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EMG TO FORCE PROCESSING I:
AN ELECTRICAL ANALOGUE OF THE
HILL MUSCLE MODEL*
A. L. HOF and J\v. V.AS DEY BERG
Laboratory for Medical Physics, State University Groningen, Bloemsingel 10, 9712 KZ Gronmgen. The
Netherlands
Aktract-A
processing method is presented by which the surfaaekctromyogram can k processed to the
muscle fora. The smoothed rectified EMG and the joint angle (proportional to the muscle length) are the
inputs ofan ekctrical analogue ofthe Hill musck model. The output of the analogue is the torque around the
joint due to the muscle force and, by multiplication with the joint rotation, the muscle work. Both can be
obtained instantaneously, irrespective of the type of contraction (isometric, isotonic, auxotonic. etc.). The
processor is described and some potentialities are shown. Methods by whieh to find the parameter values of
the musck model and methods for the evaluation of the processor performana are discussed.The rekvant
experiments, on the human calf muscles (M. triceps surae), will be described in the subsequent parts of this
series of papers.
ISTRODUCTION
Electromyograph_v
as a means to obtain
The processor imitates this behavlour of the muscle.
The two inputs of the processor are the EMG (which
can be considered a measure for the muscle’s acti-
musclejorce
Electromyography has found widespread and fruitful applications in the study of mdvements. Mostly,
however, the EMG is used only in a qualitative way,
only a few levels of activity are discriminated at sight:
nil, moderate, strong, very strong (Basmajian 1974).
A quantitative analysis is, as a rule, only used for
(quasi-) static isometric contractions. In this case it has
been shown that the mean rectified EMG signal is
linearly proportional to the muscle force (e.g. Lippold.
1952; Hof and Van den Berg, 1977; Pruim, Ten Bosch
and De Jongh, 1978).
Nevertheless a quantitative relationship was shown
to exist for some non-isometric contractions. Bigland
and Llppold (1954) found that in contractions with
constant force (isotonic) and constant velocity the
proportionality factor between mean rectified EMG
and force was dependent on the velocity of the
contraction. Their results could be explained by the
Hill force-velocity relation (Hill, 1938) by assuming
that the mean rectified EMG is proportional to the SOcalled active state (Hill, 1949).
In a series of four papers we will describe and test a
method for processing the force of a muscle from its
EMG without restrictions to any specific type of
contraction. The aim is to provide a quantitative
method for determining muscle force in kinesiological
vation)
and the joint
angle (which
we provisionally
length). The
essential part of the processor is thus an electrical
analogue of a muscle model.
assume
to be proportional
Whether
some version of a muscle model is suitable
to be incorporated
timately
to muscle
in the EMG
processor
be decided by its performance
the torque of an in uivo muscle-tendon-Jomt
from the EMG
will ul-
in predicting
system
and the joint angle. This does not alter
the fact that in designing a muscle model the avallable
physiological
knowledge
should be used as often as
possible.
Accordingly,
scription
the complete
study
by an assessment ofitsaccuracy
and Van
contains
a de-
of the processor (this paper. Part I) followed
den Berg.
additional
series of experiments
termine some parameters
is described
(Parts Ill
and IV. Hof
1981 b. c). It appeared
thar an
was needed IO de-
of the muscle model. and this
in Part II (Hof and Van den Berg. 198la).
In principle
the processor is suitable for any skeletal
muscle. In this study it has been applied to the human
calf muscle group. M. triceps surae, conasting of M.
soleus and M. gastrocnemius, the principal plantarflexors of the foot.
studies.
The Hill muscle model
In our processor
Hill
Prirlcrple $ the processor
(1938).
Wilkie
The basic diagram of the processor is given in Fig. 1.
The torque M which a muscle develops around a joint
we used the muscle model of
Complete
descriptions
(1958). Ascan be
seen in Fig. 2, it consists of three components.
parallel elastic component
is determined by the muscle’s activation and length.
can be found in
(1956) and Ritchie and Wilkie
(PEC)
The
represents the elas-
ticity of the passive muscle and the joint ligaments. The
behaviour
* Receitrd in retlised,lorfn I8 May 1981.
contractile
747
of the active muscle is determined
component
(CC)
and
by the
the series elastic
148
A. L: HOF and
Fig. I.
