\documentclass[12pt]{article}

%\usepackage{psfig}
\usepackage{epsfig}
\usepackage{amsmath}
%%%--------- Set up page size and formatting information ----------%%%
\def\newpage{\par\vfill\eject}	%new page macro
\def\tab{\ \ \ \ \ }		%tab in five spaces
\def\dtab{{\tab}{\tab}}         %double tab

\setlength{\textwidth}{6.5in}
\setlength{\textheight}{9.0in}
\setlength{\topmargin}{-0.51in}
\setlength{\oddsidemargin}{0.0in}
\parskip 2ex
\parindent=0pt
%%%----------------------------------------------------------------------%%%
\begin{document}

\centerline{\bf ME 115(a):  Homework $\#4$}
\vskip 0.15 true in
\centerline{(Due Wednesday, March 12, 2008)}
\vskip 0.5 true in

%%%----------------------------------------------------------------------%%% 
{\bf Problem $\# 1$:} (15 points) Next quarter we will extensively study the
problem of grasping--i.e., how one can grab an object with fingers in
such as way as to prevent the grasped object from slipping out of the
grasp.  Consider the arrangement in Figure \ref{fig:figures}(a) where a
planar disc is touched by 3 ``planar'' fingers.  Assume that each
finger touches the disc with {\it frictionless} point contact.  Also
assume that each finger can apply any possible force to the object.

{\bf Question:} Is the disc immobilized?  That is, are there any free motions
of the disc that can not be prevented by the fingers?  In addition to an
intuitive discussion of this question, you must back up your answer with some
analysis.

%%%-----------------%%%
\begin{figure}[h]
\centerline{
\psfig{figure=grasp.ps,width=0.5\hsize}
\psfig{figure=cylin.ps,width=0.45\hsize}
}
\vskip -0.1 true in
\caption{(a) Grasp of a disc by frictionless fingers
         (b) Schematic of a ``Cylindrical Manipulator''}

\label{fig:figures}
\end{figure}
%%%-----------------%%%
\vskip 0.1 true in 
{\bf Problem $\# 2$:} (15 points) Figure \ref{fig:figures}(b) shows a schematic of
an 3-jointed ``cylindrical'' robot manipulator.  This manipulator
consists of two revolute joints (joints $\#1$ and $\#2$) and one
prismatic joint (the third joint).  All three joint axes are vertical
and parallel to each other.

\begin{itemize}
\item Derive the Denavit-Hartenberg parameters.
\item Derive the inverse kinematic solution, assuming that
the goal is to position the tool frame origin at some desired position,
$(x_T, y_T,z_T)$.
\end{itemize}

{\bf Problem $\# 3$:}  (20 points)

{\bf Part (a):} For Manipulator (iv) in Figure 3.23, solve the inverse
kinematic problem, where the goal is to place the origin of the tool frame
at a desired position.

{\bf Part (b):} For Manipulator (iii) in Figure 3.24 (i.e., the
``Stanford manipulator''), solve the inverse kinematics problem.  That
is, given a desired position and orientation of the tool frame, find
the joint variables that place that manipulator tool frame at that
location.  Note that the ``regional'' part of this manipulator (the part
of the linkage preceding the wrist) is exactly the same as part (a).


\end{document}
