Embedded Files
Precision Coordination of multiple air vehicles
Goal 1: Derive the equations of motion for a flapping wing micro air vehicle using Kanes Equation
Goal 2: Project: Write a MATLAB program for Kane's dynamical equations and Lagrange's equations of motion, http://www.eng.auburn.edu/~marghitu/MECH7636/mech7636.htm
Goal 3: Use LaTex for writting the report and adding all the equations of motion for FWMAV
Agenda for Week 5, 8/4/12
Please look at references to add to our report numbersÂ
done from last semester (I think-I will check tonight) 1-150 is this right??
Michael reference 250-300 plus
Ivan reference 200-250
Mariela reference 150-200
Agenda for Week 1, 7/4/12
In LaTex write all 80 equations in Bolenders report that derives the equations for flight simulation in the report that is located in Drop box. I was thinking 5 equations per day so we can be done writing out this section in our report by next friday.
Michael ------equations 1-27
Ivan------equations 27-56
Mariela ------equations 56-80
How to start a document in LaTEX
How to start a document in LaTEX
http://radiantbytes.com/books/python-latex/src/chap14.html
http://gurmeet.net/computer-science/latex-tips-n-tricks-for-conference-papers/
Detexify2 - LaTeX symbol classifier
Detexify2 - LaTeX symbol classifier
http://www.youtube.com/watch?v=fpyuPDgwFOY
http://detexify.kirelabs.org/classify.html
Agenda for Week 2, 7/9/12
Finish up this section of our report In LaTex write all 80 equations in Bolenders report that derives the equations for flight simulation in the report that is located in Drop box. I was thinking 5 equations per day so we can be done writing out this section in our report by next friday.
Michael ------equations 1-27
Ivan------equations 27-54
Mariela ------equations 54-80
Book for Kanes equations in Matlab
Grant Number: V2012ur0023
https://mge-p1000.asu.edu/waeso/Â
I checked on advantage and that company is not on the vendor list. The listed Sunrise Vendors are at the followingÂ
(http://www.asu.edu/purchasing/pdf/Supplier-Listing.pdf).
There are other vendors approved by ASU, but there is no list I can send or database to lookup without having access to ASU financial systems. If WAESO does not purchase the air vehicle, do you have any other funds that your PI might have from a separate grant. We can purchase other items that can be used for the project, example books, supplies, etc…
Here are the students names and info for modeling, analysis, controls for micro air vehicles related research project:
Name: Michael Thompson
Phone #:602 373 9921
Major: Mechanical Engineering
Expected Graduation: 5/2017 (Masters/PhD)
email: mjthomp3@asu.edu
Name: Ivan Ramirez
Phone #: (928) 581-5598
Major: Civil Engineering
Expected Graduation: 5/2013
email: iramire6@asu.edu
Mariela Robledo
Phone #: 928 919 3869
Major: Chemical Engineering
Expected Graduation: 5/2013
email: mrobled1@asu.edu
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\begin{document}
\title{\Large \bf Precision Coordination of multiple air vehicless}
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\begin{figure}[h!]
\centering
\includegraphics[width=0.6\textwidth]{Vehicles.jpg}
\caption{Vehicles}
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\thispagestyle{empty}
\newpage
\author{Reserach advisor: Dr. Armando Rodriguez$^{1}$,\\
Researchers: Michael Thompson$^{2}$,
Ivan Ramirez$^{3}$,\\
Mariela Robledo$^{4}$,
\\
\footnotesize $^{1}$ Department of Electrical Engineering, Arizona State University, Tempe , AZ, USA\\
\footnotesize $^{2}$ Department of Mechanical and Aerospace Engineering, Arizona State University, Tempe , AZ, USA\\
\footnotesize $^{3}$ Department of Civil Engineering, Arizona State University, Tempe , AZ, USA\\
\footnotesize $^{4}$ Department of Chemical Engineering, Arizona State University, Tempe, AZ, USA \\
}
\renewcommand{\today}{August 10, 2012}
\date{August 10, 2012}
\maketitle
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%% ABSTRACT %%
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\clearpage
\section{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}
\clearpage
\tableofcontents
\newpage
\listoftables
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%% NOMACLATURE %%
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\begin{table}
\section{NOMENCLATURE}
\begin{tabular}{llrlr}
{Symbol} &{Descrition} &{Min} &{Max}\\
\hline
f &Frequency &$0$ &$\infty$\\
$\varphi_{m}$ &Azimuthal amplitude &$0$ &$\pi / 2 $\\
$\theta_{m}$ &Vertical amplitude &$0$ &$\pi / 2$ \\
$\eta_{m}$ &Pitching amplitude &$0$ &$\pi $ \\
$\theta_{0}$ &Vertical offset &$\theta_{m} - \pi / 2$ &$\pi / 2 - \theta_{m}$ \\
$\eta_{0}$ &Pitching offset &$\eta_{m}- \pi$ &$\pi - \eta_{m} $ \\
K &Affects the shape of $\varphi (t)$ &$0$ &$1$ \\
$C_{\eta}$ &Affects the duration of wing rotation &$0$ &$\infty$ \\
N &Multiplier of $\theta$(t ) period &$1$ &$2$ \\
$\Phi_{\theta}$ &Vertical phase offset &$- \pi$ &$\pi$ \\
$\phi_{\eta}$ &Pitching phase offset &$- \pi$ &$\pi$ \\
\hline
\end{tabular}
\end{table}
\clearpage
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%% ACKNOWLEDGEMENT %%
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\section{ACKNOWLEDGEMENT}
The authors gratefully acknowledge the contribution of National
Research Organization and reviewers' comments.
\clearpage
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%% INTRODUCTION %%
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\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].
\subsection{\bf PROBLEM STATEMENT}
The technical challenges for small rotary wing micro air vehicles span are many. The challenge of endurance and thrust to weight ratios for the propulsion systems for a meaningful mission is critical to the micro air vehicle idea \cite{c3}. In order to enable the possibility for critical mission capable vehicle to perform batteries, electric motors, and rotors are size critical tradeoff concerns that will be investigated with this design of experiments. Our concern is to examine the payload capacity for our quad rotor using statistical analysis to determine the mean weight distribution to be put on board for numerous application of the quad rotor such as Regular police operations, traffic control, crowd management, ordinary city surveillance, hazardous waste disposal,
Exploration, Search and rescue and many more. By incorporating a reliable sized load capacity
for the vehicle one ay one day in the future is able to use this quad rotor for these applications in wide spread across the nation.
\subsection{\bf BACKGROUND INFORMATION}
The field has evolved. Various institutions such as DARPA began working on MAVs in the 1990s for military missions and other applications such as surveillance, intelligence, and reconnaissance missions [51], [55], [143]. These systems are ideal for providing fast analysis and overview of indoor and outdoor areas without exposing expensive equipment and personnel to danger zones. Other institutions have begun extensive research and development of new air vehicles, smaller in size and weight while optimizing performance and functionality.
\begin{figure}[h!]
\centering
\includegraphics[width=0.45\textwidth]{stateoftheart.jpg}
\caption{MAV sneak peek at the state-of-the-art from \cite{c124}}
\end{figure}
In the future, alternative energy supplies such as fuel cells and ultra capacitors look promising for circumventing the limitations that rechargeable batteries and their size present to the minimizing of MAV weight and for improving flight capabilities [2], [108].The state-of-the-art. The state of the art for micro air vehicles have focused on MAVs/NAVs focused on challenges and designs [136]. Specifically research and development in the past years have focused on maneuverability at low speeds in confined spaces [2], [149], [150], [56].
