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\title{11.  Kinematic models of contact}
\subtitle{ Mechanics of Manipulation}
\author{Matt Mason}
\institution{Carnegie Mellon}
\email{matt.mason@cs.cmu.edu \\
http://www.cs.cmu.edu/\symbol{"7E}mason}
\slideCaption{Lecture 11. \hfill Mechanics of Manipulation }

\begin{document}

\maketitle

%=======================================================================
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% SLIDE 2
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\begin{slide}[R]{Lecture 11.  Kinematic models of contact.}
\vspace{-0.5in}
\includegraphics[height=3.0in]{TOC11}
\end{slide}

%=======================================================================
%
% SLIDE 3
%
%

\begin{slide}[R]{Outline.}

\begin{trivlist}
\item Gr\"ubler
\begin{itemize}
\item Review of kinematic mechanisms
\item Mobility and connectivity
\item Gr\"ubler's formula
\end{itemize}
\item Salisbury
\begin{itemize}
\item Taxonomy of contacts
\item Mobility and connectivity of grasp
\end{itemize}
\end{trivlist}

\end{slide}


%=======================================================================
%
% SLIDE 8
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%
\begin{slide}[R]{Kinematic mechanisms}
\begin{minipage}[t]{4cm}
\begin{trivlist}
\item[]\defterm{Link}:  a rigid body;
\item[]\defterm{Joint}:  imposes one or more constraints on the relative motion
of two links;
\item[]\defterm{Kinematic mechanism}:  a bunch of links joined by joints;
\item[]\defterm{lower pairs} joints involving positive contact area.
\end{trivlist}
\end{minipage}
\hfill
\begin{minipage}[t]{6cm}
\begin{figure}[h!]
\includegraphics[width=6cm]{lowerpairs}
%\includegraphics[height=3cm]{*file*}
\end{figure}\end{minipage}
\end{slide}

%=======================================================================
%
% SLIDE 9
%
%
\overlays{5}{
\begin{slide}[R]{Mobility and connectivity}
\begin{minipage}[t]{6cm}
\begin{trivlist}
\item[]\defterm{mobility} of a 
mechanism:  DOFs with one link fixed.
\item[]\defterm{connectivity} DOFs of one link relative to another.
\item[]What is the mobility of the five bar linkage at right?  \fromSlide{2}{Two.}
\item[]What is the connectivity of
\begin{itemize}\item[]Link 1 relative to link two?  \fromSlide{3}{One.}
\item[]Link 3 relative to link 1?  \fromSlide{4}{Two.}
\item[]Link 3 relative to link 4?  \fromSlide{5}{One.}
\end{itemize}
\end{trivlist}
\end{minipage}
\hfill
\begin{minipage}[t]{4cm}
\begin{figure}[h!]
\includegraphics[width=4cm]{mobconn}
%\includegraphics[height=3cm]{*file*}
\end{figure}\end{minipage}
\end{slide}}

%=======================================================================
%
% SLIDE 10
%
%
\begin{slide}[R]{Gr\"ubler's formula}
\begin{trivlist}
\item[]Given $n$ links joined by $g$ joints,
\item[]with $u_{i}$ constraints and $f_{i}$ freedoms at joint $i$.
(Note that $u_{i}+f_{i} = 6$.)
\item[] Assume one link is fixed and constraints are all independent.
\item[] The mobility $M$ is
\begin{align*}
        M & = 6(n-1) - \sum u_{i} \\
          & = 6(n-1) - \sum (6 - f_{i}) \\
          & = 6(n - g - 1) + \sum f_{i}
\end{align*}

\item[]Or, for a planar mechanism:
\begin{align*}
        M & = 3(n-1) - \sum u_{i} \\
          & = 3(n - g - 1) + \sum f_{i}
\end{align*}
\end{trivlist}

\end{slide}

%=======================================================================
%
% SLIDE 11
%
%
\begin{slide}[R]{Gr\"ubler:  special case for loops}
\begin{trivlist}
\item[]The previous formula works (sort of) for all mechanisms.
\item[]For loops there is a variant.
\item[] One loop:  $n$ = $g$, so
\[
        M = \sum f_{i} + 6 (-1)
        \]
\item[]Two loops:  make a second loop by adding 
$k$ links and $k+1$ joints:
\[
        M = \sum f_{i} + 6 (-2)
        \]
\item[]Every loop increases excess of joints over links by 1.  For
$l$ loops:
\[
        M = \sum f_{i} - 6 l
        \]
for a spatial linkage, and
\[
        M = \sum f_{i} - 3 l
        \]
for a planar linkage. 
\end{trivlist}

