slides.en.tex

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\usetheme{moulardthesis}

\DeclareUnicodeCharacter{00A0}{~}

\title{Numerical Optimization\\ and Robotics}
\author{Thomas Moulard}
\date{May 17, 2013}

\begin{document}

{
  \usebackgroundtemplate{%
    \includegraphics[width=\paperwidth,height=\paperheight]{%
      src/slides/demo1.jpg}}
  \begin{frame}[plain]
    \titlepage
    \note[item]{Remercier les rapporteurs, le jury}
  \end{frame}
}

%\maxFrameImage{src/slides/demo1.jpg}
%\maxFrameImage{src/slides/demo2.jpg}

% Introduction
% Animated scheme introducing contributions.
\begin{frame}[label=contrib]
  \note[item]{Architecture P-D-A}
  \note[item]{Perception: environment, sensors}
  \note[item]{Decision: goal resolution, planning}
  \note[item]{Action: actuators control}
  \note[item]{Humanoid robots misc.\ contributions}
  \note[item]{RobOptim: représentation info des pb optim}
  \note[item]{Correction de trajectoire: %
    rebouclage capteur de la partie action}
  \note[item]{Primitives de mouvement: résultats préliminaires. %
    Planification de tâche, intégration capteurs dans la planificapteurs}
  %
  \begin{changemargin}{-1cm}{-1cm}
    \begin{center}
      \tikzstyle{normalPath} = [line width=.7mm, color=black!50, -latex]

      \begin{tikzpicture}[
          auto,
          state/.style={
            rectangle,
            minimum size=6mm
          }]

        \uncover<1->{
          \node (perception) [state] {%
            \includegraphics[height=3cm]{%
              src/slides/perception.pdf}};
        }

        \uncover<1>{
          \node (perception-description)
                [state, right=of perception, yshift=-6mm, xshift=-6mm]
                {\textbf{Perception}};
        }


        \uncover<2->{
          \node (decision)
               [state,above right=of perception] {%
            \includegraphics[height=3cm]{%
              src/slides/decision.pdf}};
        }

        \uncover<2>{
          \node (decision-description)
                [state, right=of decision, yshift=6mm, xshift=-6mm]
                {\textbf{Decision}};
        }

        \uncover<3->{
        \node (action) [state,below right=of decision]{%
          \includegraphics[height=3cm]{%
            src/slides/action.pdf}};
        }

        \uncover<3>{
          \node (action-description)
                [state, left=of action, yshift=12mm, xshift=6mm]
                {\textbf{Action}};
        }

        \uncover<4->{
          \path (perception) edge[->, normalPath] (decision);
        }

        \uncover<5-8>{
          \path (decision) edge[->, normalPath] (action);
        }

        \uncover<6->{
          \path (action.south)
          edge[->, normalPath, dashed, bend right, bend left]
          node[color=black!50,above=5pt] {Real World}
          (perception.south);
        }

        \uncover<7->{
          \node (contrib) [state,left=of decision]{%
            \usebeamerfont{section}\large\textbf{Contributions}};

          \node (contrib1b) [state,right=1pt of decision]{%
            (1) RobOptim};
        }
        \uncover<7>{
          \node[thick,draw=ThoughfulBrick] (contrib1)
               [state,above=-3.4cm of decision] {%
                 \phantom{\includegraphics[height=3.5cm]{%
                     src/slides/perception.pdf}}};
        }

        \uncover<8->{
          \path (perception.east)
          edge[->, normalPath,color=ThoughfulBrick]
          node[above=5pt] {(2) Correction}
          (action.west);
        }
        \uncover<8>{
          \node[thick,draw=ThoughfulBrick] (contrib2)
               [state,above=-3.4cm of action] {%
                 \phantom{\includegraphics[height=3.5cm]{%
                     src/slides/action.pdf}}};
        }


        \uncover<9->{
          \path (decision) edge[->, normalPath,color=ThoughfulBrick]
          node[color=ThoughfulBrick,text width=3.5cm]{%
            (3) Motion Primitives} (action);
        }

        \uncover<10->{
          \node (contrib4b) [state,above=10pt of perception]{%
            (4) Integration};
        }
        \uncover<10>{
          \node[thick,draw=ThoughfulBrick] (contrib4)
               [state,above=-3.4cm of perception] {%
                 \phantom{\includegraphics[height=3.5cm]{%
                     src/slides/perception.jpg}}};
        }


      \end{tikzpicture}
    \end{center}
  \end{changemargin}
\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{slideDecision}
  \frametitle{Motion Generation}

  \note[item]{Contraintes = plus de temps de calcul ainsi
  que possiblement plus de minima locaux.}
  \note[item]{Exemple: contrainte de distances/évitement d'obstacles.}
  \note[item]{Travail effectué: représentation du problème pour
    trouver équilibre entre réalisme, sûreté et tractabilité.}

  \begin{changeleftmargin}{1.1cm}
  \begin{center}
    Which constraints should be considered?\par
    \vspace{1cm}
    \includegraphics[width=.25\paperheight]{src/slides/falling.pdf}%
    \includegraphics[width=.25\paperheight]{src/slides/burning.pdf}\par
  \end{center}

  \begin{itemize}
  \item Position, velocitiy and torque limits,
  \item (auto-)collision,
  \item \alert{balance}.
  \end{itemize}
  \end{changeleftmargin}
\end{slideDecision}

\begin{slideDecision}
  \frametitle{Balance (1)}

  \begin{changeleftmargin}{1.1cm}
    \begin{center}
      \vspace{-1cm}
      \begin{tikzpicture}[%
          force/.style={>=latex,
            draw=Thoughtless,
            fill=Thoughtless,
            ultra thick},
        ]
        \node (body) {\includegraphics[height=6cm]{%
            src/slides/walking.pdf}};

        % Floor
        \draw[ultra thick] ([yshift=5]body.south west)
        -- ([yshift=5]body.south east);

