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Loss of control biomechanics of the human arm-elbow system
LOSS OF CONTROL BIOMECHANICS OF THE HUMAN ARM-ELBOW SYSTEM N. G. DAGALAKIS, C. MUEHLHOUSE and S. WAKAMIY.~ National
Bureau of Standards,
Gaithersburg.
MD
20899, USA
and
J. C. S. University
YANG
of Maryland
Abstract-An experimental study was conducted to determine whether external disturbance oscillations. such as those that could be created by hand held tools, alter the dynamic response characteristics of the human arm-muscle system. A special arm-test frame was used to induce external sinusoidal torque oscillations of various amplitudes and frequencies, while the reaction force and angular displacement were monitored. Two different output variable frequency responses were determined using input/output cross-spectrum analysis. The angular displacement of the test frame and a component of hand reaction force were the output variables used, while the test frame torque was the input. Test results from one subject are presented in this paper. Changes in the magnitude and phase angle of the frequency responses were observed for different frequencies of the disturbance torque. These changes Indicate that the stability margin and response amplitude of the human arm-muscle system do change as a function of the frequency and amplitude of external disturbance oscillations. This suggests that at certain operating frequencies hand held tools can induce large reaction amplitudes or even loss of control.
tests and a brief discussion of the arm-elbow model analysis results are presented in this paper. It is hoped that a better understanding of the tool operators’ loss of control biomechanics could help in the better design of these tools so that they can be operated more safely. A well known example of tool operator injury because of loss of control is due to chain saw kickback. This occurs when the portion of the chain at the upper quadrant of the guidebar digs deeply into the wood, causing the saw to rotate violently upward toward the operator. Kickback accounts for one-fourth of all chain saw related injuries. Chain saw manufacturers have recognized this problem and have employed a variety of brakes, guards, and other safety devices in an effort to eliminate the problem. A more in depth understanding of the stability properties of the human neuromuscular system under various tool dynamic loading conditions could help identify tool changes which would make them safer to operate. These could include changes in the tool moment of inertia and mass distribution, operating frequencies, brakes, valve reaction times, etc. A literature search revealed that there are no-publications dealing with the specific biomechanic problem of arm-elbow loss of control. It was decided to use an armelbow test frame to simulate hand held tools and frequency response analysis to investigate arm-elbow stability. Frequency response is a classic too! of engineering analysis and it has been used by several investigators for the study and modeling of human or animal muscles in cico. Such studies have included modeling and quantitative evaluation of the frequency response characteristics of active human skeletal
INTHODUcTION
A program was initiated to investigate the mechanics of the human body motion in conjunction with loss-ofcontrol actions and reactions in the interaction between man and his environment (machines, tools, products). As a first step in understanding the biomechanical aspects of this interaction, the arm-elbow was selected for study due to its importance in the stable operation of hand held tools. The research work included the development of the necessary experimental set up, theoretical work, computer simulation and dynamic analysis and experimental work. This paper reports one aspect of the work which relates to loss-of-control, specifically, the effect of external vibrations, such as those that can be created by hand held tools, on the dynamic response characteristics of the human armelbow system. Other efforts in this program are described in Dagalakis et al. (1981), Dainis (198 I). Yang er 01. (1982) and Wakamiya (1982). It has been suggested by hand tool operators that vibrations induced by the tools themselves sometimes make the operator more susceptible to losing control. The purpose of the study described in this paper is to investigate this hypothesis and to try to quantify the effect of tool vibrations on the arm-elbow system, by concentrating on the study of the loss ofcontrol of the elbow joint flexor muscle group system first. A series of tests were conducted where the frequency response of the arm-elbow system was determined for different types of sinusoidal tool vibrations. The results of these Received 19 Sorember
1984; in recisedform
1 August 1986. 385
386
N.
G.
DAGALAKIS
muscle in cico (Zahalak and Heyman. 1979; Cannon and Zahalak, 1982). forced oscillations of steadily contracting muscle (Bethoz and Metral. 1970: Neilson, 1972; Agarwal and Gotlieb, 1977). voluntary oscillations of isometric muscle (Soechting and Roberts, 1975). ESPERIXIESTAL
SET UP
A generalized arm-elbow testing apparatus was constructed. This consists of the arm loading frame, the torque driving system, force, moment and angular displacement transducers and the necessary instrumentation. The torque driving system consisted of an electric torque motor and a current feedback power amplifier. Figure 1 shows a block diagram of this system. Figure 2 provides a photographic view of the arm frame, torque motor and instrumentation. Arm-elbow
loading frame
The loading frame with its handle and load cell constitutes the major structural component of the arm-elbow apparatus. The handle is attached to the frame beam by two octagonal strain gage bridge sensors, These sensors form a high quality load cell which can measure five components of handle loading: three components are moments, and the other two are forces transmitted in the frame tangentially and radially. An adjustable elbow strap provided both POSitioning and support for the arme.lbow under test. A more detailed description of the arm-elbow loading frame can be found in Dagalakis rt 01. (198 I). Torque driving system
The input torque excitation to the arm-elbow loading frame was provided by a permanent magnet
et a/.
d.c. electric torque motor. The torque motor was driven by a current feedback power amplifier. This was a home-made power amplifier (Yang er (I/.. 1982) designed primarily for providing large control currents to the torque motor with good amplifier linearity. The amplifier was producing 1 A of output current for every 1 V of input command signal. A summing circuit at the input could add up to three voltage signals to produce the amplifier command signals. The frequency response of the amplifier was uniform from d.c. to 100 Hz. The electric motor had a torque sensitivity constant of 0.55 Nm A-‘, so the motor could produce 5.56 Nm of torque for every 10 A ofdriving current. The electric circuit time constant of the selected motor was very small and because of the high power supply voltage capability, the response time of the torque drive system was very short. In the case ofa step input excitation. the torque motor could reach steady state in 6.4 ms. with zero angular speed. The frequency response of the combined torque driving and arm-elbow test frame systems, for the arm frame tangential force. was uniform for up to 90 Hz. tnput excitation
system
The objective of the torque driving system was to produce the torque input excitation for the arm-elbow testing frame. The frequency range of interest for these tests was &30 Hz. Since the frequency response of the amplifier is uniform in that range, the amplifier may be modeled by a simple gain of 1 A V - ’ . The amplifier input voltage command signal used in most of these experiments consisted of a d.c. component, a sine wave and a random input. The d.c. offset component of the input command signal was always set at IO V. This torque was selected rather arbitrarily,
Load Cell
Noise
venerator
I
.
-’ +
DC Power +
SUPPlY
lnpulVariable Channel
Oulput Variable Channel
Fig. I. Block diagram
of the arm-elbow
testing system with all the necessary monitoring instrumentation.
and analysis
Fig. 2. Subject
with his arm in the test frame.
3x7
Biomechanics of the human arm-elbow system
for the tests reported here, to correspond to the torque necessary to support the weight of a hand held tool. The sine wave component was varied from 0 to 5 V peak amplitude, depending on the experiment. The noise component of the signal had a uniform spectrum from 0.1 to 2000 Hz. All three components of the input voltage command signal were fed into the current feedback power amplifier, where they were summed before being transformed into an equivalent current signal. Ourpur cariables and analysis
instrumentation
In the series of experiments reported in this paper, two output variables were monitored. The arm-elbow test frame handle tangential force F, (normal to the plane of the frame) and the frame angular displacement. These outputs were selected as being relevant to the actual process of injury and loss of control. After amplification, the output signals were fed into a dual channel FFT analyser. The analyzer, which also monitored the power amplifier input command signal, calculated a linearized system frequency response and performed any other standard signal correlation analysis that was desired. The strain gage conditioner output signals were fed into a home-made voltage amplifier. The offset was adjusted to eliminate the d.c. voltage component of the load cell signal. The d.c. offset was eliminated in order to increase the accuracy of the digitization and correlation analysis of the a.c. component of the signal by the FFT analyzer. Three different upper frequency analysis ranges were used, either 50, 20 or 10 Hz. The elbow-frame angular displacement was measured by a precision potentiometer, which was mounted directly onto the steel shaft supporting the arm frame. The potentiometer output was fed into the voltage amplifier, where again the offset level was adjusted to eliminate the d.c. component of the angular displacement signal. The use of filters, in order to eliminate d.c. components of the input and output signals was avoided because of the very low modal frequencies (2-4 Hz) of the arm-elbow system under investigation. Most analog filters do not behave well at those frequencies, sometimes causing signal distortions. EXPERIMENTAL
WORK
The main objective of the experimental work reported here was to study and characterize the changes in the dynamic response of the arm-muscle system related to its stability, as a function of external disturbances, when it is subjected to different types of simulated sinusoidal tool vibrations. Since the loss of control of the actual arm-muscle system may be caused by many types and varieties of external disturbances, it is highly impractical to determine the response for every possible disturbance. However, a good measure of the transient behavior of the system may be obtained by determining its frequency response.
