Below is the simulink model used to simulate the reaction of the arm.
These are the results predicted by that model with two sets of values for B and K:
Theoretical values for B and K derived from the time constant and period of the observed data did not give similar outputs to the observed data. After some adjustment of these values (B = 0.1*B, K = 10*K), a set of outputs more similar to the observed data was generated. Both of these changes correspond to an increase in the period of the motion. It may be, therefore, that this model performs poorly when measuring changes over such a small time interval, where less than ten data points form the entire range of analysis. It is also possible that the basic theoretical foundation of the model is flawed, since the entire motion (and cessation of that motion) is consciously controlled, rather than being the result of a spring and damping. Instead of having actual damping, there are muscles pushing in the direction opposite motion, which would produce a different motion. The bicep being modeled as a spring is also somewhat problematic, since a spring would exert the most force when fully extended, whereas one's bicep exerts the least amount of force in this position.
These graphs show the actual observed response of the subject:
These graphs show very little to no overshoot in the motion of the arm in either of the two cases (weighted or empty hand). In the model, this response would correspond to an overdamped system. Based on this model, it would be expected that when the weight was added, more overshoot would be observed, since the moment of inertia would be much larger. The small change in the motion therefore seems surprising. Realistically, however, since the motion is consciously controlled, there may be no additional overshoot if the subject is still able to exert the same degree of control at the new weight. Perhaps with much more weight, such a relationship would be observed. As the subject grew more tired, overshoot might increase as well, but the 10-15 second trials performed were not nearly long enough to observe such a change.
This is the file we used to convert video-captured data to angular motion for modeling.
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