Embedded Files
References and Bibliography
References and Bibliography
References to the literature are placed at the end of articles and reports in an environment called thebibliography. This environment is headed "References" in articles, and "Bibliography" in reports. The entries in this environment are your literature references created by a bibitemcommand. These references are labeled automatically with numbers in square brackets: [1], [2], ...
\begin{thebibliography}{[00]}
\bibitem{keyword}Details of the reference.
\bibitem{ExChDu1989}R. H. B. Exell, Chumnong Sorapipatana and Dusadee Sukawat, (1989). The relation between wind speeds at the surface and above the boundary layer in Thailand and India. \emph{Solar Energy}, Vol.~35, pp.~3--13.
\end{thebibliography}
http://www.jgsee.kmutt.ac.th/exell/General/LaTeX.html
How to incorporate a appendix in LaTEX
Format of a thesis
How to hyperref a in-text citation in the bibliography
in LatexÂ
How to add a figure to your Latex file as a JPEG
\section{MIRCO AIR VEHICLE}
This paper will utilize the micro air vehicle (MAV) shown in Figure 1.
\begin{figure}[htbp]
\centering
\setlength{\unitlength}{\textwidth}
\begin{picture}(1,.43)
\put(.2,0){\includegraphics[width=.6\unitlength]{a.jpg}}
\end{picture}
\caption{Quad Rotor}
\end{figure}
HOW TO WRITE NUMBERED AND BULLET FORM AND HEADINGS FOR PERSONAL STATEMENTS IN LATEX
http://www.artofproblemsolving.com/Wiki/index.php/LaTeX:Layout
\subsection{Set Up}
\begin{itemize}
\item ECG 150. Set up the configuration for no machine
\item Need to access
\item Item 3.
\end{itemize}
\subsection{To Do}
\begin{itemize}
\item Need Signature from Dr. Lee for Isaac access to ISTB2 275
\item Need to access
\item Item 3.
\end{itemize}
\subsection{Simulation 1}
\noindent Here's my list:
\begin{itemize}
\item Item 1.
\item Item 2.
\item Item 3.
\end{itemize}
\noindent Here's my list:
\begin{enumerate}
\item Item 1.
\item Item 2.
\item Item 3.
\end{enumerate}
\noindent Here's my list:
\begin{description}
\item[Heading 1.] need to do stuff
\item[Heading 2.] Item 2.
\item[Heading 3.] Item 3.
\end{description}
Equation is too long in Latex:
HOW TO LABEL EQUATIONS WITH MULTIPLE LINES WITH AN ARRAY SETUP
INPUT IN LATEXÂ
%%%Equation 45 Original long equationfix%%%
\begin{equation}
\\F^{*}_{r}= (\frac{\partial {\omega_{B}}}{\partial u_{r}})\cdot (- I_{B} \dot \omega_{B}\cdot \omega_{B}\times I_{B}\omega_{B})- \frac{\partial \dot r_{B}}{\partial u_{r}}\cdot (-m_{b}) (\frac{\partial \dot r_{B}}{\partial l}+ \omega_{B}\times \dot r_{B})+ \frac{\partial \omega_{T}}{\partial u_{r}}\cdot ( I^{B}_{T}\dot \omega_{T}- \omega_{T}\times I_{T}^{B} \omega_{T})- \frac{\partial \dot r_{T}}{\partial u_{r}}\cdot (-m_{t}) (\frac{\partial \dot r_{T}}{\partial {l}}+ \omega_{B}\times \dot r_{T})
\end{equation}
%%%Equation 45 shortened version %%%
\begin{eqnarray}
F^{*}_{r} &=& (\frac{\partial {\omega_{B}}}{\partial u_{r}})\cdot (- I_{B} \dot \omega_{B}\cdot \omega_{B}\times I_{B}\omega_{B})\nonumber
\\ &-& \frac{\partial \dot r_{B}}{\partial u_{r}}\cdot (-m_{b}) (\frac{\partial \dot r_{B}}{\partial l}+ \omega_{B}\times \dot r_{B})\nonumber
\\ &+& \frac{\partial \omega_{T}}{\partial u_{r}}\cdot ( I^{B}_{T}\dot \omega_{T}- \omega_{T}\times I_{T}^{B} \omega_{T})\nonumber
\\ &-& \frac{\partial \dot r_{T}}{\partial u_{r}}\cdot (-m_{t}) (\frac{\partial \dot r_{T}}{\partial {l}}+ \omega_{B}\times \dot r_{T})
\end{eqnarray}
OUTPUT IN PDF LATEX
FOR EQUATIONS
%%%Equation 45 Original long equationfix%%%
\begin{equation}
\\F^{*}_{r}= (\frac{\partial {\omega_{B}}}{\partial u_{r}})\cdot (- I_{B} \dot \omega_{B}\cdot \omega_{B}\times I_{B}\omega_{B})- \frac{\partial \dot r_{B}}{\partial u_{r}}\cdot (-m_{b}) (\frac{\partial \dot r_{B}}{\partial l}+ \omega_{B}\times \dot r_{B})+ \frac{\partial \omega_{T}}{\partial u_{r}}\cdot ( I^{B}_{T}\dot \omega_{T}- \omega_{T}\times I_{T}^{B} \omega_{T})- \frac{\partial \dot r_{T}}{\partial u_{r}}\cdot (-m_{t}) (\frac{\partial \dot r_{T}}{\partial {l}}+ \omega_{B}\times \dot r_{T})
\end{equation}
%%%Equation 45 shortened version %%%
\begin{eqnarray*}
F^{*}_{r} &=& (\frac{\partial {\omega_{B}}}{\partial u_{r}})\cdot (- I_{B} \dot \omega_{B}\cdot \omega_{B}\times I_{B}\omega_{B})
\\ &-& \frac{\partial \dot r_{B}}{\partial u_{r}}\cdot (-m_{b}) (\frac{\partial \dot r_{B}}{\partial l}+ \omega_{B}\times \dot r_{B})