389
The frequency response of a system is its steadystate response to a sinusoidal input signal. The amplitude and the phase of the response are usually plotted as functions of the frequency of the disturbance signal. The transfer function is a mathematical representation of the frequency response of a system defined as the ratio of a specified output variable to a given input variable. Frequency response studies of activated human skeletal muscles have been published by several investigators (Zahalak and Heyman, 1979: Soechting and Roberts, 1975; Bethoz and Metral, 1970: Neilson, 1972; Agarwal and Gottlieb, 1977; Cannon and Zahalak, 1982). The frequency response and transfer function are probably two of the most common ways to describe the dynamic characteristics of linear systems. Since our arm-muscle system is a non-linear system. we had to restrict ourselves to small amplitude disturbance excitation and response signals around a steady state position. Thus the frequency response determined constituted a linearized model of the non-linear armmuscle system for the steady state operating conditions of each test. These steady state conditions included a dc. offset torque and a sinusoidal torque vibration. The dc. offset torque could be considered to simullate the torque created by the weight of the tool and was maintained constant for all the tests reported in this paper. The sinusoidal torque vibration could be eonsidered to simulate the vibrations caused by the operation of the tool. Three different sinusoidal torque vibration amplitudes of various frequencies up to 30 Hz were used. The frequency response obtained for each offset torque and sinusoidal tool torque vibra;tion describes characteristics (stability, speed of response etc.) of the arm-muscle system within the frequency range of data analysis. The test frame and torque drive had to be used in order to apply the excitation torque, hence their dynamic characteristics should be included in the dynamics of the arm-muscle system. The moment of inertia of the test frame with the rotor of the torque motor was 0.170 Nms’ (Yang et al., 1982). The damping coefficient due to the ball bearing friction losses was 0.0093 (Yang et al., 1982). Figure 2 shows a test subject with his arm in the test frame. The forearm and upper arm were positioned so that each was about a 45” angle from the horizontal and 90” from one another. The elbow rested on the leather strap and the frame length was adjusted so that the subject had a good grip at the handle. An offset torque, applied to the test frame shaft, was translated to an arm force Frat the handle causing flexion of the forearm from a supination position. Any desired torque disturbance could be superimposed on the offset torque. However, the disturbance torque was always kept below the level of the offset torque to avoid creating arm extension loading torques. Several male and one female subject were tested (Wakamiya, 1982). The subjects were instructed to simply try to maintain the initial 45” arm setting.
N. G. DAGALAKIS et al.
390
An offset torque of 5.56 Nm was applied for all subjects and test conditions, but two different types of disturbance torques were used. namely, sinusoidal at various amplitudes and frequencies and white random noise of small RMS amplitude. The amplitudes and frequencies of the disturbance torques used are provided in the, ‘Experimental Results-Frequency Responses with Imposed External Oscillations,’ section. As explained earlier, the frequency response of a linearized approximation of the system was determined by monitoring the input excitation variable and any desired measurable output. The frequency response estimate was given by the Wiener-Hopf equation (Bendat and Piersol, 1971). H(f)
= +_
(1)
xx
Where S,, (f) is the averaged cross-spectrum function between the input variable x and the output variable y, S,, (f) is the averaged auto-spectrum function of the input variable x, andjrepresents frequency. The dual channel FFT analyzer samples the input and output signals and saves a record of certain length T, then solves equation (I), to determine H (/).That terminates one cycle of operation, and the analyzer is ready to repeat the process. This is done several times. Each newly computed frequency response H (f) is added to the previous one and the average is displayed on the analyzer screen. The number of averages used for all the test results reported here ranged from 40 to 60. Notice that equation (1) is a complex valued relation that can be broken down into a pair of equations to give both the magnitude )H (f)land phase angle 4(/) of H (f). Equation (1) is valid for random (stationary or transient) x(r) records, so there is always need for a random signal component in the input excitation. This component can be the excitation itself or, in the case of a deterministic input signal (like the simulated sinusoidal tool vibrations torque of our tests), the random componeni can be superimposed for pure diagnostic purposes. In the final frequency response plot, the segment corresponding to the deterministic signal frequency was ignored. The noise generator used for these tests was selected because it could produce a random noise signal with uniform spectrum for as low as 0.1 Hz. This is important because it was expected that several modal frequencies of the arm-elbow system lie between 1 and 4 Hz and we wanted to make sure that the diagnostic noise input excites them ail. The input variable used initially to determine the frequency response was the torque applied to the arm frame, measured by two 45” strain gage rosettes mounted on the motor shaft. Unfortunately the sensitivity of this load-ceil was low because of the high stiffness of the steel shaft. Making the shaft hollow could increase the sensitivity, but it was estimated that it would have simultaneously lowered the natural
frequency of the first mode of the frame, bringing it close to the arm-muscle system natural frequencies. Since it had been determined that the frequency response of the torque motor and power amplifier system is uniform in the 0 to 30 Hz frequency range of interest, it was decided to use the amphfier voltage command signal as the input variable.
ESPERI>lESTAL Baseline frequency
RESLLTS
response
To study and characterize the changes in the armmuscle test frame system frequency response, it is first necessary to establish a baseline frequency response. This characterizes the arm when the test frame is held at 45” (see Fig. 2) against a steady offset flexing torque of 5.56 Nm. The baseline frequency responses, corresponding to zero amplitude sinusoidal tool vibrations disturbance, were determined. Figure 3 shows the frequency response between the voltage command signal and the test frame angular displacement, 4. Figure 4 shows the frequency response between the same input and the handle load cell force, F, The plots were produced by the FFT analyzer. To translate the non-dimensional readings (V/V ratio) into meaningful physical units it is necessary to multiply by the proper constant factors as indicated in Figs 3 and 4. The subject used for the frequency response plots of these two figures (and all other plots presented in this paper) was a 35 yr old male. The general form of the baseline frequency response was found to be the same except for minor differences in the location and magnitude of the first peak. The baseline frequency response magnitude of the arm-test frame angular displacement, r#~.vs the input torque, M (see Fig. 3) has a single, rather broad peak around 3 Hz, most likely reflecting several system modes, The peak is followed by an uninterrupted decline in magnitude. Initially at a rate of approximately - 48 db decade- ‘, the rate of decline moderates with the amplitude becoming very small around 30 Hz. The phase angle of the frequency response starts from 0” at a very low frequency and then gradually declines to - 180” around 8 Hz excitation frequency. The phase angle is always negative, meaning that the output angle always lags the input torque. The sudden jumps in the phase angle from - 180’ to + 180” at higher frequencies are due to computer calculation of 4. The noise in the frequency response at very low frequency, below 0.6 Hz, is due to the low signal to noise ratio at that frequency range. This baseline frequency response is similar to those which have been reported by Zahalak and Heyman (1979, Fig. 3; Cannon and Zahalak. 1982, Fig. 5). The magnitude of the baseline frequency response of the test frame handle force, F, (tangential component of force), vs the input torque, )M (see Fig. 4), has two peaks. The first is located around 3 Hz and the second
Biomechanics of the human arm+lbow
.*. ,?_, .-,._.
