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{\Large\bf Assignment #1}
{(Hand-in date #3)}
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\begin{document}
%\header{BlattNr}{Tutor}{Abgabedatum}{Vorlesungsname}{Namen}{Semester}{Anzahl Aufgaben}
\header{10}{}{2015-07-07}{Mobile Robots}{
\textit{Jan-Peter Hohloch}\\ \textit{Maximus Mutschler}
}{SS 15}{2}
\vspace{1cm}
\Aufgabe{Convex Hull Algorithms}{10}
\begin{enumerate}[(a)]
\item \includegraphics[width=0.7\linewidth]{A1a}
\item \includegraphics[width=0.7\linewidth]{A1b}
~\\$ \begin{tabular}{|c|c|c|c|c|c|c|c|c|c|}
1&2&3&4&5&6&7&8&9&10\\
\hline
2&3&4&5&6&7&8&9&10&9\\
1&2&2&4&4&6&7&8&9&8\\
& 1&1&2&2&4&6&7&8&7\\
&& &1&1&2&4&6&7&6\\
&&&& &1&2&4&6&4\\
&&&&&& 1&2&4&2\\
&&&&&&& 1&2&1\\
&&&&&&&& 1&\\
&&&&&&&&&\\
\end{tabular}$\\
S pushes the new point if it is on the left of the line through the two highest elements of the stack.\\
S pops elements until the new point is on the right of the resultin line.
\item \pagebreak~
\includegraphics[width=0.48\textwidth]{A1c/step1}
\includegraphics[width=0.48\textwidth]{A1c/step2}
\includegraphics[width=0.48\textwidth]{A1c/step3}
\includegraphics[width=0.48\textwidth]{A1c/step4}
\includegraphics[width=0.48\textwidth]{A1c/step5}
\includegraphics[width=0.48\textwidth]{A1c/step6}
\item Sklansky's algorithm runs in linear time, i.e. $\mathcal{O}(n)$
\item QuickHull has a worst case runningtime of $\mathcal{O}(n^2)$, when there are no points that can be ignored. A geometric example of the worst case is a cylindric obstacle because a circle doesn't allow any point to be ignored because the drawn line is always nearer to the circles center then a point on the circle.
\end{enumerate}
\pagebreak
\Aufgabe{Path Planning by dynamic programming}{10}
\begin{enumerate}[(a)]
\item $\left[ \begin{array}{cccccccccc}
-1 & -1 & -1 & -1 & -1 & -1 & -1 & -1 & -1 & -1 \\
-1 & -1 & Inf & Inf & -1 & Inf & Inf & Inf & Inf & -1 \\
-1 & Inf & Inf & Inf & -1 & Inf & Inf & Inf & Inf & -1 \\
-1 & Inf & Inf & Inf & -1 & -1 & Inf & Inf & Inf & -1 \\
-1 & Inf & -1 & Inf & Inf & -1 & -1 & Inf & -1 & -1 \\
-1 & Inf & -1 & Inf & -1 & -1 & -1 & Inf & -1 & -1 \\
-1 & Inf & -1 & Inf & Inf & Inf & Inf & Inf & Inf & -1 \\
-1 & Inf & -1 & Inf & Inf & Inf & Inf & Inf & Inf & -1 \\
-1 & -1 & -1 & -1 & -1 & -1 & -1 & -1 & -1 & -1
\end{array} \right]$
\item Point A:\\
forward1:\\
$\left[ \begin{array}{cccccccccc}
-1 & -1 & -1 & -1 & -1 & -1 & -1 & -1 & -1 & -1 \\
-1 & -1 & 6.41421 & 6.82843 & -1 & - & - & - & - & -1 \\
-1 & 5 & 5.41421 & 5.82843 & occ & - & - & - & - & occ \\
occ & 4 & 4.41421 & 5.41421 & occ & occ & - & - & - & occ \\
occ & 3 & occ & - & - & occ & occ & - & occ & occ \\
occ & 2 & occ & - & occ & occ & occ & - & occ & occ \\
occ & 1 & occ & - & - & - & - & - & - & occ \\
occ & - & occ & - & - & - & - & - & - & occ \\
occ & occ & occ & occ & occ & occ & occ & occ & occ & occ
\end{array} \right]$\\
backward1:\\
$\left[ \begin{array}{cccccccccc}
occ & occ & occ & occ & occ & occ & occ & occ & occ & occ \\
occ & occ & - & - & occ & - & - & - & - & occ \\
occ & - & - & - & occ & - & - & - & - & occ \\
occ & - & - & - & occ & occ & - & - & - & occ \\
occ & - & occ & 5.82843 & 6.82843 & occ & occ & - & occ & occ \\
