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defense-p1-utility.tex
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\begin{frame}
\frametitle{Talk Outline}
\begin{tikzpicture}[font=\small]
\tikzset{>=latex} % arrow heads
\draw[step=1,black!15,very thin,opacity=\gridopacity] (0,0) grid (12,8);
% START LEMUR
\node[fill=blue!5,draw=blue!10,rounded corners,minimum width=9.7cm,minimum height=5.8cm,anchor=north] (lemur) at (6,6.0) {};
\node[anchor=north west] at (lemur.north west) {Motion Planner};
% right side
\node[fill=blue!10,draw=blue!20,rounded corners,align=center,minimum height=1.5cm,inner sep=0pt,minimum width=6cm] (maxutil) at (6.0,4.5) {
\hspace{0.8cm} Maximizing Utility
};
%\node[fill=black!3,draw=blue!20,inner sep=2pt] at (7,5.4) {\includegraphics[width=1.3cm]{build/pvx-utility-anytime-simple}};
\node[fill=black!3,draw=blue!20,inner sep=2pt] at (3.8,4.5) {\includegraphics[width=1.0cm]{build/pvx-linear-discounting-simple}};
\only<2->{
\draw[thick,rounded corners] (maxutil.south west) rectangle (maxutil.north east);
}
\draw[->] (6.0,3.75) -- (6.0,3.0) node [pos=0.45,fill=blue!5,align=center,inner sep=2pt] {$G$};
% START LAZYSP
\node[fill=blue!10,draw=blue!20,rounded corners,minimum width=7cm,minimum height=2.5cm,anchor=north] (lazysp) at (6,3.0) {};
\node[align=center] at (4,2.4) {Lazy\\Pathfinding};
\node[draw=black!30,fill=white,inner sep=5pt] at (4,1.4) {
\includegraphics[width=2.0cm]{build/lazysp-icon}};
\node[fill=blue!20,draw=blue!30,rounded corners,align=center,minimum width=3.3cm,minimum height=2.1cm,anchor=north] (dynsp) at (7.5,2.8) {};
\node[anchor=north west] at (dynsp.north west) {\begin{minipage}{3.1cm}Dynamic Pathfinding\end{minipage}};
\node[draw=black!30,inner sep=0pt] at (7.5,1.55) {
\includegraphics[width=2.0cm]{figs/incbi-road-ne/singleshot/example-bidijkstra.png}};
% END LAZYSP
% END LEMUR
% top left side
\node[fill=blue!10,draw=blue!20,rounded corners,align=center,minimum height=1.5cm,minimum width=1.8cm,inner sep=0pt] at (6.0,7.0) {};
\node[fill=white,inner sep=0pt] at (6.0,7.0) {\includegraphics[width=1.4cm]{build/c-space-simple}};
\node[font=\scriptsize] at (5.65,6.7) {$\mathcal{C}_{\mbox{\tiny free}}$};
\draw[->] (6.0,6.25) -- (6.0,5.25) node [pos=0.55,fill=blue!5,align=center,inner sep=0pt] {\strut $\mathcal{C}$};
%\node[inner sep=4pt] (cspace) at (3.3,7.0) {$\mathcal{C}$-Space};
%\draw[->] (cspace) -- (3.3,5.25);
% left side
\draw[->] (0.5,2.0) -- (2.5,2.0);
\node[fill=white,align=center,inner sep=2pt] at (0.5,2.0)
{$q_{\ms{start}}$\\$q_{\ms{dest}}$};
% right side
\draw[->] (9.5,2.0) -- (11.3,2.0);
\node[fill=white,align=center,inner sep=2pt] at (11.5,2.0)
{$\xi$};
\end{tikzpicture}
\end{frame}
%\begin{frame}
%\frametitle{Talk Outline}
%\begin{tikzpicture}[font=\small]
%\tikzset{>=latex} % arrow heads
%\draw[step=1,black!15,very thin,opacity=\gridopacity] (0,0) grid (12,8);
%\node[fill=blue!15,minimum width=11cm] at (6,7.5) {\strut What to Optimize?};
%\node[fill=black!3,minimum width=5cm,minimum height=3cm] (utility) at (3.25,5.6) {};
%\node[anchor=north] at (utility.north) {\strut Maximizing Utility};
%\node at (3.25,5.35) {\includegraphics[width=2.5cm]{build/pvx-utility-anytime}};
