diff --git a/mr/Ub2/mr2.pdf b/mr/Ub2/mr2.pdf index f236466..d314208 100644 --- a/mr/Ub2/mr2.pdf +++ b/mr/Ub2/mr2.pdf Binary files differ diff --git a/mr/Ub2/mr2.tex b/mr/Ub2/mr2.tex index a8837f9..a4e86dc 100644 --- a/mr/Ub2/mr2.tex +++ b/mr/Ub2/mr2.tex @@ -109,6 +109,26 @@ $\Rightarrow \frac{2v+l\omega}{2}+v_l =2v \Leftrightarrow v_l=2v-\frac{2v+l\omega}{2}=\frac{2v-l\omega}{2}$\qed \end{enumerate} \Aufgabe{}{8} + \begin{enumerate}[(a)] + \item $s_l=s_r=s,\ \theta_{t-1}=\theta_t=\theta$\\ + $\mathbf{x}_t=\mathbf{x}_{t-1}+\mathbf{Rot}(z,\theta)\begin{pmatrix} + s\\0\\0 + \end{pmatrix}=\mathbf{x}_{t-1}+\begin{pmatrix} + \cos\theta & -\sin\theta & 0\\ + \sin\theta & \cos\theta & 0\\ + 0&0&1 + \end{pmatrix}\begin{pmatrix} + s\\0\\0 + \end{pmatrix}=\mathbf{x}_{t-1}+\begin{pmatrix} + s\cos\theta\\s\sin\theta\\0 + \end{pmatrix}$ + \item $r\left(\begin{pmatrix} + 2l\\l + \end{pmatrix}\right)=-\frac{3}{2} l$\\ + $\frac{\partial r}{\partial s_r}=\frac{(s_l+s_r)'(s_r-s_l)-(s_l+s_r)(s_r-s_l)'}{(s_r-s_l)^2}\cdot\frac{l}{2}=\frac{-2s_l}{(s_r-s_l)^2}\cdot\frac{l}{2}\overset{s=\overline{s}}{=}\frac{-4}{l}\cdot\frac{l}{2}=-2$\\ + $\frac{\partial r}{\partial s_l}=\frac{(s_l+s_r)'(s_r-s_l)-(s_l+s_r)(s_r-s_l)'}{(s_r-s_l)^2}\cdot\frac{l}{2}=\frac{2s_r}{(s_r-s_l)^2}\cdot\frac{l}{2}\overset{s=\overline{s}}{=}\frac{2}{l}\cdot\frac{l}{2}=1$\\ + $\Rightarrow r(\mathbf{s})\approx -\frac{3}{2}-2\left(s_l-2l\right)+\left(s_r-l\right)$ + \end{enumerate} \Aufgabe{}{6} \end{document}