Jw. VAS DES BERG
Block diagram of the processingmethod.
component (SEC), which are connected in series. The
SEC and PEC are characterized by their loadextension diagrams, the length of the SEC being
represented by 4, and that of the PEC by 4. The
torque developed by the CC is a function of three state
variables: the active state M,. the CC’length’b, and its
derivative 4,. An important-but
not unchallenged
(e.g. Jewel1 and Wilkie, 1960; Julian and Moss.
1976)-assumption
of the Hill model is that the
active state, in constrast to the resulting muscle torque.
is not influenced by mechanical events, i.e. shortening
or kngthening of the muscle. The relation between the
active state and the muscle torque is comparable to
that between the electromotive force and the output
voltage of a battery. In our model the sum of the active
states of all motor units is taken to be the active state of
the whole muscle and this signal is derived from the
EMG. In fact the presence of an active state in the Hill
model, a variable which is absent in most recent muscle
models, favours the use of this model in an EMG
processor, because the active state can be coupled in a
relatively simple way to the rectified EMG.
Force or torque
When referring to the it1 riro situation muscle
torques and joint angles will be used in this study
they can be measured and calibrated directly,
in contrast to muscle forces and lengths, which can
only be calculated if the lever arm of the muscle with
respect to the joint is known. This kver arm can be
estimated on the basis ofdata from dissections (Grieve,
et al., 1978). In the calf muscles. fortunately. this does
not involve geometrical complications because it appears from the same study that the lever arm with
respect to the ankle is quite constant over the whole
range of motion. Thus the proportionality
between
muscle length andjoint angle, as assumed above. seems
justified in this case. In the experiments to be described
the knee angle was constant. so it did not influence
gastrocnemius length. The use of the term muscle
torque. however, does not imply that only the combined torque of all muscles acting on a joint can be
measured with our method (as with the more usual
kinesiological methods); it refers to the force times
lever arm of a single muscle or synergistic muscle
group. In theoretical discussions or with reference to in
drro experiments it is often clearer to adhere to the
terms muscle force and length.
Muscle work can be obtained bx multiplying the
muscle torque with the angular velocity and integrating with respect to time.
because
DESCRIPTION OF THE PROCESSOR
PEC
I
M
c
load
Fig. 2.
Schematicdiagram ofthe Hill muscle model. See text.
The EMG is recorded by bipolar surface electrodes.
It is necessary to use separate recordings from the
soleus and gastrocnemius muscles as described by HOT
and Van den Berg (1977). Both EMGs are preamplified 100x and bandpass filtered (30-6OOHz)
(Fig. 3). The gains of the bandpass amplifiers are
variable: gr and go for soleus and gastrocnemius
respectively. After this the EMGs are full-wave rectified, added and smoothed by means of a third order
averaging filter with a time constant T, of 25ms
(Garland, Angel and Melen, 1972). The preprocessing
of the EMGs yields a voltage U(r).
The EMG power density spectrum extends from ccz
10 Hz-300 HI. being maximal from ca 50 Hz-150 Hz
Schwedyk er al., 1977 and own measurements). The high
frequency cut-oil (600 Hz) restricts the amplifier noise. The
frequenaes under 30 Hz are suppressed because this great]?
reduces the low frequent electrode artifacts. In this way also a
small part of the EMG power is filtered out. but this will not
749
EMC to force processing I
result in systematic errors as the form of the power spectrum
does not change with the strength of the contraction
(!Schwedyk ef 01.. 1977). For the choice of r, see the section
‘EMG to active state conversion’.
The angle of the ankle 4 is recorded by an
electrogoniometer ; a potentiometer strapped on the
lower leg, its axis being connected to a footplate in the
shoe (HOT 1980).
The angle I#Iis defined as follows: in upright standing 4 =
90’ and increasing r#~corresponds to plantarflexion. We use
this sign convention for &Ito obtain a positive value for the
angular velocities 6, and 4, at muscle shortening, as is usual in
musck physiology. If the SEC extension, expressed as the
angle I$., is counted positive. this gives c$, = I#I+ dC The
angular velocities are given in radians per second while the
angles are given in degrees. The only reason for this is
convenience: ‘90”’ is easier to interpret than ‘1.57 rad’.