10
15
.__ ._._ :..,* .\,d
20
25(Hz) 30
,-- Phase Angle
---./L-V-J Magnitude
L-
I
5
IO
15
20
25(Hz) 30
Fig. 3. Baseline frequency response (6/M). Fig. 4. Baseline frequency response ( F,/M ). around 12 Hz. Both peaks are very broad. Right after the first peak the amplitude decreases very sharply at approximately - 150 db decade-’ slope. The amplitude (in absolute terms) becomes almost zero around 6 Hz (i.e. nearly 30 db down), and then increases gradually to reach the second peak. The phase angle starts from 0” at low frequency and declines gradually until it reaches its lowest value of approximately - 100” around 6 Hz. Then it rises very sharply until it reaches 0”. For higher excitation frequencies the phase angle declines, then remains essentially constant between - 30” and - 40” above 20 Hz. The physical meaning of the F,/M baseline frequency response is not clear yet, future work will attempt to explain the significance of the two peaks and the dip around 6 Hz. A new baseline frequency response was established every day before conducting any series of tests. At least 50-60 averages were used for each baseline. The tests were run continuously and data were collected for 5-8 min until the subject began to feel tired. The frequency resolution of the frequency response plots shown in Figs 3 and 4 is 0.125 Hz. The baseline frequency response was reproducible. Only minor differences were observed after tests performed several hours, or days, or weeks apart. The
391
system
frequency response agrees well with the results of the dynamic and modeling analysis reported by Wakamiya (1982). According to Brunnstrom ( 1966) and Wells (1966) the main flexor muscle groups of the elbow joint are the: (1) biceps brachii, (2) brachialis. (3) brachioradialis. and (4) pronator teres. If one models each of these as a spring with a contractile element and dashpot in series (Wakamiya. 1982), it is possible to show that they contribute several uscillatory modes which, for most adults, lie in the frequency range of O-5 Hz. The result of the combined action of all these modes must be the first peak, which we see in the frequency responses of Figs 3 and 4.
Frequency
responses
with imposed
externd
oscillutions
The objective of the tests was to study and characterize the changes in the arm-frame system frequency response due to an externally imposed sinusoidal oscillation. This oscillation could be compared to that imposed on the operator of a hand held tool. The moment of inertia of the test frame and torque motor rotor could be considered to represent the moment of inertia of the tool, but no attempt was made to model any particular tool. The arm test position (see Fig. 2) used was selected as being a common hand tool operating position, and one which could result in the injury of the tool operator in the case of loss of control. Changing the arm test position and offset torque changes the stiffness of the arm musculature and the forearm weight load torque. This could result in the change of the linearized system frequency response. Future work will investigate the effect of these test conditions on the loss of stability properties of the arm+lbow system. The test procedure was the following. After an initial warm up period a baseline frequency response test was conducted using a small amplitude white noise excitation signal. A small period of rest followed, then’ the test was repeated with a sinusoidal signal of 5 V peak amplitude, corresponding to 2.78 Nm, superimposed on the same noise signal. The noise signal served as a diagnostic in order to determine the frequency response. The excitation continued for 5-10 min until the subject started to feel tired. After another period of rest, the experiment was repeated with a sinusoidal signal of 3 V peak amplitude, corresponding to 1.66 Nm, then a sinusoidal signal of 1 V peak amplitude, corresponding to 0.55 Nm. Only one sinusoidal test was conducted per day. The test frequencies 06 the sinusoidal disturbance used were the following: 2 Hz, 3Hz,4Hz,5Hz,6Hz,7H~8Hz,9Hz,lOHz,lliHz, 12 Hz, 15 Hz, 17 Hz, 20 Hz, 22 Hz, 25 Hz, 27 Hz, 30 Hz. The frequency response of the arm-ftame system with the sinusoidal signal disturbance excitation was compared with the baseline frequency response obtained during the same day of testing. Figures 5-10 are typical frequency response results from these tests, showing angular displacement, 4, to input torque, M. Figure 5 shows the frequency re-
392
N. G. DAGALAKIS er at.
laoa 90’ 0'
-90
-iao*
*
(db3 0 -10 -20 -30
180’ 90” 0” -90’ -180” Fig. 7 Fr.R. (o/M)
18
10
lS(Hz)
la!
1
(lOHr, 5V, Sine) fig. 8 Fr.R. (NM)
I
(db3 -10 IBe -20
a
r
in
5
10
15(Hz)
Figs S-8. Frequency responses (4/M 1.
sponse of the arm-frame system with a 3 Hz frequency, 1 V peak amplitude sinusoidal external disturbance along with the baseline frequency response. Figure 6 shows the results for a sinusoidal signal of the same frequency, but 5 V peak amplitude. Figures 7 and 8 show the results for a sinusoidal signal of 10 Hz frequency with 1 V and 5 V peak amplitude respectively. Figures 9 and 10 show the results for a sinusoidal signal of 30 Hz frequency with 1 V and 5 V peak amplitudes, respectively. Figures 11 and 12 show typical force, F,, to input torque, M, frequency response results. Figure 11 shows the results for a sinusoidal external disturbance excitation signal of 10 Hz and 1 V peak amplitude. Figure 12 shows the results for a sinusoidal signal of the same frequency but 5 V peak amplitude. The two cursor lines in these figures mark the 3.5 Hz and 6.5 Hz frequencies respectively. As will be explained later the frequency response characteristics at those two frequencies were used as a part of the stability analysis study.
The scale of the vertical axis of the frequency response plot of Figs 5-10 is the same as that used for the baseline plot of Fig. 3. Similarly, the vertical scale of the frequency response plots of Figs 11 and 12 is that used for the baseline of Fig. 4. The main differences in the frequency response after the introduction of the sinusoidal disturbance are increases in the peak amplitude and shifts of both the amplitude and phase angle. The magnitude of these changes varies depending on the frequency of the sinusoidal oscillation signal and its amplitude. For example, from Fig. 5 it can be seen that a 3 Hz, 1 V sinusoidal disturbance causes a maximum increase of the 4/M peak magnitude of 7.3 db as compared to the baseline. This corresponds to a 2.3 times increase in the angular displacement, 4, for the same torque excitation, M. From the phase angle plot of the same figure it can be seen that a 3 Hz, 1 V sinusoidal disturbance causes an increase in the phase lag of 4 for low frequencies. An increase of the sinusoidal disturbance peak
Biomechanics
of the human arm-elbow
393
system
,dbJI -10 -20 -30
I-=-
-180” (lOH2,
Baseline
\
(IOHz,
IV, Sine)
SV, Sine)
i’
WbO
--10
(db; -10 -20 -30
1
5
10
lS(Hz)
Figs 9 and 10. Frequency responses (4/M). Figs 11 and 12. Frequency responses (FJM).
amplitude to 5 V (2.78 Nm) casuses even more significant changes. The maximum peak increase (see Fig. 6) of the amplitude is 15.9 db as compared to the baseline, corresponding to a 6.2 times increase in the angular displacement, $ for the same torque excitation, M. The increase in the phase angle lag is equally significant. The whole phase angle curve seems to have shifted to low frequencies, increasing the phase angle lag by 45.1’ at 3.5 Hz. A similar behavior is seen for other sinusoidal disturbance frequencies (see Figs 7-10). The increase in the amplitude of the frequency response and the lag of the phase angle diminish as the frequency of the disturbance approaches 30 Hz. Another change in the frequency response, seen for most sinusoidal disturbance frequencies, is the appearance of two distinct peaks (see Figs 5,6,8, lo), one located around 2 Hz and the other around 3.5 Hz. Under these disturbance conditions, the angular displacement response is apparently dominated by these two modes. The appearance of these peaks was observed in the case of the other subjects at ap-
proximately the same frequencies. Their physical origin is not known yet. In the case of the handle force, F, to input torque, M, frequency response, the introduction of the sinusoidal disturbance also causes significant changes. For a 10 Hz, 5 V peak amplitude sinusoidal sigrtal there is a 3.5 db increase in the peak amplitude as compared to the baseline and a significant shift of t,he phase angle plot to lower frequencies. Another change observed was a shift of the peak asymptote to lower frequencies. Decreasing the peak amplitude of tihe sinusoidal disturbance to 1 V reduced the magnitude of these changes. Changes
in
the
irequency
response
As pointed out in the previous sectlon, the addition of a sinusoidal disturbance torque to the arm-frame test system alters its dynamic response characteristics. The most visible of these changes is the increase in the amplitude of the frequency response magnitude peaks. The angular displacement, 4, to torque, M, frequency response peak splits into two peaks as a
N. G.
394
DAGALAKIS et al.
result of the sinusoidal disturbance which dominates the response. Figure 13 plots the increase in the amplitude of the first of these two peaks (around 2 Hz for this subject), for three different peak amplitudes of sinusoidal disturbance. Figure 14 plots the increase in the amplitude of the second peak (around 3 Hz for this subject) for the same sinusoidal disturbance. Figure 15 plots the increase in the amplitude of the first peak of the handle force, F, to input torque, M, frequency response magnitude as a result of the sinusoidal disturbance. The increase in the arm-frame response angular displacement and force, due to the external sinusoidal disturbance, is quite high. It appears that there are two disturbance frequency ranges which have a particularly bad effect. For this subject and test frame set up, the first is located around 3 Hz and the second around 8 Hz. The first seems to be responsible for a large increase in the amplitude of the response. Similar armframe response behavior was observed with the other test subjects. Besides the increase in the peak amplitude of the
* 5V Peak Amp. She __ 0,V II . II -*,V ., . ,, ._._._
10 20 30 (Hz) Slnusoidal Dlsturbanca Fraquancy Fig.
