cartpole.texp

% simple file for working on the shared slides (by simply including one or two slides at a time).

\documentclass[serif]{beamer}
\usepackage{movie15,multicol,relsize,xstring}
\usepackage{russt}

\renewcommand{\bInlineMovies}{true}

\title{RobotLib Examples using the Cart-Pole System}
\author{The Robot Locomotion Group}

\begin{document}

\begin{frame}
\titlepage
\end{frame}

\begin{frame}
\frametitle{Dynamics (CartPolePlant)}
Equations of motion:
\begin{align*}
\ddot{x} =& \frac{1}{m_c + m_p \sin^2\theta}\left[ f+m_p \sin\theta (l \dot\theta^2 + g\cos\theta)\right] \label{eq:ddot_x}\\
\ddot{\theta} =& \frac{1}{l(m_c + m_p \sin^2\theta)} \left[ -f
 \cos\theta - m_p l \dot\theta^2 \cos\theta \sin\theta - (m_c + m_p) g \sin\theta \right] \label{eq:ddot_theta}
\end{align*}

\begin{center}
\begin{matlabFigure}[width=.5\textwidth]
cd ..;
v=CartPoleVisualizer; 
th = .6; 
x0=[0;th;0;0];
v.draw(0,x0); 
axis off
delete(title(''));
line([0;0],[0;-.5],'LineWidth',.5,'Color','k');
s=linspace(0,th,10);
h=plot(.4*sin(s),-.4*cos(s));  set(h,'LineWidth',.5,'Color','k');
text(.085,-.5,'$\theta$','Interpreter','latex','FontSize',18);
addpath([docroot,'/techdoc/creating_plots/examples/']);
[x,y]=dsxy2figxy(-.7*[1 1], .25*[1 -1]);
h=annotation('arrow',x,y);
text(-.85,0,'$g$','Interpreter','latex','FontSize',18);
text(.6*sin(th)+.05,-.6*cos(th)+.05,'$m_p$','Interpreter','latex','FontSize',18);
text(.4*sin(th),-.4*cos(th)+.1,'$l$','Interpreter','latex','FontSize',18);
line(-.55*[1,1],.1*[-1,1],'LineWidth',.5,'Color','k');
[x,y]=dsxy2figxy([-.55,0],[0,0]);
h=[h,annotation('arrow',x,y)];
text(-.45,.1,'$x$','Interpreter','latex','FontSize',18);
text(.2,.2,'$m_c$','Interpreter','latex','FontSize',18);
\end{matlabFigure}
\end{center}

\end{frame}

\begin{frame}
\frametitle{ROA for LQR Balancing Controller}

$$Q = \text{diag}([1,50,1,50]), \quad R = .1$$

\begin{columns}
\column{.5\textwidth}
\begin{center}

\only<1>{
\begin{matlabFigure}[width=.95\textwidth]
cd ..
p = CartPolePlant;
p = p.setInputLimits(-inf,inf);

Q = diag([1 50 1 50]);  R = .1;
xG = [0;pi;0;0]; uG = 0;
[c,V0] = tilqr(p,xG,uG,Q,R);

sys = feedback(p,c);
pp = sys.taylorApprox(0,xG,[],3); 

options.degL1 = 3;
options.method='levelSet';
V = regionOfAttraction(pp,xG,V0,options);

plotFunnel(V,xG,[2 4]); %,struct('inclusion','projection'));
xlabel('$\theta$','Interpreter','latex','FontSize',18);
ylabel('$\dot\theta$','Interpreter','latex','FontSize',18);

px=msspoly('x',4);
rho=double(subs(V0,px,ones(4,1)))/double(subs(V,px,ones(4,1)));

title(['$\rho = ',num2str(rho),'$'],'Interpreter','latex','FontSize',24);
\end{matlabFigure}

Taylor approx via SOS
}

\only<2>{
\begin{matlabFigureCont}[width=.95\textwidth]
options.method='bilinear';
options.degV=2;

V2 = regionOfAttraction(pp,xG,V,options);

clf;
h2=plotFunnel(V2,xG,[2 4],struct('color',[.9 .3 .2]));  
hold on;
h=plotFunnel(V,xG,[2 4]);

xlabel('$\theta$','Interpreter','latex','FontSize',18);
ylabel('$\dot\theta$','Interpreter','latex','FontSize',18);

legend([h;h2],'Original','Optimized','FontSize',18);
\end{matlabFigureCont}

Taylor approx via SOS w/ searching over quadratic Lyapunov functions.
}

\end{center}

\column{.5\textwidth}


\begin{itemize}
\item  According to a sample-base check, this
appears to be tight (sampled ROA is 2\% larger on average).  

\item Note that these controllers use a \emph{lot} of control effort.
  Revisit when verification w/ saturations is implemented.

\item<2> Optimizing quadratic $V$ doesn't help much.  

\end{itemize}

\end{columns}

\end{frame}

\end{document}