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\title{18.  Force Dual}
\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 18. \hfill Mechanics of Manipulation }

\begin{document}

\maketitle

%=======================================================================
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% SLIDE 2
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\begin{slide}[R]{Lecture 18.  Force Dual.}
\vspace{-0.5in}
\includegraphics[height=3.0in]{TOC18}
\end{slide}

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

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

\begin{trivlist}
\item Finish planar sliding.
\item Review representation of polyhedral convex cones in wrench/twist space.
\item Duality between points and lines.
\item Extension to oriented plane.
\item Examples.
\end{trivlist}

\end{slide}


%=======================================================================
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% SLIDE 4
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%
\begin{slide}[R]{Planar sliding, so far}
\begin{minipage}[t]{10cm}
\begin{trivlist}
\item[]We derived force and torque for planar sliding:
\begin{align*}
 \vv f_f & = -\mu \, \sgn(\dot\theta) \, \uu k 
 \btimes \int_R \frac{\vv r -\vv r_{\mathrm{IC}}}{|\vv r -\vv
   r_{\mathrm{IC}}|} p(\vv r) \, d\!A \\
 n_{fz} &  = - \mu \, \sgn(\dot\theta) 
        \int_R \vv r \cdot \frac{\vv r - \vv r_{\mathrm{IC}}}{| \vv r - \vv r_{\mathrm{IC}} |}
                 p(\vv r) \, d\!A
\end{align*}

\item[]We noted a simpler expression for translational sliding:
\begin{align*}
        \vv f_f & =  - \mu \, \frac{\vv v}{| \vv v| } f_0 \\
        \vv n_f & =  \vv r_0 \btimes \vv f_f 
\end{align*} 
where $\vv r_0$ is the \defterm{center of friction}
\item[]We observed that force and torque are undetermined when $p(x)$
  is undetermined.
\end{trivlist}
\end{minipage}
\hfill
\begin{minipage}[t]{4cm}
\begin{figure}[h!]
\includegraphics[width=4cm]{pushdefs}
%\includegraphics[height=3cm]{*file*}
\end{figure}\end{minipage}
\end{slide}

%=======================================================================
%
% SLIDE 5
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%
\begin{slide}[R]{Planar sliding:  limit surface}
\begin{minipage}[t]{6cm}
\begin{trivlist}
\item[]To explore mapping of planar sliding motion to force we use the
  \defterm{Limit Surface}.
\item[]Assume pressure distribution is known, and not necessarily finite.
\item[]Define \defterm{frictional load} as wrench applied by slider to ground.
\item[]Define \defterm{Limit Surface} as boundary of set of all
  possible load wrenches $\vv p^*$, constrained only to satisfy
  Coulomb's law locally.
\item[]Derive \defterm{maximum power inequality}: the frictional load
  wrench yields maximum power over all wrenches in the limit surface.
\item[]Equivalently: during slip the total frictional load wrench $\vv
  p$ lies on the limit surface, and the velocity twist $\vv q$ is
  normal to the limit surface at $\vv p$.

\end{trivlist}
\end{minipage}
\end{slide}

%=======================================================================
%
% SLIDE 6
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%
\begin{slide}[R]{Barbell Limit Surface}
\begin{figure}[h!]
\includegraphics[height=7cm]{barbellLS}
%\includegraphics[height=3cm]{*file*}
\end{figure}\end{slide}

%=======================================================================
%
% SLIDE 7
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\begin{slide}[R]{LS properties}
The barbell LS illustrates some properties that hold generally: 
\begin{trivlist}
\item[]Closed, convex, enclosing the origin of wrench space.
\item[]Symmetric when reflected through origin.
\item[]Orthogonal projection onto the $f_x$,$f_y$ plane is a circle of
  radius $\sum \mu f_n$.
\item[]Each discrete point of support yields two antipodal flat
  facets.  On each facet several loads map to one motion (rotation
  about the support point.)
\item[](No discrete points: LS is strictly convex and load-motion
  mapping is one-to-one.)
\item[]\emph{Collinear} discrete support is even weirder: vertices on
  LS where one load maps to several velocities (rotation about point
  collinear with support).
\end{trivlist}

\end{slide}

%=======================================================================
%
% SLIDE 8
%
%
\begin{slide}[R]{Revisiting representation of PCCs of wrenches and twists}
\begin{trivlist}
\item[]Why polyhedral convex cones in wrench or twist space?
\begin{itemize}
\item[]Possible wrenches resulting from frictional or frictionless
  contacts.  (Positive linear span $\pos(\{\vv w_i\})$.  Edge
  representation of a cone.)
\item[]Twists consistent with constraints.  (Intersection of half
  spaces reciprocal or repelling to the constraint $\Cap\half(\vv
  c_i)$.  Face representation of a cone.)
\end{itemize}
\item[]For 3 space (6D wrench or twist space) represent them by the
  edges or by the face normals.
\item[]For the plane (3D wrench or twist space) we can use 2D
  graphical techniques:
\begin{itemize}
\item[]Reuleaux's method.  Label rotation centers.  Equivalent to
  projection of twists to oriented plane.
\item[]Moment labeling.  Label moments.  Equivalent to $\ldots$
\item[]Force dual $\ldots$.  
\end{itemize}
\end{trivlist}