The muscle model
Figure 4 gives a block diagram of the muscle model,
a look inside the right hand block of Fig. I. The
components of the Hill model (Fig. 2) are all incorporated, the properties of the contractile component are represented separately : the active state, the
torque-angle and the torque-velocity (Hill) relation.
Ihe model has been constructed with hard-wired
electronic circuits in which voltages represent the
variables.
EMG lo acriw stale corlrersion
The EMG to active state convertor is a crucial
component of the processor ; it is the link between the
preprocessed EMG and the muscle model. Our convertor can be described mathematically as :
U(l-AI)
conversion
is correct, but we can see that it yields the
well known properties of the active state of a muscle
in various situations.
--In the quasi-static situation M,(r)
=
U(r). i.e., the
smoothed rectified EMG is linearly proportional to
the active state, in accordance with the results of
Bigland and Lippold (1954).
-The response of M,(r) to a short EMG burst, see
Figs 5 and 6(a), has the form of the active state of a
twitch (Hill, 1949; Wilkie, 1956; Jewel1 and Wilkie,
1960).
-The
response to a series of impulses, to a step
function (comparable to a tetanus) and to some
combinations of a step and an impulse are also given
in Fig. 6. Note the summation effects.
The steepness of the leading edge, corresponding to the fast
rising phase ol the active state, is determined by the time
constant r, of the EMG smoothing filter (see the section
‘Input stgnals’ and Fig. 3).
According to physiological data (Edman, 1970) r, should
be very short. On the other hand one should prefer a large 7,
to obtatn a good smoothing ofthe rectified EMG. For the calf
muscles r, = 25 ms offers a reasonable compromise. This was
checked by comparing the responses of the complete muscle
model. with the other parameters set at the experimentally
determined values (Part III), to a step input at the rectified
EMG summingpoint(Fig. 3). For T, = O(no smoothing)and
7 , = 25ms the difference could hardly be seen, but rt =
50ms Introduced a noticeable slowing down of the step
response
Both other parameters, plateau duration r2 and decay
constant 7, are free to choose The choice of an exponential
decay profile is justified by literature data (Jewel1and Wilkie.
1960).
0 2 Ar < r2
0)
MJr) = max.
2 5 Al
I
V(r) is the preprocessed EMG, already
smoothed with the time constant 7,. The ‘max.’
operator denotes that the maximal value of the
respective functions is taken over the interval indicated. The working is such that M,(r) follows V(r)
when U(r) is rising. When a relative maximum of U(r)
occurs, this maximum is held for a time 7* and followed
by an exponential decay with a time constant 7,.
Fig. 5. The plateau or decay of M,,(r) goes on until U(r)
or the pfateau resulting from a subsequent relative
in which
maximum of U(r) becomes larger than M,,(r), cf. Fig. 6.
Of course there is no direct proof that this way of
IForque-angle
relation
When a muscle is kept isometric the force developed
depends on the length of the muscle (Gordon et al..
1966). In fact this dependence reflects the properties of
the contractile component. In our notation :
M, = M,f(4,)
(isometric).
(2)
The function f(d,) is equal to 1 around the optimum
muscle length and decreases for smaller and for larger
lengths. In the body, the calf muscles as a rule cannot
attain the longer lengths because the foot dorsiflexion
is limited by the ankle ligaments, so it is sufficient to
Fig. 3. Preprocessing of the EMGs
A. L. HOF and Jw. VAN DES BERG
750
ue
OCllVe
sfate
*
pcpocessed
-
HIII
MO
b
relation
-
EMG
t
t
torqwangle
relation
r+
go
joml
angle
@e ‘-,
dtfferentlator
Gc
I
senes
elashc
-
COmpOtWlt
parallel
,
r‘
I
l------l
; _ _ _ _ _ ;dlffeyWaloc
Mp
elastic
component
-+
____-_-___
r-Lf___l
.
L 2 _ ;
mApher
L f.9:
I
:_____;
:_._____I
2
r----7
cntegmtor
J
r;
L_-____,work
*Wed
L
_~
1muscle
reset
Fig. 4. Block diagram of the muscle model. Components necessary for the calculation of muscle power and
work are in dashed lines. The multiplier is type AD-424 or AD-531 (Analog Devices).
make allowance for the effect at muscle lengths around
or shorter than optimum. Forj(4,) we chose:
I
1
/(&) =
d, < 42
$$
I-
2
for
42 c 4, < 6,
LO
(3)
M = M
(I
4c>4,.