        % Center of mass.
        \fill[radius=0.4em] ([xshift=-5.5]body.center) -- ++(0.4,0)%
        arc [start angle=0,end angle=90] -- ++(0,-0.4)%
        arc [start angle=270, end angle=180];%
        \draw[radius=0.4em] (body.center) circle;%

        % Forces
        \draw[force,->] (body.center)
        -- ++(0,-1) node[left] {$\mathcal{P}$};

        \draw[force,->] ([xshift=-15,yshift=5]body.south east)
        -- ++(0,1) node[right] {$f_c$};
      \end{tikzpicture}

      \begin{columns}
        \begin{column}{.5\paperwidth}
          \begin{equation*}
            \mathbf{x} = (\dot{x_G}, \dot{y_G}, \dot{z_G})
          \end{equation*}
        \end{column}
        \begin{column}{.5\paperwidth}
          \begin{equation*}
            \left(
            \begin{array}{c}
              \dot{\mathbf{x}}\\
              \dot{\sigma}
            \end{array}
            \right) = f(m, f_c)
          \end{equation*}
        \end{column}
      \end{columns}
    \end{center}
  \end{changeleftmargin}
\end{slideDecision}


\begin{slideDecision}
  \frametitle{Balance (2)}

  \note[item]{Critère de stabilité.}
  \note[item]{Point virtuel.}

  \begin{changeleftmargin}{1.1cm}
    \begin{center}
      ZMP in the support polygon: dynamically stable motion.

      \bigskip

      \begin{tikzpicture}[%
          auto,
          footprint/.style={%
            rectangle,
            draw=black,
            ultra thick,
            minimum size=6mm,
            text width=8mm,
            decoration={brace},
        },
        fpLeft/.style={%
          footprint,
          decoration={brace, raise=-1.05cm},
        },
        fpRight/.style={%
          footprint,
          decoration={brace, raise=1.05cm},
        },
        supportPolygon/.style={%
          draw,
          ultra thick
        },
        zmpPoint/.style={%
          fill=ThoughfulBrick,
        }]

        \matrix[ampersand replacement=\&,
          row sep=1cm, column sep=.5cm]{%
            \uncover<1-6> {
              \node[fpLeft] (l0) {};
            } \&
            \&
            \uncover<8-> {
              \node[fpLeft] (l1) {};
            }\\
%
            \uncover<1-2> {
              \node[fpRight] (r0) {};
            }\&
            \uncover<4-10> {
              \node[fpRight] (r1) {};
            } \&
            \&
            \uncover<12-> {
              \node[fpRight] (r2) {};
            }\\
          };

        \uncover<1>{
          \draw[zmpPoint] (barycentric cs:l0=1/2,r0=1/2) circle (1mm);
        }
        \uncover<2-4>{
          \draw[zmpPoint] (l0) circle (1mm);
        }
        \uncover<5>{
          \draw[zmpPoint] (barycentric cs:l0=1/2,r1=1/2) circle (1mm);
        }
        \uncover<6-8>{
          \draw[zmpPoint] (r1) circle (1mm);
        }
        \uncover<9>{
          \draw[zmpPoint] (barycentric cs:l1=1/2,r1=1/2) circle (1mm);
        }
        \uncover<10-12>{
          \draw[zmpPoint] (l1) circle (1mm);
        }
        \uncover<13>{
          \draw[zmpPoint] (barycentric cs:l1=1/2,r2=1/2) circle (1mm);
        }
        \uncover<14>{
          \draw[zmpPoint] (r2) circle (1mm);
        }


        % 1/ double support.
        \uncover<1-2> {
          \path[supportPolygon]
          (l0.north west) -- (r0.south west)
          -- (r0.south east) -- (l0.north east) -- cycle;
          }

        % 2/ single support, left
        \uncover<3> {
          \path[supportPolygon]
          (l0.north west) -- (l0.south west)
          -- (l0.south east) -- (l0.north east) -- cycle;
        }

        % 3/ double support lr
        \uncover<4-6> {
          \path[supportPolygon]
          (l0.north west) -- (l0.north east)
          -- (r1.north east) -- (r1.south east) -- (r1.south west)
          -- (l0.south west) -- cycle;
        }

        % 4/ single support, right
        \uncover<7> {
          \path[supportPolygon]
          (r1.north west) -- (r1.south west)
          -- (r1.south east) -- (r1.north east) -- cycle;
        }

        % 5/ double support rl
        \uncover<8-10> {
          \path[supportPolygon]
          (l1.north west) -- (l1.north east)
          -- (l1.south east)
          -- (r1.south east) -- (r1.south west)
          -- (r1.north west) -- (l1.north west)
          -- cycle;
        }

        % 6/ single support, left
        \uncover<11> {
          \path[supportPolygon]
          (l1.north west) -- (l1.south west)
          -- (l1.south east) -- (l1.north east) -- cycle;
        }

        % 3/ double support lr
        \uncover<12-14> {
          \path[supportPolygon]
          (l1.north west) -- (l1.north east);
          \path[supportPolygon]
          (l1.north west) -- (l1.south west);


          \path[supportPolygon]
          (r2.south west) -- (r2.south east);
          \path[supportPolygon]
          (r2.south east) -- (r2.north east);

          \path[supportPolygon]
          (l1.south west) -- (r2.south west);
          \path[supportPolygon]
          (l1.north east) -- (r2.north east);
        }
      \end{tikzpicture}

