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proptalk-backup.tex
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\begin{frame}
\frametitle{Motion Planning in Configuration Space}
\begin{tikzpicture}
\draw[step=1,black!15,very thin,opacity=\gridopacity] (0,0) grid (12,8);
\node[inner sep=0] at (3,4)
{\includegraphics[width=5cm]{figs/fridge-intro.png}};
\node[inner sep=0] at (9,5) {%
{\only<1>{\includegraphics{build/talk-act1-2d,rootsonly}}}%
{\only<2>{\includegraphics{build/talk-act1-2d,cfree}}}%
%{\only<3>{\includegraphics{build/talk-act1-2d,paths}}}%
};
% legend
\node[draw,circle,inner sep=2pt,ultra thick,fill=red!50] at (7.05, 2.0) {};
\node[draw,circle,inner sep=2pt,ultra thick,fill=green!50] at (7.5, 2.0) {};
\node[anchor=west] at (7.9,2.0) {start, goal configs};
\only<2->{
\node[anchor=west,draw,line width=1pt,fill=blue!20,minimum width=0.75cm,minimum height=0.10cm]
(Cfreebox) at (6.9, 1.5) {};
\node[anchor=west] at (7.9,1.5) {$\mathcal{C}_{\mbox{\scriptsize free}}$};
}
%\only<1>{\node at (6,1) {High-dimensional C-space (cite TLP).};}
%\only<2>{\node at (6,1.3) {\begin{minipage}{11cm}\begin{center}
% \begin{equation*}
% f_x(\pi) = x(\pi)
% \end{equation*}
% Lots of possible paths.
%
% {\bf How to find low-cost paths quickly?}%
% \end{center}\end{minipage}
%};}
\end{tikzpicture}
\end{frame}
\begin{frame}
\frametitle{Motion Planning as Best-First Search over Paths}
\begin{tikzpicture}
\draw[step=1,black!15,very thin,opacity=\gridopacity] (0,0) grid (12,8);
\node[inner sep=0] at (9,5) {%
{\only<1>{\includegraphics{build/talk-act1-2d,cfree}}}%
{\only<2>{\includegraphics{build/talk-act1-2d,paths}}}%
{\only<3>{\includegraphics{build/talk-act1-2d,traja}}}%
{\only<4>{\includegraphics{build/talk-act1-2d,firstfail}}}%
{\only<5>{\includegraphics{build/talk-act1-2d,firstfailnext}}}%
};
% legend
\node[draw,circle,inner sep=2pt,ultra thick,fill=red!50] at (7.05, 2.0) {};
\node[draw,circle,inner sep=2pt,ultra thick,fill=green!50] at (7.5, 2.0) {};
\node[anchor=west] at (7.9,2.0) {start, goal configs};
\node[anchor=west,draw,line width=1pt,fill=blue!20,minimum width=0.75cm,minimum height=0.10cm]
(Cfreebox) at (6.9, 1.5) {};
\node[anchor=west] at (7.9,1.5) {$\mathcal{C}_{\mbox{\scriptsize free}}$};
\only<2->{
\draw[dashed,thick] (6.9,1.0) -- (7.65,1.0);
\node[anchor=west] at (7.9,1.0) {candidate paths $\Pi$};
}
\only<3->{
\draw[black!30,line width=7pt,line cap=round] (6.9,0.5) -- (7.65,0.5);
\node[anchor=west] at (7.9,0.5) {path to evaluate ${\hat \pi}^*$};
}
% bfs background
\node[anchor=north,fill=black!5,rounded corners,inner sep=0.2cm,minimum height=3cm,
minimum width=5.9cm]
at (3.2,7.5) {};
% bfs highlights
\only<2,5>{\fill[green!30] (0.25,5.8) rectangle (6.15,6.35);}
\only<3,5>{\fill[green!30] (0.25,5.1) rectangle (6.15,5.8);}
\only<4>{\fill[green!30] (0.25,4.6) rectangle (6.15,5.1);}
% bfs text
\node[anchor=north,inner sep=0.2cm,minimum height=3cm]
at (3.2,7.5) {\begin{minipage}{5.5cm}
Best First Search over paths:
\algrenewcommand\algorithmicindent{0.0cm}%
\algrenewcommand\algorithmicloop{\!\!\!\!\textbf{loop}}
\begin{algorithmic}
\Loop%
\State $\Pi \leftarrow \mbox{\textsc{GetCandidatePaths}}()$
\State ${\hat \pi}^* \leftarrow \argmin\limits_{\pi \in \Pi} f(\pi)$
\State \textsc{EvalPath}$({\hat \pi}^*)$
\EndLoop
\end{algorithmic}
\end{minipage}};
%\only<2>{\node at (6,1) {Select best candidate path.};}
%\only<3>{\node at (6,1) {Evaluate.};}
\end{tikzpicture}
\end{frame}
%\begin{frame}
% \frametitle{Best-First Search over Paths}
% \begin{tikzpicture}
%
% \draw[step=1,black!15,very thin,opacity=\gridopacity] (0,0) grid (12,8);
%
% \node[inner sep=0] at (9,5) {%
% {\only<1>{\includegraphics{build/talk-act1-2d,firstfailnext}}}%
