|
|
J Biomech. Author manuscript; available in PMC 2007 January 1. Published in final edited form as: Published online 2005 October 20. doi: 10.1016/j.jbiomech.2005.08.007. | PMCID: PMC1661835 NIHMSID: NIHMS13334 |
Influence of advanced electromyogram (EMG) amplitude processors on
EMG-to-torque estimation during constant-posture, force-varying
contractions E.A. Clancy,a O. Bida,b and D. Rancourtc a Department of Electrical and Computer Engineering, Department of Biomedical Engineering, Worcester Polytechnic Institute, 100 Institute Rd., Worcester, MA, 01609, USA b Department of Electrical and Computer Engineering, Worcester Polytechnic Institute, Worcester, MA, USA c Department of Mechanical Engineering, Université de Sherbrooke, Sherbrooke, PQ, J1K 2R1, Canada Numerous studies have investigated the relationship between surface EMG
and torque exerted about a joint. Most studies have used conventional EMG
amplitude (EMGamp) processing, such as rectification followed by low pass
filtering, to pre-process the EMG before relating it to torque. Recently,
advanced EMGamp processors that incorporate signal whitening and
multiple-channel combination have been shown to significantly improve EMGamp
processing. In this study, we compared the performance of EMGamp-torque
estimators with and without these advanced EMGamp processors. Fifteen subjects
produced constant-posture, nonfatiguing, force-varying contractions about the
elbow while torque and biceps/triceps EMG were recorded. EMGamp was related to
torque using a linear FIR model. Both whitening and multiple-channel combination
reduced EMG-torque errors and their combination provided an additive benefit.
Using a 15th-order linear FIR model, EMG-torque errors with a
four-channel, whitened processor averaged 7.3% of maximum voluntary contraction
(or 78% of variance accounted for). By comparison, the equivalent
single-channel, unwhitened (conventional) processor produced an average error of
9.9% of maximum voluntary contraction (variance accounted for of 55%). In
addition, the study describes the occurrence of spurious peaks in estimated
torque when the torque model is created from data with a sampling rate well
above the bandwidth of the torque. This problem occurs when the torque data are
sampled at the same rate as the EMG data. The problem is corrected by decimating
the EMGamp prior to relating it to joint torque, in our case to an effective
sampling rate of 40.96 Hz. Keywords: EMG, EMG amplitude, Torque, EMG-torque model, Optimal sampling rate It has long been desired to relate the surface electromyogram (EMG) to the
tensions produced by individual muscles ( Inman et al., 1952). This relation would
provide a noninvasive tool for musculoskeletal assessment in various applications,
including clinical biomechanics, prosthetics control and ergonomics assessment.
However, direct mechanical verification of the estimated individual muscle tensions
is not presently possible in situ, the surface EMG is dominated by the activity of
superficial muscle fibers, and EMG recordings from the skin surface overlying one
muscle are contaminated by crosstalk arising from adjacent muscles. Although a less
specific measure, EMG-based estimates of total torque about a joint
can alleviate these short-comings and still provide valuable information about the
musculoskeletal system. First, net torque about a joint can be easily and accurately
measured in a calibration apparatus. Second, synergistic activation of muscle groups
can render the superficial muscle activity sufficient to identify total joint
torque, even if the individual torque contributions of underlying muscles can not be
discriminated. Third, certain crosstalk contributions are automatically accounted
for in the total joint torque estimate — even if they cannot be
attributed to individual muscle tension contributions (c.f., Clancy, 1991). Hence, noninvasive musculoskeletal research has often utilized EMG-torque
models (e.g., Gottlieb and Agarwal, 1971;
Thelen et al., 1994). A classic paradigm
for relating the EMG waveform to joint torque has emerged (). One or more EMG electrode recordings are made from
agonist and antagonist muscle groups, and EMG amplitude (EMGamp) estimates are
computed for each muscle group. EMGamp represents the degree of neural input to the
muscle groups. A model relationship is then constructed that relates
agonist-antagonist EMGamp's to total joint torque. Prior studies, as well as this
study, fundamentally assume that EMGamp can be identified from the
“raw” EMG waveform and that joint torque is an identifiable
function of the EMGamp's. Although individual torque contributions appear as
internal states of the model (see ), only
total joint torque contributes to the estimation error. Hence, the internal torques
need not, and are not, verified or used. Both linear ( Gottlieb and Agarwal, 1971; Thelen et al., 1994) and non-linear ( Hof
and Van den Berg, 1981; Wyss and Pollack,
1984; Zuniga and Simons, 1969)
dynamical models have been used to relate EMGamp to torque. In addition, modern
estimation techniques routinely model agonist-antagonist activity, since antagonist
muscle activity accompanies agonist contraction ( An
et al., 1983; Solomonow et al.,
