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\title{13.  Polyhedral Convex Cones}
\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 13. \hfill Mechanics of Manipulation }

\begin{document}

\maketitle

%=======================================================================
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% SLIDE 2
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\begin{slide}[R]{Lecture 13.  Cones.}
\vspace{-0.5in}
\includegraphics[height=3.0in]{TOC13}
\end{slide}

%=======================================================================
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% SLIDE 3
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\begin{slide}[R]{Outline.}
              
\begin{enumerate}
\item Positive linear span
\item Types of cones
\item Edge and face representation
\item Supplementary cones; polar
\item Representing frictionless contact
\item Cones in wrench space 
\begin{itemize}\item force closure \end{itemize}
\item Cones in velocity twist space
\end{enumerate}

\end{slide}

%=======================================================================
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% SLIDE 4
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\begin{slide}[R]{Positive linear span}
\begin{trivlist}
\item[]For now, use $n$-dimensional vector space $\mathbf{R}^{n}$.
Later, wrench space and velocity twist space. 

\item[]Let $\vv v$ be any non-zero vector in $\mathbf{R}^{n}$.  Then the set 
of vectors
\begin{equation}
        \{ k \vv v \; | \; k \geq 0 \}
\end{equation}
describes a \emph{ray}.

\item[]
Let $\vv v_{1}$, $\vv v_{2}$ be non-zero and 
non-parallel.  Then the set of positively scaled sums
\begin{equation}
        \{k_{1} \vv v_{1} + k_{2} \vv v_{2} \; | \;  k_{1}, k_{2} \geq 0 \}
\end{equation}
is a planar cone---sector of a plane.

\item[]
Generalize by defining the \emph{positive linear 
span} of a set of vectors $\{\vv v_{i}\}$:
\begin{equation}
        \pos(\{ \vv v_{i} \}) 
            = \{ {\textstyle \sum k_{i} \vv v_{i} \; | \;  k_{i} \geq 0 } \}
\end{equation}

\item[]
(The positive linear span of the empty set is the 
origin.)\end{trivlist}
\end{slide}

%=======================================================================
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% SLIDE 5
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\begin{slide}[R]{Relatives of positive linear span}
\begin{trivlist}
\item
The \emph{linear span}
\index{linear span}
\begin{equation}
        \lin(\{ \vv v_{i} \}) 
            = \{ {\textstyle \sum k_{i} \vv v_{i} \; | \;  k_{i} \in
            \mathbf{R} }\} 
\end{equation}

\item 
The \emph{convex hull}
\index{convex hull}
\begin{equation}
        \conv(\{ \vv v_{i} \}) 
           = \{ {\textstyle \sum k_{i} \vv v_{i} \; | \;  k_{i} \geq
           0, \sum k_i = 1 }\} 
\end{equation}
\end{trivlist}
\end{slide}

%=======================================================================
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% SLIDE 6
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%
\begin{slide}[R]{Varieties of cones in three space}
\begin{figure}
\includegraphics[width=0.75\linewidth]{wcones.eps}
\end{figure}

\end{slide}

%=======================================================================
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% SLIDE 7
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\begin{slide}[R]{Spanning all of $\mathbf{R}^n$}
\begin{trivlist}
\item
Theorem:
A set of vectors $\{\vv v_{i}\}$ positively spans the entire space 
$\mathbf{R}^{n}$ if and only if the origin is in the interior of the 
convex hull:
\begin{equation}
        \pos(\{\vv v_{i}\}) = \mathbf{R}^{n} \leftrightarrow
        \vv 0 \in \intoper( \conv( \{\vv v_{i}\} ))
\end{equation}


\item
Theorem:
It takes at least $n + 1$ vectors to positively span $\mathbf{R}^{n}$.

\end{trivlist}
\end{slide}

%=======================================================================
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% SLIDE 8
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\begin{slide}[R]{Representing cones}
\begin{trivlist}
\item
Two ways to represent cones:  \emph{edge representation} and
\emph{face representation}.

\item
Edge representation uses positive linear span.   Given a set of edges 
$\{ \vv e_{i} \}$, the cone is given by $\pos( \{ \vv e_{i} \})$.
\end{trivlist}
\end{slide}


%=======================================================================
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% SLIDE 9
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\begin{slide}[R]{Face representation of cones}
\begin{trivlist}
\item
First represent \emph{planar half-space}
by inward pointing normal vector $\vv n$.

\begin{equation}
        \half( \vv n ) = \{ \vv v \; | \;  \vv n \cdot \vv v \geq 0 \}
\end{equation}
(Here we use dot product, but when working with twists and wrenches we
will use reciprocal product.)