%\node[fill=black!3,minimum width=5cm,minimum height=3cm] (family) at (8.75,5.6) {};
%\node[anchor=north] at (family.north) {\strut Utility in C-Space Familes};
%\node at (8.75,5.4) {\includegraphics[width=3.0cm]{build/multiple-sets}};
%\node[fill=blue!15,minimum width=11cm] at (6,3.5) {\strut How to Optimize?};
%\node[fill=black!3,minimum width=5cm,minimum height=3cm] (lazysp) at (3.25,1.6) {};
%\node[anchor=north] at (lazysp.north) {\strut Lazy Pathfinding};
%\node[draw=black!30,fill=white,inner sep=5pt] at (3.25,1.4) {
%\includegraphics[width=3.8cm]{build/lazysp-icon}};
%\node[fill=black!3,minimum width=5cm,minimum height=3cm] (ibid) at (8.75,1.6) {};
%\node[anchor=north] at (ibid.north) {\strut Dynamic Pathfinding};
%\node[draw=black!30,inner sep=0pt] at (8.75,1.4) {
%\includegraphics[width=2.9cm]{figs/incbi-road-ne/singleshot/example-bidijkstra.png}};
%\only<2>
%{
%\draw[ultra thick] (utility.north west) rectangle (utility.south east);
%}
%\end{tikzpicture}
%\end{frame}
%\begin{frame}
% \frametitle{Maximizing Utility in Motion Planning}
% \begin{tikzpicture}[font=\small]
% \tikzset{>=latex} % arrow heads
%
% \draw[step=1,black!15,very thin,opacity=\gridopacity] (0,0) grid (12,8);
%
% \node at ( 6,4) {\includegraphics[width=5cm]{figs/xkcd-1445-efficiency.png}};
%
% \end{tikzpicture}
%\end{frame}
\begin{frame}
\frametitle{Maximizing Utility in Motion Planning}
\begin{tikzpicture}[font=\small]
\tikzset{>=latex} % arrow heads
\draw[step=1,black!15,very thin,opacity=\gridopacity] (0,0) grid (12,8);
%\only<1>{\node at (2.75,5.5) {\includegraphics{build/pvx-graph-build,drawempty}};}
\only<2>{\node at (2.75,5.5) {\includegraphics{build/pvx-graph-build,drawaxes}};}
\only<3>{\node at (2.75,5.5) {\includegraphics{build/pvx-graph-build,drawa}};}
\only<4>{\node at (2.75,5.5) {\includegraphics{build/pvx-graph-build,drawab}};}
\only<5>{\node at (2.75,5.5) {\includegraphics{build/pvx-graph-build,drawabxstar}};}
\only<6>{\node at (2.75,5.5) {\includegraphics{build/pvx-graph-build,drawabxstarc}};}
\only<7-8>{\node at (2.75,5.5) {\includegraphics{build/pvx-graph-build,drawabxstarcd}};}
\only<9->{\node at (2.75,5.5) {\includegraphics{build/pvx-graph-build,drawabxstarcdutility}};}
\only<1->{
\node[fill=blue!5,draw=blue!10,rounded corners,align=center,anchor=north,minimum width=5cm] at (8.5,7.0) {
\only<1-3>{$A_1$: a planner invocation}
\only<4->{$A_1, A_2$: planner invocations}
\only<6->{\\ $A_3$: an anytime planner}
\only<7->{\\ $A_4$: a parameterized planner}};
}
\only<2->{
\node[fill=blue!5,draw=blue!10,rounded corners,align=center] at (8.5,4.5) {
$p$: planning cost incurred during invocation\\
$x$: execution cost of resulting solution path};
}
\only<8->{
\node[fill=blue!5,draw=blue!10,rounded corners,align=center,anchor=north] at (6,3.0) {
$\arraycolsep=1.5pt \begin{array}{rc}
U(p,x): & \mbox{real-valued utility function} \\
& \mbox{which scores the invocation}
\only<9->{\\ & \mbox{(isocontours shown)}}
\end{array}$};
}
\only<8->{
\node[anchor=west,font=\scriptsize] at (1.0,0.5) {\begin{minipage}{10cm}
\hangindent=0.3cm \raggedright
\PaperPortrait\; Burns, Ruml, and Do,
``Heuristic Search when Time Matters,''
IJCAI 2013.