% };
%
% % legend
% \node[anchor=west,draw,line width=1pt,fill=blue!20,minimum width=0.75cm,minimum height=0.10cm]
% (Cfreebox) at (6.9, 1.9) {};
% \node[anchor=west] at (7.9,1.9) {$\mathcal{C}_{\mbox{\scriptsize free}}$};
% \draw[dashed,thick] (6.9,1.3) -- (7.65,1.3);
% \node[anchor=west] at (7.9,1.3) {candidate paths $\Pi$};
%
% % bfs background
% \node[anchor=north,fill=black!5,rounded corners,inner sep=0.2cm,minimum height=3cm,
% minimum width=5.9cm]
% at (3.2,7.5) {};
%
% % bfs highlights
% \only<1>{\fill[green!30] (0.25,5.8) rectangle (6.15,6.35);}
% %\only<2,4>{\fill[green!30] (0.25,5.1) rectangle (6.15,5.8);}
% %\only<3>{\fill[green!30] (0.25,4.6) rectangle (6.15,5.1);}
%
% % bfs text
% \node[anchor=north,inner sep=0.2cm,minimum height=3cm]
% at (3.2,7.5) {\begin{minipage}{5.5cm}
% Best First Search over paths:
% \algrenewcommand\algorithmicindent{0.0cm}%
% \algrenewcommand\algorithmicloop{\!\!\!\!\textbf{loop}}
% \begin{algorithmic}
% \Loop%
% \State $\Pi \leftarrow \mbox{\textsc{GetCandidatePaths}}()$
% \State ${\hat \pi}^* \leftarrow \argmin\limits_{\pi \in \Pi} f(\pi)$
% \State \textsc{EvalPath}$({\hat \pi}^*)$
% \EndLoop
% \end{algorithmic}
% \end{minipage}};
%
% \end{tikzpicture}
%\end{frame}
\begin{frame}
\frametitle{Best-First Search over Paths: Roadmaps}
\begin{tikzpicture}
\draw[step=1,black!15,very thin,opacity=\gridopacity] (0,0) grid (12,8);
\node[inner sep=0] at (9,5) {%
{\only<1>{\includegraphics{build/talk-act1-2d,firstfailnext}}}%
%{\only<1>{\includegraphics{build/talk-act1-2d,paths}}}%
{\only<2>{\includegraphics{build/talk-act1-2d,graph}}}%
%{\only<2>{\includegraphics{build/talk-act1-2d,graphfirst}}}%
%{\only<3>{\includegraphics{build/talk-act1-2d,graphfirstevaled}}}%
%{\only<4>{\includegraphics{build/talk-act1-2d,graphfirstnext}}}%
%{\only<5>{\includegraphics{build/talk-act1-2d,graph}}}%
{\only<3>{\includegraphics{build/talk-act1-2d,astara}}}%
};
% legend
\node[draw,circle,inner sep=2pt,ultra thick,fill=red!50] at (7.05, 2.0) {};
\node[draw,circle,inner sep=2pt,ultra thick,fill=green!50] at (7.5, 2.0) {};
\node[anchor=west] at (7.9,2.0) {start, goal configs};
\node[anchor=west,draw,line width=1pt,fill=blue!20,minimum width=0.75cm,minimum height=0.10cm]
(Cfreebox) at (6.9, 1.5) {};
\node[anchor=west] at (7.9,1.5) {$\mathcal{C}_{\mbox{\scriptsize free}}$};
\draw[dashed,thick] (6.9,1.0) -- (7.65,1.0);
\node[anchor=west] at (7.9,1.0) {candidate paths $\Pi$};
% bfs background
\node[anchor=north,fill=black!5,rounded corners,inner sep=0.2cm,minimum height=3cm,
minimum width=5.9cm]
at (3.2,7.5) {};
% bfs highlights
\only<2>{\fill[green!30] (0.25,5.8) rectangle (6.15,6.35);}
%\only<2,4>{\fill[green!30] (0.25,5.1) rectangle (6.15,5.8);}
%\only<3>{\fill[green!30] (0.25,4.6) rectangle (6.15,5.1);}
% bfs text
\node[anchor=north,inner sep=0.2cm,minimum height=3cm]
at (3.2,7.5) {\begin{minipage}{5.5cm}
Best First Search over paths:
\algrenewcommand\algorithmicindent{0.0cm}%
\algrenewcommand\algorithmicloop{\!\!\!\!\textbf{loop}}
\begin{algorithmic}
\Loop%
\only<1>{\State $\Pi \leftarrow \mbox{\textsc{GetCandidatePaths}}()$}
\only<2-3>{\State $\Pi \leftarrow \mbox{\textsc{GetRoadmapPaths}}()$}
\State ${\hat \pi}^* \leftarrow \argmin\limits_{\pi \in \Pi} f(\pi)$
\State \textsc{EvalPath}$({\hat \pi}^*)$
\EndLoop
\end{algorithmic}
\end{minipage}};
%\only<1-4>{
% \node at (6,1) {E.g. Probabalistic RoadMap, state lattice, etc.};
%}
\end{tikzpicture}
\end{frame}
\begin{frame}
\frametitle{RRTs as Best-First Search over Paths}
\begin{tikzpicture}
\draw[step=1,black!15,very thin,opacity=\gridopacity] (0,0) grid (12,8);
\node[inner sep=0] at (9.4,5) {%
{\only<1>{\includegraphics{build/talk-act1-2d,firstfailnext}}}%
