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\title{15.  Moment Labeling}
\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 15. \hfill Mechanics of Manipulation }

\begin{document}

\maketitle


%=======================================================================
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% SLIDE 2
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\begin{slide}[R]{Lecture 15.  Moment Labeling.}
\vspace{-0.5in}
\includegraphics[height=3.0in]{TOC15}
\end{slide}

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

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

\begin{trivlist}
\item Review:  where are we?  Where are we going?
\item Moment labeling:  practical introduction.
\item Moment labeling:  a more formal approach.
\item Relation to PCC's in wrench space.
\item Examples.
\end{trivlist}

\end{slide}


%=======================================================================
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% SLIDE 4
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\begin{slide}[R]{Where are we?  Where are we going?}
\begin{tabular}{@{}lll@{}}

\hline
  & \textbf{Project cone to}
  & \textbf{Project supplementary}\\
  & \textbf{oriented plane}
  & \textbf{cone to oriented plane}
\\ \hline
  single wrench & (acc'n center) & line of action \\
  single diff'l twist & IC & ? \\
  wrench cone & force dual & moment labeling \\ 
  diff'l twist cone & Reuleaux & ?
\\ \hline
\end{tabular}
\end{slide}

%=======================================================================
%
% SLIDE 5
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%
\begin{slide}[R]{Possible resultants for one contact.}
\begin{minipage}[t]{6cm}
\begin{trivlist}
\item[]For a frictionless contact, force is along contact normal,
  magnitude is indeterminate.
\item[]I.e., the (directed) line of force is determined.
\item[]Another definition of line of force: locus of zero moment
  points.
\end{trivlist}
\end{minipage}
\hfill
\begin{minipage}[t]{4cm}
\begin{figure}[h!]
\includegraphics[width=4cm]{unilatLOF}
%\includegraphics[height=3cm]{*file*}
\end{figure}\end{minipage}
\end{slide}

%=======================================================================
%
% SLIDE 6
%
%
\begin{slide}[R]{Another way of drawing the line of force.}
\begin{minipage}[t]{6cm}
\begin{trivlist}
\item[]If we indicate sign of moment at each point $\ldots$
\item[]$\ldots$ half plane of $\oplus$ to the left;
\item[]$\ldots$ half plane of $\ominus$ to the right;
\item[]zero moment points are the boundary.
\item[]Gives line of force \emph{and} its direction.
\end{trivlist}
\end{minipage}
\hfill
\begin{minipage}[t]{4cm}
\begin{figure}[h!]
\includegraphics[width=4cm]{unilat}
%\includegraphics[height=3cm]{*file*}
\end{figure}\end{minipage}
\end{slide}

%=======================================================================
%
% SLIDE 7
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%
\begin{slide}[R]{Reuleaux's method has \emph{two} interpretations!}
\begin{minipage}[t]{6cm}
\begin{trivlist}
\item[]The cone of differential twists reciprocal or repelling to a
  given wrench.
\item[]The cone of wrenches in the positive linear span of a given
  wrench.
\item[]\emph{Reciprocal or repelling} turns any wrench-PCC rep'n into
  a twist-PCC rep'n.
\end{trivlist}
\end{minipage}
\hfill
\begin{minipage}[t]{4cm}
\begin{figure}[h!]
\includegraphics[width=4cm]{unilat}
%\includegraphics[height=3cm]{*file*}
\end{figure}\end{minipage}
\end{slide}

%=======================================================================
%
% SLIDE 8
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%
\begin{slide}[R]{Superiority of wrench interpretation.}
\begin{minipage}[t]{6cm}
\begin{trivlist}
\item[]Actually the wrench interpretation is more useful.
\begin{itemize}
\item[]Differential twist cone is only first order approximation.
\item[]Wrench interpretation extends to include friction.
\item[]Wrench interpretation extends beyond convex cones.
\end{itemize}
\end{trivlist}
\end{minipage}
\hfill
\begin{minipage}[t]{4cm}
\begin{figure}[h!]
\includegraphics[width=4cm]{elps}
%\includegraphics[height=3cm]{*file*}
\end{figure}\end{minipage}
\end{slide}

