Problem $1.(20$pts$)$ Suppose a two$-$wheel differential drive mobile robot equipped with a $2$D

range sensor starts at position $x=1\mathrm{.}0m,y=2\mathrm{.}0m,$ with heading $=\frac{}{4}.$ A range sensor is

attached to the center of the robot. The range sensor detects an obstacle and returns a reading of

$=-\frac{}{6}$ and $d=1\mathrm{.}0m.$

What is the position of the obstacle in the global coordinate frame?

For the same robot, suppose the wheel radius is $0\mathrm{.}3$ m and the length of the axles is $1\mathrm{.}6$ m $.$ For

the wheel encoder, the total ticks per revolution is $50.$ After a while, $20$ ticks were recorded

for the left wheel, and $40$ ticks were recorded for the right wheel, will the car collide with the

obstacle? Write down all your work.