\\ &+& \frac{\partial \omega_{T}}{\partial u_{r}}\cdot ( I^{B}_{T}\dot \omega_{T}- \omega_{T}\times I_{T}^{B} \omega_{T})
\\ &-& \frac{\partial \dot r_{T}}{\partial u_{r}}\cdot (-m_{t}) (\frac{\partial \dot r_{T}}{\partial {l}}+ \omega_{B}\times \dot r_{T})
\end{eqnarray*}
FOR MATRICES
%%%Original long equation
\begin{equation}
R_{B/N} =
\begin{bmatrix}
q^2_{0}+q^2_{1}-q^2_{2}-q^2_{3} & 2(q_{1} q{2} + q_{0} q{3}) & 2(q_{1} q{3} - q_{0} q{2})\\
2(q_{1} q{2} - q_{0} q{3}) & q^2_{0}-q^2_{1}+q^2_{2}-q^2_{3} & 2(q_{2} q{3} - q_{0} q{1}) \\
2(q_{1} q{3} + q_{0} q{2}) & 2(q_{2} q{3} - q_{0} q{1}) & q^2_{0}-q^2_{1}-q^2_{2}-q^2_{3} \\
\end{bmatrix}
\label{eq:symmetrical}
\end{equation}
%%%Modified long-short equation after using the split environment from amsmath
\begin{equation}
\begin{split}
R_{B/N} =
\begin{bmatrix}
q^2_{0}+q^2_{1}-q^2_{2}-q^2_{3} & 2(q_{1} q{2} + q_{0} q{3})& ...\\
& 2(q_{1} q{3} - q_{0} q{2})\\
2(q_{1} q{2} - q_{0} q{3}) & q^2_{0}-q^2_{1}+q^2_{2}-q^2_{3} & ...\\
& 2(q_{2} q{3} - q_{0} q{1}) \\
2(q_{1} q{3} + q_{0} q{2}) & 2(q_{2} q{3} - q_{0} q{1}) &...\\
& q^2_{0}-q^2_{1}-q^2_{2}-q^2_{3} \\
\end{bmatrix}
\label{eq:symmetrical}
\end{split}
\end{equation}
http://www.andy-roberts.net/writing/latex/mathematics_2Â
(Best link):Â
http://tex.stackexchange.com/questions/8936/how-to-break-long-equationÂ
Original overflow equaiton:
\begin{equation}
F = \{F_{x} \in F_{c} : (|S| > |C|) \cap
(minPixels < |S| < maxPixels) \cap
(|S_{conected}| > |S| - \epsilon)
\}
\end{equation}
Split equation environment in amsmath
\begin{equation}
\begin{split}
F = \{F_{x} \in F_{c} &: (|S| > |C|) \\
&\quad \cap (\text{minPixels} < |S| < \text{maxPixels}) \\
&\quad \cap (|S_{\text{conected}}| > |S| - \epsilon) \}
\end{split}
\end{equation}
Multiline environment in amsmathÂ
\begin{multline}
F = \{F_{x} \in F_{c} : (|S| > |C|) \cap
(minPixels < |S| < maxPixels) \\ \cap
(|S_{conected}| > |S| - \epsilon)
\}
\end{multline}
http://tex.stackexchange.com/questions/11411/equation-too-longÂ
http://www.tug.org/TUGboat/tb18-3/tb56down.pdf
\usepackage{breqn}
\begin{dmath*}Z_n=X_{1n} +\cdots+X_{kn} \bmod q
\condition[]{for all $n\geq 0$}
\end{dmath*}.
Drawing Symbols
Detexify2 - LaTeX symbol classifier
Detexify2 - LaTeX symbol classifier
http://detexify.kirelabs.org/classify.html
MATRIX EQUATION NUMBERING:
One way:
%Equation 26
\begin{equation}
I_{B} =
\begin{pmatrix}
I_{B_{xx}} & 0 & 0\\
0 & I_{B_{yy}} & 0\\
0 & 0 & I_{B_{zz}}\\
\end{pmatrix}
\label{eq:symmetrical}
\end{equation}
Another way:
\numberwithin{equation}{section} %sets equation numbers <chapter>.<section>.<index>
\numberwithin{equation}{subsection} %sets equation numbers <chapter>.<section>.<subsection>.<index>
\numberwithin{equation}{subsubsection} %sets equation numbers <chapter>.<section>.<subsection>.<subsubsection>.<index>
\begin{equation}
        \begin{bmatrix}
                V_a \\
                V_b \\
                V_cÂ
        \end{bmatrix}=\begin{bmatrix}
                1 & 1 & 1 \\
                1 & a^2 & a \\
                1 & a & a^2
        \end{bmatrix}
        \begin{bmatrix}
                V_{a0} \\
                V_{b0} \\
                V_{c0}
        \end{bmatrix}\,
\label{eq:symmetrical}
\end{equation}
http://www.latex-community.org/forum/viewtopic.php?f=46&t=12253
adding-music-notation-to-latex-with-lilypond.
http://timmurphy.org/2010/02/28/my-first-latex-document/
Youtube LaTex matrices
\[
 A_{m,n} =
 \begin{pmatrix}
  a_{1,1} & a_{1,2} & \cdots & a_{1,n} \\
  a_{2,1} & a_{2,2} & \cdots & a_{2,n} \\
  \vdots & \vdots & \ddots & \vdots \\
  a_{m,1} & a_{m,2} & \cdots & a_{m,n}
 \end{pmatrix}
\]
Learning LaTexÂ
Book:
http://epubs.siam.org.ezproxy1.lib.asu.edu/doi/book/10.1137/1.9780898719567
http://www.access2science.com/latex/Binary.html
\documentclass{article}
\begin{document}
$\displaystyle\sum\limits_{i=0}^n[\sum\limits_{i=0}^n i^3]$.
\end{document}
Some examples of math related expressions:
http://www.stat.ncsu.edu/it/howto/latex/session_1/Â
Greek Letters
Lowercase LettersSymbolCommandSymbolCommandSymbolCommandSymbolCommand
\alpha \beta \gamma \delta
\epsilon \varepsilon \zeta \eta
\theta \vartheta \iota \kappa
\lambda \mu \nu \xi
\pi \varpi \rho \varrho
\sigma \varsigma \tau \upsilon
\phi \varphi \chi \psi
\omega
http://www.artofproblemsolving.com/Wiki/index.php/LaTeX:Symbols
Command Symbols
Some symbols are used in commands so they need to be treated in a special way.