This function is depicted in Fig. 7 (dashed line). it
must be kept in mind that increasing #, corresponds lo
shorter CC lengths. It is obvious that the length of the
SEC must be known in ordur to calculate 4, = 4 + 4,
Torque-angular velocity relation
The force-velocity relation for isotonic contractions
asgiven by Hill (1938), but written with our symbols. is
(M,+ a) (d,+ b) = Pf,
+ a&
relation holds for muscle lengths around the optimum,
i.e. when/(4,) -c 1. Abbot and Wilkie (1953) showed
(for frog muscles)that the Hill relation can beextended
to apply also to shorter muscle lengths, simply by
replacing M, by M, f(tj,).
This gives after some
rearrangement :
(4)
in which b is a constant and a = nM,, with II a
constant. Equation (4) was originally found for frog
muscles at OT, but it applies to human muscles at
body temperature as well (e.g. Wilkie 1950). This
f(k)-ndclb
’
1+&/b
for 6,<4,.
(5)
The maximum (unloaded) velocity 4, follows easily
from this asd, = (b/n)/(4J,
it also depends on/(4c).
For mere mathematical reasons equation 5 cannot
apply to all negative (stretching) velocities. We have
adopted a relation which is congruent with results
obtained by Joyce and Rack (1969) on the soieus
muscle of the cat at body temperature and with a
distributed stimulation at low frequency; relation (5)
extends into the region of negative velocities until a
maximal value of M, is attained, which is determined
by the parameter c:
M, I (1 +c)M,f(&).
(6)
At higher negative speeds M, remains constant at this
value. The complete torque-velocity relation is shown
in Fig. 8.
There is some controversy in literature about the mechani-
Fig. 5. Schematic profile of the active state after a short EMG
burst.
cal behaviour of striated muscle at negative velocities. Some
authors (Katz, 1939; Abbot and Aubert, 1952)report that the
fora increases much more strongly with negative velocity
than according to (5). Sometimes it is even reported (Abbot
and Aubert, 1952 ; Edman er ol., 1978) that the fora does not
return to its isometric value after the lengthening. It must be
taken into account. however. that these results refer lo
experiments on frog muscles or muscles fibres at 0°C with
tetanic stimulation.
751
EMG to force processing
200
Nm
h
Fig 6. Response of the active state convertor to some input signals. Parameter settingsr, = 25 ms, z2 = 30
ms,r, = 6Oms.(a)Oneimpulse,duration lOms.(b)Twosuch impulses, 100msapart.(c)Twoimpulses40ms
apart. (d),(e)and(f)Combinationsofastepand
an impulse.(g),(h)and(i),as(d).(e)and(f)
but now theinput
signal after filtering with the smoothing filter L’(r) IS recorded together with the output signal.
as-
%
Fig 7. Torque-angle relation /(d,). Dashed line: accordrng
to equation (3) with d, = 159’ and 4, - dr = 31”. Drawn
line: actualoutput of the processorcircuit, which isessentially
a diode limited amplifier. The contours are rounded because.
on purpose, only a simple diode circuit has been used. This
agrees better with the real/@,) relation than equation (3).
Series elastic componetlr
The serieselastic component (SEC) is determined by
the relation between load M, (equal to the torque of
the contractile component) and extension 4V It is
generally agreed in literature (Ritchie and Wilkie.
1958 ; Joyce and Rack. 1969) that the SEC compliance.
i.e. inverse elasticity, decreases with increasing M,.
One way to express this is to assume that the total
compliance is the sum of a constant term l/K and a
term inversely proportional to M,:
wedM,-i<l=
l
1
P(M,+M,f
+ i’
(7)
in which K’ is the effective elasticity, and 8. M, and K
are parameters. The extension of the SEC is found by
integrating
I
-5
Fig. 8. Torque-velocrt! relation, recorded with the analopue circuit. Parameter settings:b = 2.25, n = 0.25, c = 0.2.
