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\title{4.  Spatial kinematics.  Constraint.}
\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 4. \hfill Mechanics of Manipulation }

\begin{document}

\maketitle


%=======================================================================
%
% SLIDE 2
%

\begin{slide}[R]{Lecture 4.  Spatial kinematics.  Constraint.}
\vspace{-0.5in}
\includegraphics[height=3.0in]{TOC4}
\end{slide}

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

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

\begin{itemize}
\item Spherical kinematics
\begin{itemize}
  \item Euler's theorem
  \item Cones
\end{itemize}
\item Spatial kinematics
\begin{itemize}
  \item Chasles' theorem
  \item Screws and twists
  \item Axodes
\end{itemize}
\item Kinematic constraint
\begin{itemize}
  \item Overview
  \item Taxonomy and terminology
  \item Reuleaux' method for unilateral constraints
\end{itemize}
\end{itemize}

\end{slide}

%=======================================================================
%
% SLIDE 4
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%

\overlays{2}{
\begin{slide}[R]{About spherical kinematics}
\begin{itemize}
\item Why study motions of the sphere?  Because it corresponds to rotations about
a given point of $\mathbb{E}^3$.
\fromSlide{2}{
\item There is a close connection to planar kinematics.  Let the radius of the sphere approach infinity $\ldots$}
\end{itemize}%
\end{slide}}

%=======================================================================
%
% SLIDE 5
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%

\begin{slide}[R]{Two not-antipodal points enough}
Theorem 2.5:  A displacement of the sphere is completely determined by the motion of
any two points that are not antipodal.
  \\[6pt]
Proof:  Construct a coordinate frame $\ldots$
\end{slide}
%=======================================================================
%
% SLIDE 6
%
%

\begin{slide}[R]{Euler's theorem}
\begin{minipage}[t]{6cm}
Theorem 2.6:  For every spatial rotation, there is a line of fixed points.  In other
words, every rotation about a point is a rotation about a line, called
the \emph{rotation axis}. \\[6pt]
Proof:  
\begin{itemize}
\item[]
  Prove that every displacement of the sphere has a fixed point.
\item[]
  Define $A$, $\perp \! A \! A^\prime$, $B$, $B^\prime$, $\perp \! B \! B^\prime$.
\item[]
  Define $C$ to be either intersection of $\perp \! A \! A^\prime$ with $\perp \! B \! B^\prime$.
\item[]
  Let $R$ be the rotation mapping $A$ to $A^\prime$ and $C$ to itself.
\item[]
  Show $R$ maps $B$ to $B^\prime$, so $R$ is the given displacement.
\end{itemize}
\end{minipage}
\hfill
\begin{minipage}[t]{4cm}
\begin{figure}[h!]
\includegraphics[width=4cm]{euler.eps}
\end{figure}
\end{minipage}
\end{slide}

%=======================================================================
%
% SLIDE 7
%
%

\overlays{16}{
\begin{slide}[R]{Review of displacements:  planar and spherical}
\begin{minipage}[t]{6cm}
For the Euclidean plane, are there $\ldots$
\begin{itemize}
\item[]  $\ldots$ rotations that are not translations?
\fromSlide{3}{
\item[] $\ldots$ translations that are not rotations?}
\fromSlide{5}{
\item[] $\ldots$ displacements that are both rotations and translations?}
\fromSlide{7}{
\item[] $\ldots$ displacements that are neither?}
\end{itemize}
For the sphere, are there $\ldots$
\begin{itemize}
  \fromSlide{9}{
\item[]  $\ldots$ rotations that are not translations?}
  \fromSlide{11}{
\item[] $\ldots$ translations that are not rotations?}
  \fromSlide{13}{
\item[] $\ldots$ displacements that are both rotations and translations?}
  \fromSlide{15}{
\item[] $\ldots$ displacements that are neither?}
\end{itemize}
\end{minipage}
\hfill
\begin{minipage}[t]{4cm}
\vspace*{-1.5cm}
\begin{figure}[h!]
\includegraphics[width=4cm]{Venn.eps}
\end{figure}
\begin{picture}(0,0)
\fromSlide{2}{
\put(1,2){\makebox[0pt]{Lots!}}}
\fromSlide{4}{
\put(3,2){\makebox[0pt]{Some.}}}
\fromSlide{6}{
\put(2,2){\makebox[0pt]{I}}}
\fromSlide{8}{
\put(2,.85){\makebox[0pt]{None}}}
\end{picture}
\centerline{The Euclidean plane}
\begin{figure}[h!]
\includegraphics[width=4cm]{Venn.eps}
\end{figure}
\begin{picture}(0,0)
\fromSlide{10}{
\put(1,2){\makebox[0pt]{Lots!}}}
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\end{picture}
\centerline{The sphere}
\end{minipage}
\end{slide}}

