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\title{16.  Friction}
\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 16. \hfill Mechanics of Manipulation }

\begin{document}

\maketitle


%=======================================================================
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% SLIDE 2
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\begin{slide}[R]{Lecture 16.  Friction.}
\vspace{-0.5in}
\includegraphics[height=3.0in]{TOC16}
\end{slide}

%=======================================================================
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% SLIDE 3
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\begin{slide}[R]{Outline.}

\begin{trivlist}
\item Coulomb's Law.
\item Friction angle, friction cone.
\item Moment labeling of friction cone.
\item Static equilibrium problems.
\end{trivlist}

\end{slide}


%=======================================================================
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% SLIDE 4
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%
\begin{slide}[R]{How do you move things around?}
\begin{trivlist}
\item[]Kinematics, kinematic constraint.
\item[]Force.
\begin{itemize}
  \item[]Force of constraint;
  \item[]Gravity;
  \item[]Friction;
\end{itemize}
\item[]Momentum.
\end{trivlist}

\end{slide}

%=======================================================================
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% SLIDE 5
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\begin{slide}[R]{Coulomb's experiments}
\begin{minipage}[t]{6cm}
An experiment:
\begin{trivlist}
\item[]Clean surfaces, but not too clean.  Dry.  Unlubricated.
\item[]Pull on string with force $\vv f_a$, ramping up from 0.
\item[]Friction force $\vv f_f$ will balance $\vv f_a$, up to a point.
\item[]Max $\vv f_f$ when not moving:  $\mu_s m g$.
\item[]Max $\vv f_f$ when moving: $\mu_d m g$.
\end{trivlist}
From now on we will assume $\mu_s = \mu_d = \mu$.
\end{minipage}
\hfill
\begin{minipage}[t]{4cm}
\begin{figure}[h!]
\includegraphics[width=4cm]{Coulomb}
%\includegraphics[height=3cm]{*file*}
\end{figure}\end{minipage}
\end{slide}

%=======================================================================
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% SLIDE 6
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\begin{slide}[R]{Coulomb's observations}
  Coulomb conducted hundreds of experiments, and over a \emph{broad
    range} of conditions observed:
\begin{trivlist}
\item[]Frictional force is \emph{approximately} independent of contact
  area.
\item[]Frictional force is \emph{approximately} independent of
  velocity magnitude.
\item[]Coefficient of friction depends on pairs of materials.
\end{trivlist}
\begin{figure}[h!]
\centerline{
\begin{tabular}{@{}cc@{}} \hline
  \textbf{Materials} & $\mu$ \\ \hline
  metal on metal & 0.15--0.6 \\ 
  rubber on concrete & 0.6--0.9 \\
  plastic wrap on lettuce & $\infty$ \\ 
  Leonardo's number & 0.25 \\ \hline
\end{tabular}
}
\end{figure}
\end{slide}

%=======================================================================
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% SLIDE 7
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\begin{slide}[R]{Think when using Coulomb's law!}
\begin{trivlist}
\item[]It holds over a \emph{broad range}, but not nearly everywhere.
\item[]It is approximate.
\item[]Coefficients of friction tables are terrible.
\item[]How can you use something so unreliable?
\item[]But, how can you \emph{not} use it?
\end{trivlist}

\end{slide}

%=======================================================================
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% SLIDE 8
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\begin{slide}[R]{Contact modes}
\begin{minipage}[t]{6cm}
We can write Coulomb's law:

\centerline{
\begin{tabular}{@{}ccll@{}}
\\ \hline
 $\dot x$ &     $\ddot x$       &       &
\\ \hline
 $<0$     &                     & $f_t = \mu f_n$       & left sliding \\
 $>0$     &                     & $f_t = - \mu f_n$     & right sliding \\
 $=0$     &     $<0$            & $f_t = \mu f_n$       & left sliding \\
 $=0$     &     $>0$            & $f_t = - \mu f_n$     & right sliding \\
 $=0$     &     $=0$            & $|f_t| \leq |\mu f_n|$ & rest
\\ \hline
\end{tabular}
}
\end{minipage}
\hfill
\begin{minipage}[t]{4cm}
\begin{figure}[h!]
\includegraphics[width=4cm]{frblock}
%\includegraphics[height=3cm]{*file*}
\end{figure}\end{minipage}
\end{slide}

%=======================================================================
%
% SLIDE 9
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%
\begin{slide}[R]{Friction angle}
\begin{minipage}[t]{6cm}
  Block at rest on plane with angle $\alpha$:
\begin{align*}
        f_n &= m g \cos \alpha \\
        f_t &= m g \sin \alpha 
\end{align*}
At rest $|f_t| \leq \mu f_n$.  Maximum $\alpha$:
\begin{equation*}
        f_t = \mu f_n
\end{equation*}
Substituting,
\begin{align*}
        m g \sin\alpha &= \mu m g \cos\alpha \\
        \alpha &= \tan^{-1}\mu 
\end{align*}
Sometimes called the \emph{friction angle} or the 
\emph{angle of repose}.
\end{minipage}
\hfill
\begin{minipage}[t]{4cm}
\begin{figure}[h!]
\includegraphics[width=4cm]{incplane}
%\includegraphics[height=3cm]{*file*}
\end{figure}\end{minipage}
\end{slide}

%=======================================================================
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% SLIDE 10
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%
\begin{slide}[R]{Friction cone}
\begin{minipage}[t]{7cm}
Define the \defterm{friction cone} to be the set of all wrenches satisfying Coulomb's law for an object at rest, i.e.\ all the wrenches satisfying:
\begin{equation*}
        |f_t| \leq \mu |f_n| 
\end{equation*}
This set of forces describes a cone in wrench space.