{\only<2>{\includegraphics{build/talk-act1-2d,rrtstart}}}%
{\only<3>{\includegraphics{build/talk-act1-2d,rrtsample}}}%
{\only<4-5>{\includegraphics{build/talk-act1-2d,rrtcandidates}}}%
{\only<6->{\includegraphics{build/talk-act1-2d,rrtchosen}}}%
};
% legend
\node[draw,circle,inner sep=2pt,ultra thick,fill=red!50] at (7.05, 2.0) {};
\node[draw,circle,inner sep=2pt,ultra thick,fill=green!50] at (7.5, 2.0) {};
\node[anchor=west] at (7.9,2.0) {start, goal configs};
\node[anchor=west,draw,line width=1pt,fill=blue!20,minimum width=0.75cm,minimum height=0.10cm]
(Cfreebox) at (6.9, 1.5) {};
\node[anchor=west] at (7.9,1.5) {$\mathcal{C}_{\mbox{\scriptsize free}}$};
\draw[dashed,thick] (6.9,1.0) -- (7.65,1.0);
\node[anchor=west] at (7.9,1.0) {candidate paths $\Pi$};
\only<6->{
\draw[black!30,line width=7pt,line cap=round] (6.9,0.5) -- (7.65,0.5);
\node[anchor=west] at (7.9,0.5) {path to evaluate ${\hat \pi}^*$};
}
% bfs background
\node[anchor=north,fill=black!5,rounded corners,inner sep=0.2cm,minimum height=3cm,
minimum width=6.6cm]
at (3.4,7.5) {};
% bfs highlights
\only<4>{\fill[green!30] (0.1,5.9) rectangle (6.7,6.35);}
\only<5->{\fill[green!30] (0.1,5.15) rectangle (6.7,6.0);}
%\only<3>{\fill[green!30] (0.25,4.6) rectangle (6.15,5.15);}
% bfs text
\node[anchor=north,inner sep=0.1cm,minimum height=3cm]
at (3.4,7.5) {\begin{minipage}{6.3cm}\small{
Best First Search over paths:
\algrenewcommand\algorithmicindent{0.0cm}%
\algrenewcommand\algorithmicloop{\!\!\!\!\textbf{loop}}
\begin{algorithmic}
\Loop%
\only<1>{\State $\Pi \leftarrow \mbox{\textsc{GetCandidatePaths}}()$}
\only<2->{\State $\Pi \leftarrow \mbox{\textsc{GetRRTPaths}}()$}
\State ${\hat \pi}^* \leftarrow \argmin\limits_{\pi \in \Pi}
\left[ (1-\lambda) \hat{f}_x(\pi) + \lambda \hat{f}_p(\pi) \right]$
\State \textsc{EvalPath}$({\hat \pi}^*)$
\EndLoop
\end{algorithmic}
}\end{minipage}};
%\only<2>{\node at (6,1) {Select best candidate path.};}
%\only<3>{\node at (6,1) {Evaluate.};}
\only<7>{
\node[anchor=north,fill=blue!20,rounded corners,inner sep=0.2cm,
minimum width=5.9cm,align=center]
at (3.4,4)
{
\bf RRT-Connect is a\\\bf $\lambda=1$ planner!
};
}
\end{tikzpicture}
\end{frame}
\begin{frame}
\frametitle{Best-First Search over Paths: Trajectory Optimization}
\begin{center}
\includegraphics{build/talk-act1-2d,traja}
\begin{minipage}{0.65\textwidth}
\begin{algorithmic}
\Loop
\State $\Pi \leftarrow $ \textsc{GetPaths}$()$
\Comment \tikz{\node[draw,circle,inner sep=0.7pt]{\scriptsize 1};}
\State $\pi^* \leftarrow \argmin\limits_{\pi \in \Pi} f(\pi)$
\Comment \tikz{\node[draw,circle,inner sep=0.7pt]{\scriptsize 2};}
\State \textsc{EvalPath}$(\pi^*)$
\Comment \tikz{\node[draw,circle,inner sep=0.7pt]{\scriptsize 3};}
\EndLoop
\end{algorithmic}
\end{minipage}
\end{center}
\end{frame}
\begin{frame}
\frametitle{Best-First Search over Paths: Trajectory Optimization}
\begin{center}
\includegraphics{build/talk-act1-2d,trajb}
\begin{minipage}{0.8\textwidth}
\begin{algorithmic}
\Loop
\State $\Pi \leftarrow $ \textsc{GetPaths}$()$
\Comment Local neighborhood
\State $\pi^* \leftarrow \argmin\limits_{\pi \in \Pi} f(\pi)$
\Comment \tikz{\node[draw,circle,inner sep=0.7pt]{\scriptsize 2};}
\State \textsc{EvalPath}$(\pi^*)$
\Comment \tikz{\node[draw,circle,inner sep=0.7pt]{\scriptsize 3};}
\EndLoop
\end{algorithmic}
\end{minipage}
\end{center}
\end{frame}
\begin{frame}
\frametitle{Best-First Search over Paths: Trajectory Optimization}
\begin{center}
\includegraphics{build/talk-act1-2d,trajc}
\begin{minipage}{0.8\textwidth}
\begin{algorithmic}
\Loop
\State $\Pi \leftarrow $ \textsc{GetPaths}$()$
\Comment Local neighborhood
\State $\pi^* \leftarrow \argmin\limits_{\pi \in \Pi} f(\pi)$