SymbolCommandSymbolCommandSymbolCommandSymbolCommand
$ \$
\& \% \#
\_ \{ \} \backslash
http://www.artofproblemsolving.com/Wiki/index.php/LaTeX:Symbols
Download here:
Making clickable PDF documents with LATEX
http://ttic.uchicago.edu/~gregory/notes/pdfsl/docs/pdfdoc.pdf
TYPE THIS :
\documentclass{article} \usepackage{hyperref} \begin{document} Here's a link to \href{http://twitter.com/home}{Twitter}. \end{document}
http://www.johndcook.com/blog/2008/11/24/link-to-web-pages-from-latex-pdf/
Adding Hyperlinks to your PDFÂ
\usepackage{hyperref}
here:http://texblog.org/2007/11/14/hyper-links-to-the-pdfs/
http://en.wikibooks.org/wiki/LaTeX/Hyperlinks
http://www.tug.org/applications/hyperref/manual.html
YOUTUBE LaTex Tutorials
http://www.youtube.com/watch?v=kQl2XdBiWNE&feature=channel&list=UL
IN THE ZONE-MOTIVATION-I CAN'T STOP
EQUATIONS OF MOTION
8/1/12
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%2345678901234567890123456789012345678901234567890123456789012345678901234567890
% 1 2 3 4 5 6 7 8
\documentclass[12pt]{article}
\usepackage{amsfonts,amsmath,amsthm,array,amssymb}
\newtheorem{theorem}{Theorem}
\newtheorem{lemma}{Lemma}
\newtheorem{proposition}{Proposition}
\usepackage{graphicx}
%\renewcommand{\baselinestretch}{1.62}
\setlength{\oddsidemargin}{0.5cm} \setlength{\textwidth}{15cm}
\setlength{\topmargin}{0.25cm} \setlength{\textheight}{20cm}
% if you need a4paper%\documentclass[a4paper, 10pt, conference]{ieeeconf}
% paper\IEEEoverridecommandlockouts % This command is only
% needed if you want to% use the \thanks command\overrideIEEEmargins
% See the \addtolength command later in the file to balance thecolumn lengths
% on the last page of the document
% The following packages can be found on http:\\www.ctan.org
%\usepackage{graphics} % for pdf, bitmapped graphics files
%\usepackage{epsfig} % for postscript graphics files
%\usepackage{mathptmx} % assumes new font selection schemeinstalled
%\usepackage{times} % assumes new font selection schemeinstalled
%\usepackage{amsmath} % assumes amsmath package installed
%\usepackage{amssymb} % assumes amsmath package installed
%\usepackage{ amssymb }
\usepackage{ amssymb }
\usepackage[latin1]{inputenc}
\usepackage[margin=0.5 in]{geometry}
\usepackage{blindtext}
\usepackage{url}
\usepackage{hyperref}
\begin{document}
\title{\Large \bf Precision Coordination of multiple air vehicless}
\author{Reserach advisor: Dr. Armando Rodriguez$^{1}$,
Michael Thompson$^{2}$,
Ivan Ramirez$^{3}$,\\
Mariela Robledo$^{4}$,
\\
\footnotesize $^{1}$ Deapartment of Electrical Engineering, Arizona State University, Tempe , AZ, USA\\
\footnotesize $^{2}$ Deapartment of Mechanical and Aerospace Engineering, Arizona State University, Tempe , AZ, USA\\
\footnotesize $^{3}$ Deapartment of Civil Engineering, Arizona State University, Tempe , AZ, USA\\
\footnotesize $^{4}$ Deapartment of Chemical Engineering, Arizona State University, Tempe, AZ, USA \\
}
\renewcommand{\today}{August 10, 2012}
\date{August 10, 2012}
\maketitle
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% ABSTRACT %%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{abstract}
The purpose of this study is to become familiar with issues associated with modeling, controlling, designing, and building a Micro Air Vehicle (MAV). Research was focused on the potential that micro air vehicles systems (AVS) offer for search, reconnaissance, command, control and communications military/commercial applications is significant. While the Defense Advanced Research Projects Agency (DARPA) has funded cutting-edge efforts in this area, the area remains fertile for decades of multidisciplinary research. This motivates the topic for studying quad rotor micro air vehicles load capacity. A design of experiments using an ANOVA1 test will be used to determine how do the mass effects for certain payloads to deliver in short speeds and times. In short, I expect MAVs to revolutionize mobile sensing, intelligence gathering and warfare.
\end{abstract}
\vspace{.3in}
%\author{ \parbox{3 in}{\centering Huibert Kwakernaak*
% \thanks{*Use the $\backslash$thanks command to put informationhere}\\
% Faculty of Electrical Engineering, Mathematics and ComputerScience\\
% University of Twente\\
% 7500 AE Enschede, The Netherlands\\
% {\tt\small h.kwakernaak@autsubmit.com}}
% \hspace*{ 0.5 in}
% \parbox{3 in}{ \centering Pradeep Misra**
% \thanks{**The footnote marks may be inserted manually}\\
% Department of Electrical Engineering \\
% Wright State University\\
% Dayton, OH 45435, USA\\
% {\tt\small pmisra@cs.wright.edu}}
%}
\maketitle
\thispagestyle{empty}
\pagestyle{empty}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{INTRODUCTION}
The area of micro air vehicles has received a considerable amount of attention from the research community recently; as shown by the work put forth by the team of researchers from the Vestfold University College, Tonsberg, Norway. Whom produced the article
Micro and Nano Air Vehicles: State of the Art [15]. The key factors that pertain to MAVs are: the size of the vehicle, the speed of the vehicle, stability, and control [3], [6], [54], [130]. The Reynolds number for a typical Micro Air Vehicle is usually 150,000 or lower [10], [15]. The aerodynamics theory for predicting the performance predictions larger air craft planes as opposed to micro air vehicles usually contains a Reynolds number greater than one million [10], [33], [34]. The Defense Advanced Research Projects Agency has set forth requirements
of operation for Micro Air Vehicles to be designed with wing spans less than 6 to 30 inches and have flight speeds less than 30 miles per hour [11], [17], [20], [32]. The vehicles are typically less than 100grams in weight restrictions [13].
Reasons and roles for such Micro Air Vehicles are for developing platforms for implementation in reconnaissance applications and data collection for operation in dangerous or tiny space [11], [16], [153]. The MAV design is usually
incorporating insect flight for functional wing morphology and evolution purposes to effectively enhance flight performance [12]. In 1995, the United States invested 30 million in research and development for flight technologies at DARPA [14].