The circuit was built with the multiplier-divider circuit AD531 (Analog Devices~ Some calculated points are included.
M+M
= $In =
M
+ $
(8)
M,
This function is shown in Fig. 9 for some values of fi
and K.
M, has been introduced in order to ensure that the
compliance is finite for .M, + 0. If M, is chosen small with
respect to the usual range of torques, 20-2OONm, it does not
influence the compliance (7) or the o~ored elastic energy
(M, dd, appreciably and the SEC extension 4, then changes
;*irtuall~ onl! b, the addrtion of a constant factor. The latter
752
A. L. HOF and Jw. VAN DENBERG
as-
%
Y)
Ma
(Nml IO0
Fig. 9. Series-elasticcomponent. Stretch 4, as a function of torque Ma. Recorded with the processorcircuit.
consisting of a linear and a logarithmic amplifier. M, = 1 Nm. The parameters b and K can be varied with
potentiometers over rangesofO-1C@rad-‘andO-10,CKNJNm/rad.
r respectively. Somecalculated points are
included.
effectis not important to the functioning of the model. So the
choice of M,is rather arbitrary; a constant value of I Nm has
5
D(s) =
(1 +s?,,)(l
been chosen.
Parallel
elastic
component
The parallel elastic component represents the
mechanical properties of the muscle-tendon-joint
system when the muscle is not active. It may be thought
here of the passive muscle and the joint ligaments.
Literature data indicate that the passive force of these
structures is an exponential function of their length
(Buchthal er al., 1956; Bendall, 1969; Fung, 1972).
Thus the PEC was modelled as:
M, = M,exp(-
7).
(9)
The parameters are the exponential constant r$,,and a
proportionality factor M,. The expression given here
accounts only for a torque increasing with dorsiflexion
(small 4). At extreme plantarflexion the ligaments and
the antagonist muscles will restrict the ankle movement and give rise to a negative M,. Thiseffect has not
been included in the model, however.
Assembly
The lay-out of the complete model has already been
Standard operational amplifiers are
used throughout. Some details about the circuits of the
muscle model are given in the figure legends.
given in Fig. 4.
The key component oftheelcctroniccircuit for the EMG to
active state conversion, equation (1). is a ‘tapped analogue
delay’ TAD-32 (Reticon). It is a 32-stage analoguc &lay line
in which each stage is brought to the outside. Each output is
connected via a diode and a resistor to the input of an
operational amplifier. By means of this circuit the maximum
of the delayed values of U(r) is determined. The exponential
decay for Ar > z2 is performed by a peak detector circuit
connected to the last May stage.
There are two dikrcntiators in the analogue, the transfer
function of which is
+sr,,j
(101
Differentiator I is included in a feedback loop, together with
the SEC and the torque-velocity relation. In order to reduce
the open loop gain a value of 6.8 ms has been chosen for rL,.
together with T,,~ = 0.24 ms.
The second differentiator has z~* = rDI = 6.8 ms, which
reduces the noise from the electrogoniometer. This differentiator is used for the calculation of muscle work:
U’lI, = SM. 6,dr
(111
This is done electronically. First M and 4 are multiplied by
means of an analogue multiplier and afterwards they are
integrated in a standard integrator circuit. The error in the
work due 10 electronic imperfections is less than 0.5 J per
second of integration time. The integrator can be reset at each
zero-crossing of 4. In this way separate valuesfor positive and
negative work are obtained. Other signalscan also be usedfor
resetting.
Besides the circuits described thus far the whole
processor contains several simple circuits. such as
buffers, inverters and difference amplifiers. All circuits
have been mounted on printed circuit boards in a l9inch cabinet. Two processors have been built up 10
now. one of which is shown in Fig. 10.
Some
model
responses
The properties of the muscle model can be investigated by applying known test signals at one or
both inputs and studying the various state variables.
From the multitude of possible experiments we have
taken only a few.
If an impulse, or a series of impulses, is applied at the
EMG input of the model this corresponds to the
situation in which a muscle is stimulated by (a series of)
electric impulses. Figure I l(a. b, c) show the responses
of M,and M after I. 2 and 4 impulses. the muscle being
kept isometric. Figure I I(d) gives a short isometric
tetanus, at a higher stimulation rate than in Fig. I I (a.
b, c). The figures show the well known twitch fusion
effects.
