introduction.
In this laboratory exercise, we modeled the human forearm with second-order linear time invariant differential equation methods. The 2nd order system's response to an initial offset condition was captured from real life by a computerized vision recognition system that tracked colored pieces of cloth attached at the elbow and the wrist. The experiment was repeated with the hand attached to the arm in question holding a 1.5 kilogram mass. The experimental data was also modeled theoretically with a Simulink analog computer approach. While the model used is a vast simplification of the complications of life, our model did more or less a decent job matching the real life data.
The arm was modeled by a system consisting of a cylindrical mass (the main part of the arm), some viscous friction at the elbow, and a spring (the muscular elements) that returned the mass to its steady-state position. The angular displacement of the arm was measured by manipulating the X and Y coordinates of the colored patches obtained from the video feed. The steady state of the arm was chosen to be the upright position, corresponding to an angular displacement of 0 degrees. The arm was "streched" to its initial condition of approximately 45 degrees, and then "released", allowing the muscles to return it to the steady state position. However, the human control system is not perfect, and as a result there is some overshoot as the arm's movement cannot be immediately stopped. This overshoot happens to lend itself to being modeled by the second-order LTI systems used.
By fitting nice curves to the collected data, we were able to obtain the natural frequency and time constant of the human forearm system. After measuring the mass of the arm, we were able to calculate values for J (rotational inertia), B (damping constant), and K (spring constant). We used these values in our Simulink model to confirm the results.
Mathematically, the system is described by the following differential equation.
The time domain solution for this equation is given by
which is the zero-input response to some initial condition. The initial condition is the "stretching" of the arm to a 45 degrees offset. Next, we must find the rotational inertia of the forearm, J. J can be calculated from the measured length and mass of the forearm. Since the cylinder used to approximate the arm rotates from the one end, we use the form
J = 1/3 * m * L^2
When the 1.5 kg mass is added to the hand of the arm in question, the equation for the total rotational inertia becomes
J = 1/3 * m * L^2 + 1.5 * L^2
Now that we have calculated J with or without the additional mass, we can determine the constants B and K. The equations are
B = 2 * J * alpha
K = J * (Wn)^2
.
The value of alpha can be determined several ways from the fitted data. One way is to note that alpha, Wn, and Wd are in a right-triangle relationship, with Wn along the hypotenuse. Therefore, alpha = sqrt( Wn^2 - Wd^2 ). Another way is from the value of atan(Wd / alpha) in the fitted y(t) equation. Sadly, the two different methods resulted in a variety of different values for alpha, and as we were unable to discover the source of the discrepancy (even with the aid of the instructor), we simply chose one that seemed reasonable.
Fitting Curves to the Captured Data
Experimentally, we determined that the mass of Adem's right forearm was approximately 1.76 kg. Measuring his forearm with a measuring device yielded a length of 0.35 meters. Calculating the rotational inertia J with and without the added 1.5 kg mass, we get
J1 = 0.07179 ( No added mass )
A discussion of the plots follows.
There is little very little oscillation after the initial overshoot. This means that Adem's arm has enough damping to prevent large amounts of overshoot when he is not tired. Notice in the next group of graphs that adding the weight decreases the damping capabilities of his arm, since there is much greater oscillation.
Fitting Curves to the Weighted Forearm
Note that in the following graphs, there is much greater oscillation as the arm muscles are not able to stop the angular inertia with the added mass as quickly. The data is much less smooth, and resembles less the 2nd order differential equation fits used to model it. This suggests that our model breaks down under some cases. This fact is evident when looking at the calculated B and K values from the fitted curves. The values are wildly inconsistent, suggesting either that Adem is a totally unpredictable guy, or that the model simply breaks down.
Calculating the rotational inertia J with an additional 1.5 kg mass at the end of the rotational arm element, we get
J2 = 0.25554
Simulating the Forearm with Simulink
We now simulate the same differential equation (reproduced below) with Simulink.
Since we are modeling a zero-input response with non-zero initial conditions, the F(t) in the differential equation is replaced simply with 0. This is because the force from the arm's bicep is considered to be the spring force acting on the forearm, as well as the source of the viscous damping friction that stops the arm's motion. The schematic of the simulink model used is shown below: (Download)
To model the initial offset of the arm, we set the initial condition of the integrator that outputs theta to the initial value of pi / 2 = 1.57. Using the sample values of J, B, and K listed below, we obtain the following response, which is very similar in shape to the actual data obtained from the video capture device, which is also shown below. The following J,B,K values were obtained from the curve fit of the actual data, below.
J = 0.07185
B = 2.17
K = 79.53
The two plots are the same shape. However, the actual data does not exhibit the same degree of oscillation as does the Simulink model. Perhaps the human control system is then in fact better than the differential equation model when the arm is not tired. Note though that the amount of angular overshoot is the approximately the same in both the simulated arm and the actual forearm.