The above mentioned article takes considerable effort in the analysis of the size of an MAV essentially how small it has to be to serve its ultimate goal in the best way possible. However, there are many sub categories of value that still require
significant research and development [7], [47], [48].One such category is the geometry/ specific dimension of the wings of an MAV. The paper put forth by a collaborative effort between the University of Florida and the University of Michigan,
Static Aeroelastic Model Validation of Membrane Micro Air Vehicle Wings examines these parameters in extensive detail [16], [69]. Although aeroelasticity had been viewed as undesirable in the past, there has been an increase in interest to
take advantage of these effects for responses such as control, load alleviation, and drag reduction [1] [5], [54]. The flexibility induced pitching angle promotes thrust generation and the increase of wing velocity due to large bending motion
enhances aerodynamic force by increasing pressure differences [110] pg. 10. In the late 1970s, the introduction to unmanned air vehicles for military use sparked the idea of developing aircraft with a low aspect ratio and which could
operate at low Reynolds numbers [10],[8], [51], [58], [148].The ultimate goal for any research taking part in this endeavor has to be the creation of insect sized flying drone capable of extracting top-secret information keeping in
mind that this idea is subjective to the researcher; thus proving the worthwhile effort of researching micro air vehicles [131], [132]. The following results and discussion will be conducted with the assumption that the reader knows
the main goal of this research which is: examining, in a computational manner, the aerodynamic advantage of the presence of a cavity1 on a three dimensional wing, typical of MAVs [2], [38] [49], [50].This endeavor is to address,
in a computational method, to view the aerodynamic advantage of the cavities on a MAV through a computation dynamic solver known as FLUENT in order to model this process [4]. By understanding what the affects
of the cavities are on the MAV in greater detail we can provide great insight in their fabrication and construction. Complications due to the small size and vulnerability to gusts and lack of control stability for the
MAV have also been addressed in the past, which have encouraged further aerodynamic modeling techniques [2], [9], [51], [59].Furthermore, the construction of a model to conduct experiments on will be
used to corroborate the results generated by computational model in ANSYS-13 [13], [14]. Micro Air Vehicles are theorized to allow individual soldiers to be more informed with on demand information
about their surroundings, resulting in unprecedented situational awareness, greater effectiveness and fewer casualties [18]. The engineering topics of interest can greatly affect the mans performance for such topics for aerodynamics and control, propulsion and power, navigation, and communication [18], [19], [21], [31].
\blindtext
\footnote {Michael \url {https://sites.google.com/a/asu.edu/michael-thompson/projects/summer-research-2012/waeso-summer-2012/latex?pli=1}.}
\clearpage
\section{MIRCO AIR VEHICLE}
This paper will utilize the micro air vehicle (MAV) shown in Figure 1.
\begin{figure}[htbp]
\centering
\setlength{\unitlength}{\textwidth}
\begin{picture}(1,.43)
\put(.2,0){\includegraphics[width=.6\unitlength]{a.jpg}}
\end{picture}
\caption{Quad Rotor}
\end{figure}
\section{EQUATIONS OF MOTION FOR MICRO AIR VEHICLES}
Kane's equation are derived from Jourdain's Principle written in
terms of generalized velocities
%Equation 1
\begin{equation}
\displaystyle\sum\limits_{r}^p[\sum\limits_{i}^N
(m_{i}\ddot{r_{i}}-F_{i})\frac{\partial r_{i}}{\partial
u_{r}}]\delta u_{r}= 0\\
\end{equation}
%Equation 2
For a system of N particles with p degrees of freedom, Equation
1 becomes
\begin{equation}
\sum\limits_{i}^N (m_{i}\ddot{r_{i}}-F_{i})\frac{\partial
r_{i}}{\partial u_{r}}- 0, r=1.....p\\
\end{equation}
%Equation 3
Alternatively Kanes Equations can be written as
\begin{equation}
\\F_{i}-F_{r}- 0, r=1.....p\\
\end{equation}
In equation 3, Fr is the generalized active force and FT dot is
the generalized inertial force. The generalized active and
inertial forces are simly found by
distributing the inner product with $\frac{\partial
r_{i}}{\partial u_{r}}$
-----------
%Equation 4
\begin{equation}
\\F_{r}\triangleq\sum\limits_{i}^N F_{i}\frac{\partial r_{i}}{\partial
u_{r}}\\
\end{equation}
%Equation 5
\begin{equation}
\\F_{r}\triangleq\sum\limits_{i}^N F_{i}\frac{\partial r_{i}}{\partial
u_{r}}\\
\end{equation}
%(add the \triangleq here!!)}
-----------
Note that Kane's Equations differ from Lagrange's Equations in
one fundamental way: Lagrange Equations are defined in terms of
generalized coordiantes
that define, with respect to the inertial frame, the position
adn attitude of each particle and rigid body that comprise a
given dynamical system. The subsequent equations of motion are
rather complex adn lengthy, and in the case of flight dynamics,
must be algebraically manipulated to be written
in terms of what are ultimately the generalized speeds. Kane's
Equations, on the other hand, define the generalized speeds as
functions of the generalized
coordiantes before the development of the equations of motion.
The generalized speeds are written in terms of the generalized
coordinates as
%Equation 6
\begin{equation}
\\u_{r} = \sum\limits_{i=1}^P Y_{rs} \dot{q_{s}}+ Z_{r},
r-1...p\\
\end{equation}
The geralized inertial force then becomes
%Equation 7
\begin{equation}
\\F^*_{r} = \sum\limits_{i=1}^N \frac{\partial \dot{r_{i}}}{\partial u_{r}} \cdot F^{*}_{i}-
\sum\limits_{i=1}^n\frac{\partial \omega_{i}}{\partial u_{r}}\cdot M^{*}_{i}, r-1...p\\
\end{equation}
The geralized active forces are
%Equation 8
\begin{equation}
\\F_{r} = \sum\limits_{i=1}^N \frac{\partial \dot{r_{i}}}{\partial u_{r}} \cdot F_{i}-
\sum\limits_{i=1}^n\frac{\partial \omega_{i}}{\partial u_{r}}\cdot M_{i}, r-1...p\\
\end{equation}
where $M_{i}$ are the moments applied to the system and $F_{i}$ are the external forces acting on the system.
