CSCE 452/752: Homework 3 Solutions

Solve the problems below. Show your work and mark your final answers clearly. $\newcommand{\cm}{{\rm cm}} \newcommand{\secs}{{\rm sec}} \newcommand{\rads}{{\rm rad}}$

[a] Mikey is controlling a differential drive robot whose wheels are $10\cm$ apart. He drives both wheels at $5\cm/\secs$ for $5\secs$. How far does Mikey's robot travel? (3 points)

Since $v_l=v_r$, the robot moves straight ahead. \[d = vt = 5 \cdot 5 = 25 \cm \] Note that $\ell=10\cm$ does not come into play here.

[b] Raph is controlling a differential drive robot whose wheels are $10\cm$ apart. He drives both wheels for $4\secs$, resulting in a counterclockwise rotation in place by $\pi$ radians. What velocity did Raph use for the left wheel? The right wheel? (3 points)

Find the angular velocity: $\omega = \frac{\Delta \theta}{t} = \frac{\pi}{4} \frac{\rads}{\secs} $.
Since it's rotation in place, we have $v_l=-v_r$, so $\omega = \frac{v_r-v_l}{\ell} = \frac{2v_r}{10} = \frac{v_r}{5} $ and $\frac{v_r}{5} = \frac{\pi}{4}$. This means that $v_r = \frac{5\pi}{4} \cm/\secs$ and $v_l = -v_r = -\frac{5\pi}{4} \cm/\secs$.

[c] Leo's differential drive robot starts at $(x, y, \theta) = (3\cm, 7\cm, \pi)$. He drives his robot with $v_l=3\cm/\secs$ and $v_r = -3 \cm/\secs$ for $3\secs$. After this motion, the robot is at $(x^\prime, y^\prime, \theta^\prime) = (3\cm, 7\cm, \pi/2)$. What is the distance between the wheels of Leo's robot? (4 points)

Find the angular velocity, starting from $\Delta\theta = \omega t$: \begin{equation*} \omega = \frac{\Delta \theta}{t} = - \frac{\pi/2}{3} = - \frac{\pi}{6} \frac{\cm}{\secs} \end{equation*} Use $\omega$ to find $\ell$: \begin{equation*} \ell = \frac{v_r - v_l}{\omega} = \frac{-3-3}{-\pi/6} = \frac{36}{\pi} \cm \end{equation*}