\end{slide}


%=======================================================================
%
% SLIDE 12
%
%
\overlays{7}{
\begin{slide}[R]{Common sense}
\begin{minipage}[t]{6cm}
\begin{trivlist}
\item[]Example:  what is the mobility of Watt's linkage?
\item[]Planar Gr\"ubler's formula:
\begin{align*}
        M & = 3(n-1) - \sum u_{i}  = \fromSlide{2}{1}\\
        M  & = 3(n - g - 1) + \sum f_{i} = \fromSlide{3}{1} \\
        M & = \sum f_{i} - 3 l = \fromSlide{4}{1}
\end{align*}
\item[]Spatial Gr\"ubler's formula:
\begin{align*}
        M & = 6(n-1) - \sum u_{i} = \fromSlide{5}{-2} \\
        M & = 6(n - g - 1) + \sum f_{i} = \fromSlide{6}{-2} \\
        M & = \sum f_{i} - 6 l = \fromSlide{7}{-2}
\end{align*}

\item[]Why?
\end{trivlist}
\end{minipage}
\hfill
\begin{minipage}[t]{4cm}
\begin{figure}[h!]
\includegraphics[width=4cm]{watt}
%\includegraphics[height=3cm]{*file*}
\end{figure}
\begin{trivlist}
\item[]Independent constraints is a very strong assumption.
\end{trivlist}
\end{minipage}
\end{slide}}

%=======================================================================
%
% SLIDE 13
%
%
\begin{slide}[R]{Kinematic models of contact}
\begin{minipage}[t]{4cm}
\begin{trivlist}
\item[]A grasp is like a kinematic mechanism.
\item[]Assume fingers do not lift or slip.
\item[]Model each contact as a joint
\item[]Apply Gr\"ubler's formula!
\end{trivlist}
\end{minipage}
\hfill
\begin{minipage}[t]{6cm}
\begin{figure}[h!]
\includegraphics[width=6cm]{minhand}
%\includegraphics[height=3cm]{*file*}
\end{figure}\end{minipage}
\end{slide}

%=======================================================================
%
% SLIDE 14
%
%
\begin{slide}[R]{Taxonomy of contact types}
\begin{minipage}[t]{4cm}
\begin{trivlist}
\item[]In previous slide, contact was modeled as spherical joint.  Are there other possibilities?
\item[]Salisbury's PhD thesis, 1982, included a taxonomy.
\end{trivlist}
\end{minipage}
\hfill
\begin{minipage}[t]{6cm}
\begin{figure}[h!]
\includegraphics[width=6cm]{contacts}
%\includegraphics[height=3cm]{*file*}
\end{figure}\end{minipage}
\end{slide}

%=======================================================================
%
% SLIDE 15
%
%
\begin{slide}[R]{Mobility and connectivity of grasp}
\begin{trivlist}
\item[]Salisbury suggests four measures:
\begin{itemize}
        \item[$M$]
        Mobility of the entire system with the finger joints free.
        
        \item[$M^{\prime}$]     
        Mobility of the entire system, with the finger 
        joints locked.
        
        \item[$C$]      
        Connectivity of the object relative to a fixed palm, with 
        the finger joints free.
        
        \item[$C^{\prime}$]
        Connectivity of the object relative to a fixed 
        palm, with the finger joints locked.
        
\end{itemize}
\item[]If $C = 6$ then object can make general motions.
\item[]If $C^{\prime} \leq 0$ then hand can immobilize object.
\end{trivlist}

\end{slide}

%=======================================================================
%
% SLIDE 16
%
%
\begin{slide}[R]{Example:  the Salisbury hand}
\begin{minipage}[t]{4cm}
\begin{trivlist}
\item[]What is $C$?
\item[]What is $C^\prime$?
\end{trivlist}
\end{minipage}
\hfill
\begin{minipage}[t]{6cm}
\begin{figure}[h!]
\includegraphics[width=6cm]{minhand}
%\includegraphics[height=3cm]{*file*}
\end{figure}\end{minipage}
\end{slide}


%=======================================================================
%
% SLIDE last
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\begin{slide}[R]{Next:  Foundations of statics.}
\vspace{-0.5in}
\includegraphics[height=3.0in]{TOC12}
\end{slide}
\end{document}



\end{document}