The main applications have stayed the same which were to complete challenging indoor and outdoor missions for the air vehicle systems that are to gather intelligence, surveillance, and reconnaissance (ISR) missions [152], [37].
\begin{figure}[h!]
\centering
\includegraphics[width=0.6\textwidth]{airvehicledevelopment.jpg}
\caption{The air vehicle development for dimension reduction for the past years \cite{c44}}
\end{figure}
\begin{figure}[h!]
\centering
\includegraphics[width=0.6\textwidth]{reforairvehicles.jpg}
\caption{Reynolds number for aerial vehicles \cite{c44}}
\end{figure}
A critical tradeoff for vehicles is the size and speed \cite{c125}. The aerodynamics of the speed of the vehicle is dictated by the Reynolds number. The following equation describes the Reynolds number for our quad-rotor micro air vehicle, $Re= (\rho \times V \times d) /\mu$
\begin{figure}[h!]
\centering
\includegraphics[width=0.6\textwidth]{lifttodrag.jpg}
\caption{The graph represents lift to drag ratio variation with Reynolds number \cite{c44} pg.3, \cite{c64}vehicles}
\end{figure}
\begin{figure}[h!]
\centering
\includegraphics[width=0.6\textwidth]{piechartsensors.jpg}
\caption{The graph represents various sections of parts on an mav \cite{c44} pg.3.}
\end{figure}
The current trends. More focus has been placed towards rotary-wing MAV since it has the best performing technology at present and it will be the most important commercial type. However, more research is being conducted for flapping-wing solutions which are viable in the near future due to improved maneuverability and efficiency relative to NAVs [106] pg. 2. There is a large interest in technologies that can potentially replace batteries. Good candidates are ultra capacitors and fuel cells. Trends are still moving towards further size and weight reductions of communication systems. Micro and nano electromechanical systems (MNEMS) technologies can be used to provide devices, such as lighter, smaller, and less power consuming resonators and filters [133], [35].
Expansion of capabilities are being further explored with equipping NAVs with GPS, radar systems, infrared, and high-definition cameras.
Research issues and critical tradeoffs. The main issues with Air vehicle systems is that they are not scaled down versions of larger aircraft [44] pg. 2. These vehicles must be able to contain the same features for larger aircraft and in a tiny volume. In doing so, the complexity of scaling down and miniaturizing increases greatly hindering the physical and technological challenges for the vehicle.
Applications for air vehicle systems. Air Vehicle Systems can support the following tasks/situations:
\footnote {Michael \url {https://sites.google.com/a/asu.edu/michael-thompson/projects/summer-research-2012/waeso-summer-2012/latex?pli=1}.}
\vspace{.3in}
\begin{description}
\item[Applications for air vehicle systems] Air Vehicle Systems can support the following tasks/situations:
\end{description}
\begin{enumerate}
\item Regular police operations,
\item traffic control,
\item crowd management,
\item ordinary city surveillance,
\item Hazordous waste disposal,
\item Exploration,
\item Search and rescue
\end{enumerate}
\begin{figure}[h!]
\centering
\includegraphics[width=0.6\textwidth]{mavapplications.jpg}
\caption{MAV applications/missions, \cite{c52}}
\end{figure}
By using these air vehicle systems we may reduce the cost and overall enhance our nations performance in reducing crime rate.
\begin{description}
\item[Applications for air vehicle systems] Air Vehicle Systems can support the following tasks/situations:
\end{description}
A review of the following aerodynamic material regarding each of the following was looked at during the research:
\begin{enumerate}
\item Aerodynamics - computation of lift, drag, moments, \cite{c65}
\item Propulsion - computation of thrust.
\item Unsteady effects.
\item Aircraft/Rotorcraft dynamics and modes (e.g. phugoid, short-period, spiral, roll subsidence, Dutch roll, back flapping mode for helicopter, etc.)
\item Aerodynamic, stability and control derivatives.
\item Developing a 3 degree of freedom (x, z, theta) nonlinear model for the longitudinal dynamics of an aircraft.
\item Developing a 6 degree of freedom (x, y, z, yaw, pitch, roll) nonlinear model for an aircraft.
\item Wing dynamics.
\end{enumerate}
Traditionally, various aerodynamic effects had been deemed undesirable and designs had been modified to avoid these unwanted effects. In turn, these modifications negatively affected other important concepts, thus increasing interest in taking advantage of effects such as roll control, load alleviation, drag reduction, and other aero elastic phenomena. In order to analyze significant effects, there has also been an increase in use of computational fluid dynamic tools to capture non-linear aerodynamic behaviors. [1] pg 1, [144], [36], [57].
\section{DESING CONSIDERATIONS}
The quad rotor was used indoors in order to control the environment problems. We did not desire to address natural issues associated with wind disturbances, bug disturbances, and environmental impacts.
The camera we used was to control the time environment and collect the time stamps. The quad rotor was used for its easiness to fly and easy predictability concerns. The mass distribution was the washers and was used for equal distribution of weight parameters. The added on weights were varied in equal size increments to gather a critical load capacity to determine optimal velocity. A constant one meter distance was used in this design of experiment because the length was a long enough distance to gather the time and velocity for our given constraints. We started from a fixed position and ramped up the quad rotor to approximately 0.5 meters per second with an added payload. The power capacity that was used for the quad rotor was a 3.7 Volts LiPo battery. The time length of the battery for running the experiments was approximately 10-15 minutes [144]. A critical issue for the time characteristic was that the battery life was limited, so we had to conduct the experiment in the allotted time length. As such, the operation of the batter was at a full charge, thus, enabling the capability to conduct the full experiment in a timely fashion for our experiment was done. We chose to run 5 samples because It is a large enough sample to test our hypothesis. The size of the sample was able to conduct the experiments we wanted and obtain the pertinent and critical information we desired. We choose time to gather time and velocity data for its pertinent to loading characteristic for any micro air vehicle. In particular, the quad demonstrates a highly maneuverable vehicle and in order to capture the loading capacity we had to take into consideration the time and velocity variables and look at the mean distribution by using the nested ANOVA test for statistical analysis.
For out quad rotor some of the consideration for the design incorporated and examined issues and tradeoffs of the following: power budget in for MAV/NAV, actuator, communications, size, about 15 cm (above 60 cm in the 1990s, now can be less than 10 cm), battery life, flight capacity: 10-20 minutes, mission Profile to be able to carry and be able to begin flight in seconds, chemical sensors, thermal sensors, video recorder, and a payload abilities of about 52 grams from experiments, [70].
\begin{description}
\item [Design/performance specifications generated.] MAVs/NAVs are customizable, and therefore the specific desired designed and performance specifications vary according to its use. Battery life optimization and size minimization would be more desired by clients planning its use for surveillance, while others may be interested in speed, video camera, and GPS potential. The necessary sensors, size specifications, and other specifications will vary with mission needs [44] pg 15, [137].
\end{description}
\begin{description}
\item [Relevant and critical tradeoffs.] Although size reduction for micro air vehicles is desirable, there are various tradeoffs that must be made to be able to accommodate for the smaller size. For example, reducing antenna size will reduce performance, especially as performance is a function of size. Also, the transmission power cannot be reduced after a certain point without degrading the quality of the device communication. This would prove difficult for certain uses of interest for MAVs, such as surveillance. [44] pg 15.Current 2011 tradeoffs are represented here in Table 2.
\end{description}
\begin{figure}[h!]