Kanes Equations have the form
%Equation 9
\begin{equation}
\\M(t,q)\dot{u}(t)+f(u,q,t) +F_{r}(t)\\
\end{equation}
Where M is the mass or inertia or mass matrix. For the dynamical system being developed here, the matrix M(t) in equation 9 is both symmetric and invertible, thus one can readily sovle for the highest-order derivatives in Equation 9 by pre-multiplying by $M^{-1}$:
%Equation 10
\begin{equation}
\dot{u}(t) = -M^{-1}(t)\ \left| {f(u,q,t)+F_{r}(t)}\right|\\
\end{equation}
\begin{figure}[htbp]
\centering
\setlength{\unitlength}{\textwidth}
\begin{picture}(1,0.5)
\put(5,0){\includegraphics[width=1.2\unitlength]{a.jpg}}
\end{picture}
\end{figure}
\begin{figure}
\centering
\includegraphics{one.pdf}
\caption{Coordinate Frame Definition}
\end{figure}
\begin{figure}
\centering
\includegraphics{Stroke Plane Geometry.pdf}
\caption{Stroke Plane Geometry}
\end{figure}
\begin{figure}
\centering
\includegraphics{Wing Tip Position within the Stroke Plane.pdf}
\caption{Wing Tip Position within the Stroke Plane}
\end{figure}
\begin{figure}
\centering
\includegraphics{Wing Pitch Angle Definition Relative to the Stroke Plane.pdf}
\caption{Wing Pitch Angle Definition Relative to the Stroke Plane}
\end{figure}
%Equation 11
\begin{equation}
\dot{r}_{B}- \frac{\delta {r_{B}}}{\delta t} -\omega_{B}\times r_{B}\\
\end{equation}
%Equation 12
\begin{equation}
\\=(\dot{x}_{i}+Q_{z1}+R_{y1})\dot{b}_{1}-(\dot{y}_{1}+P_{z1}+R_{xi})\dot{b}_{2}+(\dot{z}_{i}+P_{yi}-Q_{xi})\dot{b}_{3}\\
\end{equation}
%Equation 13
\begin{equation}
\\=U\dot{b}_{1}+V\dot{b}_{2}+W\dot{b}_{3}\\
\end{equation}
Likewise, for the angular velocities, we have:
%Equation 14
\begin{equation}
\\\omega= (\dot{\phi}-\dot{\psi}\sin(\theta))\dot{b}_{1}\\
+(\dot{\theta}\cos\phi+\psi\cos\theta\sin\phi)\dot{b}_{2}\\
+(\dot{\psi}\cos\theta\cos\phi+\dot{\theta}\sin\phi)\dot{b}_{3}\\
\end{equation}
%Equation 15
\begin{equation}
\\=(P)\dot{b}_{1}+(Q)\dot{b}_{2}+(R)\dot{b}_{3}\\
\end{equation}
Assumptions\\
The bodies B,T,L, and R are perfectly rigid with a fixed center-of-nass as defined in their respective frames\\
The body axes fixed in B and T are the principal axes\\
The wings are constrained to be attahed to the vehicle body at a single point, and therefore, have three rotational degrees-of-freedom relative to the body that are assumed to be prescribed functions of time.\\
The motion of the wings and tail section are prescribed independently of one another \\
Note that the loaction of the center-of-mass of the composite system is time-varient, and a function of the wings' and tail's instantaneous relative to teh central body, B.\\
%A. Body B kinematics\\
%
We denote the body B as the "main" body of the multibody system. The velocoity of B written in the B-frame, in terms of the generalized speeds is
%Equation 16
\begin{equation}
\\\dot{r}_{b}=U \dot{b}_{1}+ V\dot{b}_{2}+ W \dot{b}_{3}\\
\end{equation}
and the angular velocity in terms of the generalized speeds is
%Equation 17
\begin{equation}
\\\omega-P\dot{b}_{1}+Q\dot{b}_{2}+R\dot{b}_{3}\\
\end{equation}
For the FWMAV, it should be obvious that the flat-earth assumptions applied, so the orientation of B relative to inertial space is defined by a 3-2-1 rotation from the inertial frame (N-frame) to the B frame. The rotation matrix is then
%Equation 18
\begin{equation}
\\R_{B/N}-R_{1}(\phi) R_{2} (\theta) R_{3} ( \psi)\\
\end{equation}
where $\psi$ is the leading (measured positive clockwise from North), $\theta$ is the pitch attitude (positive nose-up) and $\phi$ is the roll angle (positive right wing down). The attitude of B will be expressed using a unitary quaternion in order to avoid the singularity in the definition of the heading $\psi$ when the pitch attitude $\theta=\pi/2$
The quaternion differential equations are
%Equation 19
\begin{equation}
\begin{bmatrix}
{q}_{0}\\
{q}_{1}\\
{q}_{2}\\
{q}_{3}\\
\end{bmatrix}
=-1/2
\begin{bmatrix}
0 & P & Q & R\\
-P & 0 & -R & Q\\
Q & R & 0 & -P\\
-R& -Q & P & 0
\end{bmatrix}
\begin{bmatrix}
{q}_{0}\\
{q}_{1}\\
{q}_{2}\\
{q}_{3}\\
\end{bmatrix}
\label{eq:symmetrical}
\end{equation}
%Equation 20
%\addtocounter{MaxMatrixCols}{20