1986); as well as optimize some or all of the system identification stage to
individual experimental subjects. While the system identification models have received a great deal of
attention, advances in EMGamp processing have seen little incorporation into
EMG-torque estimation. Most EMG-torque techniques still use conventional rectify and
low pass filter processing of the raw EMG. State of the art EMGamp estimation
incorporates signal whitening and multiple channel combination (see Clancy et al., 2002, for a review). Combined,
these two techniques improved the signal to noise ratio of EMGamp estimation 187%,
compared to conventional EMGamp processing, during constant-posture, constant-force,
non-fatiguing contractions ( Clancy and Hogan,
1995). During quasi-isotonic contractions about the elbow, these techniques
reduced the EMG-torque error by one third ( Clancy and
Hogan, 1997). The advanced EMG processing techniques have also shown
improvements during dynamic contractions. Subjects used real-time EMGamp feedback to
track a 0.25 Hz bandwidth random target during constant-posture force-varying
contractions. Compared to conventional EMGamp processors, the advanced processors
reduced tracking error approximately half-way to the error achieved using direct
force feedback ( Clancy and Farry, 2000).
Preliminary EMG-torque modeling has also been conducted with these dynamic data
( Clancy et al., 2001). Initial results
suggest that the advanced EMGamp processors lead to reduced EMG-torque error;
however, the system identification stage exhibited a large proportion of
“non-convergent” trials, i.e. recordings that did not produce
usable EMG-torque models. In this report, we re-examined the dynamic EMG-torque modeling of Clancy et al. (2001). Based on the results of
prior studies, we hypothesized that the inclusion of whitening and multiple-channel
combination in amplitude estimation would each decrease the error in EMG-torque
modeling, and that their simultaneous use would reduce error further. We also
hypothesized that the issue of “non-convergent” trials was not
a fundamental limitation to utilizing the advanced EMG processors in EMG-torque
processing, but rather was a correctable artifact of our prior system identification
method. 2.1. Apparatus and experimental methods The experimental apparatus and methods have been described and pictured
in detail elsewhere ( Clancy, 1999; Clancy and Farry, 2000). Briefly, after
securing written informed consent, subjects were strapped into the seat of a
Biodex exercise machine (Biodex Medical Systems, Inc., Shirley, NY). A subject's
right arm was positioned in the plane parallel to the floor, with the shoulder
abducted 90°, the forearm oriented in the parasaggital plane, the
wrist fully supinated and the elbow flexed 90°. The wrist was rigidly
attached to the Biodex dynamometer with a cuff at the styloid process. This
constant posture was utilized for all experimental trials. The skin above the investigated muscles was cleaned with an alcohol wipe
and four EMG electrode-amplifiers (Liberty Technology model MYO115, Hopkinton,
MA) were placed over each of the biceps and triceps muscles, midway between the
elbow and the midpoint of the upper arm, centered on the muscle midline. This
placement attempted to avoid the innervation region (at the muscles' midpoints)
as well as the muscles' tendons. The two contacts (4 mm diameter, stainless
steel, separated 15 mm center-to-center) of each electrode-amplifier were
oriented along the muscle's long axis. Adjacent electrode-amplifier centers were
spaced 1.75 cm apart, transversely across the arm. The ground electrode was
applied over the acromion process. Each electrode-amplifier had a gain of 725, a
common mode rejection ratio of 90 dB at 60 Hz, and a second-order
10–2000 Hz bandpass filter. Each electrode-amplifier output was
electrically isolated, amplified, and low pass filtered (fourth-order filter at
2000 Hz). This wide bandwidth for surface EMG has been shown previously to be
advantageous when signal whitening is applied ( Clancy and Hogan, 1994; Clancy and
Farry, 2000). Recordings with the two contacts of each
electrode-amplifier shorted gave a measure of equipment noise, which averaged
2.1±1.7% of the root mean square EMG at 50% maximum voluntary
contraction (MVC). The EMG and dynamometer signals were sampled at 4096 Hz using
a 16-bit A/D converter (Computer Boards model CIO-DAS1600/16, Mansfield, MA). Fifteen healthy subjects (eight male, seven female; aged 23–65
years) each completed one experiment. Subjects initially performed two 2 s MVCs
each in flexion and extension, the averages of which were used as the subject's
MVCs for the experiment. Next, they performed a 0% MVC (rest contraction) and
separate flexion and extension 50% MVCs for five seconds, utilizing force
feedback on a computer screen. These contractions were used to calibrate the
advanced EMGamp processors ( Clancy and Farry,
2000). The subjects then performed dynamic (constant-posture,
force-varying) target tracking contractions. A computer screen displayed one of
either their elbow joint torque (the dynamometer signal) or the algebraic
difference between real-time biceps and triceps EMGamp, as a biofeedback signal.