\centering
\includegraphics[width=0.6\textwidth]{quadadvantagesdisadvantages.jpg}
\caption{Quadrotor advantages and disadvantages compared to the traditional helicopter, \cite{c138}}
\end{figure}
\clearpage
%\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}
\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 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}
%%%Equation 46 %%%
\begin{eqnarray}
F_{r}&-& \frac{\partial \omega_{B}}{\partial u_{r}}\cdot (M_{B}+ r_{T/B}\times m_{l} g^{B})\nonumber
\\ &+& (\frac{\partial \dot r_{B}}{\partial u_{r}})\cdot (F_{B}-(m_{b}+m_{l}) g^{B})\nonumber
\\ &+& (\frac{\partial \omega_{T}}{\partial u_{r}})\cdot M_{T}+ (\frac{\partial \dot r_{T}}{\partial u_{r}})\cdot F_{T}
\end{eqnarray}
%%%Equation 47%%%
\begin{equation}
M_{long}(l)
\begin{bmatrix}
\dot U\\
\dot W\\
\dot Q\\
\end{bmatrix}
+ f_{long}(u, q, l)+
\begin{bmatrix}
F_{B_{x}}- (m_{b}+ m_{l})g sin\Theta\\
F_{B_{z}}- (m_{b}+ m_{l})g cos\Theta\\
M_{B}+ m_{l}g(L_{lh}cos(\Theta | \Theta_{T}))+ x_{hb}cos\Theta\\
\end{bmatrix}
=0
\label{eq:symmetrical}
\end{equation}
%%%Equation 48 fix%%%
\begin{equation}
M_{long}(l) =
\begin{bmatrix}
m_{b} m_{l} &0 &m_{l}I_{x_{lb}}sin\Theta_{T}\\
0 &-m-m_{t} &-m_{t}(x_{hb}+ L_{x_{lb}}cos\Theta_{T} )\\
-m_{t}L_{th}sin\Theta_{T} &-m_{t}(x_{l | b}+ L_{th}cos\Theta_{T} ) &I_{yx}\cdot I_{y u}- m_{l}(L^2_{th}+ x^2_{h b}+ 2L_{th} x_{h b}cos\Theta_{T})\\
\end{bmatrix}
\label{eq:symmetrical}
\end{equation}
%%%Equation 49 fix%%%
\begin{equation}
f_{long}(u,q,l) =
\begin{bmatrix}
-(m_{b}+ m_{l})QW- m_{l}x_{hb}Q^{2}- L_{lh}m_{l}cos\Theta_{T}(Q+Q_{T})^{2}- L_{lh}m_{l}\dot Q_{T}sin\Theta_{T}\\
(m_{b}+ m_{l})QU+ m_{l}L_{lh}(Q+ Q_{T})^{2}sin\Theta_{T}- L_{lh}m_{l}\dot Q_{T}cos\Theta_{T}\\
m_{l}QU(x_{hb}+ L_{lh}cos\Theta_{T})+ m_{l}L_{lh}x_{hb}Q_{T}sin\Theta_{T}(Q_{T}+ 2Q)- m_{l}L_{lh}QWsin\Theta_{T}+ ...\\
\dot Q_{T}( I_{T_{uy}} L^{2}_{lh}m_{t}- L_{th}m_{t}x_{hb}cos\Theta_{T})\\
\end{bmatrix}
\label{eq:symmetrical}
\end{equation}
%%%Equation 50 %%%
\begin{equation}
\\\dot q_{0} = \frac{1}{2} Qq_{2}\\
\end{equation}
%%%Equation 51 %%%
\begin{equation}
\\\dot q_{1} = \frac{1}{2} Qq_{3}\\
\end{equation}
%%%Equation 52 %%%
\begin{equation}
\\\dot q_{2} = \frac{1}{2} Qq_{1}\\
\end{equation}
%%%Equation 53 %%%
\begin{equation}
\\\dot q_{3} = \frac{1}{2} Qq_{3}\\
\end{equation}
%%%Equation 54 %%%
\begin{equation}
\\\dot x = (U cos\Theta + W sin\Theta ) cos\Psi\\
\end{equation}
%%%Equation 55 %%%
\begin{equation}
\\\dot y = (U cos\Theta | W sin\Theta ) sin\Psi\\
\end{equation}
%%%Equation 56 %%%
\begin{equation}
\\\dot h = Usin\Theta \cdot W cos\Theta \\
\end{equation}
%Equation 57
\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}
%Equation 58
\begin{equation}
\\x=
\end{equation}
%Equation 59
\begin{equation}
\\x=
\end{equation}
%Equation 60
\begin{equation}
\\x=
\end{equation}
%Equation 61
\begin{equation}
\\x=
\end{equation}
%Equation 62
\begin{equation}
\\x=
\end{equation}
%Equation 63
\begin{equation}
\\x=
\end{equation}
%Equation 64
\begin{equation}
\\x=
\end{equation}
%%%Equation 65 %%%
\begin{equation}
\\r_{R/B}- r_{R/A}+ r_{A/B}\\
\end{equation}
%%%Equation 66 %%%
\begin{equation}
\\\dot r_{R}= \dot r_{B}+ \omega_{B}\times r_{A/B}+ \omega_{R}\times r_{R/A}\\
\end{equation}
%%%Equation 67 %%%
\begin{equation}
\\ \ddot r_{R}= \frac {\partial \dot r_{R}}{\partial t}+ \omega_{B}\times \dot r_{R}\\
\end{equation}
%%%Equation 68 %%%
\begin{equation}
\\\dot \omega_{R}= \frac {\partial \omega_{R}}{\partial t}+ \omega_{B}\times \omega_{R}\\
\end{equation}
%%%Equation 69 %%%
\begin{equation}
I_{R} =
\begin{bmatrix}
I_{R_{rr}} &-I_{R_{zy}} &0\\
-I_{R_{xy}} &I_{R_{yy}} &0\\
0 &0 &I_{R_{zz}}\\
\end{bmatrix}
\label{eq:symmetrical}
\end{equation}
%%%Equation 70 %%%
\begin{equation}
\\dF_{1}= {[(\frac {-c(r)}{cR}m_{r}+ m_{22}) \dot \alpha+ \rho \Gamma](\dot r_{3}\times r_{R})- m_{11} \ddot r_{R}\cdot r_{1}} dr- dF_{1}^{\Phi}
\end{equation}
%%%Equation 71 %%%
\begin{equation}
\\dF_{3}= {[(\frac {-c(r)}{\bar c R}m_{r}+ m_{11}) \dot \alpha+ \rho \Gamma](\dot r_{1}\times r_{R})+ m_{22} \ddot r_{R} \cdot r_{3}} dr-dF_{1}^{\Phi} dF_{3}^{\Phi}
\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}
\clearpage
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% literature %%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{LITERATURE SURVEY}
\clearpage
\section{\bf MODELING}
Modeling issues associated with each of the following technical areas were examined:
\begin{description}
\item[aerodynamics] For an accurate analysis of flapping flight it is necessary to resolve complex three-dimensional turbulent flow. Most of the previous studies employed either URANS or more recently DES modeling for this purpose. These methods are not particularly accurate in predicting transitioning shear layers which play an important role in the breakdown of LEV and tip vortex. Hence, LES capability is needed, which does an excellent job in capturing shear layer transition. [152] pg. 4.