\Comment $f(\pi):$ fuzzy approx
\State \textsc{EvalPath}$(\pi^*)$
\Comment \tikz{\node[draw,circle,inner sep=0.7pt]{\scriptsize 3};}
\EndLoop
\end{algorithmic}
\end{minipage}
\end{center}
\end{frame}
\begin{frame}
\frametitle{Best-First Search over Paths: Trajectory Optimization}
\begin{center}
\includegraphics{build/talk-act1-2d,trajd}
\begin{minipage}{0.8\textwidth}
\begin{algorithmic}
\Loop
\State $\Pi \leftarrow $ \textsc{GetPaths}$()$
\Comment Local neighborhood
\State $\pi^* \leftarrow \argmin\limits_{\pi \in \Pi} f(\pi)$
\Comment $f(\pi):$ fuzzy approx
\State \textsc{EvalPath}$(\pi^*)$
\Comment Re-linearize
\EndLoop
\end{algorithmic}
\end{minipage}
\end{center}
\end{frame}
\begin{frame}
\frametitle{Graph Search: Edge Execution Effort Model}
\begin{tikzpicture}
\draw[step=1,black!15,very thin,opacity=\gridopacity] (0,0) grid (12,8);
\node[draw,line width=1.0pt,fill=blue!20,minimum width=0.75cm,minimum height=0.10cm]
(Cfreebox) at (8.0, 7.5) {};
\node[right=0cm of Cfreebox] {: $\mathcal{C}_{\mbox{\scriptsize free}}$};
\node[inner sep=0] at (10.75,7) {%
\includegraphics[width=2cm]{build/talk-act1-2d,graph}
};
% bfs
\only<1>{\fill[green!30] (0.5,5.6) rectangle (5.5,6.9);}
\only<2>{\fill[green!30] (0.5,5.6) rectangle (5.5,6.15);}
\only<3>{\fill[green!30] (0.5,6.15) rectangle (5.5,6.9);}
\node at (2.5,6.75) {\begin{minipage}{5cm}
\begin{algorithmic}
\Loop%
\State $\Pi \leftarrow $ \textsc{GetPaths}$()$
\State $\pi^* \leftarrow \argmin\limits_{\pi \in \Pi} f(\pi)$
\State \textsc{EvalPath}$(\pi^*)$
\EndLoop
\end{algorithmic}
\end{minipage}
};
\only<2>{
\fill[green!30] (0.1,3.9) rectangle (5.9,4.9);
\fill[green!30] (6.1,2.5) rectangle (11.9,4.9);
}
\only<3>{
\fill[green!30] (0.1,2.8) rectangle (5.9,3.9);
\fill[green!30] (6.1,1.6) rectangle (11.9,2.5);
}
\node[inner sep=0pt,anchor=north] at (3,5.5) {
\begin{minipage}[t]{5.5cm}%
\hrule
\vspace{0.1cm}
Distance Model $\mathcal{M}_{\ms{dist}}$
\begin{algorithmic}[1]
%\only<2-3>{
\Function {$x_{\ms{\textup{dist}}}$}{$e$}
\State \Return $|| q_e(1) - q_e(0) ||$
\EndFunction
%}
\only<3>{
\Function {$\hat{x}_{\ms{\textup{dist}}}$}{$e$}
\State \Return $|| q_e(1) - q_e(0) ||$
\EndFunction
}
\end{algorithmic}
\hrule
\end{minipage}%
};
\node[inner sep=0pt,anchor=north] at (9,5.5) {
\begin{minipage}[t]{5.5cm}%
\hrule
\vspace{0.1cm}
Set Validity Model $\mathcal{M}_{\ms{valid}}$
\only<2-3>{
\begin{algorithmic}[1]
\Function {$x_{\ms{\textup{valid}}}$}{$e, \mathcal{C}_{\ms{free}}$}
\If {${\bf 1}_{\ms{free}}[q_e(t)]$}
\State \Return $0$
\Else
\State \Return $\infty$
\EndIf
\EndFunction
\only<3>{
\Function {$\hat{x}_{\ms{\textup{valid}}}$}{$e, \mathcal{C}_{\ms{free}}$}
\State \Return $0$
\EndFunction
}
\end{algorithmic}
}
\hrule
\end{minipage}%
};
\node at (6,0.5) {
\begin{minipage}[t]{11cm}
\centering%
\only<1>{What is the effort model?}%
\only<2>{Edge evaluation functions (returns execution effort)}%
\only<3>{Optimistic (admissible) estimates of execution effort}%
\only<4>{%
\vspace{-0.4cm}
\begin{equation*}
\hat{f}_x(\pi) = \sum_{e \in \pi} \left\{
\begin{array}{cl}
x[e] & \mbox{if edge } e \mbox{ evaluated} \\
\hat{x}(e) & \mbox{otherwise} \\
\end{array}
\right.
\end{equation*}
}%
\end{minipage}
};
\end{tikzpicture}
\end{frame}
\begin{frame}
\frametitle{Why Not Just Run A* Graph Search?}
\begin{tikzpicture}
\draw[step=1,black!15,very thin,opacity=\gridopacity] (0,0) grid (12,8);
\node[anchor=north] at (6,7.5)
{
\begin{minipage}[t]{11cm}
Show A* on our graph. (video?)
Define EvalPath (expand first unexpanded vertex,
evaluate all edges to get g values for all successors).
\medskip
Yes, A* will find the optimal path \emph{to execute}.
But as I mentioned earlier, we want combined planner + execution effort!