\end{minipage}};
}
\end{tikzpicture}
\end{frame}
\begin{frame}
\frametitle{Examples of Utility Functions}
\begin{tikzpicture}[font=\small]
\tikzset{>=latex} % arrow heads
\draw[step=1,black!15,very thin,opacity=\gridopacity] (0,0) grid (12,8);
\only<1->{
\node at (1.92,6.5) {\includegraphics{build/pvx-sm-firstfeas}};
\node[anchor=west] at (3.5,6.6) {\begin{minipage}{8cm}
{\bf First Feasible} --
Find a feasible solution as quickly as possible \\
irrespective of its cost.
\end{minipage}};
}
\only<2->{
\node at (1.92,4.0) {\includegraphics{build/pvx-sm-pbudget}};
\node[anchor=west] at (3.5,4.1) {\begin{minipage}{8cm}
{\bf Planning Budget} --
Return the lowest-cost path possible within a \\
specified planning resource budget.
\end{minipage}};
}
\only<3->{
\node at (1.82,1.5) {\includegraphics{build/pvx-sm-xbudget}};
\node[anchor=west] (boundedcost) at (3.5,1.6) {\begin{minipage}{8cm}
{\bf Bounded Cost} --
Find a solution as fast as possible within a \\
specified path cost bound.
\end{minipage}};
\node[anchor=north,font=\scriptsize,below=-0.1cm of boundedcost.south] {\begin{minipage}{8cm}
\hangindent=0.3cm \raggedright
\PaperPortrait\; Stern et al,
``Potential-based Bounded-cost Search and Anytime Non-Parametric A*,''
AI 2014.
\end{minipage}};
}
\end{tikzpicture}
\end{frame}
\begin{frame}
\frametitle{Algorithm Outline: Maximizing Utility over Paths}
\begin{tikzpicture}[font=\small]
\tikzset{>=latex} % arrow heads
\draw[step=1,black!15,very thin,opacity=\gridopacity] (0,0) grid (12,8);
\only<2>{
\node[fill=blue!10,draw=blue!20,rounded corners,align=center] at (6,5)
{
Central assumption: planning cost is dominated by\\
\emph{validating paths} (i.e. evaluating their cost).
};
}
\only<3->{
\node[fill=blue!5,rounded corners,anchor=north,minimum width=9.5cm,minimum height=3.6cm] (alg) at (6,7.75) {};
\begin{scope} % highlights
\clip[rounded corners] (alg.north west) rectangle (alg.south east);
\only<3>{\node[fill=blue!15,anchor=north,minimum width=9.5cm,minimum height=0.5cm,below=0cm of alg.north] {};} % pinit
\only<4>{\node[fill=blue!15,anchor=north,minimum width=9.5cm,minimum height=0.45cm,below=0.4cm of alg.north] {};} % iters
\only<5,11,15>{\node[fill=blue!15,anchor=north,minimum width=9.5cm,minimum height=0.45cm,below=0.8cm of alg.north] {};} % pathset
\only<6>{\node[fill=blue!15,anchor=north,minimum width=9.5cm,minimum height=0.9cm,below=1.2cm of alg.north] {};} % estimators
\only<7,12,16>{\node[fill=blue!15,anchor=north,minimum width=9.5cm,minimum height=0.45cm,below=2.0cm of alg.north] {};} % argmin
\only<8,17>{\node[fill=blue!15,anchor=north,minimum width=9.5cm,minimum height=0.45cm,below=2.35cm of alg.north] {};} % return
\only<9,13>{\node[fill=blue!15,anchor=north,minimum width=9.5cm,minimum height=0.45cm,below=2.75cm of alg.north] {};} % eval
\only<10,14>{\node[fill=blue!15,anchor=north,minimum width=9.5cm,minimum height=0.45cm,below=3.15cm of alg.north] {};} % updatep
\end{scope}
\node[left=0.1cm of alg.north,anchor=north] {\begin{minipage}{9cm}
\begin{algorithmic}[1]
\State $p_0 \leftarrow 0$
\Comment planning cost incurred so far
\only<4->{
\For {iteration $i \in 1, 2, \dots$}
\only<5->{
\State $\Xi_i \leftarrow$ set of paths to consider at iteration $i$
}%
\only<6->{
\State $\hat{x}_i : \Xi_i \rightarrow \mathbb{R}^+$
\Comment execution cost estimator