    \end{center}
  \end{changeleftmargin}
\end{slideDecision}

\begin{slideDecision}
  \frametitle{Balance (3)}

  \note[item]{La position du ZMP évolue
    en fonction de la position/accélération du CoM
    ainsi que de la dérivée du moment cinétique.}
  \note[item]{On veut l'inverse: spécificer la trajectoire du ZMP et
    en déduire la trajectoire de CoM.}
  \note[item]{En temps réel! Ici pas le cas.}

  \begin{changeleftmargin}{1.1cm}
  \begin{center}
    \only<1-2>{%
      \begin{equation*}
        \begin{array}{lll}
          x_{ZMP} &=& x_G -
          \frac{\tikz \node[mathbox,%
              onslide=<2->{cross out,draw=ThoughfulBrick}]{$\dot{\sigma}_y$};
            + m z_G \ddot{x}_G}{%
            m (\tikz \node[mathbox,%
              onslide=<2->{cross out,draw=ThoughfulBrick}] {$\ddot{z}_G$}; + g)}\\
          y_{ZMP} &=& y_G +
          \frac{\tikz \node[mathbox,%
              onslide=<2->{cross out,draw=ThoughfulBrick}] {$\dot{\sigma}_x$}; -
            m z_G \ddot{y}_G}{%
            m (\tikz \node[mathbox,%
              onslide=<2->{cross out,draw=ThoughfulBrick}] {$\ddot{z}_G$}; + g)}
        \end{array}
      \end{equation*}
    }
    \only<3>{%
      \begin{equation*}
        \begin{array}{lll}
          x_{ZMP} &=& x_G - \frac{z_G}{g} \ddot{x_G}\\
          y_{ZMP} &=& y_G - \frac{z_G}{g} \ddot{y_G}
        \end{array}
      \end{equation*}
    }
  \end{center}

  \begin{description}
    \item[$(x_G, y_G, z_G)$]~\\
      Center of Mass position (CoM).
    \item[$(\sigma_x, \sigma_y)$]~\\
      Kinetic Momentum around the Center of Mass.
    \item[$m$] Robot Mass.
    \item[$g$] Gravity Constant.
  \end{description}

  \end{changeleftmargin}
\end{slideDecision}

\begin{slideDecision}
  \frametitle{Motion Generation}

  \note[item]{Détailler les étapes.}
  \note[item]{Trajectoire corps complet dans la planif: pas de
    comportement réactif.}

  \begin{changeleftmargin}{1.5cm}
  \begin{center}
    \begin{enumerate}
      \item Footsteps generation.
      \item ZMP trajectory generation.
      \item Corresponding Center of Mass trajectory generation.
      \item Feet trajectories generation.
    \end{enumerate}

    \bigskip

    \alert{What about the whole-body trajectory?}
  \end{center}
  \end{changeleftmargin}
\end{slideDecision}

\maxFrameImageWidth{src/chap3-primitive-mouvement/footsteps1.jpg}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\againframe<7>{contrib}
\begin{slideDecision}
  \frametitle{Motion Planning}
  \begin{changeleftmargin}{1.1cm}
    \note[item]{Deux familles d'algos: planification vs optimisation}
    \note[item]{Contribution porte sur l'optimisation.}
    \note[item]{Planification utilisée pour trouver %
      une estimation initiale de la solution.}

    \begin{center}
      \includegraphics%
          [width=.75\linewidth]%
          {src/chap1-roboptim/hrp2-two-chairs.png}

          \begin{enumerate}
          \item Trajectory planning.
          \item Trajectory optimization.
          \end{enumerate}
    \end{center}
  \end{changeleftmargin}
\end{slideDecision}

\begin{slideDecision}
  \begin{changeleftmargin}{1.1cm}
    \frametitle{Optimization}

    \note[item]{Intuition: trouver la meilleure action parmi un nombre
      infini d'actions possibles.}
    \note[item]{Large zoologie de problème existe. Détailler.}
    \note[item]{Ensembles E et I.}

    \begin{equation*}
      \min_{\mathbf{x} \in \mathbb{R}^n} f(\mathbf{x})%
      \text{ under the constraints } \mathbf{x} \in \mathbf{X}
    \end{equation*}

    \begin{equation*}
      \mathbf{X} \equiv \left\{
      \begin{array}{l l}
        c_i (x) = 0    & \quad i \in \mathcal{E} \\
        c_j (x) \leq 0 & \quad j \in \mathcal{I} \\
      \end{array} \right.
    \end{equation*}

  \end{changeleftmargin}
\end{slideDecision}

\begin{slideDecision}
  \begin{changeleftmargin}{1.1cm}
    \frametitle{Motivation}

    \note[item]{Beaucoup de solvers, peu de formalisation.}
    \note[item]{Très utile en robotique. Applications:
      décision, action et eperception.}
    \note[item]{Unifier la représentation.}
    \note[item]{Dissocier les solvers des problèmes.}

    \begin{center}
      \begin{description}
        \item[No formal model]~\\
          Plenty of solvers wich can't inter-operate.
        \item[Genericity]~\\
          Separate solvers from optimization problems.
        \item[Versatility]~\\
          Numerous robotics applications.
      \end{description}
    \end{center}
    \vspace{1cm}

    How to provide an \alert{unified modeling} of optimization
    problems?
  \end{changeleftmargin}
\end{slideDecision}


\begin{slideDecision}
  \begin{changeleftmargin}{1.1cm}
    \frametitle{Mathematical functions (1)}

    \note[item]{Semble trivial: difficile à cause des limites
      des langage.}
    \note[item]{Nécessité de définir une relation d'ordre sur les
      fonctions. Pas naturel.}
   \note[item]{Nécessité de déterminer ce qui rend les fonctions
     compatibles/interchangeables.}
   \note[item]{INTERFACES: nécessaire de dériver les classes
     pour implémenter ses propres fonctions.}

    \begin{center}
      \footnotesize
    \begin{tikzpicture}[auto]

      \begin{interface}[text width=7cm]{Function}{0,0}
        \operation{+ compute}
      \end{interface}

      \begin{interface}[text width=7cm]{Differentiable Function}{0,-3}
        \inherit{Function}
        \operation{+ jacobian}
      \end{interface}

      \begin{interface}[text width=7cm]{Twice Differentiable Fct}{0,-6}
        %\inherit{Function dérivable}
        \operation{+ hessian}
      \end{interface}