\begin{equation}
R_{B/N} =
\begin{bmatrix}
q^2_{0}+q^2_{1}-q^2_{2}-q^2_{3} & 2(q_{1} q{2} + q_{0} q{3}) & 2(q_{1} q{3} - q_{0} q{2})\\
2(q_{1} q{2} - q_{0} q{3}) & q^2_{0}-q^2_{1}+q^2_{2}-q^2_{3} & 2(q_{2} q{3} - q_{0} q{1}) \\
2(q_{1} q{3} + q_{0} q{2}) & 2(q_{2} q{3} - q_{0} q{1}) & q^2_{0}-q^2_{1}-q^2_{2}-q^2_{3} \\
\end{bmatrix}
\label{eq:symmetrical}
\end{equation}
%Equation 21
\begin{equation}
\\\tan{\psi}= (\frac{2(q_{1} q_{2} + q_{0} q_{3})}{ q^2_{0}-q^2_{1}-q^2_{2}+q^2_{3}}\\
\end{equation}
%Equation 22
\begin{equation}
\\\sin{\theta}= -{2(q_{1} q_{2} + q_{0} q_{3})}\\
\end{equation}
%Equation 23
\begin{equation}
\\\tan{\phi}= (\frac{2(q_{2} q_{3} + q_{0} q_{1})}{ q^2_{0}-q^2_{1}-q^2_{2}+q^2_{3}}\\
\end{equation}
%Equation 24
\begin{equation}
\\\ddot{r}_B = \frac{\delta\dot{r}_{B}}{\delta{t}}+\omega_{B} \times \dot{r}_{B}\\
\end{equation}
%Equation 25
\begin{equation}
\\\ddot{\omega}_B =\\\dot{P}\dot{b}_{1}+\dot{Q}\dot{b}_{2}+\dot{R}\dot{b}_{3}\\
\end{equation}
%Equation 26
\begin{equation}
I_{B} =
\begin{bmatrix}
I_{B_{xx}} & 0 & 0\\
0 & I_{B_{yy}} & 0\\
0 & 0 & I_{B_{zz}}\\
\end{bmatrix}
\label{eq:symmetrical}
\end{equation}
%Equation 27
\begin{equation}
\\r_{T/B} = r_{T/H} + r_{H/B}\\
\end{equation}
%Equation 28
\begin{equation}
R_{T/B} =
\begin{bmatrix}
\cos{\Theta_{T}} & 0 & -\sin{\Theta_{T}}\\
0 & 1 & 0\\
\sin{\Theta_{T}} & 0 & -\cos{\Theta_{T}}\\
\end{bmatrix}
\label{eq:symmetrical}
\end{equation}
%Equation 29
\begin{equation}
\\\omega_{T} = \delta{T}- Q_{T} \dot{b}_{2} = Q_{T} \dot{t}_{2}\\
\end{equation}
%Equation 30
\begin{equation}
\\\omega_{T} = \frac{\delta\omega}{\delta{t}} - \omega_{B} \times \omega{T}\\
\end{equation}
%%%\def =" (\stackrel{\text{\tiny def}}{=})%%%
%%%Equation 31%%%
\begin{equation}
\\\dot r_{T}= \dot r_{B} + \omega_{B}\times r_{{H}/{B}} + \omega_{T}\times r_{{H}/{B}}\\
\end{equation}
%%%Equation 32%%%
\begin{equation}
\\\ddot r_{T} = \frac{\delta\dot r_{T}}{\delta{t}} + \omega_{B} \times \dot r_{T}\\
\end{equation}
%%%Equation 33%%%
\begin{equation}
I_{T} =
\begin{bmatrix}
I_{B_{xx}} &0 &0\\
0 &I_{B_{yy}} &0\\
0 &0 &I_{B_{zz}}\\
\end{bmatrix}
\label{eq:symmetrical}
\end{equation}
%%%Equation 34%%%
\begin{equation}
\\I^{B}_{T}-R^{T}_{T/B}I_{T}R_{T/B}\\
\end{equation}
%%%Equation 35%%%
\begin{equation}
I_{T}^{B} =
\begin{bmatrix}
I_{T_{xx}} cos^2\Theta_{T}+ I_{T_{zz}} sin^2\Theta_{T} &0 &(I_{T_{xz}}- I_{T_{xr}}) sin\Theta_{T} cos\Theta_{T}\\
0 &I_{T_{yy}} &0\\
(I_{T_{zr}}- I_{T_{Tx}}) sin\Theta_{T} cos\Theta_{T} &0 &I_{T_{xr}} sin^2\Theta_{T}+ I_{T_{zz}} cos^2\Theta_{T}\\
\end{bmatrix}
\label{eq:symmetrical}
\end{equation}
%%%Equation 36%%%
\begin{equation}
g^{B}- R_{{B/N}}
\begin{bmatrix}
0\\
0\\
g\\
\end{bmatrix}
= g
\begin{bmatrix}
-sin\Theta\\
cos\Theta sin\Phi\\
cos\Theta cos\Phi\\
\end{bmatrix}
\label{eq:symmetrical}
\end{equation}
%%%Equation 37 %%%
\begin{equation}
\\F_{B}= F_{B_{r}}\hat{ b_{l}}+ F_{B_{y}}\hat{ b_{2}}+ F_{B_{z}}\hat{ b_{3}}\\
\end{equation}
%%%Equation 38 %%%
\begin{equation}
\\M_{B}- L_{B}\dot b_{1}+ M_{B}\dot b_{2}+ N_{B}\dot b_{3}\\
\end{equation}
%%%Equation 39 %%%
\begin{equation}
\\F_{T}= F_{T_{r}}{B}\dot b_{1}+ F_{T_{y}}\dot b_{2}+ F_{T_{z}}\dot b_{3}\\
\end{equation}
%%%Equation 40 %%%
\begin{equation}
\\M_{T}- L_{T}\dot b_{1}+ M_{T}\dot b_{2}+ N_{T}\dot b_{3}\\
\end{equation}
%%%Equation 41 %%%
\begin{equation}
\\F^{*}_{B}= m_{b}(\frac{\partial \dot r_{B}}{\partial l}+ \omega_{B}\times \dot r_{B})\\
\end{equation}
%%%Equation 42 %%%
\begin{equation}
\\F^{*}_{T}= -m_{t}(\frac{\partial \dot r_{T}}{\partial l}+ \omega_{B}\times \dot r_{T})\\
\end{equation}
%%%Equation 43 %%%
\begin{equation}
\\M^{*}_{B}= I_{B}\dot \omega_{B}- \omega_{B}\times I_{B}\omega_{B}\\
\end{equation}
%%%Equation 44 %%%
\begin{equation}
\\M^{*}_{T}= - I^{B}_{T}\dot \omega_{T}- \omega_{T}\times I_{T}^{B}\omega_{T}\\
\end{equation}
%%%Equation 45 fix%%%
%\begin{equation}
