E12 Physical Systems Analysis:
Lab #5 The Human Transfer Function

Draft report due 1 week after lab date

Final report due 2 weeks after lab date

Lab Resources

We will be working in the robotics lab to capture data (video-based). Then you can do the analysis using Matlab.

Overall Objectives

For this lab you will be modeling the behavior of your arm as you move it between two positions. We will be modeling this as a simple 2nd order rotational system and be looking for changes in the model parameters as your arm gets tired.

The final product of this lab should be a characterization of the behavior of your arm. In particular, you are looking to see if you can characterize your arm as underdamped, critically damped, or overdamped. There is some expectation that it is not an unstable system.

As before, please work in groups of 2-3 people. Feel free to change lab groups if you wish.

Introduction

For this lab you will be investigating moving your forearm between a 90 degree bent position and a straight arm, bending only at the elbow. We will be tracking the hand and elbow using a video camera.

We can model this motion using a simple mass/spring/damper rotational system. The mass is your forearm, and the muscles provide the spring and damping forces. While this is not a strictly correct model of what is going on, it is close enough for us to characterize the system.

We can treat moving your arm from one position to another as a zero-input response to an offset. In other words, your arm wants to be stretched out, but you have bent it back to 90 degrees and are going to let it go.

The basic equation of motion is:

Taking the Laplace Transform of this expression, with initial conditions, we get the following:

Reorganizing the initial conditions to the right side, we get closer to a standard form.

Finally, we can make the substitution that B/J = 2 * alpha and K/J = natural undamped frequency.

When this system is overdamped, the roots of the denominator are real and negative. When the roots are complex conjugates of one another, the system is underdamped, and will oscillate before coming to rest. Finally, if the roots are purely complex or lie in the right half of the complex plane the system is purely oscillatory or unstable.

The time domain solution for the damped system is a pair of exponentially decreasing functions. You can identify the different time constants by factoring the polynomial. The roots will identify the time constant--the larger the root, the longer the time constant. If you have two identical real, roots, then the system is critically damped.

The time domain system for the underdamped system is:

You can find this in Laplace transform tables for a pair of complex roots. Note that the overall exponential decay is bounded by the time constant determined by alpha. The oscillation frequency is given by the damped frequency, which is related to alpha and the natural frequency. by a right-triangle relationship, with the natural frequency as the hypoteneuse.

Have fun with this lab

Tasks

  1. Determine the approximate mass of the arm of each person in your lab group. You can do this by using the density of water and approximating your arm as a cylinder, or you can try using a scale (the weight of a simply supported beam with supports at either end and no moments applied is evenly split between the supports). Assuming the mass is somewhat evenly distributed across yoru forearm, calculate the angular momentum.

  2. Use the camera capture program (the instructor will show you how to do this) to capture data of each person your lab group moving their arm from a ninety degree bend to fully extended, pausing, and then moving it back. Make the motion as quickly as possible while trying not to overshoot your target position. Repeat the sequence twice. Then add a small weight and repeat the motion twice more.

  3. The capture program will capture a position for both the elbow and the arm. Use this information to convert your data into angular motion. Plot the resulting angles and see if they appear reasonable. Do some curve fitting with the theoretical models suggested above and see if they give reasonable parameters.

  4. Build a Simulink model, plug in some reasonable parameters and see how they compare to your capture data.

  5. Try to determine the effect of adding the small weight, and of your arm(s) getting tired. Do these change the behavior of the system at all?

  6. Can you say anything about how "optimal" our control system is? From your estimates for the system parameters, can you estimate the damping and spring coefficients for this approximate?

  7. Comment on how well this model fit your data and what the causes of error could be.

Extensions

  1. Try other rotational joints.

  2. Try it with a lot more weight.

  3. Try it for a longer period of time and see if there is a noticeable effect of tiredness.

  4. Compare an athlete and non-athlete.

Data

Writeup

Make a web page that shows your models, experiments, and conclusions. Organize it so that there is a main page that gives a discription of the task and a summary of all the cases you explored. This page should also include a brief discussion of what you discovered and your conclusions, and it needs to have any extra derivations, outside information, or calculations you made. From the main page link each of the cases, as appropriate. You should combine cases to make a point--for example, somewhere in your writeup you should have a table with the results for weight/no weight.

Send the URL to Prof. Maxwell when it is ready for viewing.