      \path[draw=\umldrawcolor, open triangle 45-, ultra thick]
      (Function.south) -- (Differentiable Function.north);
      \path[draw=\umldrawcolor, open triangle 45-, ultra thick]
      (Differentiable Function.south) -- (Twice Differentiable Fct.north);
    \end{tikzpicture}
    \end{center}
  \end{changeleftmargin}
\end{slideDecision}

\begin{slideDecision}
  \begin{changeleftmargin}{1.1cm}

    \note[item]{Création d'une famille de types.}

    \frametitle{Functions (2)}

    \begin{center}
      \footnotesize
    \begin{tikzpicture}[auto]

      \begin{interface}[text width=7cm]{Twice Differentiable Function}{0,0}
        \operation{+ hessian (vector, integer): vecteur}
      \end{interface}

      \begin{interface}[template parameter=n,text width=7cm]{%
          ``n'' Times Differentiable Function}{0,-3}
        %\inherit{Function dérivable deux fois}
        \operation{+ gradient}
      \end{interface}

      \path[draw=\umldrawcolor, open triangle 45-, ultra thick]
      (Twice Differentiable Function.south) node[below right] {$n=3$}
      -- (``n'' Times Differentiable Function.north);

      \path[draw=\umldrawcolor, open triangle 45-, ultra thick]
      (``n'' Times Differentiable Function.south)
      -- ++(0, -1)
      -- ++(-1, 0) node[below] {$n > 3$}
      -- ++(0, 1);

    \end{tikzpicture}
    \end{center}
  \end{changeleftmargin}
\end{slideDecision}

\begin{slideDecision}
  \begin{changeleftmargin}{1.1cm}
    \frametitle{Problem}

    \note[item]{Defined by one cost functions and constraints.}
    \note[item]{Several kind of constraints are possible.}
    \note[item]{Utile pour différencier les comportements selon
      les types.}
    \note[item]{Pas de contraintes également possible.}

    \begin{center}
      \footnotesize
    \begin{tikzpicture}[auto]

      \begin{interface}[template parameter={F,$C_1$,$\dotsc$,$C_n$},%
          text width=7cm]{Problem}{0,0}
        \attribute{+ initial guess: vector}
        \attribute{+ bounds: (double, double)[*]}
        \attribute{+ cost: \texttt{$F$}}
        \attribute{+ constraints: (\texttt{$C_i$}, integer, integer)[*]}
      \end{interface}
    \end{tikzpicture}

    \begin{equation*}
      \begin{split}
        F <: \text{Function}\\
        \forall i, C_i <: \text{Function}
      \end{split}
    \end{equation*}
    \end{center}
  \end{changeleftmargin}
\end{slideDecision}

\begin{slideDecision}
  \begin{changeleftmargin}{1.1cm}
    \frametitle{Solver}

    \note[item]{Prend une classe de problème en entrée.}

    \begin{center}
      \footnotesize
    \begin{tikzpicture}[%
        y=.1\paperheight,
        auto]

      \node[na] at (5,-1) (P) {P};

      \begin{interface}[%
          template parameter={Problem<$F$,$C_1$,$\dotsc$,$C_n$>},
          text width=7cm]{Solver}{0,0}
        %\inherit{P}
        \attribute{+ problem: $\text{Problème}<F, C_1, \dotsc, C_n>$}
        \operation{+ solve (): vector}
      \end{interface}

      \composition{Solver}{}{1}{P}
    \end{tikzpicture}

    \begin{equation*}
      (F, C_1, \dotsc, C_n) <: Function^{n+1}
    \end{equation*}

    \end{center}
  \end{changeleftmargin}
\end{slideDecision}

\begin{slideDecision}
  \begin{changeleftmargin}{1.1cm}
    \frametitle{Numerical Differentiation}

    \note[item]{Rend la fonction utilisable par un plus grand nombre
      de solver au prix d'un (possible) ralentissement de
      l'exécution. MECHANISME EXPLICITE}

    \note[item]{Gang of Four (Erich Gamma, Richard Helm, Ralph Johnson
      and John Vlissides).}

    \begin{center}
      \footnotesize
      \begin{tikzpicture}[%
          y=.1\paperheight,
          auto]

      \node[na] at (0,0) (F) {F};

        \begin{interface}[template parameter=F,%
            text width=7cm]{FiniteDifferences}{0,-2}
          %\inherit{F}
          \attribute{- function: F}
          \operation{+ jacobian (vector): vector}
        \end{interface}

      \path[draw=\umldrawcolor, open triangle 45-, ultra thick]
      (F.south) -- (FiniteDifferences.north);
      \end{tikzpicture}
    \end{center}
    \bigskip
    \alert{Decorator} Design Pattern.
  \end{changeleftmargin}
\end{slideDecision}


\begin{slideDecision}
  \begin{changeleftmargin}{1.1cm}
    \frametitle{Properties ensured through typing}

    \note[item]{Expliquer comment un problème plus simple peut être
      résolu en tant que problème plus complexe (et l'inverse!).}

    \begin{center}
      \begin{description}
        \item[Correction]~\\
          All problems than can be built are consistent.

        \item[Genericity]~\\
          A solver can work on problems which may be ``too easy'' for them.

        \item[Efficiency]~\\
          A solver may handle problems which are ``harder'' than
          planned by relying on automatic differentiation for
          instance.
      \end{description}
    \end{center}
  \end{changeleftmargin}
\end{slideDecision}

\begin{slideDecision}
  \frametitle{RobOptim}

  \note[item]{Détailler les solvers supportés.}
  \note[item]{Couche d'abstraction.}
  \note[item]{Couche métier (fonctions).}

  \begin{changeleftmargin}{1.1cm}
  \begin{center}
    \tikzstyle{roboptim} = [align=center,
      rectangle,
      minimum size=6mm,
      minimum height=15mm,
      color=Thoughtless,
      fill=ThoughtBySome,
      line width=1pt,
      inner sep=0pt,
      outer sep=0pt,
      text width=70pt
    ]