\end{description}
\begin{description}
\item[propulsion,] Propellers aerodynamics must be efficient because the vehicle depend solely to the propeller to lift from the ground. However, propellers below 3 inches in diameter have efficiencies of about 50 percent or less. The weight factor of the vehicle, in other way, much be reduced should the power available is confined. [14] pg. 7, [58].Lateral wind velocity directly affects thrust in ambient conditions. The thrust generated by the propellers generally decrease when subjected to a lateral flow which can cause the vehicle to drop in altitude due to insufficient thrust. It was found that by increasing the throttle on the vehicle will counter this effect [82] pg. 31. Moreover, during flights in low-altitudes, quadrotors are prone to sudden wind gusts that can significantly affect their flight performance and even cause instability [92] pg. 12. Successful autonomous operation of quadcopters in windy conditions while carrying an unknown payload remains an open problem [94] pg. 928. The rotors, especially, influence the natural dynamics and power efficiency. An approximate understanding of helicopter rotor performance can be obtained from the momentum theory of rotors. An accurate model needs to be created for the flapping behavior which can be described as functions of the helicopters forward velocity and are obtained by simultaneously solving the constant and sinusoidal components of the blade centrifugal-aerodynamic-static weight moment system. [154] pg. 4-5.Aerodynamic performance is enhanced with cycloidal rotor blades according to the pivot point location. Thrust is maximized in a cycloidal rotor since the oscillating airfoil experiences the rotor rotating motion, which is related to the virtual camber effect as shown in Fig. 8 [80] pg. 60.
\end{description}
\begin{figure}[h!]
\centering
\includegraphics[width=0.6\textwidth]{virtual.jpg}
\caption{Virtual camber effect \cite{c80} pg. 62.}
\end{figure}
\begin{figure}[h!]
\centering
\includegraphics[width=0.6\textwidth]{onboard.jpg}
\caption{Onboard sensors and avionics with electronic buses, \cite{c140}}
\end{figure}
Navigation in a complicated environment is a difficult problem if the flight is desired to be autonomous. Detecting the obstacles such as buildings, holes; pedestrians, vehicles etc. for collision free autonomous flight is the crux of the design problem [95] pg. 896. A solution found for autonomous navigation of an MAV was the use of a system employs a three level sensing hierarchy, as shown in Figure 7. At the base level, the on -board IMU and processor create a very tight feedback loop to stabilize the helicopters pitch and roll, operating at 1000Hz [83] pg. 1. At the next level, the realtime odometry algorithm (laser or visual) estimates the vehicles position relative to the local environment, while an Extended Kalman Filter (EKF) combines these estimates with the IMU outputs to provide accurate state estimates of the position and velocity at 50Hz. These estimates enable the LQR controller to hover the helicopter stably in small, local environments [83] pg. 1.An approach used by California Institute of Technology is to employ a single camera as a vision sensor to detect a navigation target and generate 3D waypoints to the target for autonomous navigation of a quadcopter [90] pg. 3.
Researchers have studied insect flight and derived a method called optical flow. Optical flow enables the detection of the movement of brightness in sequentially ordered gray scaled images, e.g., a video stream.composed of grayscale images. It also gives information about the motion of the observer as well as the objects in the scene. Optical flow takes the advantage of gradients of the temporal and spatial information to calculate approximate motion vectors, i.e. the temporal derivatives lead to the detection of motion in time domain, and spatial derivatives facilitate the perception of the motion in the two dimensional (2D) coordinate system [95] pg. 896. Another area of concern in vision-based control is the Field of View (FOV). It is proposed that the control of rotation is decoupled from that of position which reduces likelihood of feature loss [96] pg. 1.
\begin{figure}[h!]
\centering
\includegraphics[width=0.6\textwidth]{schematic.jpg}
\caption{Schematic of the hierarchical sensing, control and planning system \cite{c83} pg. 1.}
\end{figure}
In the implement for the state feedback control, there are often some perturbations appearing the in the controller gain. This problem has been investigated by the Beijing Institute of Technology and it was found that the controller has robust stabilization [87] pg. 2. Another option is to use a memoryless state feedback controllers for all admissible uncertainties, therefore the resulting closed-loop system is stable and the closed-loop transfer function is extended strictly positive real [89] pg. 9, [102] pg. 2. As proposed by Sabanci University, a PI controller is designed for the along track direction and a PID controller is designed for the cross track direction. The desired acceleration vector in the XY plane of the vehicle is constructed from the controller outputs [91] pg. 4.
\begin{description}
\item[embedded-system/algorithm implementation issues.] Designing a tiny unmanned aircraft like an MAV would entail making severe compromises. At the size of about 6 inches (15 cm), the concept of stuffing an airframe with subsystem becomes extremely difficult. For example, batteries need to severe multiple functions, such as contributing to the vehicle structure. Electronic functions have to be combined and thus, a single application specific integrated circuit could be used for the entire vehicle. Mass can be reduced by thinning electronic circuitry and using interconnections printed onto the vehicle shell in the place of interconnecting wiring. The close proximity of vehicle subsystems also provides challenges such as control of heat dissipation, vibration from internal combustion engines and electromagnetic interference from electric motors. [14] pg. 7
\end{description}
The Air Force Institute of Technology have investigated a series of slightly smaller rigid and flexible wing MAVs using a closely-coupled method incorporating a Pressure Implicit with Splitting Operators (PISO) algorithm to solve the incompressible Navier-Stokes equations and a hyperelastic Moody-Rivlin model. Using the PISO algorithm, it was found that the membrane deformations at nominal to high angle of attack increased the maximum lift-to-drag ratio [81] pg. 4, [105] pg. 8. Another algorithm used by other researchers have been the simultaneous localization and mapping (SLAM) where the local state estimates are computed from visual odometry and to correct for drift in these local estimates, the estimator periodically incorporates position corrections provided by the SLAM algorithm [83] pg. 8, [84] pg. 1. A robust controller was proven to be effective with a fuzzy controller based on neuro-fuzzy learning algorithms [101] pg. 2.
\begin{description}
\item[energy management.] Promising energy sources for robotic insects include conventional batteries, fuel cells, ultra capacitors, and solar cells. At present, however, lithium polymer batteries are the only developed, commercially available technology that can satisfy the requirements of insect-sized MAVs. The energy source model is complicated by the fact that the capacity of lithium polymer batteries decreases at high discharge rates. As a result, it is difficult to estimate battery performance without a specific battery in mind; at the same time, it is important to consider derating because the high power requirements of hovering MAVs inevitably translate to high battery discharge rates. [68] pg. 4.
\end{description}
\begin{description}
\item[multidisciplinary design/optimization.] The method for optimization uses the three design tools already presented, that include the three major disciplines: aerodynamics, acoustics and structural analysis. Giving the disciplines equal importance in the task of optimization and considering them all enables the designer to approach a variety of design problems [67]. In a traditional design process the variables are given by: the pitch angle distribution, the chord distribution and the thickness ratio distribution. The main categories of concern are the following:
\end{description}
\begin{enumerate}
\item general design variables, affect the global configuration: number of propellers, engine gear ratio,
\item number of propeller blades, propeller radius, rotational speed, airspeed.
\item blade design variables, define the geometry and structure of the blade: pitch angle, chord, sweep angle, mass, dihedral angle, structural properties, [66],
\item cross-sectional variables, define the cross-sectional airfoil geometry: thickness ratio, lift coefficient [27] pg. 5
\end{enumerate}
Since only recent improvements in battery technology made it possible to power such small helicopters, the research in this field is not yet well established. Concerning control, the small size yields a system with extremely fast dynamics where the low quality of the miniaturized sensors and actuators additionally handicap the control performance-[63], pg119-120.
\begin{description}
\item[micro/nano fabrication.] due to the small size of micro/nano vehicles, the rotors are working in the low Reynolds number regime where viscous effects dominate the flow and reduce the efficiency. Since only recent improvements in battery technology made it possible to power such small helicopters, the research in this field is not yet well established. Concerning control, the small size yields a system with extremely fast dynamics where the low quality of the miniaturized sensors and actuators additionally handicap the control performance-[63] pg119-120. It was found that insect-sized MAVs are most energy effective when propelled by spinning wings [107] pg. 18.