\medskip
What is A*'s model of planning effort?
A* assumes implicit graph representation!
A* assumes expanding vertices (discovering the graph)
is expensive, and is optimal in that metric.
\medskip
There's a mismatch!
\end{minipage}
};
\end{tikzpicture}
\end{frame}
\begin{frame}
\frametitle{Explicit Graph Representation}
Our graphs are small!
Our EvalPath function can be different (e.g. bidirectional).
Taken from RRTConnect.
Maybe it will perform better?
\begin{equation*}
\hat{f}_x(\pi) = \sum_{e \in \pi} \left\{
\begin{array}{cl}
x[e] & \mbox{if edge } e \mbox{ evaluated} \\
\hat{x}(e) & \mbox{otherwise} \\
\end{array}
\right.
\end{equation*}
Similarity to front-to-front algorithms.
This is how Lazy PRM works.
\end{frame}
\begin{frame}
\frametitle{Roadmaps $\rightarrow$ Graphs}
\begin{tikzpicture}
\draw[step=1,black!15,very thin,opacity=\gridopacity] (0,0) grid (12,8);
\node[inner sep=0] at (9,5) {%
{\only<1-4>{\includegraphics{build/talk-act1-2d,graph}}}%
};
% legend
\node[draw,line width=1.5pt,fill=blue!20,minimum width=0.75cm,minimum height=0.10cm]
(Cfreebox) at (3.0, 7.0) {};
\node[right=0cm of Cfreebox] {: $\mathcal{C}_{\mbox{\scriptsize free}}$};
\node at (3,6.25) {$\Pi$ : set of candidate paths};
\only<1-2>{\fill[green!30] (0.5,3.4) rectangle (5.5,4.1);}
\only<3>{\fill[green!30] (0.5,2.9) rectangle (5.5,3.4);}
\only<4>{\fill[green!30] (0.5,3.4) rectangle (5.5,4.1);}
% bfs
\node at (3,4) {\begin{minipage}{5cm}
\begin{algorithmic}
\Loop%
\State $\Pi \leftarrow $ \textsc{GetPaths}$()$
\State $\pi^* \leftarrow \argmin\limits_{\pi \in \Pi} f(\pi)$
\State \textsc{EvalPath}$(\pi^*)$
\EndLoop
\end{algorithmic}
\end{minipage}
};
\only<2-4>{%
\node at (6,1) {\begin{minipage}{5cm}%
\begin{equation*}%
f(\pi) = {\hat f}_x(\pi) :
\left\{ \begin{array}{ll}
x(\pi) & \mbox{if } {\bf 1}_{\mbox{\scriptsize free}}(\pi) \\
\infty & \mbox{otherwise}
\end{array} \right.
\end{equation*}
\end{minipage}
};
}
\end{tikzpicture}
\end{frame}
\begin{frame}
\frametitle{Best-First Search over Paths: Graphs}
\begin{center}
\includegraphics{build/talk-act1-2d,graphfirstnext}
\begin{minipage}{0.65\textwidth}
\begin{algorithmic}
\Loop
\State $\Pi \leftarrow $ \textsc{GetPaths}$()$
\Comment \tikz{\node[draw,circle,inner sep=0.7pt]{\scriptsize 1};}
\State $\pi^* \leftarrow \argmin\limits_{\pi \in \Pi} f(\pi)$
\Comment \tikz{\node[draw,circle,inner sep=0.7pt]{\scriptsize 2};}
\State \textsc{EvalPath}$(\pi^*)$
\Comment \tikz{\node[draw,circle,inner sep=0.7pt]{\scriptsize 3};}
\EndLoop
\end{algorithmic}
\end{minipage}
\begin{equation*}
f_x(\pi) =
\left\{ \begin{array}{ll}
x(\pi) & \mbox{if } {\bf 1}_{\mbox{\scriptsize free}}(\pi) \\
\infty & \mbox{otherwise}
\end{array} \right.
\end{equation*}
\end{center}
\end{frame}
\begin{frame}
\frametitle{Graph Search: Run A*!}
\begin{tikzpicture}
\tikzset{>=latex} % arrow heads
\draw[step=1,black!15,very thin,opacity=\gridopacity] (0,0) grid (12,8);
\node[inner sep=0] at (9,5) {%
{\only<1>{\includegraphics{build/talk-act1-2d,graph}}}%
{\only<2>{\includegraphics{build/talk-act1-2d,astara}}}%
{\only<3>{\includegraphics{build/talk-act1-2d,astarb}}}%
{\only<4>{\includegraphics{build/talk-act1-2d,astarc}}}%
{\only<5>{\includegraphics{build/talk-act1-2d,astard}}}%
{\only<6>{\includegraphics{build/talk-act1-2d,astare}}}%
{\only<7>{\includegraphics{build/talk-act1-2d,astarf}}}%
{\only<8>{\includegraphics{build/talk-act1-2d,astarg}}}%
{\only<9>{\includegraphics{build/talk-act1-2d,astarh}}}%
{\only<10>{\includegraphics{build/talk-act1-2d,astari}}}%
{\only<11>{\includegraphics{build/talk-act1-2d,astarj}}}%
{\only<12>{\includegraphics{build/talk-act1-2d,astark}}}%
{\only<13>{\includegraphics{build/talk-act1-2d,astarl}}}%
{\only<14>{\includegraphics{build/talk-act1-2d,astarm}}}%
{\only<15>{\includegraphics{build/talk-act1-2d,astarn}}}%
{\only<16>{\includegraphics{build/talk-act1-2d,astaro}}}%
{\only<17>{\includegraphics{build/talk-act1-2d,astarp}}}%
{\only<18>{\includegraphics{build/talk-act1-2d,astarq}}}%
{\only<19>{\includegraphics{build/talk-act1-2d,astarr}}}%
{\only<20-27>{\includegraphics{build/talk-act1-2d,astars}}}%
};
\node at (3,5.5) {\begin{minipage}{5cm}
Execution Effort Model:
\begin{equation*}
x(e) = \left\{
\begin{array}{cl}
||e|| & \mbox{if } {\bf 1}_{\ms{free}}(e) \\
\infty & \mbox{otherwise} \\
\end{array}
\right.
