-velocity curves of their Fig. 5 can quite well be fitted with b = 1.2 rad/s over the range of velocities they covered. Wiikie (1950) gives values of h for the parameter
elbow flexors. Ifconverted
to relative length changes of
EMG IO force processing II
the biceps brachii (fibre length I4 cm) they range from 1.6 to 3.4 fibre lengths per second. For the calf muscles (fibre length 6 cm, lever arm 5 cm) these values range from 1.0 to 1.7 1,/s (corresponding to 1.2-2.0 radis), and are thus somewhat lower. This is consistent with the finding that the calf muscles consist mainly of slow muscle fibres, while the biceps contains a large amount of fast fibres (Schmalbruch. 1970). Literature values for n in human muscles are 0.20-0.48 (Wilkie, 1950, elbow flexors) and 0.29-1.3 (Binkhorst, Hoofd and Vissers, 1977, handgrip muscles). The value we found, n = 0.12, is clearly lower. For most subjects c = 0 was optimal for soleus only, and this value was also found as the common optimum for both muscles. This is the lowest possible value in the model. but in spite of this in several subjects the torque was on the average too high at negative speeds. suggesting that c had to be made still smaller. On the other hand values in the range 0.2-0.5 are suggested by the literature (Joyce and Rack, 1969; Flitney and Hirst, 1978). This contradiction was reflected in the recordings: in some contractions the measured torque was approximated best by choosing c = 0.2 or higher. in other ones the model torque was systematically too high. It seems that the latter condition goesalong with the appearance of large fluctuations in the mean rectified EMG U(r), indicated with arrows in Fig. 2. These synchronous discharges of the EMG are a well known effect (Bigland and Lippold. 1954; Lippold. et al.. 1957; Milner-Brown et al.. 1975). They will give rise to larger fluctuations in the model torque than estimated under the assumption that the EMG is a stationary noise (see below). Moreover. the active state block in the analogue will transform these plus and minus fluctuations into a systematically too high M, because of its inherent non-linear properties. The choice of c = 0 must therefore be seen as a compromise. Errors Even if the processor would work perfectly there would remain a stochastic error. This is because the EMG input signal is a stochastic signal: an amplitude modulated Gaussian noise with zero mean (Schwedyk. Balasubramanian and Scott. 1977). Part of the stochastic fluctuations will remain after the processing, setting a limit to the achievable performance. When a stationary Gaussian noise with equivalent statistical bandwidth B is rectified and subsequently filtered by a linear low-pass filter. the remaining fluctuations have a variation coefficient E = AM (r.m.s.)/M,with (Bendat and Piersol. 1971, Chapter 8): s= &
(r.m.s.)
From a measured power spectrum of the soleus surface EMG we calculated B = 29OHz. In (2) 7 is the elTective averaging time. which depends on the filtering properties of the processor after the rectifier. An analysis is complicated by the non-linearity of the processor (i.e. the muscle model).
769
As a result the filtering properties depend on the state variables M, 4 and 4. Refraining from a detailed analysis, we will give here two estimations for T,which can serve as a lower and an upper bound respectively. (a) T is determined by the properties of the active state converter. As an estimate we may adopt for this case : T =
7,
+
T2 +
53
(3)
which gives T = 25 + 30 + 60 = 115 ms, and, from (2). & = 12%. On the basis of the model properties this value of E can be expected to hold for negative velocities. (b) T is determined by the low-pass properties of the Hill model at positive velocities. The linearized Hill model (Calvert and Chapman, 1977 ; Bawa, Mannard and Stein, 1976) is a first order low-pass filter with time constant T,. The highest value for T,,,. valid at low positive velocity, is: m 7 For a first order filter it holds (Bendat and Piersol. 1971) that: T = 2 T,.
(51
Depending on the value of K used, this gives 7 between 400 and 7OOms and E between 5 and 7”$ Equation (4) explains why the random error decreases at lower values of fi and K (cf. section on EMG gain factors. etc.). Predicted values for the error in the contractions with soleus only, both for (a) and (b). are given in Table 4. It is seen that the measured r.m.s. error in several series of contractions is within the predicted range. In other series, often characterized by asymmetric peak values, some systematic error was present. A possible source of the high. systematically positive error at negative velocities (subjects I and 4) has already been discussed. It is apparen t. nevertheless. that in these experiments the accuracy of the processor in a number of cases is not limited so much by the exactness of the model and its parameters. as by the inevitable statistical fluctuations caused by the EMG signal. In the experiments with the combination of both calf muscles the EMGs of soleus and gastrocnemius are added after rectification. If both signals are uncorrelated (which they supposedly are) and if they are equally strong (which they are not always, of course I the sum signal is twice the component signals, while the r.m.s. error in the sum signal is only ,i2 times as large as the error due to one of the components, This gives, instead of (2):
In the prediction of the error for the experiments with soleus and gastrocnemius combined. Table 5. equation (6) has been used. Although in one suhiect (no. 2) the theoretical minimum is approximated. in
710
A. L. HOF and Jw. VA?; DES BERG
Subjects the measured r.m.s. error is about 2-3 times the theoretical value, while the peak values indicate systematic errors. From inspection of the recordings it appeared that subjects with large errors, like no 8, showed irregularities in the EMG: in almost identical contractions the (rectified) EMG could be considerably different. Maybe in these cases the recorded EMG was less representative for the activity of the whole muscle mm
group. Acknowledgements - The calfergometer was designed by Dr. J. Lubbers and constructed by Mr. L. Martijn. The advice of Professor H. B. K. Boom was ofgreat value in the preparation of the manuscript.