The following plot is of the forearm with the additional 1.5 kg mass. Note that the amount of oscillation is somewhat greater, and the frequency of oscillation is lower. Also, the amount of angular overshoot is larger, as it clearly became more difficult to stop the forearm's motion with the additional weight.
J = 0.25555(with 1.5 kg mass)
Simulink results were generated for other situations, but are not posted here in order to maintain website clarity. In summary, the Simulink models provide an idealized representation of human forearm, but are generally close to the real unweighted data obtained.
Results and Concluding Notes
A summary of the obtained values for J, B, and K is given below. Note that problems with Kaleidograph might be evident below.
Only the forearm:
| Run |
m1 |
m2 |
m3 |
m4 |
Wd |
Wn |
|
alpha |
J |
B |
K |
| (#1) |
2.5835 |
8.7093 |
12.883 |
0.70544 |
12.883 |
33.28323 |
.. |
15.12745 |
0.07185 |
2.173814 |
79.5935211 |
| (#2) |
2.2113 |
11.112 |
-15.888 |
2.3053 |
-15.888 |
-35.1331 |
.. |
14.34783 |
0.07185 |
2.061784 |
88.687123 |
| (#3) |
2.4165 |
12.4888 |
-15.367 |
2.3373 |
-15.367 |
-37.1344 |
.. |
14.797 |
0.07185 |
2.126329 |
99.0783018 |
With the additional 1.5 kg mass:
| Run |
m1 |
m2 |
m3 |
m4 |
Wd |
Wn |
|
alpha |
J |
B |
K |
| (#1) |
2.395 |
9.6553 |
-15.367 |
0.71507 |
10.2 |
24.429 |
.. |
11.74597 |
0.2556 |
6.00454 |
152.535956 |
| (#2) |
1.2149 |
9.7443 |
10.2 |
1.75 |
-13.541 |
-16.451 |
.. |
2.452911 |
0.2556 |
1.253928 |
69.1740797 |
| (#3) |
-0.76163 |
8.91 |
-16.683 |
1.1266 |
-16.683 |
12.70627 |
.. |
-7.93972 |
0.2556 |
-4.05879 |
41.2664618 |
It is clear that some aspects of this laboratory exercise are not quite perfect, since the different ways of calculating alpha as discussed in the introductory notes consistently yield inconsistent values for the same data values. We also experienced great difficulties in obtaining valid curve fits to our collected data. There were data runs for which we obtained K values of 65252 N/m, which would clearly indicate that Adem Kader's right bicep is the strongest in the history humankind. We realized after a very long time that Kaleidograph was performing calculations in degrees, not radians. After fixing the silly problem we arrived at agreeable curve fits. It is unfortunate that the Kaleidograph software is in want of a more 'upfront' interface in selecting the angle mode.
From the video data and the Simulink simulation results, it became clear that the addition of the mass to the hand of the arm in question increased the amount of angular overshoot and oscillation of the human muscle control system. This is not surprising, as we sensed during the execution of the experiment that it became much harder to 'immediately' stop the movement of the arm once we were holding the mass. The Simulink simulations elucidate this fact especially well.
The differential equation model, though a vast simplification it is, models the human forearm surprisingly well under normal circumstances. When adding an additional mass to the hand on the arm in question, the model seemed to break down, since none of the values calculated from our data seemed to correlate with each other any longer (Note especially the B values for the weighted arm.) A quick review of the results of other lab groups' data revealed that in fact certain people's arms were very well modeled under nearly all circumstances by the 2nd order differential equation model. Perhaps Adem is just different.
Comparison to a Non-Athlete
A quick look at the pictures of Adem and Ekua showing off their muscles reveals that Adem is an athlete (soccer) and Ekuwa is not (although we're sure she does sports). So we compared the K and B values of Adem and Ekuwa and tried to compare an athlete's arm to a non-athlete's arm lifting a 1.5kg weight.
Both the K and B values of the athlete were greater than those of the non-athlete. For Adem, Kavg =87.7 N/m and Bavg =3.8 Ns/m whereas Ekua's Kavg = 14.6 N/m and her Bavg=2.1 Ns/m. This shows that an athlete is more capable of lifting the weight faster and stopping it in a shorter time than a non-athlete (stronger biceps and triceps). If you consider the fact that we are also comparing a male to a female, you understand how strong(!) Adem is. The plots below are of Ekua lifting the weight three times, and the table holds her results in comparison with Adem's.
person alpha J B K
------------------------------------------------------------------
Ekua 5.466093 0.2456 2.684945 12.8728344
Ekua 4.774440 0.2456 2.345205 4.85241378
Ekua 2.533421 0.2456 1.244417 26.0367828
Adem 11.74597 0.2556 6.004540 152.535956
Adem 2.452911 0.2556 1.253928 69.1740797
Adem -7.93972 0.2556 -4.05879 41.2664618
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