\end{equation*}
Heuristic:
\begin{equation*}
{\hat x}(v_a, v_b) = ||v_a - v_b||
\end{equation*}
\end{minipage}
};
\only<21-27>{
\node at (3,2.25) {Planning Effort Allocation:};
\draw[] (1,1.5) rectangle (11,2);
}
\only<22-27>{
\draw[fill=yellow] (1.1,1.5) rectangle (1.45,2);
\node[align=center,inner sep=0] at (0.9,0.8) {\small computing};
\node[align=center,inner sep=0] at (0.9,0.51) {\small vertex};
\node[align=center,inner sep=0] at (0.9,0.2) {\small successors};
\draw[->] (1,1) -- (1.25,1.45);
}
\only<23-27>{
\draw[fill=blue] (1.55,1.5) rectangle (1.95,2);
\node[align=center,inner sep=0] at (2.8,0.8) {\small sorting};
\node[align=center,inner sep=0] at (2.8,0.48) {\small open};
\node[align=center,inner sep=0] at (2.8,0.2) {\small list};
\draw[->] (2.6,1) -- (1.95,1.45);
}
\only<24-27>{
\draw[fill=red] (2.05,1.5) rectangle (10.9,2);
\node[align=center,inner sep=0] at (6,0.65) {\small edge};
\node[align=center,inner sep=0] at (6,0.35) {\small evaluations};
\draw[->] (6,1) -- (6,1.45);
}
\only<25-27>{
\draw[line width=2pt] (1.275,1.75) circle (0.5cm);
}
\only<26-27>{\node[align=center,fill=white,opacity=0.95] at (7.3,2.9)
{{\footnotesize Weighted A*}\\{\scriptsize (Pohl 1970)}};}
\only<27>{\node[align=center,fill=white,opacity=0.95] at (10.2,2.9)
{{\footnotesize Partial Expansion A*}\\{\scriptsize(Yoshizumi etal. 2000)}};}
\end{tikzpicture}
\end{frame}
\begin{frame}
\frametitle{Graph Search Mismatch}
\begin{tikzpicture}
\draw[step=1,black!15,very thin,opacity=\gridopacity] (0,0) grid (12,8);
\node[inner sep=0pt] at (6,7) {\begin{minipage}{12cm}\centering
A* efficiently addresses {\bf large graphs}, usually represented {\bf implicitly}
(with {\bf inexpensive edge costs}), by expanding the {\bf fewest vertices}.
\end{minipage}};
\only<2-4>
{
\node[inner sep=0pt] at (2,4.5) {\includegraphics[width=1in]{figs/rubik.png}};
\node[inner sep=0pt,anchor=north] at (2,2.5)
{\begin{minipage}{4cm}\centering
$4.3 \times 10^{19}$ vertices
\tiny{from user Booyabazooka, Wikipedia}
\end{minipage}};
}
\only<3-4>
{
\node[inner sep=0pt] at (6,4.5) {\includegraphics[width=1in]{figs/15puzzle.png}};
\node[inner sep=0pt,anchor=north] at (6,2.5)
{\begin{minipage}{4cm}\centering
$2.1 \times 10^{13}$ vertices
\end{minipage}};
}
\only<4>
{
\node[inner sep=0pt] at (10,4.5) {\includegraphics[width=1in]{figs/goldberg-northwest.png}};
\node[inner sep=0pt,anchor=north] at (10,2.5)
{\begin{minipage}{4cm}\centering
1.6M vertices
\tiny{
Goldberg, Harrelson, Kaplan, Werneck.
``Efficient Point-to-Point Shortest Path Algorithms''
}
\end{minipage}};
}
\end{tikzpicture}
\end{frame}
\begin{frame}
\frametitle{Graph Search Mismatch}
\begin{tikzpicture}
\draw[step=1,black!15,very thin,opacity=\gridopacity] (0,0) grid (12,8);
\node[inner sep=0pt] at (6,7) {\begin{minipage}{12cm}\centering
A* efficiently addresses {\bf large graphs}, usually represented {\bf implicitly}
(with {\bf inexpensive edge costs}), by expanding the {\bf fewest vertices}.