%=======================================================================
%
% SLIDE 9
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%
\begin{slide}[R]{Possible resultants of \emph{two} contacts.}
\begin{minipage}[t]{6cm}
\begin{trivlist}
\item[]Remember how to construct resultant of two forces in the plane?
\item[]We can represent this set of resultants by a cone drawn in the plane.
\end{trivlist}
\end{minipage}
\hfill
\begin{minipage}[t]{4cm}
\begin{figure}[h!]
\includegraphics[width=4cm]{trianglecone}
%\includegraphics[height=3cm]{*file*}
\end{figure}\end{minipage}
\end{slide}

%=======================================================================
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% SLIDE 10
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%
\overlays{3}{
\begin{slide}[R]{Some more challenging problems}
\begin{minipage}[t]{6cm}
\begin{trivlist}
\item[]How do we deal with these cases:
\begin{itemize}
\item Possible resultants of two \emph{parallel} contacts?
\item Possible resultants of three contacts?
\end{itemize}
\fromSlide{2}{\item[]Let's try Reuleaux's method and reinterpret.}
\end{trivlist}
\end{minipage}
\hfill
\begin{minipage}[t]{4cm}
\begin{figure}[h!]
\includegraphics[width=4cm]{twoharder}
%\includegraphics[height=3cm]{*file*}
\end{figure}
\fromSlide{3}{
\begin{figure}[h!]
\includegraphics[width=4cm]{twosolns}
\end{figure}}
\end{minipage}
\end{slide}}


%=======================================================================
%
% SLIDE 11
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%
\begin{slide}[R]{Interpreting the labeled regions.}
\begin{minipage}[t]{6cm}
\begin{trivlist}
\item[]The set of all forces that go between $\oplus$ and $\ominus$,
  in the right direction.
\item[]They can graze, but cannot go through the interior.
\end{trivlist}
\end{minipage}
\hfill
\begin{minipage}[t]{4cm}
\begin{figure}[h!]
\includegraphics[width=4cm]{labelinterp}
%\includegraphics[height=3cm]{*file*}
\end{figure}\end{minipage}
\end{slide}

%=======================================================================
%
% SLIDE 12
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%
\begin{slide}[R]{More formally}
\begin{trivlist}
\item[]Given a set of contact wrenches $w_i$,
\item[]Let $\oplus_i$ and $\ominus_i$ be the points of nonnegative and
  nonpositive moments, respectively, for contact $w_i$,
\item[]Let $\oplus$ be the intersection of all nonnegative regions
\[ \oplus = \cap \oplus_i  \]
\item[]and let $\ominus$ be the intersection of all nonpositive regions
\[ \ominus = \cap \ominus_i  \]
\item[]Then the possible resultants $\pos(\{w_i\})$ is the set of all
  wrenches making nonnegative moments with all points in $\oplus$ and
  nonpositive moments with all points in $\ominus$.
\end{trivlist}

\end{slide}

%=======================================================================
%
% SLIDE 13
%
%
\begin{slide}[R]{Why does it work?}
\begin{trivlist}
\item[]If, for example, all $w_i$ give positive moments with respect
  to some point,
\item[]Then so does any wrench of the form
\[ \sum k_i w_i \]
if all the $k_i$ are nonnegative.
\item[]So, represent every $w_i$ by where it \emph{can't} go,
\item[]Intersect to determine where \emph{none} of the $w_i$ can go,
\item[]That's where wrenches $\pos(\{w_i\})$ cannot go.
\end{trivlist}

\end{slide}

%=======================================================================
%
% SLIDE 14
%
%
\begin{slide}[R]{Examples}
\begin{trivlist}
\item[]Sliding friction (preview).
\item[]Two facing cones.
\item[]Disk in concrete.
\item[]GOAT against step.

\end{trivlist}

\end{slide}


%=======================================================================
%
% SLIDE last
%

\begin{slide}[R]{Next:  Friction.}
\vspace{-0.5in}
\includegraphics[height=3.0in]{TOC16}
\end{slide}
\end{document}



\end{document}
