Commit f88c4cdc authored by Florent Lamiraux's avatar Florent Lamiraux
Browse files

[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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