For the Two$-$Link$-$Planar Robot $($TLPR$)$ shown in

It is desired to move its end effector in the x$\_(0)-$y$\_(0)$ plane along a

vertical straight line parallel to the y$\_(0)-$axis such that it moves from

$(0\mathrm{.}5,0)$ to $(0\mathrm{.}5,0\mathrm{.}7)$ in $5$ seconds following a cubic polynomial

trajectory: y$($t$)=$a$\_(0)+$a$\_(1)$t$+$a$\_(2)$t$^(2)+$a$\_(3)$t$^(3)$ with initial and final

velocities equal to zero. Thus, the trajectory is given as:

y$($t$)=0\mathrm{.}084$t$^(2)-0\mathrm{.}0112$t$^(3).$ The trajectory generator is given as

a MATLAB function in the attached SIMUUNK environment and

titled "Trajectory Generator". This function generates both x$($t$)$

and y$($t$).$

To find the corresponding joint trajectories. The inverse kinematic

problem is solved for this robot and is given as a MATLAB function in the attached SIMULINK environment and titled "Inverse

Kinematics". Its inputs are x$($t$)$ and y$($t$)$ and its outputs are $\backslash$theta $\_(1)($t$)$ and $\backslash$theta $\_(2)($t$)-$

To make sure that the inverse kinematics is correct and to compare your control results with the desired task$-$space task, the

forward kinematics for this manipulator is given as the MATLAB function "Forward Kinematics". Its inputs are $\backslash$theta $\_(1)($t$)$ and $\backslash$theta $\_(2)($t$)$ and

its cutputs are x$($t$)$ and y$($t$).$

You can use any of these functions as you wish and as you find necessary without needing to change anything. You can combine

the signals by using SIMULINK MUXes and DEMUXes as you find suitable.

The robot dynamic parameters are given in the attached "param$\_$TLPR$.$m$"$ and they are defined as global parameters to pass

them to the S$-$function used. Make sure that you run the file every time you start MATLAB.

The robot dynamic equations $($Eqs$.6\mathrm{.}86-6\mathrm{.}91)$ are given as a MATLAB s$-$function called "sfunc$\_$TLPR$".$ This s$-$function has two

inputs corresponding to the control torques applied at the two joints: $\backslash$tau $\_(1)$ and $\backslash$tau $\_(2).$ It has four outputs corresponding to the states:

$\backslash$theta $\_(1)($t$),\backslash$theta $\_(2)($t$),\backslash$theta $\_(1)^()($t$),$ and $\backslash$theta $\_(2)^()($t$).$ The initial conditions corresponding to the initial end effector position $(0\mathrm{.}5,0)$ are already

calculated and given in the structure of the s$-$function. Your goal is to find control laws for $\backslash$tau $\_(1)$ and $\backslash$tau $\_(2)$ to achieve the desired

tracking task. Make sure that "sfunc$\_$TLPR$"$ is always in your MATLAB working directory. Don't change its name or content.

Two independent joint models are developed for this robot based on the dynamic model $($Eq$5.6\mathrm{.}86-6\mathrm{.}91)$ and given

as:

J$\_(1)\backslash$theta $\_(1)^()+$d$\_(1)(\backslash$theta $\_(1),\backslash$theta $\_(2),\backslash$theta $\_(1)^(),\backslash$theta $\_(2)^()(,)/($b$)$ar $(\backslash$theta $)\_(1),\backslash$theta $\_(2)^())=\backslash$tau $\_(1)$

J$\_(2)\backslash$theta $\_(2)^()+$d$\_(2)(\backslash$theta $\_(1),\backslash$theta $\_(2),\backslash$theta $\_(1)^(),\backslash$theta $\_(2)^()(,)/($b$)$ar $(\backslash$theta $)\_(1)(,)/($b$)$ar $(\backslash$theta $)\_($z$))=\backslash$tau $\_(2)$

Where

J$\_(1)=$m$\_(1)$l$\_($c$1)^(2)+$m$\_(2)($l$\_(1)^(2)+$l$\_($cz$)^(2))+$I$\_(1)+$I$\_(2)$

J$\_(2)=$m$\_(2)$l$\_($c$2)^(2)+$I$\_(2)$

And

d$\_(1)=2$m$\_(2)$l$\_(1)$l$\_($c$2)$c$\_(2)\backslash$theta $\_(1)^()+$d$\_(12)\backslash$theta $\_(2)^()+2$h$\backslash$theta $\_(1)^()\backslash$theta $\_(2)^()+$h$\backslash$theta $\_(2)^()^(2)+$g$\_(1)$

d$\_(2)=$d$\_(21)\backslash$theta $\_(1)^()-$h$\backslash$theta $\_(1)^()^(2)+$g$\_(2)$

You don't have to add a disturbance term to any controller simulation here as the disturbance is already

incorporated in the nonlinear dynamics within the s$-$function. Further, the process transfer functions shown in

different places in the textbook $($e$.$g$.$ Figure $8\mathrm{.}12$ on page $283$ and $8\mathrm{.}17$ on page $289)$ are for designing controllers not

for simulating them. Use only the provided s$-$function without any change for all your simulations.

Notice that the robot has neither motor dynamics nor damping.

Required:

You are asked to design, test, simulate, and discuss the performance of $4$ controllers corresponding to the above mentioned

and detailed task:

give

one degree of freedom$)$ or

$-$D$($s$)/($r$)$

$$

$($two degree of freedom$).$ Don't add the

disturbance term as it is already in the dynamics. Don't simulate in the presence of the process transfer function but

instead using the s$-$function. As said, the process transfer function is just for developing the controller. Assume any

natural frequency and damping you find suitable. Make sure the closed$-$loop controllers work well and that the

tracking task of the end effector can be achieved.

npare the performance with the above controller $($in

s$1.$

Design a multivariable nonlinear controller $($one central controller in the form of matrices for the whole robot$)$

according to the PD with gravity compensation scheme given

u$=-$K$\_($P$)$tilde$($q$)-$K$\_($D$)$q$^()+$g$($q$)$

Find the gain matrices as you find

suitable. Make sure the closed$-$loop controller works well and that the tracking task of the end effector can be

achieved. Compare the results with the above controllers.

Please note the following while preparing your report:

Solve as much as you can with an acceptable system performance. This is a learning experience designed

for you. The more you practice the more you learn.

Submit "your work" only and don't risk your semester$-$long hard work.

Show all necessary solution steps.

Assume whatever you find necessary.

Explain your work and results with