\end{minipage}};
\only<2-3>{
\node[inner sep=0pt] at (3.5,4.5) {\includegraphics[width=4cm]{figs/fridge-intro.png}};
\node[inner sep=0pt,anchor=north] at (3.5,3)
{\begin{minipage}{4cm}\centering
Our problem:
$\sim10$k vertices
\end{minipage}};
}
\only<3>{
\node[inner sep=0pt] at (8.5,4.5) {\includegraphics[width=4cm]{figs/prm.png}};
\node[inner sep=0pt,anchor=north] at (8.5,3)
{\begin{minipage}{4cm}\centering
Probabalistic RoadMap
\tiny{Kavraki, Svestka, Latombe, Overmars, 1996}
\end{minipage}};
}
\only<2-3>{
\node[inner sep=0pt] at (6,1) {\begin{minipage}{12cm}\centering
We can represent our graphs {\bf explicitly};
{\bf evaluating graph edges} is expensive,
whereas entire {\bf graph search queries} are inexpensive.
\end{minipage}};
}
\end{tikzpicture}
\end{frame}
\begin{frame}
\frametitle{Roadmaps: Accounting for Spatial Coherence in $\mathcal{C}_{\ms{free}}$}
\begin{tikzpicture}
\draw[step=1,black!15,very thin,opacity=\gridopacity] (0,0) grid (12,8);
\node[inner sep=0] at (9.4,5) {
\includegraphics{build/talk-act1-2d,graphfirstevaled}
};
\node[anchor=north,fill=black!5,rounded corners]
at (3.4,7.5) {\begin{minipage}{6.3cm}\small{
Best First Search over paths:
\algrenewcommand\algorithmicindent{0.0cm}%
\algrenewcommand\algorithmicloop{\!\!\!\!\textbf{loop}}
\begin{algorithmic}
\Loop%
\State $\Pi \leftarrow \mbox{\textsc{GetRoadmapPaths}}()$
\State ${\hat \pi}^* \leftarrow \argmin\limits_{\pi \in \Pi}
\left[ (1-\lambda) \hat{f}_x(\pi) + \lambda \hat{f}_p(\pi) \right]$
\State \textsc{EvalPath}$({\hat \pi}^*)$
\EndLoop
\end{algorithmic}
}\end{minipage}};
\only<2->{
\node[anchor=north,fill=blue!20,rounded corners]
at (3.4,4.7) {\begin{minipage}{6.3cm}\small{
\begin{itemize}
\item Roadmap: maintain sparsity
\item BFS: be optimistic
\item Densify roadmap on failure
\end{itemize}
}\end{minipage}};
}
\only<3->{
\node[anchor=north,inner sep=0.3cm,fill=blue!20,rounded corners]
at (6,2.2) {\begin{minipage}{10cm}\small{\centering
{\bf What if we optimized ensemble effort \emph{in expectation}?}
\begin{equation*}
{\hat \pi}^* \leftarrow \argmin\limits_{\pi \in \Pi}
\mbox{E} \left[ (1-\lambda) \hat{f}_x(\pi) + \lambda \hat{f}_p(\pi) \right]
\end{equation*}
}\end{minipage}};
}
\end{tikzpicture}
\end{frame}
\begin{frame}
\frametitle{Alternative: Optimization in Expectation}
\begin{tikzpicture}
\draw[step=1,black!15,very thin,opacity=\gridopacity] (0,0) grid (12,8);
\node[inner sep=0pt] at (6,6)
{\includegraphics{build/example-in-expectation}};
\only<2->{
\node[anchor=north,fill=blue!20,rounded corners] at (6,4) {\begin{minipage}{10cm}
{\bf Research Question Q5:}\\
Is there a sufficiently expressive and efficient probabilistic
model of $\mbox{P}_{\ms{free}}(q)$ to allow for minimization
of ensemble effort in expectation at each iteration?
\medskip
Starting point: Gaussian Processes for binary classification.
\end{minipage}};
}
\end{tikzpicture}
\end{frame}
\begin{frame}
\frametitle{Efficiency in Graph Search}
\begin{tikzpicture}
\draw[step=1,black!15,very thin,opacity=\gridopacity] (0,0) grid (12,8);
% example (FIRST) problems
\only<4-14>
{
\node[inner sep=0pt] at (6.5,7) {\includegraphics[width=1.5cm]{figs/rubik.png}};
\node[inner sep=0pt,anchor=north,align=center,font=\scriptsize] at (6.5,6) {$4.3 \times 10^{19}$\\vertices};
}
\only<5-14>
{
\node[inner sep=0pt] at (8.5,7) {\includegraphics[width=1.5cm]{figs/15puzzle.png}};
\node[inner sep=0pt,anchor=north,align=center,font=\scriptsize] at (8.5,6) {$2.1 \times 10^{13}$\\vertices};
}
\only<6-14>
{
\node[inner sep=0pt] at (10.5,7) {\includegraphics[width=1.5cm]{figs/goldberg-northwest.png}};
\node[inner sep=0pt,anchor=north,align=center,font=\scriptsize] at (10.5,6) {1.6M\\vertices};
}
% example problems SECOND
\only<17->
{
\node[inner sep=0pt] at (2.0,7) {\includegraphics[width=2.5cm]{figs/fridge-intro.png}};
\node[inner sep=0pt,anchor=north,align=center,font=\scriptsize] at (2.0,6) {$\sim10$k\\vertices};
}
\only<18->
{
\node[inner sep=0pt] at (5.0,7) {\includegraphics[width=2.5cm]{figs/prm.png}};
\node[inner sep=0pt,anchor=north,align=center,font=\scriptsize] at (5.0,6)
{Motivation: PRM};
% cite
%\node[anchor=east,font=\scriptsize] at (12,0.5)
% {$^\dag$ Kavraki, Svestka, Latombe, Overmars, 1996.};
}
% draw graph box
\only<1-2>{
\node[draw,align=center,minimum height=1.3cm,anchor=south east] at (5,3.5) {
Graph\\$G = (V,E)$\\\includegraphics[width=1cm]{build/roadmap-2d-simple}
};
}
\only<3-15>{
\node[draw,align=center,minimum height=1.3cm,anchor=south east] at (5,3.5) {
Graph\\$G = (V,E)$\\\includegraphics[width=3cm]{build/roadmap-2d-simple}
};
}
% grow using a*!