\State $\hat{p}_i : \Xi_i \rightarrow \mathbb{R}^+$
\Comment additional planning cost estimator%
}%
\only<7->{
\State $\xi_i = \argmax_{\xi \in \Xi_i}
U\!\left( \; p_{i-1} \! + \! \hat{p}_i(\xi), \; \hat{x}_i(\xi) \; \right)$
\label{line:outline-argmax}
}%
\only<8->{
\State \Return $\xi_i$
if $\xi_i$ fully evaluated, i.e. $\hat{p}(\xi_i) = 0$
}%
\only<9->{
\State evaluate $\xi_i$
\Comment incurs planning cost $\Delta p_i$
}%
\only<10->{
\State $p_i \leftarrow p_{i-1} + \Delta p_i$
}%
\EndFor
}
\end{algorithmic}
\end{minipage}};
}
\only<1-4>{\node at (3.5,2.0) {\includegraphics{build/cspace-utility-intro,start}};}
\only<5-6>{\node at (3.5,2.0) {\includegraphics{build/cspace-utility-intro,candidatesa}};}
\only<7-8>{\node at (3.5,2.0) {\includegraphics{build/cspace-utility-intro,candidatea}};}
\only<9-10>{\node at (3.5,2.0) {\includegraphics{build/cspace-utility-intro,candidateaevaled}};}
\only<11>{\node at (3.5,2.0) {\includegraphics{build/cspace-utility-intro,candidatesb}};}
\only<12>{\node at (3.5,2.0) {\includegraphics{build/cspace-utility-intro,candidateb}};}
\only<13-14>{\node at (3.5,2.0) {\includegraphics{build/cspace-utility-intro,candidatebevaled}};}
\only<15>{\node at (3.5,2.0) {\includegraphics{build/cspace-utility-intro,candidatesfinal}};}
\only<16->{\node at (3.5,2.0) {\includegraphics{build/cspace-utility-intro,candidatefinal}};}
\only<3>{\node at (9.0,2.0) {\includegraphics{build/p-estimates-build,start}};}
\only<4-6>{\node at (9.0,2.0) {\includegraphics{build/p-estimates-build,iterlabel}};}
\only<7-8>{\node at (9.0,2.0) {\includegraphics{build/p-estimates-build,candidatea}};}
\only<9>{\node at (9.0,2.0) {\includegraphics{build/p-estimates-build,candidateadelta}};}
\only<10-11>{\node at (9.0,2.0) {\includegraphics{build/p-estimates-build,candidateaeval}};}
\only<12>{\node at (9.0,2.0) {\includegraphics{build/p-estimates-build,candidateb}};}
\only<13>{\node at (9.0,2.0) {\includegraphics{build/p-estimates-build,candidatebdelta}};}
\only<14>{\node at (9.0,2.0) {\includegraphics{build/p-estimates-build,candidatebeval}};}
\only<15-16>{\node at (9.0,2.0) {\includegraphics{build/p-estimates-build,prefinal}};}
\only<17>{\node at (9.0,2.0) {\includegraphics{build/p-estimates-build,final}};}
\end{tikzpicture}
\end{frame}
\begin{frame}
\frametitle{Aside: Simplified Bidirectional RRT-Connect}
\begin{tikzpicture}[font=\small]
\tikzset{>=latex} % arrow heads
\draw[step=1,black!15,very thin,opacity=\gridopacity] (0,0) grid (12,8);
\node[fill=blue!5,rounded corners,anchor=north,minimum width=9.5cm,minimum height=3.6cm] (alg) at (6,7.75) {};
\begin{scope} % highlights
\clip[rounded corners] (alg.north west) rectangle (alg.south east);
%\only<2>{\node[fill=blue!15,anchor=north,minimum width=9.5cm,minimum height=0.5cm,below=0cm of alg.north] {};} % pinit
%\only<3>{\node[fill=blue!15,anchor=north,minimum width=9.5cm,minimum height=0.45cm,below=0.4cm of alg.north] {};} % iters
\only<4>{\node[fill=blue!15,anchor=north,minimum width=9.5cm,minimum height=0.45cm,below=0.8cm of alg.north] {};} % pathset
%\only<5>{\node[fill=blue!15,anchor=north,minimum width=9.5cm,minimum height=0.9cm,below=1.2cm of alg.north] {};} % estimators
\only<5->{\node[fill=blue!15,anchor=north,minimum width=9.5cm,minimum height=0.45cm,below=2.0cm of alg.north] {};} % argmin
