CMAC2 Project

Introduction

The shoulder joint and the elbow joint of this arm are each limited to 180 degrees of travel. As a result, the arm has a teardrop-shaped "workspace" of points it can reach. The forward kinematics problem is to calculate the position of the end of the arm (the "hand") given the shoulder and elbow joint angles. This is a straightforward bit of math, solved by the arm_position function. The inverse kinematics problem is harder: given a point we want the arm to reach, what joint angles should be used to accomplish that?

To solve this problem, you will train a CMAC on random pairs of joint angles (theta1,theta2) and the associated point (x,y) that the arm reaches. After training, the CMAC should be able to estimate the joint angles need to reach any target (x,y) in the workspace. We are actually constructing two CMACs: one to estimate theta1, and another to estimate theta2.

1. Download the code from the matlab/cmac2 directory, or grab the zip file . This code runs, but the train function is incomplete; you will have to fill in the missing details.

2. Run the initialize function to set up the required data structures and initialize the display. You will see the arm drawn as a blue line (upper arm) and a green line (forearm).

3. Click in the figure window. A red asterisk will appear to mark the location (x,y) of the mouse click. Surrounding it will be a set of black squares. These are the bins the CMAC is using to encode the value associated with (x,y). By clicking repeatedly in the window, you will note that nearby points have overlapping sets of bins; that's how the CMAC is able to approximate a function from sparse data. The 1000 bins are generated randomly; the cordinates of their centers are stored in the variable bin_centers.

4. Type train to train the CMAC. The algorithm will run through 5000 points, initially doing nothing (you will have to fill in the details), and then displaying all the training data.

5. Fill in the missing bits of the train function. Note that there is a learning rate eta set up for you in initialize, and the set of bins to be used for storing each data point is set up for you in the variable b. This should be used to index the variables named theta1_memory and theta2_memory. If your learning algorithm is correct, the error plot in figure 2 should show a rapid decline, and clicking inside the workspace in Figure 1 should bring the arm close to the target location.

6. Look at the clickhandler function for a hint on how to use the bins.

What to Hand In (Hardcopy Only, Please)

• The source listing of your modified train function.

• A plot of the training error.

• Answers to the following questions:
  1. What happens when you click far outside the workspace? Why do you get this result?

  2. What happens when you click outside but close to the workspace? Why do you get this result?

  3. After training, what fraction of the bins have nonzero values?