    \begin{tabular}{cccl}
      \tikz \node[roboptim] (cfsqp) {CFSQP}; &
      \tikz \node[roboptim] (ipopt) {IPOPT}; &
      \tikz \node[roboptim] (cminpack) {CMinPack}; &
      \parbox[l][1.5cm][l]{2cm}{%
        \vspace{-.6cm}\rotatebox{-90}{\footnotesize \textbf{Solver}}}\\
      \multicolumn{3}{c}{%
        \tikz \node[roboptim,
          text width=235pt] (core) {RobOptim Core};
      } &
      \parbox[l][1.5cm][l]{2cm}{%
        \vspace{-.6cm}\rotatebox{-90}{\footnotesize \textbf{Interface}}}\\
      \tikz \node[roboptim] (traj) {RobOptim Trajectory}; &
      \tikz \node[roboptim] (post) {RobOptim Posture}; &
      \tikz \node[roboptim] (others2) {\ldots}; &
      \parbox[l][1.5cm][l]{2cm}{%
        \vspace{-.6cm}\rotatebox{-90}{%
          \footnotesize \textbf{Application}}}\\
    \end{tabular}
  \end{center}
  \end{changeleftmargin}
\end{slideDecision}

\begin{slideDecision}
  \frametitle{Applications}

  \note[item]{Application initiale: optimisation de la trajectoire
    de marche d'un robot humanoïde.}

  \note[item]{Puis: planification de points de contact en
    multi-contact, planification dans un environnement non-rigide.}

  \begin{center}
    \includegraphics[height=.33\paperheight]{%
      src/chap1-roboptim/straight-line-obstacle.png}
    \includegraphics[height=.33\paperheight]{%
      src/slides/agent-067.jpg}\par
    \includegraphics[height=.33\paperheight]{%
      src/slides/multi-014.jpg}
    \includegraphics[height=.33\paperheight]{%
      src/slides/multi-015.jpg}

    Figures 2, 3, 4: K. Bouyarmane et collab.
  \end{center}
\end{slideDecision}

\fullFrameVideo{src/slides/multi-014.jpg}{vid/karim-multi-contact.mp4}{}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\againframe<8>{contrib}

\begin{slideAction}
  \frametitle{Controller}

  \note[item]{Deux types de tâches. Point6D et CoM.}
  \note[item]{Peut également ajouter butées, vitesse
    maximum mais pas d'inégalités en pratique.}
  \note[item]{Ici tâches ``de base'' mais on peut également
    ajouter des tâches de priorité moindre: tête.}
  \note[item]{Il manque le rebouclage capteur. Comment faire?}

  \begin{changeleftmargin}{1.1cm}
  \begin{center}

    \begin{tikzpicture}[%
        >=latex,
        node distance=1em,
        fixed/.style={
          text width=120pt}]
      \node [state,fixed] (com) {Center of Mass};
      \node [state,fixed, below=of com] (lf) {Left Foot};
      \node [state,fixed, below=of lf] (rf) {Right Foot};
      \node [state,fixed, below=of rf] (posture) {Posture};

      \node [left=of com] (fort) {High Priority};
      \node [left=of posture] (faible) {Low Priority};
      \draw[->, line width=1.05]
      (fort.south) -- (faible.north);
    \end{tikzpicture}
  \end{center}
  \end{changeleftmargin}
\end{slideAction}

\fullFrameVideo{src/slides/over.jpg}{vid/humanoids11-lbaudouin3.mp4}{%
    \tiny{%
      \vspace{-.6cm}%
      ~\includegraphics[height=.25cm]{src/slides/idea.pdf}~%
      Real-time Replanning Using 3D Environment for Humanoid Robot,
      Proc. of Humanoids'11.%
    }%
}

\begin{slideAction}
  \frametitle{Motivation}

  \note[item]{FIXME: expliquer le code couleur des feet}
  \note[item]{Pourquoi y a-t-il une dérive après la correction?}


  \begin{center}
    \begin{tikzpicture}[>=latex]
        \def\w{0.5}
        \def\h{0.75}

        \foreach \dy in {0., 1., 2., 3.}
                 {
                   % left step planned
                   \filldraw[pattern=north east lines] (0., \dy)
                   rectangle (0. + \w, \dy + \h);

                   % right step planned
                   \filldraw[pattern=north west lines] (1., \dy + 0.5)
                   rectangle (1. + \w, \dy + \h + 0.5);
                 }

        % before correction
        \def\driftx{0.1}
        \def\drifty{0.05}
        \def\drifttheta{5.}

        \def\dy{0}
        % left step planned
        \draw<2->[style=solid, color=red, rotate=\drifttheta * \dy,line width=1.25]
        (0. + \driftx * \dy, \drifty * \dy + \dy)
        rectangle (0. + \driftx * \dy + \w, \drifty * \dy + \dy + \h);

        % right step planned
        \draw<3->[style=solid, color=red, rotate=\drifttheta * \dy + \drifttheta / 2.,line width=1.25]
        (1. + \driftx * \dy + \drifty / 2., \drifty * \dy + \drifty / 2. + \dy + 0.5)
        rectangle (1. + \driftx * \dy + \drifty / 2. + \w, \drifty * \dy + \drifty / 2. + \dy + \h + 0.5);

        \def\dy{1}
        % left step planned
        \draw<4->[style=solid, color=red, rotate=\drifttheta * \dy,line width=1.25]
        (0. + \driftx * \dy, \drifty * \dy + \dy)
        rectangle (0. + \driftx * \dy + \w, \drifty * \dy + \dy + \h);

        % right step planned
        \draw<5->[style=solid, color=red, rotate=\drifttheta * \dy + \drifttheta / 2.,line width=1.25]
        (1. + \driftx * \dy + \drifty / 2., \drifty * \dy + \drifty / 2. + \dy + 0.5)
        rectangle (1. + \driftx * \dy + \drifty / 2. + \w, \drifty * \dy + \drifty / 2. + \dy + \h + 0.5);


        \def\dy{2}
        \draw<6->[style=solid, color=red,line width=1.25, rotate=\drifttheta] (0. + \driftx, \dy + \drifty)
        rectangle (0. + \w + \driftx, \dy + \h + \drifty);