13.
10 20 Slnusoldal Dlaturbanca Frequency
Fig. 14.
30 (Hz)
* 5V Paak Amp. Sine sjy.. , .--
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.I
I
,. ------
Sinusoidal Diaturbanca Frequency
Fig. 15.
frequency response magnitude caused by the sinusoidal disturbance, a significant increase in the phase angle lag was also observed. Both of these changes could affect the stability properties of the arm-frame system. For this reason it was decided to measure the stability margin of this system before and after the application of the sinusoidal disturbance. One common measure of stability margin, used in the Systems and Controls field, is the phase margin. This measure of relative stability is equal to the additional phase lag required before a system becomes unstable. For instance, consider the arm reaction torque to a test frame input torque. If the reaction torque lags the input torque by more than 180”. then its effect will be superimposed on that of the input torque, resulting in an increase of the angular displacement. If this continues for cycle after cycle of input excitation, it could lead to a loss of stability. The distance between the handle support and the shaft of the test frame was 40 cm. Multiplying this by the tangential component of the handle force, F,will give the reaction arm torque. This multiplication corresponds to 4.9 db, which should be added to the scale of the magnitude (vertical axis) of Figs 4, 11 and 12, to convert it to decibels of Nm/Nm ratio. In Fig. 12 the 0 db line of this torque-to-torque ratio scale is marked with a horizontal dashed line. The intersection of this line with the baseline F,/M frequency response (point no. 1) marks the excitation frequency for which the magnitude of the arm reaction torque is equal to that of the frame input torque. The corresponding phase angle is 80.6’) giving a phase margin = 180” - 80.6” = 99.4”. When a sinusoidal disturbance of 10 Hz frequency and 5 V peak amplitude is applied, the F,/M frequency response changes as shown in Fig. 12. The 0 db point no. 1 now shifts to point no. 2 and the corresponding phase margin becomes 81.6”. This amounts to a decrease in the phase margin stability of 17.8 % with respect to the baseline phase margin. Figure 16 shows the decrease in the phase margin versus the frequency of the sinusoidal disturbance for peak amplitudes of 5 and 3 V. The decrease in the phase margin for a sinusoidal disturbance of 1 V was very small and is not shown here.
395
Biomechanics of the human arm-elbow system W)i 1OOi
* 5V Peak Amp. Sine ~ ,3V I/ --
. ,”
.,
.I
~----
. SV Peak Amp. Sine ?3V . . . --
m 10 Sinusoidal Disturbance Frequency
Sinusoidal Disturbance Frequency
/
* 5V Peak Amp. Sine -
Sinusoidal Disturbance Frequency
Fig. 17.
(HZ)
Fig. 18.
Fig. 16.
Again we see that two frequency ranges of sinusoidal disturbance can cause a significant decrease in the stability margin of the arm-frame system. For this particular subject and test frame, the first frequency range is around 2 Hz and the second around 10 Hz. Although the phase margin is a good index of the stability margin of a system, a more meaningful measure (from a biomechanics point of view) is the phase lag itself. The shift of the frequency response phase angle to lower frequencies due to the sinusoidal disturbance means that the human arm responds more promptly above a certain characteristic frequency of excitation, while below that frequency it responds more sluggishly. From the F,/ M frequency response experimental results it is seen that, for most sinusoidal disturbance frequencies. this characteristic frequency lies around 5 Hz. To demonstrate this shift in the character of the frequency response we measured the change in the phase angle for two frequencies. At 3.5 Hz, located below this characteristic frequency, and at 6.5 Hz, located above it. Figure I7 shows the percent decrease of the phase angle lag of the Fr/M frequency response at 6.5 Hz with respect to the baseline phase angle. Figure I8 shows the percent increase of the phase angle lag of the F,/M frequency response at 3.5 Hz with respect to the baseline phase angle. Figure I9 shows the percent increase of the phase angle lag of the d/M
30
o $2
(%)j 100.
px
“Z o(E :r 5 5i iE Bo sl;;i
mz
:B
* 5V Peak Amp. Sine __
%_3V.x .,I/ >I
I / . J
II 3,
, ,,
-____.
1:) / i’ “&, j
Frequency
Fig. 19.
frequency response at 3.5 Hz. At 6.5 Hz the phase angle of the 4/M frequency response was very close to a - 180” asymptote so no shift could be observed.
CONCLUSIONS
An experimental study has been conducted to determine whether external sinusoidal disturbance oscillations (simulating tool vibrations) alter the dynamic response characteristics of the human armmuscle system. The results of this study indicate that the maximum amplitude of the response, as well as the stability margin of the system, do change. The changes depend on the frequency and amplitude of the sinusoidal disturbance. Two sinusoidal disturbance frequency ranges seem to have a particularly strong effect on the dynamic response characteristics. In general, the stability margin of the system seems to decrease for low frequency excitation as a result of the sinusoidal disturbance. Since a decrease in stability margin means that the system becomes less stable, the results indicate that when the human arm system is subjected to certain low frequency sinusoidal vibrations, the neuromuscular system may become more susceptible to losing control. At this stage of the investigation it is not clear why the arm-muscle system becomes more susceptible to lose
396
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DAGALAKIS er al.
control when it is subjected to external tool vibrations of those frequencies. One possible explanation is that two different groups of muscles resonate at those
frequencies. When any of them is excited to oscillate, it causes a reduction in the ability of the arm-muscle system to react to emergencies. Since it is known that active muscle stiffness increases with mean muscle moment (Cannon and Zahalak, 1982). it should be expected that these critical tool vibration frequencies will increase as the d.c.
offset torque is increased. This means that heavier tools might resonate these muscle groups at higher frequencies. This hypothesis
will be tested at future stages of
our investigation.
subjected to a sinusoidal and trapezoidal variation offorce. J. Appl. PhJsio[. 29, 378-384. Brunnstrom. S. 11966) Clinical Kinesioloyy. 2nd edn. p, 58. F. A. Davis, Philadelphia. P.1. Cannon, S. C. and Zahalak. G. I. (19S2) The mechanical behavior of active human skeletal muscle in small oscillations. J. Biomechanics 15. I I l-121. Dagalakis, N., Yang. J.. Muehlhouse, C.. Sushinsky. G. and Wakamiya. S. (1981) An arm elbow apparatus for studying the biomechanical aspects of loss-of-control. 1981 Advances in Bioenyineering. pp. 6 l-65. .American Society of Mechanical Engineers. New York. Dainis. A. (1981) Evaluation and calibration of the SELSPOT system. Annuui Conference of the .4mericun Socier.r of Biomechonics. Case Western University. Cleveland. OH. Neilson, P. D. (1972) Frequency response characteristic of the tonic stretch reflexes of biceps brachii muscle in intact man. Med.
Acknowledgemenr-This research effort was supported by the National Engineering Laboratory of the National Bureau of Standards. REFERENCES
Agarwal, G. C. and Gottlieb. G. L. (1977) Oscillation of the human ankle ioint in resoonse to applied sinusoidal torque at the foot. J. Physih. 268, ISlzi76. Bendat, J. S. and Piersol, A. G. (197 I) Random Dora: Analysis and Measuremenl Procedures, Wiley-Interscience, New York. Bethoz, A. and Metral, S. (1970) Behavior ofa muscular group
Biol.
Engng
IO, 460372.
