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CompliantPassiveWalking.tex

CompliantPassiveWalking.tex — TeX document, 9Kb

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\documentclass[hyperref={pdfpagelabels=false}]{beamer}
% This file is a solution template for:

% - Talk at a conference/colloquium.
% - Talk length is about 20min.
% - Style is ornate.



%%%% PREAMBLE
\mode<presentation>
{
    \usetheme{Warsaw}
    \useinnertheme{rectangles}
    \setbeamercovered{transparent}
    % or whatever (possibly just delete it)
}
\usepackage[english]{babel}
\usepackage[latin1]{inputenc}
\usepackage{times}
\usepackage[T1]{fontenc}
\usepackage{multimedia}
\usepackage{amsmath}
\usepackage{setspace}
%\usepackage{graphics}
%  Note: the encoding and the font should match. If T1 % does not look nice, try deleting the line with the fontenc.


%%%% TITLE PAGE CONTENTS
\title{Effects of Compliant Point-Feet on a Passive Dynamic Walker}
%\title[Short Paper Title] % (optional, use only with long paper titles)
%\subtitle
%{Include Only If Paper Has a Subtitle}

\author{Thomas R. Johnston}
% Multi-Author example:
%\author[Author, Another] % (optional, use only with lots of authors)
%{F.~Author\inst{1} \and S.~Another\inst{2}}
% - Give the names in the same order as the appear in the paper.
% - Use the \inst{?} command only if the authors have different
%   affiliation.
\institute[University of California, Davis] 
{
    Department of Biomedical Engineering\\
    Sports Biomechanics Lab
}
\date{April 28, 2011}
%\date[CFP 2003] % (optional, should be abbreviation of conference name)
%{Conference on Fabulous Presentations, 2003}
% - Either use conference name or its abbreviation.

%\subject{Multibody Dynamics}

% If you have a file called "university-logo-filename.xxx", where xxx
% is a graphic format that can be processed by latex or pdflatex,
% resp., then you can add a logo as follows:
%\pgfdeclareimage[height=0.5cm,width=0.5cm]{logo}{seal_blue-gold.jpg}
%\logo{\pgfuseimage{logo}}



%% Delete this, if you do not want the table of contents to pop up at
%% the beginning of each subsection:
%\AtBeginSubsection[]
%{
%  \begin{frame}<beamer>{Outline}
%    \tableofcontents[currentsection,currentsubsection]
%  \end{frame}
%}


% If you wish to uncover everything in a step-wise fashion, uncomment
% the following command: 
%\beamerdefaultoverlayspecification{<+->}


\begin{document}

%%%% TITLE PAGE FRAME
\begin{frame}
    \titlepage
\end{frame}

%%%% TABLE OF CONTENTS
\begin{frame}{Outline}
    \tableofcontents
    % You might wish to add the option [pausesections]
\end{frame}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\section{Passive Dynamic Walking From 30,000 Feet}
\subsection{Definition \& applications}
%%%% First Frame of Subsection ``Definition & Applications''
%\begin{frame}{Make Titles Informative. Use Uppercase Letters.}{Subtitles are optional.}
  % - A title should summarize the slide in an understandable fashion
  %   for anyone how does not follow everything on the slide itself.
\begin{frame}{Definition \& Applicaitons}
    \begin{center}
	\movie[loop, width=7cm,height=5cm,label=walker_movie,showcontrols=true,poster]{}{Passive_Dynamic_Walking_Nagoya_Inst_Tech_June_2005.mp4}
    \end{center}
\end{frame}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Important results}
    \begin{frame}{Important Results from the Literature}
	\begin{itemize}
	    \item preferred choice of step length (Donelan et al 2001), width, and frequency (Kuo 2001)
	    \pause 
		\begin{itemize}
		    \item optimal actuation strategies
		\end{itemize}
	    \pause 
	    \item effects of model features and parameter variations
	\end{itemize}
    \end{frame}



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Basic model types}
	\begin{frame}{Inverted Pendulum v. Spring-Mass Model}
	    \begin{center}
		\begin{figure}
		    \includegraphics[height=2.4cm]{Inverted_Pendulum_Model.jpeg} \\
%		\pause
		    \includegraphics[height=3.1cm]{Spring_Mass_Model.jpeg}
		\end{figure} 
	    \end{center}
	\end{frame}



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Model limitations and current objectives}
    \begin{frame}{Model Limitations and Current Objectives}
	\begin{itemize}
	    \item Inverted Pendulum Model
		\begin{itemize}
		    \item double support impossible
		    \item pattern of GRFs inaccurate
		\end{itemize}
	    \item Spring Mass Model
	    \begin{itemize}
		\item most are completely conservative
		\item choice of visco-elastic parameters seems arbitrary
	    \end{itemize}
	\end{itemize}
	\pause
	\begin{itemize}
	    \item Compare: stability, energy cost per step, and speed 
 	\end{itemize}
   \end{frame}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\section{The Models Used Here}
\subsection{Kinematic description and assumptions}
    \begin{frame}{Rigid Model (RM)}
	\begin{center}
	    \includegraphics[height=6cm]{MassiveFeetWalker_LowRes.png}
	\end{center}
    \end{frame}


\begin{frame}{Compliant Foot Model (CF)}
    \begin{center}
	\includegraphics[height=6cm]{CompliantFeetWalker_LowRes.png}
    \end{center}
\end{frame}