\end{slide}

%=======================================================================
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% SLIDE 9
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%
\begin{slide}[R]{Roadmap to graphical techniques}
\includegraphics[width=6cm]{RoadMap1}\hfill%
\includegraphics[width=4cm]{RoadMap2}
\end{slide}

%=======================================================================
%
% SLIDE 9.5
%
%
\begin{slide}[R]{Duality in the projective plane}
\begin{minipage}[t]{10cm}
\begin{trivlist}
\item[]Recall that for the projective plane there is a duality between
  \emph{point} and \emph{line}
\item[]We can make that concrete by defining a mapping $D$.
\begin{itemize}
\item[]Define $D(l)$ of a line $l$ to be the point $p$ such that $Op \ldots$
\item[]Define $D(p)$ of a point $p$ to be the locus of $D(l)$ for all
  $l$ through $p$.
\item[]Note $D(p)$ is a line, and $D(D(p))$ is $p$.
\item[]Note what happens at infinity.
\item[]Note it depends totally on choice of scale and origin.
\end{itemize}
\item[]Check out the movies.
\end{trivlist}
\end{minipage}
\end{slide}


%=======================================================================
%
% SLIDE 10
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%
\begin{slide}[R]{Construction of force dual}
\begin{minipage}[t]{4cm}
\begin{figure}
\includegraphics{dualdef.eps}
\label{fig:dualdef}
\end{figure}
\end{minipage}
\hfill
\begin{minipage}[t]{6cm}
\begin{trivlist}
\item
Given a (directed) line of force, and an origin;

\item
Construct perpendicular through origin;

\item
Take point on perpendicular, at distance inverse to moment arm;

\item
Note the sign of the moment.
\end{trivlist}
\end{minipage}

\end{slide}


%=======================================================================
%
% SLIDE 11
%
%
\begin{slide}[R]{Dual of a signed point}
\begin{minipage}[t]{4cm}
\begin{figure}
\includegraphics[width=4cm]{dualdual.eps}
\label{fig:dualdual}
\end{figure}
\end{minipage}
\hfill
\begin{minipage}[t]{6cm}
\begin{trivlist}
\item
We defined map of directed line to a signed point.

\item
Extend definition to map signed points to something.
\begin{itemize}
\item
  Given signed point $P$, let $\{l\}$ be the set of directed lines
  through $P$.

\item Define $P^\prime$ is defined to be $\{l^\prime\}$, with a direction
determined by the sign of $P$.

\item Note that  A simple geometric
\begin{itemize}
\item
        $P^\prime$ is a directed line

\item
        $P^{\prime\prime} = P$

\end{itemize}
Hence the transformation is \emph{dual}.
\end{itemize}
\end{trivlist}
\end{minipage}


\end{slide}


%=======================================================================
%
% SLIDE 12
%
%
\begin{slide}[R]{Representing wrench cones}
\begin{minipage}[t]{6cm}
\begin{figure}
\includegraphics[width=6cm]{dualexamp.eps}
\label{fig:dualexamp}
\end{figure}
\end{minipage}
\hfill
\begin{minipage}[t]{4cm}
The method:
\begin{enumerate}
\item  Choose origin and unit length.

\item  Construct dual of each line of action.

\item  Take the convex hull.
\end{enumerate}
\end{minipage}
\end{slide}


%=======================================================================
%
% SLIDE 13
%
%
\begin{slide}[R]{Zigzag locus}
\begin{minipage}[t]{6cm}
\begin{figure}
\includegraphics{dualperm.eps}
\index{zigzag locus}
\label{fig:dualperm}
\end{figure}
\end{minipage}
\hfill
\begin{minipage}[t]{4cm}
\begin{trivlist}
\item
Force dual can represent \emph{non-convex cones}!

\item
Example:  The set of contact normals.

\item
Also known as the set of frictionless contact forces.

\item
Force dual is called the \emph{zigzag locus}.
\end{trivlist}
\end{minipage}
\end{slide}



%=======================================================================
%
% SLIDE last
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\begin{slide}[R]{Next:  Wrap up graphical methods.}
\vspace{-0.5in}
\includegraphics[height=3.0in]{TOC19}
\end{slide}
\end{document}



\end{document}