Soechting, J. F. and Roberts. W. J. (19731Transfer characteristici between EMG activity and muscle tension under isometric conditions in man. J. PhrrioL 70. 779-793. Wakamiya, S. K. (1982) Development of a mathemetical model of the human arm. M.S. Thesis, University of Maryland, Mechanical Engineering Department. Wells, K. F. (1966) Kinesiology, p. 26:. W.B. Saunders. Philadelohia. PA. Yang, J. er-ol. (1982) Design. construction of an arm-elbow apparatus. University of ,Maryland Technical Report No. 82-8. Zahalak, G. 1. and Heyman, S. J. (1979) A quantitative evaluation of the frequency response characteristics of active human skeletal muscle in cico. J. biomech. Enyny 101, 28-37.
Bureau of Standards,
Gaithersburg.
MD
20899, USA
and
J. C. S. University
YANG
of Maryland
Abstract-An experimental study was conducted to determine whether external disturbance oscillations. such as those that could be created by hand held tools, alter the dynamic response characteristics of the human arm-muscle system. A special arm-test frame was used to induce external sinusoidal torque oscillations of various amplitudes and frequencies, while the reaction force and angular displacement were monitored. Two different output variable frequency responses were determined using input/output cross-spectrum analysis. The angular displacement of the test frame and a component of hand reaction force were the output variables used, while the test frame torque was the input. Test results from one subject are presented in this paper. Changes in the magnitude and phase angle of the frequency responses were observed for different frequencies of the disturbance torque. These changes Indicate that the stability margin and response amplitude of the human arm-muscle system do change as a function of the frequency and amplitude of external disturbance oscillations. This suggests that at certain operating frequencies hand held tools can induce large reaction amplitudes or even loss of control.
tests and a brief discussion of the arm-elbow model analysis results are presented in this paper. It is hoped that a better understanding of the tool operators’ loss of control biomechanics could help in the better design of these tools so that they can be operated more safely. A well known example of tool operator injury because of loss of control is due to chain saw kickback. This occurs when the portion of the chain at the upper quadrant of the guidebar digs deeply into the wood, causing the saw to rotate violently upward toward the operator. Kickback accounts for one-fourth of all chain saw related injuries. Chain saw manufacturers have recognized this problem and have employed a variety of brakes, guards, and other safety devices in an effort to eliminate the problem. A more in depth understanding of the stability properties of the human neuromuscular system under various tool dynamic loading conditions could help identify tool changes which would make them safer to operate. These could include changes in the tool moment of inertia and mass distribution, operating frequencies, brakes, valve reaction times, etc. A literature search revealed that there are no-publications dealing with the specific biomechanic problem of arm-elbow loss of control. It was decided to use an armelbow test frame to simulate hand held tools and frequency response analysis to investigate arm-elbow stability. Frequency response is a classic too! of engineering analysis and it has been used by several investigators for the study and modeling of human or animal muscles in cico. Such studies have included modeling and quantitative evaluation of the frequency response characteristics of active human skeletal
INTHODUcTION
A program was initiated to investigate the mechanics of the human body motion in conjunction with loss-ofcontrol actions and reactions in the interaction between man and his environment (machines, tools, products). As a first step in understanding the biomechanical aspects of this interaction, the arm-elbow was selected for study due to its importance in the stable operation of hand held tools. The research work included the development of the necessary experimental set up, theoretical work, computer simulation and dynamic analysis and experimental work. This paper reports one aspect of the work which relates to loss-of-control, specifically, the effect of external vibrations, such as those that can be created by hand held tools, on the dynamic response characteristics of the human armelbow system. Other efforts in this program are described in Dagalakis et al. (1981), Dainis (198 I). Yang er 01. (1982) and Wakamiya (1982). It has been suggested by hand tool operators that vibrations induced by the tools themselves sometimes make the operator more susceptible to losing control. The purpose of the study described in this paper is to investigate this hypothesis and to try to quantify the effect of tool vibrations on the arm-elbow system, by concentrating on the study of the loss ofcontrol of the elbow joint flexor muscle group system first. A series of tests were conducted where the frequency response of the arm-elbow system was determined for different types of sinusoidal tool vibrations. The results of these Received 19 Sorember
1984; in recisedform
1 August 1986. 385
386
N.
G.
DAGALAKIS
muscle in cico (Zahalak and Heyman. 1979; Cannon and Zahalak, 1982). forced oscillations of steadily contracting muscle (Bethoz and Metral. 1970: Neilson, 1972; Agarwal and Gotlieb, 1977). voluntary oscillations of isometric muscle (Soechting and Roberts, 1975). ESPERIXIESTAL
SET UP
A generalized arm-elbow testing apparatus was constructed. This consists of the arm loading frame, the torque driving system, force, moment and angular displacement transducers and the necessary instrumentation. The torque driving system consisted of an electric torque motor and a current feedback power amplifier. Figure 1 shows a block diagram of this system. Figure 2 provides a photographic view of the arm frame, torque motor and instrumentation. Arm-elbow
loading frame
The loading frame with its handle and load cell constitutes the major structural component of the arm-elbow apparatus. The handle is attached to the frame beam by two octagonal strain gage bridge sensors, These sensors form a high quality load cell which can measure five components of handle loading: three components are moments, and the other two are forces transmitted in the frame tangentially and radially. An adjustable elbow strap provided both POSitioning and support for the arme.lbow under test. A more detailed description of the arm-elbow loading frame can be found in Dagalakis rt 01. (198 I). Torque driving system
The input torque excitation to the arm-elbow loading frame was provided by a permanent magnet
et a/.
d.c. electric torque motor. The torque motor was driven by a current feedback power amplifier. This was a home-made power amplifier (Yang er (I/.. 1982) designed primarily for providing large control currents to the torque motor with good amplifier linearity. The amplifier was producing 1 A of output current for every 1 V of input command signal. A summing circuit at the input could add up to three voltage signals to produce the amplifier command signals. The frequency response of the amplifier was uniform from d.c. to 100 Hz. The electric motor had a torque sensitivity constant of 0.55 Nm A-‘, so the motor could produce 5.56 Nm of torque for every 10 A ofdriving current. The electric circuit time constant of the selected motor was very small and because of the high power supply voltage capability, the response time of the torque drive system was very short. In the case ofa step input excitation. the torque motor could reach steady state in 6.4 ms. with zero angular speed. The frequency response of the combined torque driving and arm-elbow test frame systems, for the arm frame tangential force. was uniform for up to 90 Hz. tnput excitation
system
The objective of the torque driving system was to produce the torque input excitation for the arm-elbow testing frame. The frequency range of interest for these tests was &30 Hz. Since the frequency response of the amplifier is uniform in that range, the amplifier may be modeled by a simple gain of 1 A V - ’ . The amplifier input voltage command signal used in most of these experiments consisted of a d.c. component, a sine wave and a random input. The d.c. offset component of the input command signal was always set at IO V. This torque was selected rather arbitrarily,
Load Cell
Noise
venerator
I
.
-’ +
DC Power +
SUPPlY
lnpulVariable Channel
Oulput Variable Channel
Fig. I. Block diagram
of the arm-elbow
testing system with all the necessary monitoring instrumentation.
and analysis
Fig. 2. Subject
with his arm in the test frame.
3x7
Biomechanics of the human arm-elbow system
for the tests reported here, to correspond to the torque necessary to support the weight of a hand held tool. The sine wave component was varied from 0 to 5 V peak amplitude, depending on the experiment. The noise component of the signal had a uniform spectrum from 0.1 to 2000 Hz. All three components of the input voltage command signal were fed into the current feedback power amplifier, where they were summed before being transformed into an equivalent current signal. Ourpur cariables and analysis
instrumentation
In the series of experiments reported in this paper, two output variables were monitored. The arm-elbow test frame handle tangential force F, (normal to the plane of the frame) and the frame angular displacement. These outputs were selected as being relevant to the actual process of injury and loss of control. After amplification, the output signals were fed into a dual channel FFT analyser. The analyzer, which also monitored the power amplifier input command signal, calculated a linearized system frequency response and performed any other standard signal correlation analysis that was desired. The strain gage conditioner output signals were fed into a home-made voltage amplifier. The offset was adjusted to eliminate the d.c. voltage component of the load cell signal. The d.c. offset was eliminated in order to increase the accuracy of the digitization and correlation analysis of the a.c. component of the signal by the FFT analyzer. Three different upper frequency analysis ranges were used, either 50, 20 or 10 Hz. The elbow-frame angular displacement was measured by a precision potentiometer, which was mounted directly onto the steel shaft supporting the arm frame. The potentiometer output was fed into the voltage amplifier, where again the offset level was adjusted to eliminate the d.c. component of the angular displacement signal. The use of filters, in order to eliminate d.c. components of the input and output signals was avoided because of the very low modal frequencies (2-4 Hz) of the arm-elbow system under investigation. Most analog filters do not behave well at those frequencies, sometimes causing signal distortions. EXPERIMENTAL
WORK
The main objective of the experimental work reported here was to study and characterize the changes in the dynamic response of the arm-muscle system related to its stability, as a function of external disturbances, when it is subjected to different types of simulated sinusoidal tool vibrations. Since the loss of control of the actual arm-muscle system may be caused by many types and varieties of external disturbances, it is highly impractical to determine the response for every possible disturbance. However, a good measure of the transient behavior of the system may be obtained by determining its frequency response.