\subsection{Compliant element details}
\begin{frame}{Compliant Element Details - Part 1}
    \begin{center}
	\includegraphics[width=8cm]{ImpactTestRig.png} \\ 
	\footnotesize Klute et al. 2004: Heel-region properties of prosthetic feet and shoes
    \end{center}
\end{frame}

\begin{frame}{Compliant Element Details - Part 2}
    \begin{columns}
	\column{0.6\textwidth}
	    \includegraphics[width=6cm]{ImpactTestRig.png}
    	\column{0.4\textwidth}
	    \begin{minipage}[c][.5\textheight][c]{\linewidth}
		\begin{itemize}
			\item[]<1> $a = 2.35E+6$ \\
			\item[]<1> $b = 1.72$ \\
			\item[]<1> $c = 2.00E+4$ \\
			\item[]<1> $d = 0.91$ \\
			\item[]<1> $e = 1.00$
		\end{itemize}
	    \end{minipage}
    \end{columns}
    \begin{center}
    	$F = a \cdot x^b + c \cdot sign(\dot x) \cdot x^d \cdot |\dot x|^e$
    \end{center}
\end{frame}



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Methods}
\begin{frame}{Methods - Overview}
    \begin{itemize}
	\item Finding fixed points
	\begin{itemize}
	    \item Poincar\'{e} map
	    \item Netwon-Raphson algorithm
	\end{itemize}
	\item Stability test
	    \begin{itemize}
		\item Floquet multipliers (eigenvalues)
		\item Gait Sensitivity Norm (GSN) (Hobbelen and Wisse 2007)
	    \end{itemize}
	\item Energy cost per step
	    \begin{itemize}
		\item[] $$ {\varDelta}E = {\Delta}PE = \sum\limits_{j=1}^2 \left( \: \int\limits_{t=0}^{t=t_f} \mathbf{F}_j \cdot \dot{\mathbf{x}}_j  \:dt \right)  $$
	    \end{itemize}
    \end{itemize}
\end{frame}


\begin{frame}{Stride Map - Compliant Model}
    $ \mathbf{v}_{n+1} = S \left( \mathbf{v}_n \right),  \; $   where $\mathbf{v}_n = \left[ \theta, \; \phi, \; \dot{\theta}, \; \dot{\phi} \right]^T $ \\
    \begin{enumerate}
    	\item $ \int EOM_{SS} $ \\
	\item detect foot strike \\
	\item calculate post-impact states  \\
	\item $ \int EOM_{DS} $ 
    \end{enumerate}
\end{frame}

\begin{frame}{Gait Sensitivity Norm - 1}
    \begin{spacing}{.05}
	\begin{center}
	    \includegraphics[height=4cm]{GSN_Block_Diagram.png}
	\end{center}
	\begin{align*}
	    \Delta{\mathbf{v}}_{n+1} &= A \Delta{\mathbf{v}} + B \mathbf{e}_n \\ 
	    \Delta{\mathbf{g}}_{n} &= C \Delta{\mathbf{v}} + D \mathbf{e}_n
	\end{align*}
    \end{spacing}
\end{frame}

\begin{frame}{Gait Sensitivity Norm - 2}
    $$ {\Bigg\Arrowvert \frac{\partial \mathbf{g}}{\partial \mathbf{e}} \Bigg\Arrowvert}_2 = \sqrt{trace \left( D^T D \right) + \sum_{k=0}^\infty  trace \left( B^T \left( A^T \right)^k C^T C \, A^k B \right) }  $$ 
\end{frame}









%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Results}
\begin{frame}{Switching to Matlab}
	Matlab plots go here
\end{frame}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Future Work}
\begin{frame}{Future Model Enhancements}
	\begin{itemize}
	    \item anthropomorphic mass/inertia properties
	    \item wobbling torso mass
	    \item non-point feet with a distributed 
	    \item optimize spring-damper parameters?
	\end{itemize}
\end{frame}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{The End}
    \begin{center}
	\movie[loop, width=7cm,height=5cm,label=walker_movie2,showcontrols=true,poster]{}{improved_01kneed_human.mp4}
    \end{center}
\end{frame}


%\begin{frame}{Definition \& Applicaitons}
%    \begin{center}
%	\movie[loop, width=7cm,height=5cm,label=walker_movie,showcontrols=true,poster]{}{Passive_Dynamic_Walking_Nagoya_Inst_Tech_June_2005.mp4}
%    \end{center}
%\end{frame}



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% All of the following is optional and typically not needed. 
\appendix
\section<presentation>*{\appendixname}
\subsection<presentation>*{For Further Reading}

\begin{frame}[allowframebreaks]
  \frametitle<presentation>{For Further Reading}
    
  \begin{thebibliography}{10}
    
  \beamertemplatebookbibitems
  % Start with overview books.

  \bibitem{Author1990}
    A.~Author.
    \newblock {\em Handbook of Everything}.
    \newblock Some Press, 1990.
 
    
  \beamertemplatearticlebibitems
  % Followed by interesting articles. Keep the list short. 

  \bibitem{Someone2000}
    S.~Someone.
    \newblock On this and that.
    \newblock {\em Journal of This and That}, 2(1):50--100,
    2000.
  \end{thebibliography}
\end{frame}

\end{document}


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