occ & - & occ & 6.82843 & occ & occ & occ & - & occ & occ \\
occ & - & occ & 7.82843 & 8.24264 & - & - & - & - & occ \\
occ & - & occ & 8.82843 & 9.24264 & 9.65685 & - & - & - & occ \\
occ & occ & occ & occ & occ & occ & occ & occ & occ & occ
\end{array} \right]$\\
forward2:\\
$\left[ \begin{array}{cccccccccc}
occ & occ & occ & occ & occ & occ & occ & occ & occ & occ \\
occ & occ & - & - & occ & - & - & 15.6569 & 16.0711 & occ \\
occ & - & - & - & occ & - & - & 14.6569 & 15.0711 & occ \\
occ & - & - & - & occ & occ & - & 13.6569 & 14.0711 & occ \\
occ & - & occ & - & - & occ & occ & 12.6569 & occ & occ \\
occ & - & occ & - & occ & occ & occ & 11.6569 & occ & occ \\
occ & - & occ & - & - & 9.24264 & 10.2426 & 11.2426 & 12.2426 & occ \\
occ & - & occ & - & - & - & 10.6569 & 11.6569 & 12.6569 & occ \\
occ & occ & occ & occ & occ & occ & occ & occ & occ & occ
\end{array} \right]$\\
backward2:\\
$\left[ \begin{array}{cccccccccc}
occ & occ & occ & occ & occ & occ & occ & occ & occ & occ \\
occ & occ & - & - & occ & 17.6569 & 16.6569 & - & - & occ \\
occ & - & - & - & occ & 16.6569 & 15.6569 & - & - & occ \\
occ & - & - & - & occ & occ & 14.6569 & - & - & occ \\
occ & - & occ & - & - & occ & occ & - & occ & occ \\
occ & - & occ & - & occ & occ & occ & - & occ & occ \\
occ & - & occ & - & - & - & - & - & - & occ \\
occ & - & occ & - & - & - & - & - & - & occ \\
occ & occ & occ & occ & occ & occ & occ & occ & occ & occ
\end{array} \right]$\\
forward3\\
$\left[ \begin{array}{cccccccccc}
occ & occ & occ & occ & occ & occ & occ & occ & occ & occ \\
occ & occ & - & - & occ & - & - & - & - & occ \\
occ & - & - & - & occ & - & - & - & - & occ \\
occ & - & - & - & occ & occ & - & - & - & occ \\
occ & - & occ & - & - & occ & occ & - & occ & occ \\
occ & - & occ & - & occ & occ & occ & - & occ & occ \\
occ & - & occ & - & - & - & - & - & - & occ \\
occ & - & occ & - & - & - & - & - & - & occ \\
occ & occ & occ & occ & occ & occ & occ & occ & occ & occ
\end{array} \right]$\\
backward3\\
$\left[ \begin{array}{cccccccccc}
occ & occ & occ & occ & occ & occ & occ & occ & occ & occ \\
occ & occ & - & - & occ & - & - & - & - & occ \\
occ & - & - & - & occ & - & - & - & - & occ \\
occ & - & - & - & occ & occ & - & - & - & occ \\
occ & - & occ & - & - & occ & occ & - & occ & occ \\
occ & - & occ & - & occ & occ & occ & - & occ & occ \\
occ & - & occ & - & - & - & - & - & - & occ \\
occ & - & occ & - & - & - & - & - & - & occ \\
occ & occ & occ & occ & occ & occ & occ & occ & occ & occ
\end{array} \right]$
\\
Point B:\\
forward1:\\
$\left[ \begin{array}{cccccccccc}
occ & occ & occ & occ & occ & occ & occ & occ & occ & occ \\
occ & occ & - & - & occ & 2.41421 & 1 & 1.41421 & 2.41421 & occ \\
occ & - & - & - & occ & 1.41421 & - & 1 & 2 & occ \\
occ & - & - & - & occ & occ & - & - & - & occ \\
occ & - & occ & - & - & occ & occ & - & occ & occ \\
occ & - & occ & - & occ & occ & occ & - & occ & occ \\
occ & - & occ & - & - & - & - & - & - & occ \\
occ & - & occ & - & - & - & - & - & - & occ \\
occ & occ & occ & occ & occ & occ & occ & occ & occ & occ
\end{array} \right]$\\
backward1:\\
$\left[ \begin{array}{cccccccccc}
occ & occ & occ & occ & occ & occ & occ & occ & occ & occ \\