\end{description}
\begin{description}
\item[vehicle performance prediction.] an increase in flight time will require improvement in one or more of the model parameters. As may be expected, the performance of aerodynamic components has a significant effect on flight time: if less power can be used to generate a given amount of lift, this translates to relaxed requirements on the actuator and the power electronics, a consequent reduction in the mass of these subsystems, and increased room for the battery. Global optimization methods have to be applied in order to enhance and accurately predict performance. [68] pg. 7.
\end{description}
\begin{description}
\item[Low fidelity model.] reduced order models typically produce adequate results and provide near-exact solutions to approximate problems. Itis best to use low fidelity modeling for faster results and reduced complexity [24] pg. 57.
\end{description}
\begin{description}
\item[Higher fidelity model required.] A higher fidelity model is required for footprint maximization. Higher-fidelity modeling offers more accurate solutions and these solutions become attainable with advances in numerical methods and computational power. High fidelity models provide an approximate solution to an exact problem. Current, more powerful, numerical methods can efficiently provide accurate solutions to large systems of nonlinear Ordinary Differential Equations (ODEs) [24] pg. 57, [147].
\end{description}
\begin{description}
\item[Higher fidelity models be used to validate lower fidelity models.] various research efforts have shown that inaccurate and/or misleading solutions can result from using low fidelity modeling. Reduced-order modeling can compromise safety by predicting diminished footprints. In order to definitively determine feasibility, high fidelity results are needed during real missions even when using low-fidelity-based methods. [24] pg. 56.A scenario is depicted in High Fidelity Real-Time Trajectory Optimization for Reusable Launch Vehicles shows the difference between high fidelity and low fidelity models.
\end{description}
\begin{figure}[h!]
\centering
\includegraphics[width=0.6\textwidth]{maneuverability.jpg}
\caption{The graph represents maneuverability with low fidelity solutions from \cite{c24}}
\end{figure}
Consider two specific examples indicated by Case 1 and Case 2. First, assume the vehicle is originally planning to land at Air Field 2 and experiences a failure corresponding to the IC mark. For the low-fidelity model, it is no longer viable to reach Air Field 2, but it can reach the alternate, Air Field 1. In this case, even though it falsely thinks it cannot reach Air Field 2, it still may have a feasible alternative. In comparison, the high-fidelity model can reach both. Second, assume the vehicle is originally heading to Air Field 3 and experiences a failure corresponding to the IC mark. Now, the low-fidelity model provides incorrect information, a FALSE POSITIVE that it can reach Air Field 3 when in fact, it cannot, so it will unsuccessfully attempt to make it. This case will end in, the very least, an emergency situation that could have been prevented simply by using the high-fidelity model. This is important now more than ever since current research is developing on-board, re-planning and retargeting guidance schemes [24] pg. 58.
\section{ANALYSIS OF MODEL}
The model was analyzed by looking at various parameters and consideration were given to traditional aerospace concepts such as the following below.
\begin{description}
\item[Traditional-aero-propulsion-structural considerations,]the lift-to-drag curves to round out at lower values with regard to thrust for the experimental set up was taken into consideration [148]. Thrust measurements of the MAV were taken into account and by the rotors in and out of ground effect. The micro air vehicle may be hung from a rope from the ceiling and the height deviation may be taken into account for experimental set up for thrust from the ground is measured by use of a simple measuring stick. The necessary components will aid to support the systematic investigation of the challenges associated with hardware limitations and task requirements.
\end{description}
\begin{figure}[h!]
\centering
\includegraphics[width=0.6\textwidth]{6dof.jpg}
\caption{The graph represents a 6 DOF high fidelity model for solutions from \cite{c17}, \cite{c55}}
\end{figure}
The MAV moves in an earth-fixed inertial reference frame E defined by the basis vectors $(e1, e2, e3)$.The earth-fixed reference frame E is shown in Figure 10. Six Degrees Of Freedom (DOF) are necessary to fully describe the MAVs position and orientation from a specified point. The first three DOFs define the distance from the MAV to the fixed-reference frame. The other three DOFs are Euler angles and define the rotation between the fixed-reference frame and the MAV body-fixed reference frame [17] pg.14,[154], [42].
\begin{figure}[h!]
\centering
\includegraphics[width=0.6\textwidth]{table6dof.jpg}
\caption{Represents a 6 DOF high fidelity model for a MAV from \cite{c17}}
\end{figure}
Both are model-based: The state estimate uses a model of the system to make predictions of the current state based on the past state and input, and the optimization problem uses a model to evaluate the result of prospective control policies over a finite planning horizon, and thus select the best policy based on the corresponding costs [86] pg. 6. The stability derivatives are obtained from experimental methods and through the use of the standard aircraft equations of motion [42] pg. 11. Ideally a model incorporates the nonlinear gyroscopic effects resulting from both the rigid body rotation in space and the four propulsion rotors rotation. Secondary effects include rotor acceleration, aerodynamic drag, the gearbox and the motor rotation [94] pg. 928.
\begin{description}
\item[critical control relevant considerations.] one of the first consideration is to consider the electronic speed controllers for the control of the mav [138] pg 19, [39]. In addition, to one controller for the mav, it may be necessary to have micro controllers on board. The microcontroller will read in the sonar on its analog digital converter ADC and also outputs.
\end{description}
pulse width modulated (PWM) signals to the ESCs to control the motors [138] pg 20. Controllers have been derived using theoretical models of quadrotor aerodynamics with non-zero free-stream velocities based on helicopter momentum and blade element theory, validated with static tests and flight data. These controllers were implemented on the Stanford Testbed of Autonomous Rotorcraft for Multi-Agent Control (STARMAC), which demonstrated significant improvements over existing methods [98] pg. 14.
stability, the stability requirements for a nonlinear controller that uses altitude and navigation will improve stability and tracking [138] pg 34, [115], [72]. The quadrotor will diverge into instability. The four actuators and six degrees of freedom (roll, pitch, yaw, x, y, z positions), the quadrotor is an under actuated system [138] pg 34. The quadrotor are designed to be unstable like helicopters [138] pg 34, [62], [76]. A quadrotor has a horizontal displacement between its masts and center of gravity. When the craft rolls and pitches, the rotors experience a vertical velocity, leading to a change in the inflow angle [97] pg. 4. Stabilization of a quadrotor platform can be approached in a straightforward manner. Altitude is controlled by increasing or decreasing throttle to all four motors. Pitch is controlled by increasing throttle on Q1 and decreasing throttle on Q2 or vice-versa. Roll is controlled the same way, using Q3 and Q4. Yaw is controlled by increasing throttle on both Q2 and Q1 while decreasing throttle on Q3 and Q4 or vice versa, depending upon the desired direction of rotation. These four degrees of freedom are measured using an inertial measurement unit that can measure pitch, roll, altitude, and yaw [103] pg. 49, [109] pg. 4.
\begin{figure}[h!]
\centering
\includegraphics[width=0.6\textwidth]{quadrotor6dof.jpg}
\caption{The figure represents a model of a quadrotor with its associated 6 degrees of freedom. \cite{c138} pg 17.}
\end{figure}
The attached frame to the earth, relative to a fixed origin, is denoted by the aircraft in the figure above.
performance, the estimated flight performance for forward flight for a quad rotor is determined if the required predetermined requirements are met [154] pg 78.