        % --- after correction ---

        % left step planned
        \draw<7->[style=solid, color=green,line width=1.25] (0., \dy)
        rectangle (0. + \w, \dy + \h);

        \uncover<7->{
        \draw[smooth,rounded corners=1ex,<-,thick]
        (-0.3, \dy) -- (-0.3 - 0.05, \dy + 0.2) -- (-0.3 - 0.2, \dy + 0.4)
        node[midway,left]
        {
          $\delta \mathbf{x}$
        };
        }


        \draw<7->[style=solid, color=green,line width=1.25] (0., \dy)
        rectangle (0. + \w, \dy + \h);
        \draw<7->[style=solid, color=green,line width=1.25] (1., \dy + 0.5)
        rectangle (1. + \w, \dy + \h + 0.5);
        \def\dy{3}
        \draw<7->[style=solid, color=green,line width=1.25] (0., \dy)
        rectangle (0. + \w, \dy + \h);
        \draw<7->[style=solid, color=green,line width=1.25] (1., \dy + 0.5)
        rectangle (1. + \w, \dy + \h + 0.5);

        % right step planned
        \def\dy{2}
        \draw<8->[style=solid, color=orange,line width=1.25, rotate=\drifttheta * 0.5] (1. + \driftx * .5, \dy + 0.5 + \drifty * 0.5)
        rectangle (1. + \driftx * 0.5 + \w, \dy + \drifty * 0.5 + \h + 0.5);

        \def\dy{3}
        % left step planned
        \draw<9->[style=solid, color=orange,line width=1.25, rotate=\drifttheta * 1.] (0. + \driftx * 1., \dy + \drifty * 1.)
        rectangle (0. + \w + \driftx * 1., \dy + \h + \drifty * 1.);

        % right step planned
        \uncover<10-> {
          \draw[style=solid, color=orange,line width=1.25, rotate=\drifttheta * 1.5] (1. + \driftx * 1.5, \dy + 0.5 + \drifty * 1.5)
        rectangle (1. + \w + \driftx * 1.5, \dy + \h + 0.5 + \drifty * 1.5);
        }

      \end{tikzpicture}
      \end{center}
\end{slideAction}


\begin{slideAction}
  \frametitle{Real-Time Control}

  \note[item]{Actionement: nécessité d'envoyer une commande
    à une période fixe. Contraint le temps de calcul.}

  \begin{changeleftmargin}{1.1cm}
  \begin{center}
    Controller: algorithm computing actuators positions at a
    pre-determined frequency (200Hz for HRP-2).

    \bigskip

    \begin{itemize}
    \item Handle task execution as planned by the decisional components.
    \item Allow reactive motion adaptation.
    \end{itemize}

    \bigskip

    \alert{Critical} component for any robot.
  \end{center}
  \end{changeleftmargin}
\end{slideAction}


\begin{slideAction}
  \frametitle{Control Scheme}

  \note[item]{Ajout du rebouclage capteur en déformant
  les références des tâches.}
  \note[item]{Localisation externe.}
  \note[item]{Intérêt du fonctionnement par tâche.}

  \begin{changeleftmargin}{1.1cm}
  \begin{center}
    \begin{tikzpicture}[%
        >=latex]
      \node[state] (gen) {Génération};
      \node[state, below=of gen] (corr) {Correction};
      \node[state, below=of corr, align=left,inner sep=0.1ex,
        shape=rectangle split,
        rectangle split part align=left,
        rectangle split parts=5] (ctrl) {Contrôleur%
        \nodepart{two}\footnotesize{Center of Mass}
        \nodepart{three}\footnotesize{Left Foot}
        \nodepart{four}\footnotesize{Right Foot}
        \nodepart{five}\footnotesize{posture}};
      \node[state,right=2cm of corr] (loc) {Localization};

      %\draw[-,style=dashed] (2,-3) -- (2,3);

      \uncover<3-> {
        \draw[->] (loc.west) -- (corr.east)
        node [at start,left,font=\tiny,yshift=0.2cm]
        {$\hat{\mathbf{x}}(t)$};
      }

      \uncover<2-> {
        \draw[->] (gen.south) -- (corr.north)
        node [at start,right,font=\tiny,yshift=-0.2cm]
        {$(\gamma_{\text{feet}}(t), \gamma_{\text{com}}(t))$};
      }

      \uncover<3-> {
        \draw[->] (corr.south) -- (ctrl.north)
        node [at start,right,font=\tiny,yshift=-0.25cm]
        {$(\gamma'_{\text{feet}(t)}, \gamma'_{\text{com}}(t))$};
      }

      \uncover<4-> {
        \draw[->] (ctrl.south) --++ (0,-1)
        node [at start,right,font=\tiny,yshift=-0.25cm]
        {$\dot{\mathbf{q}}(t)$};
      }

      \draw[<-] (gen.north) --++ (0,1) node [left,font=\tiny,yshift=-0.25cm]
           {$\mathbf{S}$};
    \end{tikzpicture}%
    %
    \footnotesize{%
      \hspace{-5cm}%
      \includegraphics[height=.5cm]{src/slides/idea.pdf}~%
      Trajectory Following for Legged Robots, Proc. of Biorob'12.%
    }%
  \end{center}
  \end{changeleftmargin}
\end{slideAction}

\begin{slideAction}
  \frametitle{Temporal Synchronization}

  \begin{center}
    \vspace{-.5cm}
    \begin{tikzpicture}
      \begin{axis}[
          grid=major,
          legend style={
            at={(0.5,-0.2)},
            anchor=north,
            legend columns=3,
            cells={anchor=west},
            font=\footnotesize,
            rounded corners=2pt,
            draw=white
          },
          xlabel=temps ($s$),
          ylabel=position en $x$ ($m$),
          no markers,
          yticklabel pos=lower
          ]

        \addplot[
          Thoughtless, thick,
          dashed
        ] table [
          x expr=(\thisrowno{0}-0)*0.005,
          y expr=\thisrowno{1}+0.04
          ]{dat/feet_follower_feet-follower-com.dat};
        \addlegendentry{Center of Mass}