%\only<7,16>{\node[fill=blue!15,anchor=north,minimum width=9.5cm,minimum height=0.45cm,below=2.35cm of alg.north] {};} % return
%\only<8,12>{\node[fill=blue!15,anchor=north,minimum width=9.5cm,minimum height=0.45cm,below=2.75cm of alg.north] {};} % eval
%\only<9,13>{\node[fill=blue!15,anchor=north,minimum width=9.5cm,minimum height=0.45cm,below=3.15cm of alg.north] {};} % updatep
\end{scope}
\node[left=0.1cm of alg.north,anchor=north] {\begin{minipage}{9cm}
\begin{algorithmic}[1]
\State $p_0 \leftarrow 0$
\Comment planning cost incurred so far
\For {iteration $i \in 1, 2, \dots$}
\State $\Xi_i \leftarrow$ set of paths to consider at iteration $i$
\State $\hat{x}_i : \Xi_i \rightarrow \mathbb{R}^+$
\Comment execution cost estimator
\State $\hat{p}_i : \Xi_i \rightarrow \mathbb{R}^+$
\Comment additional planning cost estimator
\State $\xi_i = \argmax_{\xi \in \Xi_i}
U\!\left( \; p_{i-1} \! + \! \hat{p}_i(\xi), \; \hat{x}_i(\xi) \; \right)$
\label{line:outline-argmax}
\State \Return $\xi_i$
if $\xi_i$ fully evaluated, i.e. $\hat{p}(\xi_i) = 0$
\State evaluate $\xi_i$
\Comment incurs planning cost $\Delta p_i$
\State $p_i \leftarrow p_{i-1} + \Delta p_i$
\EndFor
\end{algorithmic}
\end{minipage}};
\only<1>{\node at (3.7,2.0) {\includegraphics{build/rrt-build,stonly}};}
\only<2>{\node at (3.7,2.0) {\includegraphics{build/rrt-build,stexisting}};}
\only<3>{\node at (3.7,2.0) {\includegraphics{build/rrt-build,target}};}
\only<4-5>{\node at (3.7,2.0) {\includegraphics{build/rrt-build,otherpaths}};}
\only<6->{\node at (3.7,2.0) {\includegraphics{build/rrt-build,all}};}
\only<6->{\node at (8.7,2.0) {\includegraphics{build/pvx-rrt}};}
\only<7>{
\node[fill=white,opacity=0.8,text opacity=1.0,inner sep=5pt,rounded corners] at (6,4.2) {
\begin{tikzpicture}
\node[fill=black!5,draw=black!15,rounded corners,align=center]
{RRT-Connect implicitly maximizes utility at each iteration\\
using the first-feasible utility function.};
\end{tikzpicture}
};
}
\end{tikzpicture}
\end{frame}
\begin{frame}
\frametitle{Maximizing Utility over Roadmaps}
\begin{tikzpicture}[font=\small]
\draw[step=1,black!15,very thin,opacity=\gridopacity] (0,0) grid (12,8);
%\node[fill=blue!15,minimum width=11cm] at (6,7.5) {\strut How to Optimize?};
%\node[fill=blue!15,minimum width=11cm] at (6,6.75) {\strut What to Optimize?};
\node[fill=blue!5,rounded corners,anchor=north,minimum width=9.5cm,minimum height=3.6cm] (alg) at (6,7.75) {};
\begin{scope} % highlights
\clip[rounded corners] (alg.north west) rectangle (alg.south east);
%\only<2>{\node[fill=blue!15,anchor=north,minimum width=9.5cm,minimum height=0.5cm,below=0cm of alg.north] {};} % pinit
%\only<3>{\node[fill=blue!15,anchor=north,minimum width=9.5cm,minimum height=0.45cm,below=0.4cm of alg.north] {};} % iters
\only<2-3>{\node[fill=blue!15,anchor=north,minimum width=9.5cm,minimum height=0.45cm,below=0.8cm of alg.north] {};} % pathset
%\only<5>{\node[fill=blue!15,anchor=north,minimum width=9.5cm,minimum height=0.9cm,below=1.2cm of alg.north] {};} % estimators
%\only<5->{\node[fill=blue!15,anchor=north,minimum width=9.5cm,minimum height=0.45cm,below=2.0cm of alg.north] {};} % argmin
%\only<7,16>{\node[fill=blue!15,anchor=north,minimum width=9.5cm,minimum height=0.45cm,below=2.35cm of alg.north] {};} % return
%\only<8,12>{\node[fill=blue!15,anchor=north,minimum width=9.5cm,minimum height=0.45cm,below=2.75cm of alg.north] {};} % eval
