Consider the $3$PRR planar parallel robot which is discussed in our lectures as shown below. The formulation of this robot is given in the lecture notes. This robot is used to draw shapes with the aid of a pen placed at point $A.$ The robot is driven by three linear actuators. The constant parameters of the robot are, $a_1=b_1=c_1=20,m_b=m_c=10,d=45,=90,=60,=120,{}_p=0$

In order to draw the desired shape by the pen at $A,$ the motion of A with respect to $O$ is specified as vec$(r{)}_A=r_1$vec$(u{)}_1^{(0)}+r_2$vec$(u{)}_2^{(0)}$ where $r_1$ and $r_2$ are specified positions as functions of time $t.$

The kinematic equations relating the motion of $A$ to the joint variables are:

$a_1$Cos${}_{a1}+s_1=r_1$

$a_1$Sin$(theta{)}_{a1}=r_2$

$s_2$Cos$+b_1$Cos${}_{b1}=r_{1B}$

$s_2$Sin$+b_1$Sin$(theta{)}_{b1}=r_{2B}$

$d+s_3$Cos$+c_1$Cos${}_{c1}=r_{1C}$

$s_3$Sin$+c_1$Sin$(theta{)}_{c1}=r_{2C}$

where the expressions for $r_{1B},r_{2B},r_{1C},r_{2C}$ are given in the lecture notes.

It is desired to draw a counter$-$clockwise circle with radius $r=2$ with a constant angular velocity of $=ra\frac{d}{s}.$ For this purpose, vec$(r{)}_A$ is specified as:

vec$(r{)}_A=[15+$rCos$(t)]vec(u{)}_1^{(0)}+[15+$rSin$(t)]vec(u{)}_2^{(0)}$

We would like to calculate the linear actuator displacements $s_1,s_2,s_3$ so that the actuators can be controlled to draw the desired circle. Perform the following computations:

i$)$ Plot the desired circle from time $t=0$ to $2$ s by using an increment of $0\mathrm{.}01$ s $.$

ii$)$ Apply Matlab's fsolve function to calculate the values of $\stackrel{}{x}={\left[\begin{array}{ccccc}{s}_1& {}_{a1}& s_2& {}_{b1}& s_3(theta{)}_{c}1\\ ?& ?& ?& ?& ?\end{array}\right]}^T$ for time $t=0$ to $2$ s with an increment of $0\mathrm{.}1$ s $.$ To start the solution, choose the initial guesses at time $t=0$ as ${\stackrel{}{x}}_0=$ transpose$([2512010605150$