$1.$ A robot is moving in a $\backslash (2$ D $\backslash )$ planar world coordinate plane. Starting from the initial point, the robot has reached point $\backslash ((7,9)\backslash )$ with a heading angle of $\backslash (15^\{\backslash$circ$\}\backslash )$ counterclockwise direction relative to the x $-$axis of the world coordinate. The robot's body coordinate frame is oriented in such a way that its x $-$axis aligns with the direction of the robot's heading angle, and the y $-$axis is to the left of the robot. A sonar sensor$,$ which is placed at $\backslash ((4\backslash$sqrt$\{6\},4\backslash$sqrt$\{2\})\backslash )$ with respect to robot body coordinate frame with an orientation angle $\backslash (30^\{\backslash$circ$\}\backslash )$ from the robot heading in counterclockwise direction, has detected a point object $\backslash (2\backslash$sqrt$\{2\}\backslash )$ directly in front of the itself i$.$e$.\backslash ((2\backslash$sqrt$\{2\},0)\backslash )$ meters. A robotic arm is also placed on the robot. The end$-$effector of this arm is located at $\backslash ((4,-4)\backslash )$ with respect to robot body coordinate frame and aligning its y $-$axis with x $-$axis of body frame i$.$e$.$ rotated $\backslash (90^\{\backslash$circ$\}\backslash )$ clockwise direction.

$($a$)$ Clearly plot the world frame $($W$),$ body frame $($B$),$ sensor frame $($S$),$ robotic arm frame $($M$),$ and the obstacle. Both computer or hand plot are acceptable.

$($b$)$ A map is generated in the world coordinate frame by the information coming from the robot. Where should this object be located in the map i$.$e$.\backslash ($ P$\_\{$o b j$\}^\{$W$\}\backslash )?$

$($c$)$ The robot arm has the capability to reach given points in its end$-$effector's coordinate frame. What is the position of the obstacle in the robot arm's coordinate plane $\backslash ($ P$\_\{$o b j$\}^\{$M$\}\backslash )$ for the robot manipulator to be able to reach the obstacle?