389
The frequency response of a system is its steadystate response to a sinusoidal input signal. The amplitude and the phase of the response are usually plotted as functions of the frequency of the disturbance signal. The transfer function is a mathematical representation of the frequency response of a system defined as the ratio of a specified output variable to a given input variable. Frequency response studies of activated human skeletal muscles have been published by several investigators (Zahalak and Heyman, 1979: Soechting and Roberts, 1975; Bethoz and Metral, 1970: Neilson, 1972; Agarwal and Gottlieb, 1977; Cannon and Zahalak, 1982). The frequency response and transfer function are probably two of the most common ways to describe the dynamic characteristics of linear systems. Since our arm-muscle system is a non-linear system. we had to restrict ourselves to small amplitude disturbance excitation and response signals around a steady state position. Thus the frequency response determined constituted a linearized model of the non-linear armmuscle system for the steady state operating conditions of each test. These steady state conditions included a dc. offset torque and a sinusoidal torque vibration. The dc. offset torque could be considered to simullate the torque created by the weight of the tool and was maintained constant for all the tests reported in this paper. The sinusoidal torque vibration could be eonsidered to simulate the vibrations caused by the operation of the tool. Three different sinusoidal torque vibration amplitudes of various frequencies up to 30 Hz were used. The frequency response obtained for each offset torque and sinusoidal tool torque vibra;tion describes characteristics (stability, speed of response etc.) of the arm-muscle system within the frequency range of data analysis. The test frame and torque drive had to be used in order to apply the excitation torque, hence their dynamic characteristics should be included in the dynamics of the arm-muscle system. The moment of inertia of the test frame with the rotor of the torque motor was 0.170 Nms’ (Yang et al., 1982). The damping coefficient due to the ball bearing friction losses was 0.0093 (Yang et al., 1982). Figure 2 shows a test subject with his arm in the test frame. The forearm and upper arm were positioned so that each was about a 45” angle from the horizontal and 90” from one another. The elbow rested on the leather strap and the frame length was adjusted so that the subject had a good grip at the handle. An offset torque, applied to the test frame shaft, was translated to an arm force Frat the handle causing flexion of the forearm from a supination position. Any desired torque disturbance could be superimposed on the offset torque. However, the disturbance torque was always kept below the level of the offset torque to avoid creating arm extension loading torques. Several male and one female subject were tested (Wakamiya, 1982). The subjects were instructed to simply try to maintain the initial 45” arm setting.
N. G. DAGALAKIS et al.
390
An offset torque of 5.56 Nm was applied for all subjects and test conditions, but two different types of disturbance torques were used. namely, sinusoidal at various amplitudes and frequencies and white random noise of small RMS amplitude. The amplitudes and frequencies of the disturbance torques used are provided in the, ‘Experimental Results-Frequency Responses with Imposed External Oscillations,’ section. As explained earlier, the frequency response of a linearized approximation of the system was determined by monitoring the input excitation variable and any desired measurable output. The frequency response estimate was given by the Wiener-Hopf equation (Bendat and Piersol, 1971). H(f)
= +_
(1)
xx
Where S,, (f) is the averaged cross-spectrum function between the input variable x and the output variable y, S,, (f) is the averaged auto-spectrum function of the input variable x, andjrepresents frequency. The dual channel FFT analyzer samples the input and output signals and saves a record of certain length T, then solves equation (I), to determine H (/).That terminates one cycle of operation, and the analyzer is ready to repeat the process. This is done several times. Each newly computed frequency response H (f) is added to the previous one and the average is displayed on the analyzer screen. The number of averages used for all the test results reported here ranged from 40 to 60. Notice that equation (1) is a complex valued relation that can be broken down into a pair of equations to give both the magnitude )H (f)land phase angle 4(/) of H (f). Equation (1) is valid for random (stationary or transient) x(r) records, so there is always need for a random signal component in the input excitation. This component can be the excitation itself or, in the case of a deterministic input signal (like the simulated sinusoidal tool vibrations torque of our tests), the random componeni can be superimposed for pure diagnostic purposes. In the final frequency response plot, the segment corresponding to the deterministic signal frequency was ignored. The noise generator used for these tests was selected because it could produce a random noise signal with uniform spectrum for as low as 0.1 Hz. This is important because it was expected that several modal frequencies of the arm-elbow system lie between 1 and 4 Hz and we wanted to make sure that the diagnostic noise input excites them ail. The input variable used initially to determine the frequency response was the torque applied to the arm frame, measured by two 45” strain gage rosettes mounted on the motor shaft. Unfortunately the sensitivity of this load-ceil was low because of the high stiffness of the steel shaft. Making the shaft hollow could increase the sensitivity, but it was estimated that it would have simultaneously lowered the natural
frequency of the first mode of the frame, bringing it close to the arm-muscle system natural frequencies. Since it had been determined that the frequency response of the torque motor and power amplifier system is uniform in the 0 to 30 Hz frequency range of interest, it was decided to use the amphfier voltage command signal as the input variable.
ESPERI>lESTAL Baseline frequency
RESLLTS
response
To study and characterize the changes in the armmuscle test frame system frequency response, it is first necessary to establish a baseline frequency response. This characterizes the arm when the test frame is held at 45” (see Fig. 2) against a steady offset flexing torque of 5.56 Nm. The baseline frequency responses, corresponding to zero amplitude sinusoidal tool vibrations disturbance, were determined. Figure 3 shows the frequency response between the voltage command signal and the test frame angular displacement, 4. Figure 4 shows the frequency response between the same input and the handle load cell force, F, The plots were produced by the FFT analyzer. To translate the non-dimensional readings (V/V ratio) into meaningful physical units it is necessary to multiply by the proper constant factors as indicated in Figs 3 and 4. The subject used for the frequency response plots of these two figures (and all other plots presented in this paper) was a 35 yr old male. The general form of the baseline frequency response was found to be the same except for minor differences in the location and magnitude of the first peak. The baseline frequency response magnitude of the arm-test frame angular displacement, r#~.vs the input torque, M (see Fig. 3) has a single, rather broad peak around 3 Hz, most likely reflecting several system modes, The peak is followed by an uninterrupted decline in magnitude. Initially at a rate of approximately - 48 db decade- ‘, the rate of decline moderates with the amplitude becoming very small around 30 Hz. The phase angle of the frequency response starts from 0” at a very low frequency and then gradually declines to - 180” around 8 Hz excitation frequency. The phase angle is always negative, meaning that the output angle always lags the input torque. The sudden jumps in the phase angle from - 180’ to + 180” at higher frequencies are due to computer calculation of 4. The noise in the frequency response at very low frequency, below 0.6 Hz, is due to the low signal to noise ratio at that frequency range. This baseline frequency response is similar to those which have been reported by Zahalak and Heyman (1979, Fig. 3; Cannon and Zahalak. 1982, Fig. 5). The magnitude of the baseline frequency response of the test frame handle force, F, (tangential component of force), vs the input torque, )M (see Fig. 4), has two peaks. The first is located around 3 Hz and the second
Biomechanics of the human arm+lbow
.*. ,?_, .-,._.
