Commit f88c4cdc by Florent Lamiraux

### [slides] Add slides on PRM*, kPRM*.

parent 6b31c9b8
Pipeline #15154 passed with stage
in 2 minutes and 22 seconds
 ... ... @@ -23,7 +23,6 @@ \usepackage{color} \usepackage{tikz} \usetikzlibrary{arrows,positioning} \usepackage{algorithm} \usepackage{algorithmic} \usepackage{hyperref} \usepackage{multimedia} ... ...
 %\section {Random methods} % % History % \begin{frame}{History} \begin{itemize} \item Before the 1990's: mainly a mathematical problem \begin{itemize} \item real algebraic geometry, \item decidability: Schwartz and Sharir 1982, \begin{itemize} \item Tarski theorem, Collins decomposition, \end{itemize} \end{itemize} \pause \item from the 1990's: an algorithmic problem \begin{itemize} \item random sampling (1993), \item asymptotically optimal random sampling (2011). \end{itemize} \end{itemize} \end{frame} % % random methods % ... ... @@ -393,3 +415,115 @@ \end{itemize} \end{itemize} \end{frame} % % asymptotically optimal random sampling % \begin{frame} {Asymptotically optimal random methods} Asymptotically optimal variants of PRM and RRT exist: \begin{itemize} \item when the number of nodes tends to infinity, \item the solution computed by the algorithm tends to the optimal collision-free path. \end{itemize} \end{frame} % % PRM* % \begin{frame} {PRM*} \parbox{.49\linewidth} { \begin{algorithmic} \STATE \textbf{PRM} \STATE{$V\gets\emptyset$, $E\gets\emptyset$} \FOR{$i\in\{1,\cdots,n\}$} \STATE{$\conf_{rand}\gets$ {SampleFree}$_i$} \STATE{$U\gets G.near(\conf_{rand},r)$} \FORALL{$\conf\in U$ in order of increasing $\|\conf-\conf_{rand}\|$} \IF{$\conf_{rand}$ and $\conf$ in different connected components} \STATE{TryConnect($\conf_{rand},\conf$)} \ENDIF \ENDFOR \ENDFOR \end{algorithmic} } \pause \parbox{.49\linewidth} { \begin{algorithmic} \STATE \textbf{PRM*} \STATE{$V\gets$ {SampleFree}$_{i=1,\cdots,n}$, $E\gets\emptyset$} \FORALL{$\conf\in V$} \STATE{$U\gets G.near(\conf,{\color{red}r^{*}})\setminus\conf$} \FORALL{$\conf'\in U$} \STATE{TryConnect($\conf,\conf'$)} \ENDFOR \ENDFOR \end{algorithmic} {\color{red}$r^{*}$}=$\gamma_{PRM}(\log(n)/n)^{\frac{1}{d}}$ } \end{frame} % % kPRM* % \begin{frame} {kPRM*} \parbox{.49\linewidth} { \begin{algorithmic} \STATE \textbf{kPRM} \STATE{$V\gets\emptyset$, $E\gets\emptyset$} \FOR{$i\in\{1,\cdots,n\}$} \STATE{$\conf_{rand}\gets$ {SampleFree}$_i$} \STATE{$U\gets G.nearest(\conf_{rand},k)$} \FORALL{$\conf\in U$ in order of increasing $\|\conf-\conf_{rand}\|$} \IF{$\conf_{rand}$ and $\conf$ in different connected components} \STATE{TryConnect($\conf_{rand},\conf$)} \ENDIF \ENDFOR \ENDFOR \end{algorithmic} } \pause \parbox{.49\linewidth} { \begin{algorithmic} \STATE \textbf{kPRM*} \STATE{$V\gets$ {SampleFree}$_{i=1,\cdots,n}$, $E\gets\emptyset$} \FORALL{$\conf\in V$} \STATE{$U\gets G.nearest(\conf,{\color{red}k^{*}})\setminus\conf$} \FORALL{$\conf'\in U$} \STATE{TryConnect($\conf,\conf'$)} \ENDFOR \ENDFOR \end{algorithmic} \begin{eqnarray*} {\color{red}k^{*}}&=&k_{PRM}(\log(n)),\\ k_{PRM}&>&e(1+\frac{1}{d}) \end{eqnarray*} } \end{frame} % % PRM*, kPRM* % \begin{frame} {PRM*,kPRM*} Note that \begin{itemize} \item PRM* and kPRM* are not iterative anymore, \item making them iterative is not trivial. \end{itemize} \end{frame} % % RRT*, RMG % \begin{frame} There exists also asymptotically optimal variants of RRT \begin{itemize} \item RRG, RRT* \end{itemize} but they are specific to a given problem $(\conf_{init},\conf_{goal})$. \end{frame}
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