Image$-$based homing for robots can be implemented by generating heading com$-$

mand inputs to a steering system based on the following guidance algorithm. Sup$-$

pose the robot shown in Fig. $3-($a$)$ is to go from point $R$ to a target $T($as shown

in Fig. $(3-($b$)).$ If $R_x,R_y,$ and $R_z$ are vectors from the robot to each landmark, $x,Y,$

$Z,$ respectively, and $T_x,T_y,$ and $T_z$ are vectors from the target to each landmark,

respectively, then heading commands would drive the robot to minimize $R_x-T_x,$

$R_y-T_y,$ and $R_z-T_z$ simultaneously, since the differences will be zero when the

robot arrives at the target.

Figure $3$: Illustration of image$-$based robot homing

If Fig. $4$ represents the control system that steers the robot, represent each block$-$the

controller, wheels, and vehicle. Assuming $a=3,b=2,$ and $c=K_1=1,$ find the cor$-$

responding state$-$space representation in the phase$-$variable form assuming output

is the first state variable.

Reminder : The robotic application means to make this exercise flavorful, one does

not need to understand the application at all to solve this homework problem.