10
15
.__ ._._ :..,* .\,d
20
25(Hz) 30
,-- Phase Angle
---./L-V-J Magnitude
L-
I
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20
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Fig. 3. Baseline frequency response (6/M). Fig. 4. Baseline frequency response ( F,/M ). around 12 Hz. Both peaks are very broad. Right after the first peak the amplitude decreases very sharply at approximately - 150 db decade-’ slope. The amplitude (in absolute terms) becomes almost zero around 6 Hz (i.e. nearly 30 db down), and then increases gradually to reach the second peak. The phase angle starts from 0” at low frequency and declines gradually until it reaches its lowest value of approximately - 100” around 6 Hz. Then it rises very sharply until it reaches 0”. For higher excitation frequencies the phase angle declines, then remains essentially constant between - 30” and - 40” above 20 Hz. The physical meaning of the F,/M baseline frequency response is not clear yet, future work will attempt to explain the significance of the two peaks and the dip around 6 Hz. A new baseline frequency response was established every day before conducting any series of tests. At least 50-60 averages were used for each baseline. The tests were run continuously and data were collected for 5-8 min until the subject began to feel tired. The frequency resolution of the frequency response plots shown in Figs 3 and 4 is 0.125 Hz. The baseline frequency response was reproducible. Only minor differences were observed after tests performed several hours, or days, or weeks apart. The
391
system
frequency response agrees well with the results of the dynamic and modeling analysis reported by Wakamiya (1982). According to Brunnstrom ( 1966) and Wells (1966) the main flexor muscle groups of the elbow joint are the: (1) biceps brachii, (2) brachialis. (3) brachioradialis. and (4) pronator teres. If one models each of these as a spring with a contractile element and dashpot in series (Wakamiya. 1982), it is possible to show that they contribute several uscillatory modes which, for most adults, lie in the frequency range of O-5 Hz. The result of the combined action of all these modes must be the first peak, which we see in the frequency responses of Figs 3 and 4.
Frequency
responses
with imposed
externd
oscillutions
The objective of the tests was to study and characterize the changes in the arm-frame system frequency response due to an externally imposed sinusoidal oscillation. This oscillation could be compared to that imposed on the operator of a hand held tool. The moment of inertia of the test frame and torque motor rotor could be considered to represent the moment of inertia of the tool, but no attempt was made to model any particular tool. The arm test position (see Fig. 2) used was selected as being a common hand tool operating position, and one which could result in the injury of the tool operator in the case of loss of control. Changing the arm test position and offset torque changes the stiffness of the arm musculature and the forearm weight load torque. This could result in the change of the linearized system frequency response. Future work will investigate the effect of these test conditions on the loss of stability properties of the arm+lbow system. The test procedure was the following. After an initial warm up period a baseline frequency response test was conducted using a small amplitude white noise excitation signal. A small period of rest followed, then’ the test was repeated with a sinusoidal signal of 5 V peak amplitude, corresponding to 2.78 Nm, superimposed on the same noise signal. The noise signal served as a diagnostic in order to determine the frequency response. The excitation continued for 5-10 min until the subject started to feel tired. After another period of rest, the experiment was repeated with a sinusoidal signal of 3 V peak amplitude, corresponding to 1.66 Nm, then a sinusoidal signal of 1 V peak amplitude, corresponding to 0.55 Nm. Only one sinusoidal test was conducted per day. The test frequencies 06 the sinusoidal disturbance used were the following: 2 Hz, 3Hz,4Hz,5Hz,6Hz,7H~8Hz,9Hz,lOHz,lliHz, 12 Hz, 15 Hz, 17 Hz, 20 Hz, 22 Hz, 25 Hz, 27 Hz, 30 Hz. The frequency response of the arm-ftame system with the sinusoidal signal disturbance excitation was compared with the baseline frequency response obtained during the same day of testing. Figures 5-10 are typical frequency response results from these tests, showing angular displacement, 4, to input torque, M. Figure 5 shows the frequency re-
392
N. G. DAGALAKIS er at.
laoa 90’ 0'
-90
-iao*
*
(db3 0 -10 -20 -30
180’ 90” 0” -90’ -180” Fig. 7 Fr.R. (o/M)
18
10
lS(Hz)
la!
1
(lOHr, 5V, Sine) fig. 8 Fr.R. (NM)
I
(db3 -10 IBe -20
a
r
in
5
10
15(Hz)
Figs S-8. Frequency responses (4/M 1.
sponse of the arm-frame system with a 3 Hz frequency, 1 V peak amplitude sinusoidal external disturbance along with the baseline frequency response. Figure 6 shows the results for a sinusoidal signal of the same frequency, but 5 V peak amplitude. Figures 7 and 8 show the results for a sinusoidal signal of 10 Hz frequency with 1 V and 5 V peak amplitude respectively. Figures 9 and 10 show the results for a sinusoidal signal of 30 Hz frequency with 1 V and 5 V peak amplitudes, respectively. Figures 11 and 12 show typical force, F,, to input torque, M, frequency response results. Figure 11 shows the results for a sinusoidal external disturbance excitation signal of 10 Hz and 1 V peak amplitude. Figure 12 shows the results for a sinusoidal signal of the same frequency but 5 V peak amplitude. The two cursor lines in these figures mark the 3.5 Hz and 6.5 Hz frequencies respectively. As will be explained later the frequency response characteristics at those two frequencies were used as a part of the stability analysis study.
The scale of the vertical axis of the frequency response plot of Figs 5-10 is the same as that used for the baseline plot of Fig. 3. Similarly, the vertical scale of the frequency response plots of Figs 11 and 12 is that used for the baseline of Fig. 4. The main differences in the frequency response after the introduction of the sinusoidal disturbance are increases in the peak amplitude and shifts of both the amplitude and phase angle. The magnitude of these changes varies depending on the frequency of the sinusoidal oscillation signal and its amplitude. For example, from Fig. 5 it can be seen that a 3 Hz, 1 V sinusoidal disturbance causes a maximum increase of the 4/M peak magnitude of 7.3 db as compared to the baseline. This corresponds to a 2.3 times increase in the angular displacement, 4, for the same torque excitation, M. From the phase angle plot of the same figure it can be seen that a 3 Hz, 1 V sinusoidal disturbance causes an increase in the phase lag of 4 for low frequencies. An increase of the sinusoidal disturbance peak
Biomechanics
of the human arm-elbow
393
system
,dbJI -10 -20 -30
I-=-
-180” (lOH2,
Baseline
\
(IOHz,
IV, Sine)
SV, Sine)
i’
WbO
--10
(db; -10 -20 -30
1
5
10
lS(Hz)
Figs 9 and 10. Frequency responses (4/M). Figs 11 and 12. Frequency responses (FJM).
amplitude to 5 V (2.78 Nm) casuses even more significant changes. The maximum peak increase (see Fig. 6) of the amplitude is 15.9 db as compared to the baseline, corresponding to a 6.2 times increase in the angular displacement, $ for the same torque excitation, M. The increase in the phase angle lag is equally significant. The whole phase angle curve seems to have shifted to low frequencies, increasing the phase angle lag by 45.1’ at 3.5 Hz. A similar behavior is seen for other sinusoidal disturbance frequencies (see Figs 7-10). The increase in the amplitude of the frequency response and the lag of the phase angle diminish as the frequency of the disturbance approaches 30 Hz. Another change in the frequency response, seen for most sinusoidal disturbance frequencies, is the appearance of two distinct peaks (see Figs 5,6,8, lo), one located around 2 Hz and the other around 3.5 Hz. Under these disturbance conditions, the angular displacement response is apparently dominated by these two modes. The appearance of these peaks was observed in the case of the other subjects at ap-
proximately the same frequencies. Their physical origin is not known yet. In the case of the handle force, F, to input torque, M, frequency response, the introduction of the sinusoidal disturbance also causes significant changes. For a 10 Hz, 5 V peak amplitude sinusoidal sigrtal there is a 3.5 db increase in the peak amplitude as compared to the baseline and a significant shift of t,he phase angle plot to lower frequencies. Another change observed was a shift of the peak asymptote to lower frequencies. Decreasing the peak amplitude of tihe sinusoidal disturbance to 1 V reduced the magnitude of these changes. Changes
in
the
irequency
response
As pointed out in the previous sectlon, the addition of a sinusoidal disturbance torque to the arm-frame test system alters its dynamic response characteristics. The most visible of these changes is the increase in the amplitude of the frequency response magnitude peaks. The angular displacement, 4, to torque, M, frequency response peak splits into two peaks as a
N. G.
394
DAGALAKIS et al.
result of the sinusoidal disturbance which dominates the response. Figure 13 plots the increase in the amplitude of the first of these two peaks (around 2 Hz for this subject), for three different peak amplitudes of sinusoidal disturbance. Figure 14 plots the increase in the amplitude of the second peak (around 3 Hz for this subject) for the same sinusoidal disturbance. Figure 15 plots the increase in the amplitude of the first peak of the handle force, F, to input torque, M, frequency response magnitude as a result of the sinusoidal disturbance. The increase in the arm-frame response angular displacement and force, due to the external sinusoidal disturbance, is quite high. It appears that there are two disturbance frequency ranges which have a particularly bad effect. For this subject and test frame set up, the first is located around 3 Hz and the second around 8 Hz. The first seems to be responsible for a large increase in the amplitude of the response. Similar armframe response behavior was observed with the other test subjects. Besides the increase in the peak amplitude of the
* 5V Peak Amp. She __ 0,V II . II -*,V ., . ,, ._._._
10 20 30 (Hz) Slnusoidal Dlsturbanca Fraquancy Fig.