%=======================================================================
%
% SLIDE 8
%
%

\overlays{10}{
\begin{slide}[R]{Preview of spatial displacements}
\begin{minipage}[t]{6cm}
For the Euclidean plane, are there $\ldots$
\begin{itemize}
\item[]  $\ldots$ rotations that are not translations?
\fromSlide{3}{
\item[] $\ldots$ translations that are not rotations?}
\fromSlide{5}{
\item[] $\ldots$ displacements that are both rotations and translations?}
\fromSlide{7}{
\item[] $\ldots$ displacements that are neither?}
  \fromSlide{9}{
\item[] $\ldots$ displacements that are not screws?}
\end{itemize}
  \fromSlide{10}{
\centerline{No.  Chasles' theorem}}
\end{minipage}
\hfill
\begin{minipage}[t]{4cm}
\vspace*{1.5cm}
\begin{figure}[h!]
\includegraphics[width=4cm]{Venn.eps}
\end{figure}
\begin{picture}(0,0)
\fromSlide{2}{
\put(1,2){\makebox[0pt]{Lots!}}}
\fromSlide{4}{
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\fromSlide{6}{
\put(2,2){\makebox[0pt]{I}}}
\fromSlide{8}{
\put(2,.85){\makebox[0pt]{Screws!}}}
\end{picture}
\centerline{Euclidean three space}
\end{minipage}
\end{slide}}

%=======================================================================
%
% SLIDE 9
%
%

\begin{slide}[R]{Chasles's theorem}

Theorem 2.7:
Every spatial  displacement is the composition of a rotation about some
axis, and a translation along the same axis.
\\[6pt]

Proof: 
\begin{itemize}
\item[] Assume arbitrary displacement $D$ is given.
\item[]  Use theorem 2.2 to decompose $D = R \circ T$.
\item[]  Decompose $T$ into components parallel to
and perpendicular to axis of $R$:  $D = R \circ T_\perp \circ T_\parallel$.
\item[] Note that $R \circ T_\perp$ is planar!  Every plane perpendicular to
rotation axis is mapped rigidly to itself.
\item[] If $R \circ T_\perp$ is a translation the theorem follows immediately.
Otherwise $R \circ T_\perp$ is a rotation about some axis parallel to the
rotation axis of $R$.
\item[] So $D = ( R \circ T_\perp) \circ T_\parallel$ is the desired decomposition.
\end{itemize}
%
\end{slide}

%=======================================================================
%
% SLIDE 10
%
%

\begin{slide}[R]{Screws.}
\begin{minipage}[t]{6cm}
\begin{itemize}
\item[]
A \emph{screw}
is a line in space with an associated pitch, which is a ratio of 
linear to angular quantities.