        \addplot[
          ThoughtYouWere,
          dashed,
        ] table[
          x index=0,
          y index=4,
          x expr=(\thisrowno{0}-0)*0.005,
          y expr=\thisrowno{4}+0.04
        ] {dat/feet_follower_feet-follower-left-ankle.dat};
        \addlegendentry{Left Foot}

        \addplot[
          Thoughtless2, thick,
          dashed,
        ] table[
          x index=0,
          y index=4,
          x expr=(\thisrowno{0}-0)*0.005,
          y expr=\thisrowno{4}+0.04
        ] {dat/feet_follower_feet-follower-right-ankle.dat};
        \addlegendentry{Right Foot}

        \addplot[
          Thoughtless, thick,
        ] table[
          x expr=(\thisrowno{0}-0)*0.005
        ]{dat/feet_follower_correction-com.dat};
        \addlegendentry{Center of Mass (corr.)}

        \addplot[
          ThoughtYouWere, thick,
        ] table[
          x index=0,
          y index=4,
          x expr=(\thisrowno{0}-0)*0.005,
        ] {dat/feet_follower_correction-left-ankle.dat};
        \addlegendentry{Left Foot (corr.)}

        \addplot[
          Thoughtless2, thick,
        ] table[
          x index=0,
          y index=4,
          x expr=(\thisrowno{0}-0)*0.005,
        ] {dat/feet_follower_correction-right-ankle.dat};
        \addlegendentry{Right Foot (corr.)}
      \end{axis}

      \def\w{0.5}
      \def\h{0.75}
      \def\dx{8.}

      \def\dy{.5}
      % left step planned 1
      \filldraw[pattern=north east lines,%
        draw=ThoughtYouWere] (\dx + 0., \dy + 0.5)
      rectangle (\dx + 0. + \w, \dy + \h + 0.5);

      % left step planned 1 - drift
      \uncover<2-> {
        \draw[style=solid,draw=ThoughtYouWere,%
          line width=1.25] (\dx, \dy - 0.3 + 0.5)
        rectangle (\dx + \w, \dy + \h - 0.3 + 0.5);
      }

      \filldraw[pattern=north east lines,%
        draw=Thoughtless2] (\dx + 1., \dy + 0.5)
      rectangle (\dx + 1. + \w, \dy + \h + 0.5);
      \draw<2->[style=solid,color=Thoughtless2,%
        line width=1.25] (\dx + 1., \dy - 0.3 + 0.5)
      rectangle (\dx + \w + 1., \dy + \h - 0.3 + 0.5);


      % right step planned 1
      \filldraw[pattern=north east lines,%
        draw=Thoughtless2] (\dx + 1., \dy + 3.)
      rectangle (\dx + 1. + \w, \dy + \h + 3.)
      node [pos=0.5,anchor=north,font=\tiny,yshift=0.75cm] {pas 1};

      % right step planned 1 - corrected
%      \uncover<2-> {
%        \draw[style=solid,draw=Thoughtless2,line width=1.25]
%        (\dx + 1., \dy + 3.-0.3)
%        rectangle (\dx + 1. + \w, \dy + \h + 3. -0.3);
%      }

      \uncover<3-> {
        \draw[style=solid,draw=Thoughtless2,%
          line width=1.25] (\dx + 1., \dy + 3.)
        rectangle (\dx + 1. + \w, \dy + \h + 3.);
      }

      \def\dy{3.}
      % left step planned 2
      \filldraw[pattern=north east lines,%
        draw=ThoughtYouWere] (\dx + 0., \dy + 0.5)
      rectangle (\dx + 0. + \w, \dy + \h + 0.5)
      node [pos=0.5,anchor=north,font=\tiny,yshift=0.75cm] {pas 2};

      % left step planned 2 - corrected
      \uncover<4-> {
        \draw[style=solid,draw=ThoughtYouWere,%
          line width=1.25] (\dx + 0., \dy + 0.5)
      rectangle (\dx + 0. + \w, \dy + \h + 0.5);
      }
    \end{tikzpicture}
  \end{center}
\end{slideAction}

\begin{slideAction}
  \frametitle{Center of Mass Trajectory}

  \note[item]{Ensure stability.}
  \note[item]{Linear operator.}
  \note[item]{Here delta $E^2$.}

  \begin{changeleftmargin}{1.1cm}
    \begin{itemize}
    \item Linear model.
    \item Suited for local correction.
    \item Real-time analytical solution.
    \end{itemize}
    
    \begin{equation*}
      \mathbf{z} = \mathbf{c} - \frac{z_c}{g} \ddot{\mathbf{c}}
    \end{equation*}
    
    \begin{equation*}
      \mathbf{c}^{(n)}(t) = \mathbf{c}(t)  + \sum_{i=1}^n \Delta^{t_i}(t)
    \end{equation*}
  \end{changeleftmargin}
\end{slideAction}

\fullFrameVideo{src/slides/demo.jpg}{vid/biorob12.mp4}{}

%%%%%%%%%%%%%%%%%%%%%%%%%%
\againframe<9>{contrib}

\fullFrameVideo{src/slides/door.jpg}{vid/hrp2-opening-door.mp4}{}

\begin{slideAction}
  \frametitle{Motion Primitives}

  \note[item]{Nécessaire d'adapter la pile de tâches
    à l'opération en cours. Dépend du scénario.}
  \note[item]{La localisation également diffère selon
    les cas.}
  \note[item]{La localisation globale parfaite d'un robot est
    difficile mais surtout souvent inutile. L'erreur de localisation
    doit décroître avec l'acomplissement de la tâche.}
  \note[item]{Il est nécessaire de planifier les tâches et
    l'asservissement de façon jointe.}
  \note[item]{Actuellement planification corps complet puis extraction
    des tâches. Choix de l'asservissement manuel.}


  \begin{changeleftmargin}{1.1cm}
  \begin{center}
    \begin{tikzpicture}[%
        node distance=1em,
        inner sep=0.1ex,
        outer sep=0.1ex,
        fixed/.style={
          text width=40pt}]