The EMGamp difference provided a biofeedback signal that was similar in
character to the torque feedback, albeit with increased variance. The distinct
biofeedback signals were not necessary for this EMG-torque study, but were
utilized for a companion study conducted with these same experimental trials
( Clancy and Farry, 2000). For this
EMG-torque work, each biofeedback signal produced a torque profile with similar
bandwidth and amplitude characteristics, thus, the post hoc EMG-to-torque
analysis did not distinguish between them. The computer screen also displayed a “target” which
traversed back and forth across the screen in a random fashion. Subjects were
instructed to follow this target as best as possible. Precise tracking
performance was not required for a successful experiment. In fact, the purpose
of the target was to ensure that subjects would produce a torque signal whose
frequency content spanned the frequency range of interest (0–2 Hz).
The random profile followed a (continuous-valued) uniform random distribution
with a bandwidth of either 0.25 Hz (slow tracking) or 1 Hz (fast tracking), over
a range spanning 50% MVC extension to 50% MVC flexion. Subjects took a few
minutes (followed by a 3 minute rest) to become familiar with the tracking task
as none had any prior experience. Subjects then completed 15 slow tracking
trials (three sets of five) and 15 fast tracking trials (three sets of five),
each of 30 s duration. Each set used each of the five biofeedback signals, with
the order of presentation randomized within each set. The subject's arm was
removed from the wrist cuff between all recording trials to allow 2–3
minutes of rest to avoid fatigue. 2.2. Data analysis All data analysis was performed off-line using MATLAB (The Mathworks,
Natick, MA). Four different EMGamp processors were contrasted. In each case, an
EMGamp estimate was produced separately for the biceps and triceps muscle groups
(refer to ). For all processors, the
EMG data were high-pass filtered at 15 Hz using a zero-phase,
10 th-order, FIR filter; detection was performed with an absolute
value operation; and the smoothing stage was omitted, since smoothing was
incorporated within the ensuing analysis step. Processor 1 was the
“conventional” single-channel, unwhitened processor (i.e.,
the digitally high-pass filtered, rectified signal). For each muscle group, an
electrode located centrally on the muscle was selected. Processor 2 was a
single-channel, whitened processor (utilizing the same electrode channels as
Processor 1). Prior to rectification, each channel was whitened using the
adaptive whitening technique of Clancy and Farry
(2000), implemented in a stand-alone MATLAB toolbox ( Clancy, 2004). Processor 3 was a
four-channel, unwhitened processor. After rectification, the four EMG signals
from a muscle group were normalized in magnitude and ensemble averaged.
Processor 4 was a four-channel, whitened processor. Each channel was adaptively
whitened prior to rectification, and then normalized and ensemble averaged. Because the raw EMG were sampled at 4096 Hz, the EMGamp estimates were
also produced at 4096 Hz. The flexion (biceps) and extension (triceps) EMGamp's
formed the inputs to a system identification model in which elbow joint torque
was the output. The torque output, however, has signal power over a
much lower band of frequencies than raw EMG. Therefore, the
EMGamp's could be decimated prior to the ensuing system identification. Various
integer-valued downsampling rates from 1–900 were evaluated. In each
case, the amplitude estimates were first low pass filtered with a cut-off
frequency equal to half the new sampling rate (8th-order Butterworth
filter applied in the forward, then the backward time directions to achieve zero
phase). Note that decimation must occur after an EMG amplitude
estimate has been formed since high-pass filtering, whitening, rectification and
channel combination utilize the full bandwidth of the raw EMG signal. The decimated flexion [F(k), where k is the downsampled
discrete-time sample index] and extension [E( k)] EMGamp's were
related to torque [T( k)] via the dynamic, linear, FIR (a.k.a.,
moving average) model ( Ljung, 1999)
where the ei are extensor model coefficients, the fi are flexor model coefficients and nb is the model order.
A train-test evaluation paradigm was utilized in which the model coefficients
were fit to the data from a training trial and then used to
“predict” the torque from a distinct test trial.