%\only<9,13>{\node[fill=blue!15,anchor=north,minimum width=9.5cm,minimum height=0.45cm,below=3.15cm of alg.north] {};} % updatep
\end{scope}
\node[left=0.1cm of alg.north,anchor=north] {\begin{minipage}{9cm}
\begin{algorithmic}[1]
\State $p_0 \leftarrow 0$
\Comment planning cost incurred so far
\For {iteration $i \in 1, 2, \dots$}
\State $\Xi_i \leftarrow$ set of paths to consider at iteration $i$
\State $\hat{x}_i : \Xi_i \rightarrow \mathbb{R}^+$
\Comment execution cost estimator
\State $\hat{p}_i : \Xi_i \rightarrow \mathbb{R}^+$
\Comment additional planning cost estimator
\State $\xi_i = \argmax_{\xi \in \Xi_i}
U\!\left( \; p_{i-1} \! + \! \hat{p}_i(\xi), \; \hat{x}_i(\xi) \; \right)$
\label{line:outline-argmax}
\State \Return $\xi_i$
if $\xi_i$ fully evaluated, i.e. $\hat{p}(\xi_i) = 0$
\State evaluate $\xi_i$
\Comment incurs planning cost $\Delta p_i$
\State $p_i \leftarrow p_{i-1} + \Delta p_i$
\EndFor
\end{algorithmic}
\end{minipage}};
\only<3->{
\fill[white,path fading=fade down] (1,6.5) rectangle (11,6.0);
\fill[white] (1,6.0) rectangle (11,4);
\node at (2.25,3.25) {\includegraphics[width=3.0cm]{build/roadmap-stack-onramps}};
\node[anchor=north,fill=black!3,rounded corners] at (8,5.5) {\begin{minipage}{7cm}
\raggedright
Focus: Roadmap methods
\medskip
Advantages:
\begin{itemize}
\item Well-studied asymptotic completeness\\
and optimality properties
\item Easy to progressively densify
\item Common structure across queries\\
(can load roadmaps from disk)
\end{itemize}
\end{minipage}};
}
\end{tikzpicture}
\end{frame}
\begin{frame}
\frametitle{Maximizing Linear Utilities with Additive Estimators}
\begin{tikzpicture}[font=\small]
\draw[step=1,black!15,very thin,opacity=\gridopacity] (0,0) grid (12,8);
\tikzset{>=latex} % arrow heads
%\node[fill=blue!15,minimum width=11cm] at (6,7.5) {\strut How to Optimize?};
%\node[fill=blue!15,minimum width=11cm] at (6,6.75) {\strut What to Optimize?};
\node[fill=blue!5,rounded corners,anchor=north,minimum width=9.5cm,minimum height=3.6cm] (alg) at (6,7.75) {};
\begin{scope} % highlights
\clip[rounded corners] (alg.north west) rectangle (alg.south east);
%\only<2>{\node[fill=blue!15,anchor=north,minimum width=9.5cm,minimum height=0.5cm,below=0cm of alg.north] {};} % pinit
%\only<3>{\node[fill=blue!15,anchor=north,minimum width=9.5cm,minimum height=0.45cm,below=0.4cm of alg.north] {};} % iters
\only<3>{\node[fill=blue!15,anchor=north,minimum width=9.5cm,minimum height=0.45cm,below=0.8cm of alg.north] {};} % pathset
\only<3>{\node[fill=blue!15,anchor=north,minimum width=9.5cm,minimum height=0.9cm,below=1.10cm of alg.north] {};} % estimators
\only<4,8->{\node[fill=blue!15,anchor=north,minimum width=9.5cm,minimum height=0.45cm,below=2.0cm of alg.north] {};} % argmin
%\only<7,16>{\node[fill=blue!15,anchor=north,minimum width=9.5cm,minimum height=0.45cm,below=2.35cm of alg.north] {};} % return
\only<5-7>{\node[fill=blue!15,anchor=north,minimum width=9.5cm,minimum height=0.45cm,below=2.75cm of alg.north] {};} % eval
%\only<9,13>{\node[fill=blue!15,anchor=north,minimum width=9.5cm,minimum height=0.45cm,below=3.15cm of alg.north] {};} % updatep
\end{scope}
\node[left=0.1cm of alg.north,anchor=north] {\begin{minipage}{9cm}
\begin{algorithmic}[1]
\State $p_0 \leftarrow 0$