The power required for hovering with the above equation. The rotors hovering flight velocity.
\begin{figure}[h!]
\centering
\includegraphics[width=0.6\textwidth]{forwardflight.jpg}
\caption{The graph represents the required total power for forward flight from \cite{c154} pg 80.}
\end{figure}
For this particular paper the flight performance for the arm of the vehicle (from the drag analysis) for the U-arm model had a better performance. The U-arm has a lower induced and profile power, because it needs to lift less weight and so the rotor can spin slower. For velocities above the parasitic power, proportional to the drag, penalizes the slim-arm, and the U-arm is a better option. For very high velocities, well above this project requisites, greater rotor angular velocities are needed which leads to the increase of the induced velocity proportional to the models weight, and thus the slim-arm becomes again the more economic alternative, as figure 6.4 demonstrates.
\begin{figure}[h!]
\centering
\includegraphics[width=0.6\textwidth]{climbing.jpg}
\caption{The graph represents the required total power for climbing from \cite{c154} pg 80.}
\end{figure}
The state estimate uses a model of the system to make predictions of the current state based on the past state and input, and the optimization problem uses a model to evaluate the result of prospective control policies over a finite planning horizon, and thus select the best policy based on the corresponding costs.[86] pg. 6. The stability derivatives are obtained from experimental methods and through the use of the standard aircraft equations of motion [42] pg. 11
\begin{figure}[h!]
\centering
\includegraphics[width=0.6\textwidth]{samari.jpg}
\caption{The figure represents the Samara MAV from \cite{c60}.}
\end{figure}
\begin{figure}[h!]
\centering
\includegraphics[width=0.6\textwidth]{weightdata.jpg}
\caption{Represents weight data in terms of percent for the Samara MAV and a hobby rotorcraft from \cite{c60} pg3.}
\end{figure}
Here lays an exhaustive list of mission critical issues for the following criterias relative to micro air vehicles; Power and endurance, the motors power consumption of trying to change a motors rotational speed without a specific requirement will affect the desired rotation speed of the motor.
\begin{figure}[h!]
\centering
\includegraphics[width=0.6\textwidth]{powerconsumption.jpg}
\caption{Represents the power consumption}
\end{figure}
The need for further optimization is a desired need to reduce the power consumption for the batteries. Capabilities, reliability, the maximum capabilities for the thrust setting is in place for preventing possible burnout of the motors. A big issues in micro air vehicles is the power supplies capabilities of handling far more current than can be required [138] pg 96.
Predictability, to be noted the unpredictability
for the sensors attached for mavs. In addition,
rims should not be used for stabilization, but
rather for assisting the attitude controller for
improved efficiency. [138] pg 93, [77], [78],
[79]. The analysis examined the following:
Equilibrium (flight condition) analysis, for
equilibrium flight the landing gear design must
be in the lower position. To prevent tilting of the
MAV and sustain large accelerations the landing
gear should have an inverted V shape [154] pg
70. For equilibrium analysis the components of
the air craft matter the most because they are the
parts that touch down on the ground first.
\begin{figure}[h!]
\centering
\includegraphics[width=0.6\textwidth]{equilibriumlanding.jpg}
\caption{Equilibrium pitch landing analysis, \cite{c154} pg. 70.}
\end{figure}
The landing gear depicted in the above
figures are the first component to touch ground
first. The hovering of the MAV is critical to the
angle at which it is at , otherwise it may pitch
and fall [154]. Furthermore, the angle of the mav
is critical to avoid the force of inversion to fall
and crash.
Linearization of nonlinear model, the
structural and operational components are
related to the nonlinear model. The nonlinearity
models associated with the quadrotor is shown
below in the give figure [138] pg. 56, [61].
\begin{figure}[h!]
\centering
\includegraphics[width=0.6\textwidth]{BlockDiagram.jpg}
\caption{Block diagram of the nonlinear passive complementary filter \cite{c138} pg. 56.}
\end{figure}
The matrix denotes the creation of
a skew-symmetric matrix generated using the
vector inside the parenthesis [138] pg. 56. The
measurements, for the accelerometers are
measured in the inertial frame shown in the
diagram.
Analysis of linear model - time and frequency domain, the analog sampling prevented the implementation of a software filter
that could run at a high frequency for a sharp noise cutoff since floating point filtering is too computationally intensive [154]. Other control
loops only ran at 350 Hz. Thus, software filtering was only minimally effective compared to the current sensor suite [154], [43].
Analysis of linear models along a mission profile, for a linear model to aid in mission scenarios we may be add a video feed.
\begin{figure}[h!]
\centering
\includegraphics[width=0.6\textwidth]{cameramodule.jpg}
\caption{represents camera properties \cite{c154} pg 56.}
\end{figure}
The advanced missions may have a camera module for the superior MAV. This accessory component is an extensive component may add as the micro-camera for surveillance.
\clearpage
\section{MODEL-BASED TRADE STUDIES}
Some model-based trade studies was conducted to identify critical tradeoffs. The following were examined:1. changes in size, 2. geometry, weight, etc.
The total mass of the vehicle should be kept as low as possible, since added weight will increase power consumption. Reduction in size show difficulties of scaling down batteries and motors while maintaining acceptable performance. Smaller geometry poses a problem with system integration in such a small volume [44] pg. 4, [53], [54] . An MAV needs hardware that is carefully chosen to meet the ensuing size and weight constraints, while providing sufficient flight endurance [100] pg.6. Some have used a C compiler for ease of integration. CCS Inc. has developed one called the PCWH for a PIC microcontroller which is easy to use with Windows based Integrated Development Environment[88]pg.7.
\begin{figure}[h!]
\centering
\includegraphics[width=0.6\textwidth]{systemsofairvehicles.jpg}
\caption{shows the different systems in an Air Vehicle System \cite{c44} pg4.}
\end{figure}
An MAV needs hardware that is carefully chosen to meet the ensuing size and weight constraints, while providing sufficient flight endurance [100] pg.6. Some have used a C compiler for ease of integration. CCS Inc. has developed one called the PCWH for a PIC microcontroller which is easy to use with Windows based Integrated Development Environment [88] pg. 7. propulsion MAV or even a trirotor. The difficulty presented with the trirotor is that according to literature, few studies have been carried out for such configuration. It has three rotors with constant pitch propellers; two fixed rotors turning in opposite directions and one rotor that can be tilted to control the yaw displacement. [99] pg. 1. It was best to choose a quadcopter because of the multitude of research and information available on such a configuration.
\clearpage
\section{VEHICLE OPTIMIZATION ISSUES}
This study examined various optimization issues associated with algorithms for static and dynamic optimization.
\begin{description}
\item [Vehicle trade studies for vehicle optimization.] for further research the optimization of the propeller design for given requirements will be looked upon. In order to optimize the propeller the blade element theory for lifting heavier masses. As such, the optimized propeller will be looked upon with the micro air vehicle quad copter coupled with an optimize [58]. Prototype based on optimized design will be tested experimentally and results will be compared with analytical results, [40].
\end{description}
\clearpage
\section{MODEL VALIDATION ISSUES}
This study examined the following validation issues associated with micro air vehicles:
\begin{description}
\item [Validate the model(s) being used.], In order to validate the models of the developed miniature rotorcraft that is small enough for indoor operation one must account for the capabilities of the micro air vehicle. From a components standpoint the components limitation
s were integrated onboard the vehicle based on understanding of the fundamental rotorcraft performance characteristics. The customized airframe and propulsion allowed for a variety of sensors that can be mounted in different configurations, an onboard processor for the implementation of estimation and control algorithms, and communication capabilities to send and receive data from a remote computer [116]. To integrate the various hardware components a software architecture was developed that can support more computational intensive applications such as video processing, path planning, state estimation and control augmentation [93] page12, [117], [45], [56].
