Project 1

1. Write a Matlab function file for \(f(\theta)\). The parameters \(L_1, L_2, L_3, \gamma, x_1, x_2, y_2\) are fixed constants, and the strut lengths \(p_1,p_2,p_3\) will be known for a given pose. To test your code, set the parameters \(L_1 = 2, L_2 = L_3 = \sqrt 2, \gamma = \frac{p}{2} , p_1 = p_2 = p_3 = \sqrt 5\) from Figure 1.15. Then, substituting \(\theta = -\frac{p}{4}\) or \( \theta = \frac{p}{4}\), corresponding to Figures 1.15(a, b), respectively, should make \(f(\theta) = 0\).

2. Plot f\((\theta)\) on \([-\pi,\pi]\). As a check of your work, there should be roots at \(\pm\frac{\pi}{4}\).

3. Reproduce Figure 1.15. The Matlab commands
>> plot([u1 u2 u3 u1],[v1 v2 v3 v1],'r'); hold on
>> plot([0 x1 x2],[0 0 y2],'bo'
will plot a red triangle with vertices (u1,v1),(u2,v2),(u3,v3) and place small circles at the strut anchor points (0,0),(0,x1),(x2,y2). In addition, draw the struts.

4. Solve the forward kinematics problem for the planar Stewart platform specified by \[x_1 = 5, (x_2, y_2) = (0,6),L_1 = L_3 = 3,L_2 = 3\sqrt 2,\gamma = \frac{\pi}{4},p_1 = p_2 = 5,p_3 = 3.\] Begin by plotting \(f(\theta)\). Use an equation solver to find all four poses, and plot them.

5. Change strut length to \(p_2 = 7\) and re-solve the problem. For these parameters, there are six poses.

\(p_2 = 7.01\)

6. Find a strut length \(p_2\), with the rest of the parameters as in Step 4, for which there are only two poses.

\(p_2 = 4\)