Prediction referred to passing the EMGamp's from the test trial through the
EMGamp-torque model calibrated in the training trial to predict the joint torque
recorded during the test trial. An error signal was formed as the difference
between the predicted and actual test trial torque. For training, optimal fit
coefficients were determined from the overdetermined system of Eq. 1 via linear least squares
( Ljung, 1999). For testing, the 15
trials at a given tracking speed were organized as three sets of five
contractions. Within a set, fit coefficients were trained to one trial, then
tested on the four remaining trials. Thus, for a given processor-decimation
combination, a total of 180 error signals were available (15 subjects
× 3 sets per subject × 1 training trial per set
× 4 test trials per training trial). One second of data from the
beginning and end of each error signal was removed (trimmed), since these data
were corrupted by the startup transients of the various processing filters. EMG-torque error was investigated with two metrics. All errors were
normalized to twice the torque at 50% flexion MVC, denoted %MVC F.
First, the mean absolute error (MAE) was computed for each trial. Second, the
percent variance accounted for (%VAF), defined as ( Kearney, et al., 1997): [where N is the sample duration of the trimmed
error sequence, Error(k)] was computed for each trial.
Statistical comparisons (performed using SAS, SAS Institute, Cary, NC) were made
using the MAE results only, since the %VAF results were derived from the same
error signals. MAE data were analyzed separately for each speed for model orders
1–20 (responses stabilized thereafter). The natural logarithm of the
natural logarithm of each MAE value was used as the response value, since this
transformation succeeded in removing the substantial right skewness in the data.
To account for the correlation across model order, a multivariate mixed linear
model was fit to the data, comprised of a fixed effect due to the EMG processor
and random effects due to the subject, subject-EMG processor interaction,
training set, training set-EMG processor interaction and test set. The model
response was a 20-dimensional vector consisting of ln(ln(MAE)) at models orders
1–20. 3.1. Decimation In the absence of decimation (i.e., when the downsampling factor equaled
one), some of the prediction torque sequences exhibited a few large spikes
— errors that were several orders of magnitude above
the test torque, but lasting only a few samples in duration. Although the spikes
occurred infrequently, their magnitude caused the overall MAE (and %VAF) to be
unrealistic. This error is what caused trials to be considered
“non-convergent” in our prior EMG-torque work ( Clancy et al., 2001). On closer inspection during this study, we were able to establish that
the errors were related to the sampling rate. Even at the fast tracking
bandwidth, 99.9% of the power in the joint torque signal occurred below 4 Hz.
Our raw EMG sampling rate of 4096 Hz represents (un-aliased) power out to 2048
Hz. When the system identification model of Eq. 1 determines the fit coefficients, it has no signal
above 4 Hz which it can use to calibrate a model — but noise will
exist above this frequency. When decimation was omitted, our system
identification method was producing models with unrealistically high gain at
frequencies above 4 Hz (e.g., gains 100,000 times the passband gain). Thus, a
small amount of noise power at frequencies above 4 Hz in a test trial caused a
noise spike in the predicted torque. Although the occurrence was infrequent, the
result was disastrous. Ljung (1999) describes this
problem of spurious model performance when the system identification model is
grossly oversampled. Our solution to this problem was to decimate
after forming the EMG amplitude estimate. As we progressively
lowered the effective EMGamp sampling rate, the spikes reduced both in
occurrence rate and magnitude. Decimating by a factor of 100 (effective EMGamp
sampling rate of 40.96 Hz) extinguished all spikes. Decimating with rates above
100 eliminated all spikes, but decreased EMGamp-torque performance. This
excessive decimating clearly discarded signal bandwidth. The optimal sampling
rate of 40.96 Hz is approximately 10 times the highest signal frequency, which
is precisely the “rule-of-thumb” rate recommended by Ljung (1999). In addition, this rate still
captured all of the signal power. This optimal decimation factor of 100 was used
in all further analyses and results. 3.2. Comparison of EMGamp processors In general, each EMGamp-torque processor captured much of the dynamics
exhibited in the actual torque ().