\end{description}
\begin{description}
\item [Determine the limitations of the model(s).] The limitations encountered for the rotorcraft platform examine a miniature plan for development in the future research of micro air vehicles [111]. The proposed aircraft have many limitations regarding the hardware and energy requirements. he developed MAV operation in indoor and confined areas that will provide the necessary components to support the systematic investigation of the challenges associated with hardware limitations and task requirements [93] page12, [145] pg.3.
\end{description}
\clearpage
\section{PLAN FOR MINIATURIZATION}
The following plan for miniaturization was addressed.
Understand relevant issues and tradeoffs associated with miniaturization, Investigations are currently being performed to see how the contributions from various parts scale when the overall weight is reduced. As shown in Figure [7], if the size is decreased, electronics still account for about 13 percent of the total weight, while motors, actuators and battery increase relatively. This reflects the difficulties of scaling down batteries and motors while maintaining acceptable performance. When the system is miniaturized, the airframe has the greatest reduction in percentage. This requires careful selection of shape and materials due to the ultra limited weight budget of the NAV in order to optimize the airframe. [44] pg. 3
\begin{figure}[h!]
\centering
\includegraphics[width=0.6\textwidth]{weightpercent.jpg}
\caption{The weight budget for a 197gMAV and 15g MAV, from \cite{c[44}.}
\end{figure}
Development plan for miniaturization; i.e. developing a Nano Air Vehicle (NAV). This plan is adopted from the development of the SAMARAI Nano Air Vehicle from Lockheed Martin Advanced Technology Laboratories (LMATL). [155]. Key observations about the Nano Air Vehicle problem as outlined by LM ATL. Nature has demonstrated the required aerodynamic principles in an existing system of exactly the right scale [112]. A vehicle with the required endurance and range cannot depend on electric power for propulsion given current battery technology. Helicopter flight principles show us how to achieve forward flight from a rotating lift-producing airfoil. A vehicle with significant mechanical complexity will not be sufficiently robust for battlefield transport and operation.
1. Configuration Design must reflect the consideration of these observations.
2. Layout of the vehicle is is complicated by the interdependence of the sizing and placement for
structure, sensor, propulsion, control components and payload. These elements must be located to obtain specific inertia characteristics for stable flight.
3. The weight must be allocated accordingly. As an Example, the weight budget of the SAMARAI NAV is shown below.
\begin{figure}[h!]
\centering
\includegraphics[width=0.6\textwidth]{weightnav.jpg}
\caption{Represents the weight budget for a SAMARI NAV.}
\end{figure}
4. Sizing of the rotor and analysis of the basic flight power requirements can be done using standard helicopter methodology accounting for induced, profile and parasite power in order to evaluate performance and power required.
5. Equilibrium must be achieved in stability control. Equilibrium is defined by the sum of aerodynamic, propulsion and inertial moments on the rotor about the feather axis. The blade feather axis can be stabilized by inertial forces that arise from shaping the vehicle mass and inertia distribution.
6. Forward flight can be achieved by the application of cyclic lift to control rolling and pitching moments from advancing and retreating blades.
7. 6-DOF Rigid-body Simulation can be performed with parameters similar to SAMARAI simulation, [146], [73].
8. Flight control system should take commands from the operator and produce control signals for lift modulation and engine throttle to maintain stable hover or forward flight direction and velocity as shown in Figure
\begin{figure}[h!]
\centering
\includegraphics[width=0.6\textwidth]{maincontrol.jpg}
\caption{A schematic for the flight control of the lift modulation and engine thrust, \cite{c[41}.}
\end{figure}
\begin{figure}[h!]
\centering
\includegraphics[width=0.6\textwidth]{sensorsgyros.jpg}
\caption{Control System Schematic, \cite{c122}.}
\end{figure}
9. Guidance and Navigation subsystem should include but not be limited to an image sensor, magnetometer, processor and supporting components [113], [118] [119], [120], [121]. 10. Feasibility and validation should be performed through analysis and testing.
\clearpage
\section{BUDGET}
Table 8. Budget for an autonomous quadcopter with full sensors and assembly, [142], [53], [68].
\begin{figure}[h!]
\centering
\includegraphics[width=0.6\textwidth]{budgetpart.jpg}
\caption{Budget for an autonomous quadcopter with full sensors and assembly, \cite{c142}, \cite{c53}, \cite{c68}.}
\end{figure}
In order for a quadrotor Project based on the MikroKopter development community. Additionally, the project maintains its own website, \url{www.cmuquadrotor.com}, and a wiki with documentation on the projects design decisions and software, [142].
\clearpage
\section{TIMETABLE}
A timetable was planned and set forth to conduct with the professor for the involvement of cutting edge research to allow for to finish the outline set forth. The outline allows for goals to be more engaged in the STEM discipline.
The timetable plan for the MAVs are characterized by a 15-20 cm length/width and weighing no less than 20 grams. While several MAVs have been successfully demonstrated, many research and practical implementation issues still remain. As such, the following is a summarized research plan for the major tasks/phases of the project.
\subsection{\bf LOW PASS FILTER}
\subsection{\bf CFD ANALYSIS FOR FLAPPING WING}
\subsection{\bf CONTROL SYSTEM FOR AVITRON}
\subsection{\bf EOM FOR FLAPPING AIR VEHICLE}
\clearpage
\section{Computational Fluid Dynamics}
\begin{description}
\item [CFD analysis and control systems]. Interdisciplinary approach for CFD analysis and control theory
are crucial in the design process for computing forces and moments. In the case of biorobotics for mechanical flapping fins
CFD analysis and control theory are used together for yaw regulation \cite{c250}. The paper provides
an interdisciplinary approach and parameterizes the hydrodynamic forces and moments caused by the
flapping foil. The CFD algorithms inplemented are used for precision control \cite{c250}. Periodic forces and moments
may be obtained using CFD through Fourier series and used for control systems. In the case of generating useful 3D data CFD
will serve as a great computational model as opposed to 2D \cite{c254}.
\end{description}
\begin{description}
\item [Direct Numerical Simulation (DNS) and Large Eddy Simulation (LES).] Through DNS and LES
pulsatile flow in a planar channel may be used to simulate this type of flow \cite{251}, \cite{255}. This is due to the Reynolds number being
very large in such a way that the turbulence scales are accurately solved. A general view for using DNS and LES is for understanding
the computational fluid dynamics of the flow downstream \cite{c251}. For tip-leakage flow problems LES may be used to study the tip gap \cite{c252}.
An analysis was done with various methodologies in simulation where tip clearance flows showed great importance. A flow field with a moving wall
simulation can lead to great understanding of the mean flow field, turbulence parameters, vorticity effects of the tip gap. The end wall vortex flows
are an indication of the flow and can lead to greater understanding with LES and DNS simulations. For numerical schemes such as Runge Kutta and Central differene schem \cite{c255}.
\end{description}
\clearpage
\subsection{Rotation Matrices}
The application of roation matrices are used for calculation of the unit quaternion vector \cite{c
}. A camera is typically needed
for a position vector and a camera rotation is represented by the quaternion. In such cases to find the rotation vector, the center of mass position is
subtracted out.