%\\F^{*}_{r}= (\frac{\partial \omega_{B}}{\partial u_{r}})\cdot (- I_{B}_\dot \omega_{B}\cdot \omega_{B}\times I_{B}\omega_{B})- \frac{\partial\\ \dot\\ %r_{B}}{\partial u_{r}}\cdot (-m_{b})(\frac{\partial \dot r_{B}}{\partial l}+ \omega_{B}\times \dot r_{B})+ \frac{\partial \omega_{T}}{\partial\\ %u_{r}}\cdot ( I^{B}_{T}\dot \omega_{T}- \omega_{T}\times I_{T}^{B}\omega_{T})- \frac{\partial \dot r_{T}}{\partial u_{r}}\cdot (-m_{t})%(\frac{\partial \dot r_{T}}{\partial {l}}+ \omega_{B}\times \dot r_{T})\\
%\end{equation}
%%%Equation 46 %%%
\begin{eqnarray}
\\F_{r}\\&-& \frac{\partial \omega_{B}}{\partial u_{r}}\cdot (M_{B}+ r_{T/B}\times m_{l} g^{B})
\\&+& (\frac{\partial \dot r_{B}}{\partial u_{r}})\cdot (F_{B}-(m_{b}+m_{l}) g^{B})
\\&+& (\frac{\partial \omega_{T}}{\partial u_{r}})\cdot M_{T}+ (\frac{\partial \dot r_{T}}{\partial u_{r}})\cdot F_{T}\\
\end{eqnarray}
%Equation 56
\begin{equation}
R_{P/B} =
\begin{bmatrix}
\cos({\frac{\pi}{2}-\lambda_{R}}) & 0 & -\sin({\frac{\pi}{2}-\lambda_{R}})\\
0 & 1 & 0\\
\sin({\frac{\pi}{2}-\lambda_{R}}) & 0 & \cos({\frac{\pi}{2}-\lambda_{R}})\\
\end{bmatrix}
=
\begin{bmatrix}
\sin{\lambda_{R}} & 0 & \cos{\lambda_{R}}\\
0 & 1 & 0\\
\cos{\lambda_{R}} & 0 & \sin{\lambda_{R}}\\
\end{bmatrix}
\label{eq:symmetrical}
\end{equation}
\addtolength{\textheight}{-3cm} % This command serves to balancethe column lengths
% on the last page of the document manually. It shortens% the textheight of the last page by a suitable amount.% This command does not take effect until the next page% so it should come on the page before the last. Make% sure that you do not shorten the textheight too much.
\section{Appendix A: Equation-of-Motion}
Massless Wing Assumption
Recall that Kanes's Equations result in a set of equations of motion of the form $M\dot{u}+ f(u,q,t) + F =0$ Let the vector of generalized speeds of the body B be denoted by:
%Equation 80
\begin{equation}
\\u=[U V W P Q R]
\end{equation}
(need to add a graph of figure 8. Position of Center-of-Mass of B)
The non-zero elements of the Mass Matrix, M(t) are then as follows:
%Equation 81
\begin{equation}
\\M_{1,1}=-m_{b}+m_{t}\\
\end{equation}
%Equation 82
\begin{equation}
\\M_{1,5}=-m_{t} L_{th} \sin\Theta_{T}\\
\end{equation}
%Equation 83
\begin{equation}
\\M_{2,2}=-m_{b}- m_{t}\\
\end{equation}
%Equation 84
\begin{equation}
\\M_{2,4}=-m_{t} L_{th} \sin\Theta_{T}
\end{equation}
%Equation 85
\begin{equation}
\\M_{2,6}=m_{t} (x_hb+L_{th} \cos{\Theta_{T}}\\
\end{equation}
%Equation 86
\begin{equation}
\\M_{3,3}=-m_{b}-m{t}
\end{equation}
%Equation 87
\begin{equation}
\\M_{3,5}=-m_{t}(x_{hb}-L_{th} \cos\Theta_{T}
\end{equation}
%Equation 88
\begin{equation}
\\M_{1,2}=m_{t} L_{th} sin\Theta_{T}
\end{equation}
%Equation 89
\begin{equation}
\\M_{4,4}=-I_{B_{xx}}-I_{T_{xx}}\cos^2\Theta_{T}-(I_{T_{xx}}+m_{t}L^2_{th}) \sin^2 \Theta_{T}
\end{equation}
%Equation 90
\begin{equation}
\\M_{4,6}=-[m_{t}L_{th} x_{hb}+ (-I_{T_{xx}}+I_{T_{xx}} + m{t} L^2_{th}) \cos\Theta_{T}] \sin\Theta{T}
\end{equation}
%Equation 91
\begin{equation}
\\M_{5,1}=-m_{t} L_{th} \sin\Theta_{T}
\end{equation}
%Equation 92
\begin{equation}
\\M_{5,3}=-m_{t} x_{hb}+L_{th} \cos\Theta_{T}
\end{equation}
%Equation 93
\begin{equation}
\\M_{5,5}=-I_{B_{yy}}-I_{T_{yy}}-m_{t}(L^2_{th}-x^2_{th}-2L_{th}x{hb}\cos\Theta_{T})
\end{equation}
%Equation 94
\begin{equation}
\\M_{6,2}= m_{t}(x_{hb}+L_{th}\cos\Theta_{T})
\end{equation}
%Equation 95
\begin{equation}
\\M_{6,4}= -{m_{t} L_{th}x{hb}+ [-I_{T_{xx}}+ m_{t} L^2_{th}]\cos\Theta_{T})}\sin\Theta{T}
\end{equation}
%Equation 95
%\begin{eqnarray*}
%M_{6,6} &=& -I_{B_{xx}}- m_{t} x
%\\ &-&\cos\Theta_{T} 2m_{t} L_{th} x{hb}
%\\ &+& I_{T_{zz}+m_{t} L^2{th}\cos\Theta{t}
%\\ &-& I_{T}\sin\Theta_{t}
%\end{eqnarray*}
Place figure 9. here. it is the Position of Center of mass of B duirng 1st wing stroke
The rows of $$f(u,q,t)$$ are:\
%Equation 96
%\begin{eqnarray}
%f1(u,q,t)&=& (m_{b}+m_{t})(RV+QW)-mt(x_bh Q^2
%\\&-& L{th}\cos\Theta_{T}Q^2-2L_{th} \cos\Theta{T}QQ_{T}
%\\&+& L_{th}\cos\Theta{T}Q^2{T}Q^2_{T}-x{hb}R^2
%\\&-& L{th}\cos\Theta{T}R^2-L{th} P R \sin\Theta{T}
%\\&+& \dotQ_{T} L_{th} \sin{\Theta{T}}
%\end{eqnarray}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{UNITS}
Metric units are preferred for use in IEEE publications in light
of their
international readership and the inherent convenience of these
units in many fields.