\item[]
A \emph{twist}
is a screw plus a scalar magnitude, giving a rotation about the screw axis
plus a translation along the screw axis.  The rotation angle is the
twist magnitude, and the translation distance is the magnitude times
the pitch.  Thus the pitch is the ratio of translation to rotation.
\end{itemize}

\end{minipage}
\hfill
\begin{minipage}[t]{4cm}
\begin{figure}[h!]
\includegraphics[width=4cm]{screw}
\end{figure}
\end{minipage}
\end{slide}

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

\begin{slide}[R]{Analogous to centrodes $\ldots$}
\begin{minipage}[t]{6cm}
\begin{itemize}
\item[]On the sphere $\ldots$
\begin{itemize}
\item[]  Plotting the instantaneous rotation axis in the fixed and moving
frames gives \emph{fixed and moving cones}.
\end{itemize}
\item[]In three space $\ldots$
\begin{itemize}
\item[]Plotting the instantaneous screw axis in the fixed and moving
frames gives \emph{fixed and moving axodes}.
\end{itemize}
\end{itemize}
\end{minipage}
\hfill
\begin{minipage}[t]{4cm}
\begin{figure}[h!]
\includegraphics[height=2.5cm]{cones}
\end{figure}
\begin{figure}[h!]
\includegraphics[height=2.5cm]{axodes}
\end{figure}
\end{minipage}
\end{slide}

%=======================================================================
%
% SLIDE 12
%
%

\begin{slide}[R]{Kinematic constraint}
One of the best manipulation tricks!
\begin{itemize}
\item[]In simple cases, freedoms and constraints are just a matter of counting unknowns and equations.
\begin{eqnarray*}
 \mbox{nominal DOFs} & & \\
- \mbox{independent constraints} & & \\
= \mbox{DOFs} & &
\end{eqnarray*}
\item[]Things to worry about:
\begin{itemize}
\item[]If an equation reduces DOFs by 1, does an inequation reduce DOFs by 1/2?
\item[]Identifying dependencies and singular cases.
\item[]Constraints on velocity versus on configuration.
\end{itemize}
\end{itemize}
\end{slide}


%=======================================================================
%
% SLIDE 13
%
%

\begin{slide}[R]{Constraint in general}
%\begin{minipage}[t]{6cm}
Consider constraints of the form
\[
        f(q,\dot q, t) = 0
\]
or
\[  
        f(q, \dot q, t) \geq 0
\]
where
\begin{eqnarray*}
        q & \in & Q \quad\quad\mbox{configuration space, e.g. $(x, y, \theta)$} \\
        \dot q & \in & TQ \quad\quad\mbox{tangent space, e.g. $(\dot x, \dot y, \dot\theta)$} \\
        t & = & \mbox{time}
\end{eqnarray*}
\end{slide}

%=======================================================================
%
% SLIDE 14
%
%

\begin{slide}[R]{Constraint:  taxonomy and examples}
\begin{minipage}[t]{6cm}
\begin{description}
\item[bilateral] \quad \\ Expressed as an equation.  Two sided.
\begin{align*}
       y &= 0 \\
       \theta &= 0 
\end{align*}

\item[unilateral]  \quad \\ Expressed as an inequation.  One sided.
\begin{align*}
        y &\geq 0 \\
        y + 2 \sin\theta &\geq 0 \\
        y + 2 \sin\theta + \cos\theta &\geq 0 \\
        y + \cos\theta &\geq 0 
\end{align*}

%\item[scleronomic]\quad \\ Independent of $t$.  Stationary.
%\item[rheonomic]\quad \\ Depends on $t$.
%\begin{align*}
%        &x \sin(2 \pi t) - y \cos(2 \pi t) = 0 \\
%        &\theta = 2 \pi t 
%\end{align*}