EMG IO force processing I
753
,
,
I I(d)
Fig. 10. The EMG processor unit. Parameters areset by the porentiometers
a~ the front. Also shown are the
pre-amplifier box (lobecarried on the back ofthesubject in walkingexperiments)with theelectrodesand the
electroponiometer.
Fig 11. Acitve state M, (upper trace)and isometric muscle torque M (lower trace) of the model as a response
IO(a) one impulse, (b) two impulses 100 ms apart, (c j four impulses 100 ms apart, and (d I a series of impulses
50ms apart. Parameters: r, = 25ms, r2 = 30ms. 7> = 60ms, b = 2_25rad,s. tt = 075. c = 0.2. /I =
IOrad- ‘. K = lOOONm/rad,/(r#t,) = I. Vertical scale: 25 Nmjdiv., horizontal scale: 50 ms dtv. for (a- c). 100
msidiv. for (d).
oscillating (1)
isotonic
-slow (c)
-fast (c)
-negative speed (c. s)
isometric
-fixed angle (c. t)
-variable angle (c)
-twitches (r)
passive rotation (c. s)
parameter
X
9.
EMG
X
%
72
active
stale
X
73
X
X
4,
h-92
torqueangle
X
x
~--
X
b
X
”
torque-velocity
X
c
X
B
SEC
X
K
X
Mm
PEC
X
4.
Table 1. Outline of the parameter estimation procedure. In the first column the instruments are indicated with which the respective contractions can be performed: c = au
ergometer, I = torqueplate, s = spring-flywheel apparatus
A. L. HOF and Jw. VAN DEN BERG
156
DISCUSSION
Parameters
The processor can only work if all model parameters
are known. Table 1 gives a survey of the thirteen
parameters in question and an indication in which
types of contraction they can be estimated, as can be
inferred from the properties of the Hill model. The
parameters fl and M, have not been listed, as they have
been given fixed values (25 ms and I Nm respectively)
as argued in the relevant sections.
The gain factors g1 and g,,, determining the ratio
between static torque and EMG voltage, can be
determined in quasi-static contractions as described
previously (Hofand Van den Berg, 1977). It turned out
that their values have to be determined again for each
subject and for each electrode placement. In the same
type of experiment, but now performed at a series of
different angles of the ankle, the torque-angle relation
can be found, giving 4, and 4, - &2. Another way to
find these parameters is to let the subject perform a
very slow isotonic contraction.
In fast movements there is nearly always a complex
interaction between the SEC and the CC because the
CC length is partly determined by the extension of the
SEC. In order to separate the effects due to these two
components it is necessary to perform experiments
with isotoniccontractions. In isotoniccontractions
the
torque is constant during some time and thus the
extension of the SEC remains the same. On the basis of
these contractions at different velocities the parameters of the torque-angle as well as those of the
torque-velocity relation can be estimated. The relevant
experiments, performed on a calf ergometer. will be
described in Part II of this series (Hof and Van den
Berg, 1981a).
The properties of the SEC can be assessed by means
of a method, not involving EMG, given by Cavagna
(1970). In Table 1 this method is indicated as
‘oscillating’.
The values for the active state time constants TV and
T3 can be inferred from very short isometric contractions, ‘twitches’. The estimation of the parameters
of the active state and the SEC will be given in Part III
(Hof and Van den Berg, 1981b), on the basis of
experiments with a torque plate.
The model
With the Hill muscle model the mechanical properties of striated muscle (and its heat production) can be
described in a relatively simple but accurate way. It is
the basis of several models: simplified versions (Bawa,
Mannard and Stein, 1976; Chapman and Harrower,
1977), or more complicated ones (Bahler. 1968;
Glantz, 1977; Hatze, 1978).
A recent set of theories on the function of the
contractile component is based on the sliding filament
theory of A.F. Huxley (1957,1974). The main purpose
of these theories is not to describe the mechanical
properties of the muscle, but to explain them in terms
of the molecular structure; moving crossbridges which
cause the muscle filaments to slide past each other. For
example, it is possible to derive Hill’s force-velocity
relation. There are some effects, e.g. the transient force
after a very quick stretch or shortening (Ford, Huxley
and Simmons, 1977; Sfenen, Blange and Schnerr,
1978) which can be explained by sliding filament
theories, but not in a simple way by the Hill model in
the form presented here. As muscles in situ and in
natural movements never will be subject to these high
velocities, it was not considered necessary to resort to
the - often very complicated - siiding filament
models.