Each wrench is applied at the contact point.  The dihedral angle is
$2 \tan^{-1}  \mu$.  

Then we can state Coulomb's law: 
%\centerline{
\begin{tabular}{@{}ll@{}}
 For left sliding & $\vv f_n + \vv f_t \in $ right edge of friction cone \\
 For right sliding & $\vv f_n + \vv f_t \in $ left edge of friction cone \\
 For rest & $\vv f_n + \vv f_t \in $ friction cone 
\end{tabular}%}
\end{minipage}
\hfill
\begin{minipage}[t]{3cm}
\begin{figure}[h!]
\includegraphics[width=3cm]{fcone}
%\includegraphics[height=3cm]{*file*}
\end{figure}\end{minipage}
\end{slide}

%=======================================================================
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% SLIDE 11
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%
\begin{slide}[R]{Pipe clamp design problem}
\begin{minipage}[t]{4cm}
\begin{trivlist}
\item[]Why does pipe clamp work?
\item[]Let diameter be 2 cm.
\item[]Let length be 2 cm.
\item[]Assume $\mu$ of 0.25.
\item[]Find min moment arm.
\item[]Extend to woodpecker toy?
\end{trivlist}
\end{minipage}
\hfill
\begin{minipage}[t]{6cm}
\begin{figure}[h!]
\includegraphics[width=6cm]{clamp}
%\includegraphics[height=3cm]{*file*}
\end{figure}\end{minipage}
\end{slide}

%=======================================================================
%
% SLIDE 12
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%
\begin{slide}[R]{Sliding rod}
\begin{minipage}[t]{7cm}
If we consider normal velocities.  and accelerations:

\centerline{
\begin{tabular}{@{}cccccl@{}}
 \\ \hline
  $\dot p_{cn}$ & $\ddot p_{cn}$ & $\dot p_{ct}$ & $\ddot p_{ct}$ & & 
\\ \hline
 $<0$ &      &       &          &                        & impact \\    
 $>0$ &      &       &          &                        & separation \\
 $=0$ & $<0$ &       &          &                        & impact \\
 $=0$ & $>0$ &       &          &                        & separation \\
 $=0$ & $=0$ & $<0$  &          & $f_t = \mu f_n$        & left sliding \\
 $=0$ & $=0$ & $>0$  &          & $f_t = - \mu f_n$      & right sliding \\
 $=0$ & $=0$ & $=0$  & $<0$     & $f_t = \mu f_n$        & left sliding \\
 $=0$ & $=0$ & $=0$  & $>0$     & $f_t = -\mu f_n$       & right sliding \\
 $=0$ & $=0$ & $=0$  & $=0$     & $|f_t| \leq |\mu f_n|$ & fixed
\\ \hline
\end{tabular}}
\vspace{0.25cm}
We're assuming pointy contact.  Rolling is more complicated.
\end{minipage}
\hfill
\begin{minipage}[t]{3cm}
\begin{figure}[h!]
\includegraphics[width=3cm]{rod}
%\includegraphics[height=3cm]{*file*}
\end{figure}\end{minipage}
\end{slide}

%=======================================================================
%
% SLIDE 13
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%
\begin{slide}[R]{Moment labeling of friction cone}
\begin{minipage}[t]{6cm}
\begin{trivlist}
\item[]Friction cone is positive linear span of left edge unit vector and right edge unit vector.
\end{trivlist}
\end{minipage}
\hfill
\begin{minipage}[t]{4cm}
\begin{figure}[h!]
\includegraphics[width=4cm]{grfconeL16.eps}
%\includegraphics[height=3cm]{*file*}
\end{figure}\end{minipage}
\end{slide}

%=======================================================================
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% SLIDE 14
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%
\begin{slide}[R]{Examples}
\begin{trivlist}
\item[]Block on table.
\item[]Wedged plank and piranha.
\item[]Triangle and three fingers.
\end{trivlist}

\end{slide}

%=======================================================================
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% SLIDE 15
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%
\begin{slide}[R]{What exactly does any of this prove?}
\begin{trivlist}
\item[]Force closure versus stability.
\item[]Force closure versus first order form closure.
\end{trivlist}

\end{slide}


%=======================================================================
%
% SLIDE last
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\begin{slide}[R]{Next:  Planar sliding.}
\vspace{-0.5in}
\includegraphics[height=3.0in]{TOC17}
\end{slide}
\end{document}



\end{document}
