Q$13.$ Computer$-$aided modeling and manufacturing, robotics and virtual reality applications require precise control of spatial

position along a prescribed motion path. In several problems in these fields, it is preferable to maintain constant speed along

the path.

The shown path, defined betwen $t=0$ and $t=1$ is given by

$P=\{\begin{array}{c}x(t)=0\mathrm{.}5+0\mathrm{.}3t+3\mathrm{.}9t^2-4\mathrm{.}7t^3\\ y(t)=1\mathrm{.}5+0\mathrm{.}3t+0\mathrm{.}9t^2-2\mathrm{.}7t^3\end{array}$

As seen, points defined by evenly spaced parameter values such as $t=0,\frac{1}{4},$

$\frac{1}{2},\frac{3}{4}$ and $1$ divide the path into non$-$equal partititons. The task is to apply

numerical integration to determine the $t$ values that will divide this path into $N$

partitions of equal length.

Recall from ME $210$ that the arc length of the path from $t_1$ to $t_2$ is

${\displaystyle \underset{t_1}{\overset{t_2}{}}}\sqrt[\phantom{2}]{[x^{\text{'}}(t){]}^2+[y^{\text{'}}(t){]}^2}dt$

Write a MATLAB code to partition the given path into $N$ subpaths of equal length. In your code use MATLAB's built$-$in integral

fuction for numerical integration. Explain the logic used in your code. Use your code for $N=4$ and $N=10,$ and provide the

obtained $t$ values that separate each partition and the resulting partition lengths.