\item
Consider a cone with face normals $\{\vv n_i\}$.  Then the cone is the
intersection of the half-spaces:
\begin{equation}
        \cap \{ \half( \vv n_{i} ) \}
\end{equation}
\end{trivlist}
\end{slide}

%=======================================================================
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% SLIDE 10
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\begin{slide}[R]{Supplementary cone; polar}
\begin{minipage}[t]{4cm}
\begin{figure}
\includegraphics[width=4cm]{suppcone.eps}
\end{figure}
\end{minipage}
\hfill
\begin{minipage}[t]{6cm}
\begin{trivlist}
\item
\emph{Supplementary} cone $\supp( V )$ (also known as \emph{polar})
comprises the 
vectors that make non-negative dot products with vectors in $V$:
\begin{equation}
  \{ u \in \mathbf{R}^n \; | \; u \cdot v \geq 0 \; \forall v \in V  \}
\end{equation}

\item
The supplementary cone's \emph{edges} are the original cone's \emph{face
normals}, and vice versa.  So if
\begin{equation}
        V = \pos( \{ \vv e_{i} \} ) = \cap \{ \half( \vv n_{i} ) \}
\end{equation}
then
\begin{equation}
        \supp( V ) = \pos( \{ \vv n_{i} \} ) = \cap \{ \half( \vv e_{i} ) \}
\end{equation}
\end{trivlist}
\end{minipage}
\end{slide}

%=======================================================================
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% SLIDE 11
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%

\begin{slide}[R]{Frictionless contact}
\begin{trivlist}
\item Characterize contact by \emph{set of possible wrenches}. 

\item Assume uniquely determined contact normal.

\item Assume frictionless contact can give arbitrary magnitude force
      along inward-pointing normal.

\item Then a frictionless contact gives a ray in wrench space,
      $\pos(\vv w )$, where $\vv w = (\vv c, \vv c_0)$ is the contact
      screw.

\end{trivlist}
\end{slide}

%=======================================================================
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% SLIDE 12
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\begin{slide}[R]{Two contacts}
\begin{trivlist}
\item
Given two frictionless contacts $\vv w_1$ and 
$\vv w_2$,  total wrench is the sum of possible positive scalings of
$\vv w_{1}$ and $\vv w_{2}$:
\begin{equation}
        k_1 \vv w_{1} + k_2 \vv w_2;  k_1, k_2 \geq 0
\end{equation}
i.e.  the positive linear span $\pos( \{ \vv w_{1}, \vv 
w_{2} \})$.

\item
Generalizing:

Theorem:  If a set of frictionless contacts on a rigid body is described by 
the contact normals $\vv w_{i} = (\vv c_{i}, \vv c_{0i})$ then the 
set of all possible wrenches is given by the positive linear 
span $\pos( \{ \vv w_{i} \} )$.

\end{trivlist}

\end{slide}

%=======================================================================
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% SLIDE 13
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\begin{slide}[R]{Force closure}
\begin{trivlist}
\item
  Definition: \defterm{Force closure} means that the set of possible
  wrenches exhausts all of wrench space.

\item
It follows from 
theorem~? that a frictionless force closure 
requires at least 7 contacts.  Or, since planar wrench space is only 
three-dimensional, frictionless force closure in the plane 
requires at least 4 contacts.
\end{trivlist}
\end{slide}


%=======================================================================
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% SLIDE 14
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\begin{slide}[R]{Example wrench cone}
\begin{minipage}[t]{6cm}
\begin{figure}[h!]
\includegraphics[width=6cm]{fclosex.eps}
\end{figure}
\end{minipage}
\hfill
\begin{minipage}[t]{4cm}
\begin{trivlist}
\item Construct unit magnitude force at each contact.
\item Write screw coords of wrenches.
\item Take positive linear span.
\item Exhausts wrench space?
\end{trivlist}
\end{minipage}
\end{slide}

%=======================================================================
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% SLIDE x15
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\begin{slide}[R]{Cones in velocity twist space}
\begin{trivlist}
\item Cannot use finite displacement twists.  They are not vectors.
\item Velocity twists are vectors!
\item Let $\{ \vv w_{i} \}$ be a set of contact normals.
\item Let $W = \pos( 
\{ \vv w_{i} \} )$
be set of possible wrenches.
\item First order analysis:  velocity twists $T$ must be \emph{reciprocal
      or repelling} to contact wrenches:  $T = \supp(W)$.
\end{trivlist}

\end{slide}

%=======================================================================
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% SLIDE last
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\begin{slide}[R]{Next:  The Oriented Plane.}
\vspace{-0.5in}
\includegraphics[height=3.0in]{TOC14}
\end{slide}
\end{document}



\end{document}