Joyce, G. C. and Rack, P. M. H. (1969) Isotonic lengthening and shortening movements ofcat soleus muscle. J. Physiol.
Land. 204.475-491. Lippold, 0. C. J.. Naylor, P. F. D. and Treadwell, E. E. E. (1952) A dynamometer for the human calf muscles. 1. sri. Instrum. 29, 365-366. Lippold 0. C. J., Redfcam, J. W. T., V&o. 1. (1957) The rhythmical activity of groups of motorunits in the voluntary contraction of muscle. J. Physiol., Land. 137,473487. Milncr-Brown, H. S., Stein, R. B., Lee, R. G. (1975)Synchronixation of human motorunits: possible roles ofexercise and supra spinal reflexes. EEG clin. Neurophysiol. 38,245-2X Schmalbntch, H. (1970)Dic quergcstrciftm Muskelfascm des Mmschcn. Erg&n. Anat. EnrwGesch. 43, Heft 1. Shwcdyk, E., Balasubramanian, R. and Scott, R. N. (1977) A nonstationary model for thcclcctromyogram. IEEE Truns. Biohfed. Engng 24,417-424. Wilkic. D. R. (1950) The relation between force and velocity in human muscle. J. Physiol. Land. 110, 249-280.
REFERENCES
Bawa. P., Mannard, A. and Stein, R. B. (1976) Predictions and experimental tests of a viscoclastic muscle model using elastic and inertial loads. Biol. Cybernet. 22, 139-147. Bcndat, J. S.. Piersol, A. G. (1971) Rondom Dara, Analysis ond Meosuremenr. Wiley-Intcrscima, New York. Bigland, B. and Lippold. 0. C. J. (1954) The relation between fora, velocity and integrated electrical activity in human muscles. J. Physiol. Land. 123, 214-224. Binkhorst, R. A., Hoofd. L. and Visscrs, A. C. A. (1977) Temperature and force-velocity relationship of human muscles. 1. appl. Physiol. 42,471-475. Calvert, T. W., Chapman, A. E. (1977) The relationship between the surface EMG and fora transients in muscle : Simulation and experimental studies. Proc. IEEE 65, 682-689. Crisp, J. D. C. (1972) Properties of tendon and skin. In Biomechanics. its Foundations and Objecriws, (Edited by Y. C. B. Fung et al.) pp. 14l- 180, hen tia-Hall, Englewood ClilTs,New Jersey. Flitney, F. W., Hirst, D. G. (1978) Cross-bridge detachment and sarcomere ‘give’ during stretch of active frog’s muscle. J. Physiol. Land. 276,449-465.
Fung, Y. C. B. (1972) Stress-strain-history relations of soft tissues in simple elongation. In Biomechunics. its Foundarions und Objectives, (Edited by Y. C. B. Fung et al.) pp. 181-208, Prentia-Hall, Engkwood ClitTs,New Jersey. Gottlieb, G. L. and Agarwal, G. C. (1978) Dcpmdcna of human ankle compliance on joint angle. J. Biomechanics 11, 177-181. Grieve, D. W., Cavanagh, P. R. and Pheasant, S. (1978) Prediction of gastrocnemius length from knee and ankle posture. In Biomechonics VI (Edited by E. Asmusscn and K. Jdrgcnsen) pp. 405-412, University Park Press, Baltimore. Hof. A. L. and Van den Berg, Jw. (1977) Linearity betweenthe weighted sum of the EMGs of the human triceps surac and the total torque. J. Biomechanics 10, 529-540. Hof, A. L. and Van den Berg, Jw. (1981a) EMG to fora processing I: An electrical analogue of the Hill muscle model. J. Biomechonics 14, 747-758. Hof, A. L. and Van den Berg, Jw. (1981b) EMG to fora processing III. Estimation of model parameters for the human triceps surae muscle and assessment of the accuracy by means of a torque plate. J. Biomechanics 14, 771-785.
NOMENCLATURE b B C
I(#) f(6,)
AM M, M, M, n
velocity parameter torque-velocity relation (5) (rad/s) equivalent statistical bandwidth of EMG signal (Hz) parameter torque-velocity relation at negative vclocities (6) torque-angle relation measured as a function of thC angle of the ankle torque-angle relation as function of CC-length 4, (form as used in the model) torque-EMG ratio gastrocnemius ditto for soleus linear elasticity parameter SEC (8) (Nm/rad) resting length of muscle (cm) total torque. as computed by the processor = .U, + M, (Nm) r.m.s. error in M = til’T){~M - M,Ydr (Nm) parameter of SEC = 1 Nm torque measured with the calfergometer (Nm) proportionality constant of PEC (9) (Nml parameter torque-velocity relation time (s) effective averaging time (s) rectified and smoothed EMG, gain set by g1 and gB tNm1 work done by the muscle = IM@dr (J) logarithmic elasticity parameter SEC (8) &ad- ‘ ) = AM/M_ coefficient of variation, in isotonic contraction time constant EMG smoothing filter = 25 ms plateau duration active state (ms) time constant exponential decay of active state (ms) time constant of linearized Hill muscle model (ms) angle of the ankle joint (deg) = d&dr (rad/s) angle corresponding to CC-length = 4 + 4. (deg) SEC stretch (rad) intrinsic velocity of muscle = (b/n)f(#,) &ad/s) exponential constant PEC (deg) parameters torque-angle relation (3) (degi parameters torque-angle relation before correction for the stretch of the SEC 4,: 4, = d’, + d,(M) (deg).