\Comment planning cost incurred so far
\For {iteration $i \in 1, 2, \dots$}
\State $\Xi_i \leftarrow$ set of paths to consider at iteration $i$
\State $\hat{x}_i : \Xi_i \rightarrow \mathbb{R}^+$
\Comment execution cost estimator
\State $\hat{p}_i : \Xi_i \rightarrow \mathbb{R}^+$
\Comment additional planning cost estimator
\only<1-8>{\State $\xi_i = \argmax_{\xi \in \Xi_i}
U\!\left( \; p_{i-1} \! + \! \hat{p}_i(\xi), \; \hat{x}_i(\xi) \; \right)$
\label{line:outline-argmax}}%
\only<9->{\State $\xi_i = \argmax_{\xi \in \Xi_i}
U\!\left( \; \hat{p}_i(\xi), \; \hat{x}_i(\xi) \; \right)$
\label{line:outline-argmax}}%
\State \Return $\xi_i$
if $\xi_i$ fully evaluated, i.e. $\hat{p}(\xi_i) = 0$
\State evaluate $\xi_i$
\Comment incurs planning cost $\Delta p_i$
\State $p_i \leftarrow p_{i-1} + \Delta p_i$
\EndFor
\end{algorithmic}
\end{minipage}};
\only<2->{
\node[fill=black!5,draw=black!20,rounded corners] at (3.0,3.0)
{\begin{minipage}{4.9cm}
We consider \emph{linear utility functions}:
\vspace{-0.2cm}
\[
-U(p,x) = \lambda \, p + (1 - \lambda) \, x
\]
\end{minipage}};
}
\only<10->{
\node[fill=black!5,draw=black!20,rounded corners] at (8.7,3.0) {\begin{minipage}{5.6cm}
We consider \emph{additive estimators}:
\vspace{-0.2cm}
\[
\xi: \mbox{path on a roadmap graph } G
\]
\vspace{-0.5cm}
\[
\hat{p}_i(\xi) = \sum_{e \in \xi} \hat{p}_i(e)
\;\;\mbox{and}\;\;
\hat{x}_i(\xi) = \sum_{e \in \xi} \hat{x}_i(e)
\]
\end{minipage}};
}
\only<3>{\node at (8.5,2) {\includegraphics[width=3.75cm]{build/pvx-linear-discounting-build,beforeeval}};}
\only<4>{\node at (8.5,2) {\includegraphics[width=3.75cm]{build/pvx-linear-discounting-build,selectcandidate}};}
\only<5>{\node at (8.5,2) {\includegraphics[width=3.75cm]{build/pvx-linear-discounting-build,newcandidate}};}
\only<6>{\node at (8.5,2) {\includegraphics[width=3.75cm]{build/pvx-linear-discounting-build,newshared}};}
\only<7-8>{\node at (8.5,2) {\includegraphics[width=3.75cm]{build/pvx-linear-discounting-build,newother}};}
\only<9>{\node at (8.5,2) {\includegraphics[width=3.75cm]{build/pvx-linear-discounting-build,newaxes}};}
\only<11->{
\draw[->,very thick] (5.65,1.9) -- (5.65,1.4);
\node[fill=black!5,draw=black!20,rounded corners] at (5.8,0.8) {\begin{minipage}{5cm}
Shortest path problem with:
\vspace{-0.2cm}
\[
w_i(e) = \lambda \, \hat{p}_i(e) + (1 - \lambda) \, \hat{x}_i(e)
\]
\end{minipage}};
}
\end{tikzpicture}
\end{frame}
% all this data cumulative over the multistep-prescribed problem
\pgfplotstableread{
lambda searchmean evalmean
lam00 0.13717146918 1.566855428440001
lam20 0.13591121493999997 1.5040972085999997
lam50 0.12633906594 1.3774433666
lam80 0.11458146473999997 1.2423356615
lam99 0.10054155609999996 0.9362152972200001
}{\datasvetimeherbbinnom}
\pgfplotstableread{
lambda searchmean evalmean
lam00 0.9183584741400002 4.73433351068
lam20 0.8734410922000001 4.4884742077799995
lam50 0.82153580024 4.091855509880002
lam80 0.75752757384 3.39455290978
lam99 0.7430251334600001 3.0377397256199994
}{\datasvetimeworkcell}
\begin{frame}
\frametitle{Search vs. Evaluation Planning Time Comparison}
\begin{tikzpicture}[font=\small]
\draw[step=1,black!15,very thin,opacity=\gridopacity] (0,0) grid (12,8);
\tikzset{>=latex} % arrow heads
\node at (8.2,7.4) {
\protect\tikz{\protect\node[fill=red!20,draw=black]{};}\;graph search vs.