\only<7-14>{
\fill[white,fill opacity=0.9] (1.8,3.6) rectangle (4.9,6.7);
\begin{scope}
\clip (1.8,3.6) rectangle (4.9,6.7);
\only<7>{\clip[rotate around={15:(2.0,4.8)}] (2.0,4.8) ellipse (0.5cm and 0.25cm);}
\only<8>{\clip[rotate around={15:(2.0,4.8)}] (2.0,4.8) ellipse (1.0cm and 0.50cm);}
\only<9>{\clip[rotate around={15:(2.0,4.8)}] (2.0,4.8) ellipse (1.4cm and 0.70cm);}
\only<10>{\clip[rotate around={15:(2.0,4.8)}] (2.0,4.8) ellipse (1.6cm and 0.80cm);}
\only<11>{\clip[rotate around={15:(2.0,4.8)}] (2.0,4.8) ellipse (1.9cm and 0.85cm);}
\only<12>{\clip[rotate around={15:(2.0,4.8)}] (2.0,4.8) ellipse (2.1cm and 1.05cm);}
\only<13-14>{\clip[rotate around={15:(2.0,4.8)}] (2.0,4.8) ellipse (2.2cm and 1.10cm);}
\node[align=center,minimum height=1.3cm,anchor=south east] at (5,3.5) {
\includegraphics[width=3cm]{build/roadmap-2d-simple}};
\only<14>{\draw[rotate around={15:(2.0,4.8)},line width=4pt]
(2.0,4.8) ellipse (2.2cm and 1.10cm);}
\end{scope}
}
\only<16->{
\node[draw,align=center,minimum height=1.3cm,anchor=south east,font=\scriptsize] at (5,3.5) {
Graph\\$G = (V,E)$\\\includegraphics[width=0.5cm]{build/roadmap-2d-simple}
};
}
% draw edge eval box
\only<1-2>{
\node[draw,align=center,anchor=south west] at (7,3.5)
{\\Edge Weight\\$x : E \rightarrow \mathbb{R}^+$\\};
}
\only<3-15>{
\begin{scope}[font=\tiny]
\node[draw,align=center,anchor=south west] at (7,3.5)
{\\Edge Weight\\$x : E \rightarrow \mathbb{R}^+$\\};
\end{scope}
}
\only<16->{
\node[draw,align=center,anchor=south west,inner sep=1cm] at (7,3.5)
{\\Edge Weight\\$x : E \rightarrow \mathbb{R}^+$\\};
}
\draw[->,line width=1pt] (4,3.25) -- (4,2.75);
\draw[->,line width=1pt] (8,3.25) -- (8,2.75);
\node[draw,align=center,minimum height=1.0cm,minimum width=5cm,anchor=north]
at (6,2.5) {\\$GS(G,x)$};
% clocks
\only<2-13>{
% graph
\node[fill=white,circle,inner sep=-2pt] at (5,3.5) {\LARGE%
\only<2-6>{\showclock{0}{0}}%
\only< 7>{\showclock{0}{10}}%
\only< 8>{\showclock{0}{20}}%
\only< 9>{\showclock{0}{30}}%
\only<10>{\showclock{0}{40}}%
\only<11>{\showclock{0}{50}}%
\only<12>{\showclock{1}{00}}%
\only<13>{\showclock{1}{10}}%
};
% edge weight
\node[fill=white,circle,inner sep=-2pt] at (7,3.5) {\LARGE%
\only<2-6>{\showclock{0}{0}}%
\only< 7>{\showclock{0}{5}}%
\only< 8>{\showclock{0}{10}}%
\only< 9>{\showclock{0}{15}}%
\only<10>{\showclock{0}{20}}%
\only<11>{\showclock{0}{25}}%
\only<12>{\showclock{0}{30}}%
\only<13>{\showclock{0}{35}}%
};
% graph search
\node[fill=white,circle,inner sep=-2pt] at (6,2.5) {\LARGE%
\only<2-6>{\showclock{0}{0}}%
\only< 7>{\showclock{0}{5}}%
\only< 8>{\showclock{0}{10}}%
\only< 9>{\showclock{0}{20}}%
\only<10>{\showclock{0}{40}}%
\only<11>{\showclock{1}{20}}%
\only<12>{\showclock{2}{40}}%
\only<13>{\showclock{5}{20}}%
};
}
\only<14>{
% graph
\node[fill=green,circle,inner sep=-2pt] at (5,3.5) {\LARGE\showclock{1}{10}};
% edge weight
\node[fill=white,circle,inner sep=-2pt] at (7,3.5) {\LARGE\showclock{0}{35}};
% graph search
\node[fill=white,circle,inner sep=-2pt] at (6,2.5) {\LARGE\showclock{5}{20}};
}
\only<15-18>{
% graph
\node[fill=white,circle,inner sep=-2pt] at (5,3.5) {\LARGE\showclock{0}{0}};