For all EMG-torque processors (,
), a progressive increase in
performance resulted as model order was increased up to about
10–15th order. Little or no additional improvement
occured above a model order of 15–20. The median results were
substantively better than the mean results, suggesting that a few large errors
overly influenced the distribution of errors. For the fast tracking speed
trials, the best EMG-torque model (multiple-channel, whitened EMGamp; system
identification order greater than 15) produced an average error of 7.3%
MVCF with a %VAF of 78%. Similar results were found for the slow
tracking speed data (). For the fast tracking speed, multivariate statistical tests indicated
highly significant differences due to the subject-EMG processor interaction
(Wilk's Lambda F = 2.27, 840 and 1397.8 degrees of freedom, p
< 0.0001). However, interaction plots and the size of the sums of
squares (one or more orders of magnitude lower than those for EMG processor)
indicated a multivariate test for EMG processor was valid. This test indicated
highly significant differences as well (Wilk's Lambda F = 3.58, 60 and 69.454
degrees of freedom, p < 0.0001). Follow-up pairwise
contrasts for EMG processor (with Holmes stepdown Bonferroni adjustment) found
all EMG processor combinations to be significantly different (each
p < 0.0174). To identify which model orders
exhibited differences in EMG processors, univariate comparisons were conducted
for each model order using Tukey pairwise comparisons (Bonferroni adjusted to an
overall confidence level of 0.95). For twelfth-order models and above,
significant differences consistently resulted when comparing the following
processor combinations: single-channel unwhitened to multiple-channel
unwhitened, single-channel unwhitened to multiple-channel whitened, and
single-channel whitened to multiple-channel whitened. Comparison of
single-channel unwhitened to multiple-channel whitened was highly significant in
every multivariate and univariate test. For the slow tracking speed,
substantially similar results were found, except that the follow-up pairwise
contrasts for EMG processors was significant for all EMG processor combinations
except when comparing single-channel whitened to
multiple-channel whitened (p = 0.19). 4.1. EMG-torque estimation The objective of this research was to investigate the incorporation of
recent advances in EMGamp processors into EMG-torque estimation algorithms. It
was anticipated that higher fidelity (i.e., lower noise) EMGamp processing would
reduce the errors in EMG-torque prediction. Indeed, our results show that both
whitening and multiple-channel combination of the EMG — methods
previously shown to improve EMGamp estimates — lead to reduced
EMG-torque prediction errors. Even lower errors result when the two techniques
are used in combination. Using 15 th-order and higher linear FIR
models, EMG-torque errors with a four-channel, whitened processor produced an
average error of 7.3% MVC (%VAF of 78%) at the fast tracking speed. By
comparison, the equivalent single-channel, unwhitened (conventional) processor
produced an average error of 9.9% MVC (%VAF of 55%). In addition, our work
highlights the oversampling problem that occurs when EMG data (typical bandwidth
from 20–500 Hz) are simultaneously sampled with torque data (typical
bandwidth < 5-10 Hz). In these cases, it is common to sample all data
at the highest required rate. Thus, the torque data are grossly oversampled. In
this circumstance, the EMGamp-torque model transfer function produces spurious
gains at frequencies above the typical band of torque data. This issue was
resolved by decimating after EMG amplitude estimation, to a rate of 40.96 Hz
— a rate that is approximately ten times the highest torque
frequency, as recommended by Ljung
(1999). 4.2. Study limitations Our primary interest was the influence of different EMGamp processors on
EMG-torque prediction performance. As such, we limited our study in several
manners so we could isolate the effect of EMGamp without the complexity of less
restrictive EMG-torque models. One would expect that the benefits shown here of
improved EMGamp processors would transfer to many other EMG-torque modeling
problems. Indeed, it now seems justified to progressively release these
restrictions and validate these benefits in more general applications. First,
our experimental design consisted of constant-posture nonfatiguing contractions
about the elbow, while most “real-life” contractions are
more fully dynamic. Second, the mechanical model for the elbow treated the joint
as a simple hinge, with only one agonist (biceps) and one antagonist (triceps)
muscle group. These assumptions simplified the system identification to a
two-input (flexion, extension EMGamp), one-output (torque) system. In actuality,
each muscle acts in a unique manner and other muscles (e.g., brachialis,
brachioradialis) contribute to torque about the joint. However, crosstalk and
the limited spatial resolution of conventional surface EMG constrain the ability
to monitor closely spaced muscles, or even distinct compartments within a muscle
group. Contributions of muscles that act synergistically with the biceps/triceps
are accounted for in the model. However, muscles that do not act synergistically
with the biceps/triceps see their contributions contribute to the system
identification error. Third, we limited our EMGamp-torque model to a linear FIR
form. Linear IIR forms (c.f., Gottlieb and
Agarwal, 1971) often require lower model orders. Nonlinear models can
potentially capture additional subtle behavior in an EMG-torque relationship
such as the electromechanical delay between action potential activation and
muscle fiber contraction, and any systematic differences in the EMG-torque
relation between concentric and eccentric contractions. Fourth, the contraction
bandwidth was limited to a 1 Hz tracking target, since subjects were unable to
track random targets with wider bandwidths. Supported by the National Institute for Occupational Safety and Health (NIOSH) under
grant R03-OH007829. Statistical consultation was provided by Joseph D. Petruccelli,
Department of Mathematical Sciences, WPI. - An KN, Cooney WP, Chao EY, Askew LJ, Daube JR. Determination of forces in extensor pollicis longus and flexor
pollicis longus of the thumb. Journal of Applied Physiology: Respiratory and Environmental Exercise
Physiology. 1983;54:714–719.