      \matrix[ampersand replacement=\&,
        row sep=.5cm, column sep=.5cm]{%
        \node {%
          \includegraphics[height=1.5cm]{src/slides/doorstate1.jpg}};\&
        \node {%
          \includegraphics[height=1.5cm]{src/slides/doorstate2.jpg}};\&
        \node {%
          \includegraphics[height=1.5cm]{src/slides/doorstate3.jpg}};\&
        \node {%
          \includegraphics[height=1.5cm]{src/slides/doorstate4.jpg}};\\
        \node[state,fixed] {\tiny{Balance}};\&
        \node[state,fixed] {\tiny{Balance}};\&
        \node[state,fixed] {\tiny{Balance}};\&
        \node[state,fixed] {\tiny{Balance}};\\
%
        \node [state,fixed] {\tiny{Posture}};\&
        \node [state,fixed] {\tiny{Right arm}};\&
        \node [state,fixed] {\tiny{Left arm}};\&
        \node [state,fixed] {\tiny{Left arm}};\\
%
        \&
        \node [state,fixed] {\tiny{Posture}};\&
        \node [state,fixed] {\tiny{Right arm}};\&
        \node [state,fixed] {\tiny{Posture}};\\
%
        \&
        \&
        \node [state,fixed] {\tiny{Posture}};\&
        \\

        \node[state,fixed,color=ThoughtBySome,fill=Thoughtless2]
             {\tiny{Door}};\&
        \node[state,fixed,color=ThoughtBySome,fill=Thoughtless2]
             {\tiny{Knob}};\&
        \node[state,fixed,color=ThoughtBySome,fill=Thoughtless2]
             {\tiny{Knob}};\&
        \node[state,fixed,color=ThoughtBySome,fill=Thoughtless2]
             {\tiny{Knob}};\\
      };
    \end{tikzpicture}
  \end{center}
  \end{changeleftmargin}
  \vbox{%
    \tikz \node[state,
      color=ThoughtBySome,fill=Thoughtless2] {\tiny{Amers}};
    \tikz \node[state] {\tiny{Tâche}};
  }
\end{slideAction}

%%%%%%%%%%%%%%%%%%%%%%%%%%
\againframe<10>{contrib}

\begin{slidePerception}
  \frametitle{Localization}

  \note[item]{Parler d'intégration logicielle.  Mentioner ROS
    rapidement.}

  \begin{changeleftmargin}{1.1cm}
  \begin{description}
  \item[Motion Capture]~\\
    Precision: $\pm 1$ cm. Managed environment required.
  \item[Vision-Based Object Tracking]~\\
    Precision: a few centimeters. Local method.
  \item[Visual Odometry]~\\
    Low precision. Global method.
  \end{description}
  \end{changeleftmargin}
\end{slidePerception}

\maxFrameImageHeight{src/chap4-integration/shelf.png}

%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{slidePerception}
  \frametitle{Software Contribution (1)}

  \begin{changeleftmargin}{1.1cm}
    ROS Stacks:
    \begin{itemize}
    \item ViSP Tracker. BSD.
    \item Motion Analysis Mocap. BSD.
    \item RCPDF. BSD.
    \item RobotModelPy. BSD.
    \end{itemize}
  \end{changeleftmargin}
\end{slidePerception}

\begin{slidePerception}
  \frametitle{Software Contribution (2)}

  \begin{changeleftmargin}{1.1cm}
    \begin{itemize}
      \item Full port of HRP-2 architecture to
        ROS: control, sensors (IMU, vision), planning.
      \item Low-level software and Humanoid Motion Planning
        packages (HPP) various enhancements.
      \item C++: \textasciitilde{} 100k modified lines\\
        Python: \textasciitilde{} 5k modified lines.
    \end{itemize}

    \bigskip

    \includegraphics[height=.5cm]{src/slides/idea.pdf}~%
    Coordinate Frames for Humanoid Robots, REP 120.

  \end{changeleftmargin}
\end{slidePerception}


\maxFrameImageHeight{src/chap3-primitive-mouvement/calibextrinsic.jpg}

\begin{frame}
  \frametitle{Conclusion}

  \note[item]{Versatility.}
  \note[item]{Integration.}
  \note[item]{Scénarii complexes pour étendre les capacités des robots
    humanoïdes.}
  \note[item]{Faire le lien entre la planification et l'exécution.}
  \note[item]{Démontre le fossé entre robotique et simulation
    robotique.}

  \begin{itemize}
  \item Complete architecture: perception, decision, action.
  \item Footstep placement correction for robut trajectory execution.
  \end{itemize}

  \bigskip

  \begin{itemize}
  \item Validate arms position.
  \item Online footstep placement validation.
  \item Enhance robot localization. Switch between different
    localization strategies.
  \end{itemize}
\end{frame}

\maxFrameImage{src/slides/hrp2-thx.jpg}

\begin{frame}[plain]
  \begin{changemargin}{-1cm}{-1cm}
    \begin{center}
      \vspace{-0.1cm}
      \includegraphics[width=.95\paperwidth,height=\paperheight,%
        keepaspectratio]{src/chap4-integration/archi.pdf}
    \end{center}
  \end{changemargin}
\end{frame}
%\maxFrameImage{src/chap4-integration/archi.pdf}


%
%\begin{slidePerception}
%  \frametitle{De la bibliothèque au composant}
%
%  FIXME
%\end{slidePerception}

%\maxFrameImageHeight{src/slides/demo2.jpg}

%\maxFrameImageHeight{src/chap4-integration/footsteps2.jpg}
%\maxFrameImageHeight{src/chap4-integration/disparity-1.jpg}
%\maxFrameImageHeight{src/chap4-integration/disparity-2.jpg}
%\maxFrameImageHeight{src/chap4-integration/hrp2_urdf.jpg}
%\maxFrameImageHeight{src/chap4-integration/rviz-full.jpg}



\end{document}