13.
10 20 Slnusoldal Dlaturbanca Frequency
Fig. 14.
30 (Hz)
* 5V Paak Amp. Sine sjy.. , .--
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.I
I
,. ------
Sinusoidal Diaturbanca Frequency
Fig. 15.
frequency response magnitude caused by the sinusoidal disturbance, a significant increase in the phase angle lag was also observed. Both of these changes could affect the stability properties of the arm-frame system. For this reason it was decided to measure the stability margin of this system before and after the application of the sinusoidal disturbance. One common measure of stability margin, used in the Systems and Controls field, is the phase margin. This measure of relative stability is equal to the additional phase lag required before a system becomes unstable. For instance, consider the arm reaction torque to a test frame input torque. If the reaction torque lags the input torque by more than 180”. then its effect will be superimposed on that of the input torque, resulting in an increase of the angular displacement. If this continues for cycle after cycle of input excitation, it could lead to a loss of stability. The distance between the handle support and the shaft of the test frame was 40 cm. Multiplying this by the tangential component of the handle force, F,will give the reaction arm torque. This multiplication corresponds to 4.9 db, which should be added to the scale of the magnitude (vertical axis) of Figs 4, 11 and 12, to convert it to decibels of Nm/Nm ratio. In Fig. 12 the 0 db line of this torque-to-torque ratio scale is marked with a horizontal dashed line. The intersection of this line with the baseline F,/M frequency response (point no. 1) marks the excitation frequency for which the magnitude of the arm reaction torque is equal to that of the frame input torque. The corresponding phase angle is 80.6’) giving a phase margin = 180” - 80.6” = 99.4”. When a sinusoidal disturbance of 10 Hz frequency and 5 V peak amplitude is applied, the F,/M frequency response changes as shown in Fig. 12. The 0 db point no. 1 now shifts to point no. 2 and the corresponding phase margin becomes 81.6”. This amounts to a decrease in the phase margin stability of 17.8 % with respect to the baseline phase margin. Figure 16 shows the decrease in the phase margin versus the frequency of the sinusoidal disturbance for peak amplitudes of 5 and 3 V. The decrease in the phase margin for a sinusoidal disturbance of 1 V was very small and is not shown here.
395
Biomechanics of the human arm-elbow system W)i 1OOi
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. ,”
.,
.I
~----
. SV Peak Amp. Sine ?3V . . . --
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Sinusoidal Disturbance Frequency
/
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Sinusoidal Disturbance Frequency
Fig. 17.
(HZ)
Fig. 18.
Fig. 16.
Again we see that two frequency ranges of sinusoidal disturbance can cause a significant decrease in the stability margin of the arm-frame system. For this particular subject and test frame, the first frequency range is around 2 Hz and the second around 10 Hz. Although the phase margin is a good index of the stability margin of a system, a more meaningful measure (from a biomechanics point of view) is the phase lag itself. The shift of the frequency response phase angle to lower frequencies due to the sinusoidal disturbance means that the human arm responds more promptly above a certain characteristic frequency of excitation, while below that frequency it responds more sluggishly. From the F,/ M frequency response experimental results it is seen that, for most sinusoidal disturbance frequencies. this characteristic frequency lies around 5 Hz. To demonstrate this shift in the character of the frequency response we measured the change in the phase angle for two frequencies. At 3.5 Hz, located below this characteristic frequency, and at 6.5 Hz, located above it. Figure I7 shows the percent decrease of the phase angle lag of the Fr/M frequency response at 6.5 Hz with respect to the baseline phase angle. Figure I8 shows the percent increase of the phase angle lag of the F,/M frequency response at 3.5 Hz with respect to the baseline phase angle. Figure I9 shows the percent increase of the phase angle lag of the d/M
30
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Fig. 19.
frequency response at 3.5 Hz. At 6.5 Hz the phase angle of the 4/M frequency response was very close to a - 180” asymptote so no shift could be observed.
CONCLUSIONS
An experimental study has been conducted to determine whether external sinusoidal disturbance oscillations (simulating tool vibrations) alter the dynamic response characteristics of the human armmuscle system. The results of this study indicate that the maximum amplitude of the response, as well as the stability margin of the system, do change. The changes depend on the frequency and amplitude of the sinusoidal disturbance. Two sinusoidal disturbance frequency ranges seem to have a particularly strong effect on the dynamic response characteristics. In general, the stability margin of the system seems to decrease for low frequency excitation as a result of the sinusoidal disturbance. Since a decrease in stability margin means that the system becomes less stable, the results indicate that when the human arm system is subjected to certain low frequency sinusoidal vibrations, the neuromuscular system may become more susceptible to losing control. At this stage of the investigation it is not clear why the arm-muscle system becomes more susceptible to lose
396
N. G.
DAGALAKIS er al.
control when it is subjected to external tool vibrations of those frequencies. One possible explanation is that two different groups of muscles resonate at those
frequencies. When any of them is excited to oscillate, it causes a reduction in the ability of the arm-muscle system to react to emergencies. Since it is known that active muscle stiffness increases with mean muscle moment (Cannon and Zahalak, 1982). it should be expected that these critical tool vibration frequencies will increase as the d.c.
offset torque is increased. This means that heavier tools might resonate these muscle groups at higher frequencies. This hypothesis
will be tested at future stages of
our investigation.
subjected to a sinusoidal and trapezoidal variation offorce. J. Appl. PhJsio[. 29, 378-384. Brunnstrom. S. 11966) Clinical Kinesioloyy. 2nd edn. p, 58. F. A. Davis, Philadelphia. P.1. Cannon, S. C. and Zahalak. G. I. (19S2) The mechanical behavior of active human skeletal muscle in small oscillations. J. Biomechanics 15. I I l-121. Dagalakis, N., Yang. J.. Muehlhouse, C.. Sushinsky. G. and Wakamiya. S. (1981) An arm elbow apparatus for studying the biomechanical aspects of loss-of-control. 1981 Advances in Bioenyineering. pp. 6 l-65. .American Society of Mechanical Engineers. New York. Dainis. A. (1981) Evaluation and calibration of the SELSPOT system. Annuui Conference of the .4mericun Socier.r of Biomechonics. Case Western University. Cleveland. OH. Neilson, P. D. (1972) Frequency response characteristic of the tonic stretch reflexes of biceps brachii muscle in intact man. Med.
Acknowledgemenr-This research effort was supported by the National Engineering Laboratory of the National Bureau of Standards. REFERENCES
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Soechting, J. F. and Roberts. W. J. (19731Transfer characteristici between EMG activity and muscle tension under isometric conditions in man. J. PhrrioL 70. 779-793. Wakamiya, S. K. (1982) Development of a mathemetical model of the human arm. M.S. Thesis, University of Maryland, Mechanical Engineering Department. Wells, K. F. (1966) Kinesiology, p. 26:. W.B. Saunders. Philadelohia. PA. Yang, J. er-ol. (1982) Design. construction of an arm-elbow apparatus. University of ,Maryland Technical Report No. 82-8. Zahalak, G. 1. and Heyman, S. J. (1979) A quantitative evaluation of the frequency response characteristics of active human skeletal muscle in cico. J. biomech. Enyny 101, 28-37.