occ & occ & - & - & occ & 2 & - & - & - & occ \\
occ & - & - & - & occ & 1 & - & - & - & occ \\
occ & - & - & - & occ & occ & 1 & 1.41421 & 2.41421 & occ \\
occ & - & occ & - & - & occ & occ & 2.41421 & occ & occ \\
occ & - & occ & - & occ & occ & occ & 3.41421 & occ & occ \\
occ & - & occ & 7.82843 & 6.82843 & 5.82843 & 4.82843 & 4.41421 & 4.82843 & occ \\
occ & - & occ & 8.24264 & 7.24264 & 6.24264 & 5.82843 & 5.41421 & 5.82843 & occ \\
occ & occ & occ & occ & occ & occ & occ & occ & occ & occ
\end{array} \right]$\\
forward2:\\
$\left[ \begin{array}{cccccccccc}
occ & occ & occ & occ & occ & occ & occ & occ & occ & occ \\
occ & occ & - & 12.8284 & occ & - & - & - & - & occ \\
occ & - & - & 11.8284 & occ & - & - & - & - & occ \\
occ & - & - & 10.8284 & occ & occ & - & - & - & occ \\
occ & - & occ & 9.82843 & 10.2426 & occ & occ & - & occ & occ \\
occ & - & occ & 8.82843 & occ & occ & occ & - & occ & occ \\
occ & - & occ & - & - & - & - & - & - & occ \\
occ & - & occ & - & - & - & - & - & - & occ \\
occ & occ & occ & occ & occ & occ & occ & occ & occ & occ
\end{array} \right]$\\
backward2:\\
$\left[ \begin{array}{cccccccccc}
occ & occ & occ & occ & occ & occ & occ & occ & occ & occ \\
occ & occ & 13.8284 & - & occ & - & - & - & - & occ \\
occ & 13.8284 & 12.8284 & - & occ & - & - & - & - & occ \\
occ & 12.8284 & 11.8284 & - & occ & occ & - & - & - & occ \\
occ & 13.2426 & occ & - & - & occ & occ & - & occ & occ \\
occ & 14.2426 & occ & - & occ & occ & occ & - & occ & occ \\
occ & 15.2426 & occ & - & - & - & - & - & - & occ \\
occ & 16.2426 & occ & - & - & - & - & - & - & occ \\
occ & occ & occ & occ & occ & occ & occ & occ & occ & occ
\end{array} \right]$\\
forward3\\
$\left[ \begin{array}{cccccccccc}
occ & occ & occ & occ & occ & occ & occ & occ & occ & occ \\
occ & occ & - & - & occ & - & - & - & - & occ \\
occ & - & - & - & occ & - & - & - & - & occ \\
occ & - & - & - & occ & occ & - & - & - & occ \\
occ & - & occ & - & - & occ & occ & - & occ & occ \\
occ & - & occ & - & occ & occ & occ & - & occ & occ \\
occ & - & occ & - & - & - & - & - & - & occ \\
occ & - & occ & - & - & - & - & - & - & occ \\
occ & occ & occ & occ & occ & occ & occ & occ & occ & occ
\end{array} \right]$\\
backward3\\
$\left[ \begin{array}{cccccccccc}
occ & occ & occ & occ & occ & occ & occ & occ & occ & occ \\
occ & occ & - & - & occ & - & - & - & - & occ \\
occ & - & - & - & occ & - & - & - & - & occ \\
occ & - & - & - & occ & occ & - & - & - & occ \\
occ & - & occ & - & - & occ & occ & - & occ & occ \\
occ & - & occ & - & occ & occ & occ & - & occ & occ \\
occ & - & occ & - & - & - & - & - & - & occ \\
occ & - & occ & - & - & - & - & - & - & occ \\
occ & occ & occ & occ & occ & occ & occ & occ & occ & occ
\end{array} \right]$
\pagebreak
\item ~
Point A:\\
\includegraphics[width=0.7\linewidth]{A2/gradientAb}\\
Point B:\\
\includegraphics[width=0.7\linewidth]{A2/gradientBb}
\pagebreak
\item path a:\\
$\left[ \begin{array}{cc}
8 & 2 \\
7 & 2 \\
6 & 2 \\
5 & 2 \\
4 & 3 \\
5 & 4 \\
6 & 4 \\
7 & 5 \\
7 & 6 \\
7 & 7 \\
8 & 8
\end{array} \right]$\\
path b:\\
$\left[ \begin{array}{cc}
3 & 7 \\
4 & 7 \\
5 & 8 \\
6 & 8 \\
7 & 8 \\
8 & 8
\end{array} \right]$
\end{enumerate}
\end{document}
Jan-Peter Hohloch