$3$ Bug Algorithms $(30$ points$)$

In class, we will soon talk about different bug algorithms used for simple robotic motion planning. We will analyze each algorithm's performance on a few example scenarios in this problem.

$3\mathrm{.}1$ Lower and Upper Bounds

Imagine that you are a robot wanting to go from a start point to an end point. Between the start and the end point, there are the following obstacles:

A building with a perimeter of $400$ meters

$5$ cars, each with a perimeter of $8$ meters

The straight line distance between the start and end point is $150$ meters. What are the upper and lower bounds for distance traveled using the Bug $1$ and Bug $2$ algorithms? Assume the goal line intersects the building and each car. Provide the following answers in your report:

$($i$)$ Upper and lower bounds for the Bug $1$ algorithm

$($ii$)$ Upper and lower bounds for the Bug $2$ algorithm