$5\mathrm{.}9$ Problems

Consider a differential$-$drive robot that uses wheel encoders only. The wheels are a distance $d$ apart, and each wheel has radius $r.$ Suppose this robot uses only its encoders to attempt to describe a square, with sides of length $1000r,$ returning to the origin. For each of range error, turn error, and drift error, supposing an error rate of $10\%,$ compute the worst$-$case effect of each type of error on the different between the final actual robot position and the original position in both position and orientation.