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co21 029081 11074? II 50200’0 C Pereamon Prcs Ltd 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. REFERESCES Abbott, B. C. and Aubert, X. M. (1951) The force exerted by active striated muscle during and after change of length. J. Physiol. Lend. 117, 77-86. Abbott, B. C. and Wilkie, D. R. (1953) The relation between velocity of shortening and the tension-length curve of skeletal muscle. J. Phpsiol. Lund. 120, 214-223. Bahler, A. S. 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(1950) The relation between force and velocit! in human muscle. J. Phyiol. Lond. 110, 249. 280. Wilkie. D. R. (1956) Mechanical properties of muscle Br med Bull 12. 177-183. 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)