The project description can be found here. The diagram below, taken from
Numerical Analysis by Sauer, shows the setup for the setup of the Stewart
Platform. The platform consists of three hydraulic struts, p1, p2, and p3, and
a traingular figure which the platform holds. The configuration on the
traingular platform is specficied is by angles gamma and theta. The three hydraulic
struts are fixed at (0, 0), (x1, 0), and (x2, y2)
respectively. The goal of the project was to calculate the x,y, and theta
values given the strut lengths p1, p2, and p3.
The figure below shows a graph of f(theta). The
roots of the function correspond to the possible angles of theta. Each root
corresponds to a unique pose of the traingular object.
Figure 1.15 from the project description was replicated in order to check if the
plotting code worked properly. The replicated figures are shown below:
The parameters were then swithced (see step 4 on the project description)
and the new equation of f(theta) produced four roots, each corresponding to a
unique pose of the traingular object. In order to solve the precise
roots of f(theta), we programmed the bisection
method into MatLab. The bisection method requires a given interval which
contains the root of the function. By looking at the graph of f(theta), we computed
these intervals and the precise value of the roots were found. Then, we programmed
a MatLab function called coordinates which
inputs the roots of f(theta) and outputs the verticies of the traingular figure.
Finally, we edited this code to plot the struts, fixed points,
and the traingular object. A few of the poses collected are shown below.
When the length of the second parameter was changed to p2=7, the function
of theta now contained six roots which corresponded to six unique poses. Once
again our group used the bisection method to find the roots and the coordinates
function to find and plot the figure. The six roots can be seen in the figure
below.
The six poses are shown below:
The final step of this project was to find a possible p2 strut length
which corresponded to the traingular figure having exactly two poses. This was
done by examining the graph of f(theta) while altering the length of strut 2.
The graph of f(theta) with p2=9 is shown below.
The two associated poses are shown below:
