Biomedical Engineering Reference
In-Depth Information
Figure 9.19
Analog or digital solution to solve for tendon force
F
t
for any given
F
ce
and for any muscle.
B /M
and
K /B
ratios are known if the twitch time
T
for that
muscle is known. See the text for complete details.
damper and spring elements. Thus, the equations of motion are:
x
1
+
x
2
=
L
0
=
constant
F
ce
=
M x
1
+
B x
1
+
Kx
1
(9.5
a
)
The tendon force
F
t
is seen only if the series elastic spring increases its
length beyond resting length, or:
F
t
=
Kx
1
(9.5
b
)
In analog or digital form, this second-order differential equation can be
solved for
x
1
(see Figure 9.19) to predict the tendon force for any given
EMG. The final part of the model is to determine
K
,
B
, and
M
for the muscle.
Fortunately, we do not have to measure them separately. Rather, we can make
use of the fact that the twitch waveform is quite close to that of a critically
damped second-order system (Milner-Brown et al., 1973
a
). The duration of
the twitches is sufficiently long, such that, in comparison, the motor unit
action potential can be considered as an impulse. For a mass-spring-damper
system that is critically damped, the twitch time,
T
, (see Section 9.0.5) allows
us to calculate the correct ratios,
T
=
2
M /B
or
B /M
=
2
/T
.Also,
B
=
2
√
MK
. Thus,
K /B
1
/
2
T
. Therefore, the differential equation as
modeled in Figure 9.19 can be solved by knowing the values for
B /M
and
K /B
, which are automatically known if we know
T
for any given muscle.
The only remaining parameter that is needed to curve-fit the model is a
“gain”
A
to model the relationship between the EMG (in microvolts) and the
contractile element tension
F
ce
(in newtons).
Figure 9.20 shows the results of comparisons between the predicted muscle
tension
F
t
and that measured experimentally. The biceps muscle EMG was
recorded along with elbow flexor tension from a force transducer attached
firmly to the wrist via a cuff. The smoother curve is the transducer output,
while the noisier signal is the predicted
F
t
. The difference between measured
and predicted curves is minimal and can be partially attributed to several
measurement and assumption errors. First, we are not measuring the tendon
tension directly but via a transducer attached to the wrist. The recorded tension
=
B /
4
M
=
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