\documentclass[a4paper,12pt]{scrartcl}
\usepackage[ngerman]{babel}
\usepackage{graphicx} %BIlder einbinden
\usepackage{amsmath} %erweiterte Mathe-Zeichen
\usepackage{amsfonts} %weitere fonts
\usepackage[utf8]{inputenc} %Umlaute & Co
\usepackage{hyperref} %Links
\usepackage{ifthen} %ifthenelse
\usepackage{enumerate}
\usepackage{pdfpages}
\usepackage{algpseudocode} %Pseudocode
\usepackage{dsfont} % schöne Zahlenräumezeichen
\usepackage{amssymb, amsthm} %noch stärker erweiterte Mathe-Zeichen
\usepackage{tikz} %TikZ ist kein Zeichenprogramm
\usetikzlibrary{trees,automata,arrows,shapes}

\pagestyle{empty}


\topmargin-50pt

\newcounter{aufgabe}
\def\tand{&}

\newcommand{\makeTableLine}[2][0]{%
  \setcounter{aufgabe}{1}%
  \whiledo{\value{aufgabe} < #1}%
  {%
    #2\tand\stepcounter{aufgabe}%
  }
}

\newcommand{\aufgTable}[1]{
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}

\def\header#1#2#3#4#5#6#7{\pagestyle{empty}
\begin{minipage}[t]{0.47\textwidth}
\begin{flushleft}
{\bf #4}\\
#5
\end{flushleft}
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\begin{minipage}[t]{0.5\textwidth}
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#6 \vspace{0.5cm}\\
%                 Number of Columns    Definition of Columns      second empty line
% \begin{tabular}{|*{5}{C{1cm}|}}\hline A1&A2&A3&A4&$\Sigma$\\\hline&&&&\\\hline\end{tabular}\\\vspace*{0.1cm}
\aufgTable{#7}
\end{flushright}
\end{minipage}
\vspace{1cm}
\begin{center}
{\Large\bf Assignment #1}

{(Hand-in date #3)}
\end{center}
}



%counts the exercisenumber
\newcounter{n}

%Kommando für Aufgaben
%\Aufgabe{AufgTitel}{Punktezahl}
\newcommand{\Aufgabe}[2]{\stepcounter{n}
\textbf{Exercise \arabic{n}: #1} (#2 Punkte)\\}




\begin{document}
    %\header{BlattNr}{Tutor}{Abgabedatum}{Vorlesungsname}{Namen}{Semester}{Anzahl Aufgaben}
    \header{4}{}{2015-19-05}{Mobile Robots}{
    	\textit{Jan-Peter Hohloch}\\ \textit{Maximus Mutschler}
    }{SS 15}{3}
    \vspace{1cm}

	\Aufgabe{}{8}
	\begin{enumerate}[(a)]
	\item $v_w = \cos\left(\theta_w-\phi\right)\cdot v_R  = v_R \cdot \underbrace{\left(\cos\left(\theta_w\right)\cdot \cos\left(\phi\right) + \sin\left(\theta_w\right) \cdot \sin\left(\phi\right)\right)}_{T}$
	\item \begin{align*}
	v_1&= v_R \cdot \left(\sin\left(\frac{\pi}{2}+\frac{\pi}{6}\right)\cdot \cos\left(\phi\right) + \cos\left(\frac{\pi}{2}+\frac{\pi}{6}\right) \cdot \sin\left(\phi\right)\right)-l ^R\omega\\
	&=v_R \cdot \left(\frac{\sqrt{3}}{2} \cos\left(\phi\right)- \frac{1}{2} \sin\left(\phi\right)\right)-l ^R\omega\\
	v_2&= v_R \cdot \left(\sin\left(\frac{\pi}{2}-\frac{\pi}{2}\right)\cdot \cos\left(\phi\right) + \cos\left(\frac{\pi}{2}+\frac{\pi}{2}\right) \cdot \sin\left(\phi\right)\right)-l ^R\omega\\
	&=v_R \cdot \sin\left(\phi\right) -l^R\omega\\
	v_3&= v_R \cdot \left(\sin\left(\frac{\pi}{2}+\frac{5\pi}{6}\right)\cdot \cos\left(\phi\right) + \cos\left(\frac{\pi}{2}+\frac{5\pi}{6}\right) \cdot \sin\left(\phi\right)\right)-l ^R\omega\\
	&=v_R \cdot \left(-\frac{\sqrt{3}}{2} \cos\left(\phi\right)- \frac{1}{2} \sin\left(\phi\right)\right)-l ^R\omega\\
	^Rv_x &= v_R * \cos\left(\phi\right)\\
	^Rv_y &= v_R * \sin\left(\phi\right)\\
	\\ \Rightarrow
	\\ \begin{pmatrix}
	v_{w,1} \\
	v_{w,2}\\
	v_{w,3}\\
	\end{pmatrix} &=
	\begin{pmatrix}
	\frac{\sqrt{3}}{2} & -\frac{1}{2} &-l\\
		1 & 0 &-l\\
			-\frac{\sqrt{3}}{2} & -\frac{1}{2} &-l\\
	\end{pmatrix} \cdot
	\begin{pmatrix}
	^Rv_x \\
	^Rv_y\\
	^R\omega\\
	\end{pmatrix}
\end{align*}