- Clancy EA. Stochastic modeling of the relationship between the surface
electromyogram and muscle torque. Massachusetts Institute of Technology; Cambridge, MA, USA: 1991. pp. 343–345. Ph.D. thesis.
- Clancy EA. Electromyogram amplitude estimation with adaptive smoothing
window length. IEEE Transactions on Biomedical Engineering. 1999;46(6):717–729. [PubMed]
- Clancy EA. EMG amplitude estimation toolbox: User's guide. Alpha version 0.04. 2004. Available on-line at: http://ece.wpi.edu/~ted/emg_tool.htm. Accessed 23
September 2004.
- Clancy EA, Bouchard S, Rancourt D. Estimation and application of EMG amplitude during dynamic
contractions. IEEE Engineering in Medicine and Biology. 2001;20(6):47–54.
- Clancy EA, Farry KA. Adaptive whitening of the electromyogram to improve amplitude
estimation. IEEE Transactions on Biomedical Engineering. 2000;47(6):709–719. [PubMed]
- Clancy EA, Hogan N. Single site electromyograph amplitude estimation. IEEE Transactions on Biomedical Engineering. 1994;41:159–167. [PubMed]
- Clancy EA, Hogan N. Multiple site electromyograph amplitude estimation. IEEE Transactions on Biomedical Engineering. 1995;42:203–211. [PubMed]
- Clancy EA, Hogan N. Relating agonist-antagonist electromyograms to joint torque
during isometric, quasi-isotonic, nonfatiguing contractions. IEEE Transactions on Biomedical Engineering. 1997;44(10):1024–1028. [PubMed]
- Clancy EA, Morin EL, Merletti R. Sampling, noise-reduction and amplitude estimation issues in
surface electromyography. Journal of Electromyography and Kinesiology. 2002;12:1–16. [PubMed]
- Gottlieb GL, Agarwal GC. Dynamic relationship between isometric muscle tension and the
electromyogram in man. Journal of Applied Physiology. 1971;30:345–351. [PubMed]
- Hof AL, Van den Berg Jw. EMG to force processing I: An electrical analogue of the Hill
muscle model. Journal of Biomechanics. 1981;14(11):747–758. [PubMed]
- Inman VT, Ralston HJ, Saunders JBCM, Feinstein B, Wright EW. Relation of human electromyogram to muscular tension. Electroencepholography and Clinical Neurophysiology. 1952;4:187–194.
- Kearney RE, Stein RB, Parameswaran L. Identification of intrinsic and reflex contributions to human
ankle stiffness dynamics. IEEE Transactions on Biomedical Engineering. 1997;44(6):493–504. [PubMed]
- Ljung L. System Identification: Theory for the User. Prentice-Hall; Upper Saddle River, NJ: 1999. pp. 80–83. 203–206, 444–449.
- Solomonow M, Guzzi A, Baratta R, Shoji H, D'Ambrosia R. EMG-force model of the elbow's antagonistic muscle pair: The
effect of joint position, gravity and recruitment. American Journal of Physical Medicine. 1986;65:223–244. [PubMed]
- Thelen DG, Schultz AB, Fassois SD, Ashton-Miller JA. Identification of dynamic myoelectric signal-to-force models
during isometric lumbar muscle contractions. Journal of Biomechanics. 1994;27:907–919. [PubMed]
- Wyss UP, Pollak VA. Surface electromyogram (EMG)/ muscle force: A muscle model based
on RMG peaks. Engineering in Medicine. 1984;13(1):27–33. [PubMed]
- Zuniga EN, Simons DG. Nonlinear relationship between averaged electromyogram potential
and muscle tension in normal subjects. Archives of Physical Medicine and Rehabilitation. 1969;50:613–620. [PubMed]
|
The publisher's final edited version of this article is available at 