\end{description}
\end{minipage}
\hfill
\begin{minipage}[t]{4cm}
\begin{figure}[h!]
  \includegraphics[width=4cm]{kinb1}
\end{figure}
\begin{figure}[h!]
  \includegraphics[width=4cm]{kinb3}
\end{figure}
%  \subfigure[Unilateral]{\psfig{file=kinb3.eps}
%     \label{fig:kinb3}}\qquad
%  \subfigure[Nonholonomic]{\psfig{file=kinb4.eps}
%     \label{fig:kinb4}}
%\end{figure}
\end{minipage}
\end{slide}
%=======================================================================
%
% SLIDE 15
%

\begin{slide}[R]{Constraint:  taxonomy and examples}
\begin{minipage}[t]{6cm}
\begin{description}
\item[scleronomic]\quad \\ Independent of $t$.  Stationary.
\item[rheonomic]\quad \\ Depends on $t$.
\begin{align*}
        &x \sin(2 \pi t) - y \cos(2 \pi t) = 0 \\
        &\theta = 2 \pi t 
\end{align*}
\item[holonomic] \quad \\ Independent of $\dot q$ and bilateral.
\[
        f(q,t) = 0
\]
\item[nonholonomic] \quad \\
\end{description}
\end{minipage}
\hfill
\begin{minipage}[t]{4cm}
\begin{figure}[h!]
  \includegraphics[width=4cm]{kinb2}
\end{figure}
\begin{figure}[h!]
  \includegraphics[width=4cm]{kinb4}
\end{figure}
\end{minipage}
\end{slide}

%=======================================================================
%
% Slide 16
%

\begin{slide}[R]{Analysis of planar constraints using velocity centers}
\begin{minipage}[t]{6cm}
\begin{itemize}
\item[]Bilateral:  recall technique from previous lecture.  Construct perpendicular to
allowed velocity at point.  IC must be at intersection of perpendiculars.
\item[] Extension to unilateral.  Perpendicular to constraint divides plane into positive IC's, negative IC's, and IC's of either sign.
\end{itemize}
\end{minipage}
\hfill
\begin{minipage}[t]{4cm}
\vspace{-1.5cm}
\begin{figure}[h!]
\includegraphics[height=4cm]{astroidcent}
\end{figure}
\begin{figure}[h!]
\includegraphics[height=3cm]{unilat}
\end{figure}\end{minipage}
\end{slide}

%=======================================================================
%
% Slide 17
%

\overlays{6}{
\begin{slide}[Dissolve]{Multiple unilateral constraints (Reuleaux's method)}
\begin{minipage}[t]{6cm}
\begin{itemize}
\item[]
Can this triangle move?
\fromSlide{2}{
\item[]
Construct positive and negative half-planes for each contact.}
\fromSlide{4}{
\item[]
Keep consistently labelled points.}
\fromSlide{6}{
\item[]
Triangle can rotate CW about any $-$ point.}
\end{itemize}
\end{minipage}
\hfill
\begin{minipage}[t]{4cm}
\onlySlide*{1}{
\begin{figure}[h!]
\includegraphics[width=4cm]{uniex2i}
\end{figure}}
\onlySlide*{2}{
\begin{figure}[h!]
\includegraphics[width=4cm]{uniex2i}
\end{figure}}
\onlySlide*{3}{
\begin{figure}[h!]
\includegraphics[width=4cm]{uniex2ii}
\end{figure}}
\onlySlide*{4}{
\begin{figure}[h!]
\includegraphics[width=4cm]{uniex2ii}
\end{figure}}
\fromSlide{5}{
\begin{figure}[h!]
\includegraphics[width=4cm]{uniex2}
\end{figure}}
\end{minipage}
\end{slide}}

%=======================================================================
%
% SLIDE 18
%

\begin{slide}[R]{But watch for false positives}
\begin{minipage}[t]{5cm}
\begin{figure}[h!]
\includegraphics[width=5cm]{bifalse}
\end{figure}
\end{minipage}
\hfill
\begin{minipage}[t]{5cm}
\begin{figure}[h!]
\includegraphics[width=5cm]{elps}
\end{figure}
\end{minipage}
\end{slide}

%=======================================================================
%
% SLIDE 15
%

\begin{slide}[R]{Next:  spherical, spatial kinematics; start on constraint.}
\vspace{-0.5in}
\includegraphics[height=3.0in]{TOC5}
\end{slide}
\end{document}



\end{document}