In discussing the choice of the muscle model it must
be stressed that we are concerned with a problem of a
practical nature: how to use the surface EMG to
determine the muscle torque in natural movements.
This means that the requirements of the model are
restricted in two respects. Because of the noise-like
properties of the EMG input signal, there will always
remain stochastic fluctuations in the torque output
signal. This sets a limit to the achievable accuracy. The
restriction to natural movements means that the
velocities and the accelerations are limited by the
maximal velocity of the muscle and the mass of the
limbs.
Evaluation of the processing
method
An evaluation of the processing method asdescribed
here of course requires experimental data. These data
were obtained with proper test methods and their
description has been divided over two papers, Part III :
experiments on a torque plate and Part IV: experiments on a spring-flywheel apparatus. In these experiments (as well as in those of Part II) human subjects
had to perform contractions on the respective instruments, which permitted measurement of the exerted
joint torque. As care was taken that only the muscle
under investigation (M. triceps surae) was active, the
measured torque M, can be compared to the muscle
torque calculated by the processor. 54. in some
specific types of contractions which cover essentially
the whole range of possible muscular activity.
The accuracy of the processor will be given in three
different measures :
(a) The error in the work IMddt. when compared to
the measured work JM,ddt. This is a practical measure because the processor will often be used to
calculate muscle work and therefore knowledge of the
accuracy of this quantity is desirable.
(b) The error in the integrated torque J’Mdr, when
compared to the measured value lM,dr. This quantity
gives information about the mean processor error over
a contraction.
(c)The peak error, the hghest value of (M - M,) over a
series of contractions. It gives some idea of the
maximum error in M which might occur when using
the processor.
With these experiments the accuracy of the processing method can be assessed. This will give a basis for
757
EMG to force processingI
discussing
the various assumptions
muscle model as it has been pr&nted
involved
in the
Hof, A. L. (1980) EMG
10 force processing An electrical
analogue of the calf muscles for the assessmentof their force
and work.
Thesis
University
of Groningen.
The
Netherlands.
here.
AcknoH,ledgeme,trs-We
wish to thank G. Andries and E.
Albronda for their work on the design and building of the
ekctronics and Prof. Dr. H. B. K. Boom for his valuable
advice in the preparation of the manuscript.
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NOMENCLATCRE
= nM,lNm)
b
velocity parameter
torque-velocity
(rad/s)
parameter torque-velocity
velocities (6)
relation
relation
(5)
at negafilme
torque angle relation
torque-EMG
ratio gastrocnemius
torque-EMG
ratio soleus
linear elasticity parameter SEC (81 (Nm rad)
e&ctive elasticity SEC at .21, (Nm rad)
total torque, as computed by the processor =
\f,
‘f,,
M,+M,
(Nm)
torque due to the CC (Nm)
parameter of SEC (8) = 1 Nm (Nrn~
total ankle torque. as measured bv mechanical
means (Nm)
active state (Nm)
torque from the PEC (Nm I
proportionality
constant of PEC (9 I (km I
758
n
s
u
72
73
A. L. HOF
parameter torque-velocity
Laplaceoperator
and Jw.
relation
time (s)
rectified and smoothed EMG. gain set by g‘ and
9( Wm)
work done by the muscle = jM&lr (J)
logarithmicclasticityparameter
SEC(8)(rad-t)
time constant EMG smoothing filter = 25 ms
(ms)
plateau duration active state (ms)
time constant exponential decay of active state
(ms)
VAN DEN BERG
time constants dilkrentiators (10) (ms)
angk of the ankle joint, see ‘input signals’ (deg)
= d&dr @ad/s)
angle corresponding IO CC-length = 4 +&
(dcg)
= d&./dt (rad/s)
SEC stretch, see (8) (rad)
intrinsic velocity ol tnusck = (b:rr\/(t$,) (rad/s)
exponential constant PEC (dcg)
parameters torque-angle relation 13) (deg)