\protect\tikz{\protect\node[fill=blue!20,draw=black]{};}\;edge evaluation
};
\node[fill=black!5,draw=black!20,rounded corners,minimum width=11cm,minimum height=3cm] at (6.0,5.5) {};
\node at ( 2.5,5.5) {
\includegraphics[width=1.7cm]{figs/herbbin/step0cropped.png}
\includegraphics[width=1.7cm]{figs/herbbin/step01cropped.png}};
\begin{scope}[shift={(5.5,5.0)}]
\begin{axis}[
width=7.0cm,
height=3.2cm,
xbar stacked,
bar width=7,
%xmin=0,xmax=10,
xmin=0,
%xtick pos=bottom,
%symbolic y coords={E, F, R, A, B, W, P},
%yticklabels from table={\datasvetimeherbbinnom}{lambda},
yticklabels={$0.00$,,,,$0.99$},
ylabel={$\lambda$},
ytick=data,
%ytick pos=left,
xmajorgrids,
xmajorticks=true,
%ticklabel style={font=\footnotesize},
enlarge y limits={abs=0.2cm},
ylabel near ticks,
ylabel shift=-0.5cm,
ylabel style={rotate=-90},
xlabel={Average S/E Planning Time (s)},
xlabel near ticks,
]
%\node[circle,fill=white,inner sep=1pt,text=black!40] at (axis cs:40,-5.6) {\scriptsize 40};
%\node[circle,fill=white,inner sep=1pt,text=black!40] at (axis cs:60,-5.6) {\scriptsize 60};
%\node[circle,fill=white,inner sep=1pt,text=black!40] at (axis cs:80,-5.6) {\scriptsize 80};
\addplot[color=black,fill=red!20]
table[y expr=-\coordindex,x=searchmean]
{\datasvetimeherbbinnom};
\addplot[color=black,fill=blue!20]
table[y expr=-\coordindex,x=evalmean]
{\datasvetimeherbbinnom};
\end{axis}
\end{scope}
\node[fill=black!5,draw=black!20,rounded corners,minimum width=11cm,minimum height=3cm] at (6.0,2.2) {};
\node at ( 2.5,2.2) {
\includegraphics[width=3.5cm]{videos/workcell-cropped.png}};
\begin{scope}[shift={(5.5,1.7)}]
\begin{axis}[
width=7.0cm,
height=3.2cm,
xbar stacked,
bar width=7,
%xmin=0,xmax=10,
xmin=0,
%xtick pos=bottom,
%symbolic y coords={E, F, R, A, B, W, P},
%yticklabels from table={\datasvetimeherbbinnom}{lambda},
yticklabels={$0.00$,,,,$0.99$},
ylabel={$\lambda$},
ytick=data,
%ytick pos=left,
xmajorgrids,
xmajorticks=true,
%ticklabel style={font=\footnotesize},
enlarge y limits={abs=0.2cm},
ylabel near ticks,
ylabel shift=-0.5cm,
ylabel style={rotate=-90},
xlabel={Average S/E Planning Time (s)},
xlabel near ticks,
]
%\node[circle,fill=white,inner sep=1pt,text=black!40] at (axis cs:40,-5.6) {\scriptsize 40};
%\node[circle,fill=white,inner sep=1pt,text=black!40] at (axis cs:60,-5.6) {\scriptsize 60};
%\node[circle,fill=white,inner sep=1pt,text=black!40] at (axis cs:80,-5.6) {\scriptsize 80};
\addplot[color=black,fill=red!20]
table[y expr=-\coordindex,x=searchmean]
{\datasvetimeworkcell};
\addplot[color=black,fill=blue!20]
table[y expr=-\coordindex,x=evalmean]
{\datasvetimeworkcell};
\end{axis}
\end{scope}
\end{tikzpicture}
\end{frame}