	\end{enumerate}
\pagebreak
	\Aufgabe{}{8}
	\begin{enumerate}[(a)]
			\item $	^bg=
			 ^b\mathbf{R}_w \cdot \begin{pmatrix}
			 0 \\ 0\\ -g
			 \end{pmatrix}
			$
			\item $^ba_m = ^bg+^ba_l\\
			$
			if  $ ^ba_m=0$ \\
			$
			^ba_l = -^bg
			$
			\item static $\Rightarrow ^ba_l=0$\begin{align*}
			^ba_m &= b_g+^ba_l\\ &=  ^b\mathbf{R}_w \cdot \begin{pmatrix}
			0 \\ 0\\ -g
			\end{pmatrix} \\
			&=
			\begin{pmatrix}
			\sin(\theta)g \\-\cos(\theta)*\sin(\phi)g \\ -\cos(\theta)*\cos(\phi)g
			\end{pmatrix}\\
			^ba_{m,x}&=	\sin(\theta)g \\
			\theta &= \sin^{-1}(\frac{^ba_{m,x}}{g})\\
			^ba_{m,y} &= -g\cdot \sin(\phi)\cdot \sqrt{1-\left(\frac{^ba_x}{g}\right)^2} \\
			\phi&= \sin^{-1} \left(-\frac{^ba_y}{g*\sqrt{1-\left(\frac{^ba_x}{g}\right)^2}}\right)
			 \end{align*}
	\end{enumerate}\pagebreak
	\Aufgabe{}{8}
	\begin{enumerate}[(a)]
	\item \begin{align*}
		\frac{g*M}{RT_0} &=: \beta\\
		\left[\beta\right] &= \dfrac{1}{m}\\
		\beta& \approx 1.1854 *10^{-4} \dfrac{1}{m}\\
		p_0 &= 101325 Pa\\
		p(a)&= p_0 \cdot exp(-\beta a)\\
		p(a)_{lin}&= p(\overline{a}) + p'(\overline{a})\cdot (\overline{a}-a)\\
		p'(\overline{a})& = -\beta\cdot p_0 \cdot exp(-\beta\overline{a})\\
		p(a)_{lin}&= p_0 \cdot exp(-\beta\overline{a}) -\beta\cdot p_0 \cdot exp(-\beta\overline{a})\cdot (a-\overline{a})\\
		&= p_0 \cdot exp(-\beta\overline{a})(1-\beta(a-\overline{a}))
	\end{align*}

	\item \begin{align*}
	\overline{a}_1&= 100m \\
p(a)_{model} &=	p_0 \cdot exp(-\beta100m)(1-\beta(a-100m))\\
	p(a)_{model} &= 101317.937 - 11.86959596*a \\
	\overline{a_2}&= 10000m\\
p(a)_{passenger} &=	p_0 \cdot exp(-\beta 10000m)(1-\beta(a-10000m))\\
	p(a)_{passenger} &= 67675.77342 - 3.670864739*a\\
	\end{align*}
	\item
    	\begin{math}
    	   \sigma_p= 5Pa\\
           a(p)=\frac{p-p_0e^{-\beta\bar{a}}-p_0\beta\bar{a}e^{-\beta\bar{a}}}{-p_0\beta e^{-\beta\bar{a}}}=-\frac{p}{\beta p_0}\cdot e^{\beta\bar{a}}+\beta^{-1}+\bar{a}
        \end{math}
        \begin{align*}
           a(p\pm\sigma_p)&=-\frac{p\pm\sigma_p}{\beta p_0}\cdot e^{\bar{a}\beta}+\beta^{-1}+\bar{a}\\
           &=-\frac{p}{\beta p_0}\cdot e^{\bar{a}\beta}\pm\frac{\sigma_p}{\beta p_0}\cdot e^{\bar{a}\beta}+\beta^{-1}+\bar{a}\\
           \sigma_a&=\frac{\sigma_p}{\beta p_0}\cdot e^{\bar{a}\beta}
        \end{align*}
        $\Rightarrow \sigma_{a_{model}}=\frac{5}{\beta p_0}\cdot e^{100\beta}\approx 0.42124,\\
        \sigma_{a_{passenger}}=\frac{5}{\beta p_0}\cdot e^{10000\beta}\approx 1.3621$
	\end{enumerate}



\end{document}