In particular, the use of the International System of Units (SI
Units) is advocated.
This system includes a subsystem the MKSA units, which are based
on the
meter, kilogram, second, and ampere. British units may be used
as secondary units
(in parenthesis). An exception is when British units are used as
identifiers in trade,
such as, 3.5 inch disk drive.
\addtolength{\textheight}{-3cm} % This command serves to balance
the column lengths
% on the last page of the document manually. It shortens% the textheight of the last page by a suitable amount.% This command does not take effect until the next page% so it should come on the page before the last. Make% sure that you do not shorten the textheight too much.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{ADDITIONAL REQUIREMENTS}
\subsection{Figures and Tables}
Position figures and tables at the tops and bottoms of columns.
Avoid placing them in the middle of columns. Large figures and
tables
may span across both columns. Figure captions should be below
the figures;
table captions should be above the tables. Avoid placing figures
and tables
before their first mention in the text. Use the abbreviation
``Fig. 1'',
even at the beginning of a sentence.
Figure axis labels are often a source of confusion.
Try to use words rather then symbols. As an example write the
quantity ``Inductance",
or ``Inductance L'', not just.
Put units in parentheses. Do not label axes only with units.
In the example, write ``Inductance (mH)'', or ``Inductance L
(mH)'', not just ``mH''.
Do not label axes with the ratio of quantities and units.
For example, write ``Temperature (K)'', not ``Temperature/K''.
\subsection{Numbering}
Number reference citations consecutively in square brackets
\cite{c1}.
The sentence punctuation follows the brackets \cite{c2}.
Refer simply to the reference number, as in \cite{c3}.
Do not use ``ref. \cite{c3}'' or ``reference \cite{c3}''.
Number footnotes separately in superscripts\footnote{This is a
footnote}
Place the actual footnote at the bottom of the column in which
it is cited.
Do not put footnotes in the reference list.
Use letters for table footnotes (see Table I).
\subsection{Abbreviations and Acronyms}
Define abbreviations and acronyms the first time they are used
in the text,
even after they have been defined in the abstract. Abbreviations
such as
IEEE, SI, CGS, ac, dc, and rms do not have to be defined. Do not
use
abbreviations in the title unless they are unavoidable.
\subsection{Equations}
Number equations consecutively with equation numbers in
parentheses flush
with the right margin, as in (1). To make your equations more
compact
you may use the solidus (/), the exp. function, or appropriate
exponents.
Italicize Roman symbols for quantities and variables, but not
Greek symbols.
Use a long dash rather then hyphen for a minus sign. Use
parentheses to avoid
ambiguities in the denominator.
Punctuate equations with commas or periods when they are part of
a sentence:
$$\Gamma_2 a^2 + \Gamma_3 a^3 + \Gamma_4 a^4 + ... = \lambda
\Lambda(x),$$
where $\lambda$ is an auxiliary parameter.
Be sure that the symbols in your equation have been defined
before the
equation appears or immediately following.
Use ``(1),'' not ``Eq. (1)'' or ``Equation (1),''
except at the beginning of a sentence: ``Equation (1) is ...''.
\begin{figure}[thpb]
\centering
%\includegraphics[scale=1.0]{figurefile}
\caption{Inductance of oscillation winding on amorphous
magnetic core versus DC bias magnetic field}
\label{figurelabel}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{CONCLUSIONS AND FUTURE WORKS}
\subsection{Conclusions}
This is a repeat.
Position figures and tables at the tops and bottoms of columns.
Avoid placing them in the middle of columns. Large figures and
tables
may span across both columns. Figure captions should be below
the figures;
table captions should be above the tables. Avoid placing figures
and tables
before their first mention in the text. Use the abbreviation
``Fig. 1'',
even at the beginning of a sentence.
Figure axis labels are often a source of confusion.
Try to use words rather then symbols. As an example write the
quantity ``Inductance",
or ``Inductance L'', not just.
Put units in parentheses. Do not label axes only with units.
In the example, write ``Inductance (mH)'', or ``Inductance L
(mH)'', not just ``mH''.
Do not label axes with the ratio of quantities and units.
For example, write ``Temperature (K)'', not ``Temperature/K''.
\subsection{Future Works}
This is a repeat.
Position figures and tables at the tops and bottoms of columns.
Avoid placing them in the middle of columns. Large figures and
tables
may span across both columns. Figure captions should be below
the figures;
table captions should be above the tables. Avoid placing figures
and tables
before their first mention in the text. Use the abbreviation
``Fig. 1'',
even at the beginning of a sentence.
Figure axis labels are often a source of confusion.
Try to use words rather then symbols. As an example write the
quantity ``Inductance",
or ``Inductance L'', not just.
Put units in parentheses. Do not label axes only with units.
In the example, write ``Inductance (mH)'', or ``Inductance L
(mH)'', not just ``mH''.
Do not label axes with the ratio of quantities and units.
For example, write ``Temperature (K)'', not ``Temperature/K''.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{ACKNOWLEDGMENTS}
The authors gratefully acknowledge the contribution of National
Research Organization and reviewers' comments.
\cite{[15] Francis Barnhart, Michael Cuipa,}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
References are important to the reader; therefore, each citation
must be complete and correct. If at all possible, references
should be commonly available publications.
% \begin{thebibliography}{15}
% \bibitem{latex-book}
% Leslie Lamport, \cit{\LaTeX: A Document Preparation System,}
Addison-Wesley, 1986.
% \end{thebibliography}
\begin{thebibliography}{15}
\bibitem{c1}
J.G.F. Francis, The QR Transformation I, {\it Comput. J.}, vol.
4, 1961, pp 265-271.
\bibitem{c2}
H. Kwakernaak and R. Sivan, {\it Modern Signals and Systems},
Prentice Hall, Englewood Cliffs, NJ; 1991.
\bibitem{c3}
D. Boley and R. Maier, "A Parallel QR Algorithm for the
Non-Symmetric Eigenvalue Algorithm", {\it in Third SIAM
Conference on Applied Linear Algebra}, Madison, WI, 1988, pp.
A20.
\end{thebibliography}
\end{document}
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