\clearpage
\subsection{FLUENT Solver}
\clearpage
\subsection{Dynamic Meshing}
\clearpage
\subsection{User Define Function}
\clearpage
\section{Structures}
structures, Mechanical stiffness of the rotor is crucial in reducing the angular deflection of the blade due to aerodynamic moments under operating conditions. Stiffness is improved by maximizing material shear modulus, material bulk and torque factor, q, of the cross-sectional shape. Typical high-lift airfoils are too thin to obtain sufficient structural strength. It is necessary to increase bulk to reduce the moment while attempting to preserve the blades lift performance. The sensitivity of the steady - state angle to variation in the system attributes must also be considered. [153] pg. 3, [139] pg. 5.
sensing and actuation for guidance, navigation and control, MAV need a flight control system that can maintain its course in the face of turbulence or sudden wind gust. Line of sight stabilization is an essential issue if the MAV has an imaging mission. The challenge here is the miniaturization of the necessary electronics like MEMS. [14] pg. 7, [71]. A MEMS rate gyro contains a small vibrating lever. When the lever undergoes an angular rotation, Coriolis effects change the frequency of the vibration, thus detecting the rotation [85] pg. 21.
Actuation is needed for flight control, for making the vehicle turn, for moving the sensors, for movable cameras or for building useful tools such as micro pliers for picking up samples. Similarly as for flapping wing and rotor systems, linear actuators are theoretically the most suitable solution for this application. Although there are a lot of studies of new materials and new concepts for linear actuators, all the existing prototypes have limited maximum elongation and/or long response time that limit the applicability on board AVS. [44] pg. 13. The sensors allow minimizing the error rate of the quad copter [141].
\subsection{Materials}
\clearpage
\section{DESING OF EXPERIMENTS}
im doing refe \cite{c150}.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% PROCEDURE %%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\clearpage
\section{PROCEDURE}
Our statistical method of choice to run by inherent design of experiments was the nested ANOVA1 test. The reasons for using this test is due both of our variables being dependent on each other. Time and velocity are related to one another, so we naturally chose the nested anova1 statistical test.
A study was carried out to determine how much does the load mass amount effects thetime and speed deviation for the micro air vehicle quad rotor. The following times and speed deviation data was taken. The times and speed is given in the results section and a figure is shown in figure 2. Our null hypothesis tests if the masses added are the same effects in speed and time. Our alternative hypothesis is that the various payload masses will affect the time and speed of the vehicles performance.
\begin{figure}[h!]
\centering
\includegraphics[width=0.6\textwidth]{collectingdataquad.jpg}
\caption{Collecting the data using a quad-rotor with various payloads measuring the speed and time from the left tip to right tip of the white board in Gold Water Center laboratory 379.}
\end{figure}
\begin{figure}[h!]
\centering
\includegraphics[width=0.6\textwidth]{speedtestingquad.jpg}
\caption{Speed/Payload Testing, Collecting the data using a quad-rotor with various payloads measuring the speed and time in Gold Water Center laboratory 379. Video from the experiments can be viewed online on the this website: \url{https://sites.google.com/a/asu.edu/michael-thompson/projects/modeling-analysis-control-and-design-of-micro-air-vehicles-and-nano-air-vehicles}}
\end{figure}
\clearpage
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% MIRCO AIR VEHICLE %%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{MIRCO AIR VEHICLES}
This paper will utilize the micro air vehicle (MAV) shown in Figure 1.
\begin{figure}[htbp]
\centering
\setlength{\unitlength}{\textwidth}
\begin{picture}(1,.25)
\put(.2,0){\includegraphics[width=.7\unitlength]{Avitron.jpg}}
\end{picture}
\caption{Flapping Air Vehile}
\end{figure}
\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 Air Vehicle}
\end{figure}
\begin{table}
\section{Flapping Air Vehicle}
\begin{center}
\begin{tabular}{llrlr}
{Parameter} & {Descrition}\\
\hline
Type&Flapping\\
Motor Type& \\
Gross Weight & 8.5 grams \\
2.4 ghz radio system &ange up to 100 yards\\
Flight time(1 fully charge)& 6minutes\\
Wing Span& \\
Wing's radius& \\
Wing's weight& \\
\hline
\end{tabular}
\end{center}
\end{table}
We are now importing the Avitron Micro RC Ornithopter from France. Designed by Edwin Van Ruymbeke, this little bird is a technological marvel of miniaturizationing a mere 8.4 grams making this a true "micro aerial vehicle" (MAV). It comes ready to fly including its own 2.4 ghz radio system with a range up to 100 yards. Features include a patented compact planetary flapping unit, a built in electronic speed controller and lipo battery, and a proprietary turning system. Charging is fast..directly off a built in transmitter charger, taking only 12 minutes to charge for flights up to 6 minutes long. The Avitron comes with one spare set of wings, its own carrying case for safe transport, and a 30 day warranty. Kits are in stock and we ship Avitrons within one day of payment, often the same day.
\begin{table}
\section{Quad-Rotor Air Vehicle}
\begin{center}
\begin{tabular}{llrlr}
{Parameter} & {Descrition}\\
\hline
Type &Quad-Copter\\
Main Rotor Diameter &5.5 in (140mm)\\
Gross Weight & 2.65 oz (75.0 g)\\
Length & 11.5 in (292mm)\\
Motor Size & (4) 8.5mm brushed\\
\hline
\end{tabular}
\end{center}
\end{table}
Micro Air Vehicle in Gold Water Center 379 at Arizona state university
Table 1: Properties of the MAV
\clearpage
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% RESULTS %%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{RESULTS}
A study was carried out to determine if mass amount effects the time and height deviation for the micro air vehicle quad rotor. The following times and height deviation data will be taken. The results will be given in the table showed below:
\begin{figure}[h!]
\centering
\includegraphics[width=01\textwidth]{resultsquad.jpg}
\caption{Represents the mass distribution and velocity profile for the MAV Quad-Copter at various times and velocities with various payloads on the vehicle}
\end{figure}
The testing hypothesis here is how is the mass amount effecting the time and velocity deviation for the micro air vehicle quad rotor.
We will be using the ANOVA1 Test to perform the analysis of variance for the mean distribution of the velocity and time.
\clearpage
\section{PREVIOUS WORK}
This research is an extension of the computational study of transient Couette flow over an embedded cavity surface from the University of Alabama. Dr. Amy Lang and Dr. Will worked on creating computational models to identify the mechanism of how butterfly scales geometry influences the low drag coefficient of its flight. These computational models simulated the flow over a butterfly wing in a 2D model [46]. Computational fluid analysis has become more popular in conducting research, especially for modeling purposes. In this computational study, these models aid in the understanding of the grounds for low drag coefficient in insect flight [57]. Previously, 2D models have been designed for such purposes, and these 2D embedded cavities can be used to model the flow over a wing in a 3D sense. Traditional methods of testing had analyzed the couette flow by having the top wall move in one direction as the bottom wall stays in place. In this case, couette flow will be analyzed from a stationary top wall and a bottom, left, and right wall moving in the same direction. Fundamentally, the work presented in this paper would be used for present research being done in micro air vehicles for security purposes, such as optical sensor MAVs and potentially, chemical sensors in these vehicles [4].
Creating the 3D models add real life perspective to the aerodynamics of wings. We created the model in ANSYS 13 to simulate the flapping motion of insect flight for application towards micro-air-vehicles. As such here we are trying to take advantage of the 3D geometry that would be created prior to the application of the C.F.D. software [3]. The cavities along the 3D models will aid in gathering drag data and flow visualization data to be able to study trends of micro and nano air vehicle flight performance [15] with computational fluid dynamic technology; which takes advantage of the 3D geometry for application of the C.F.D. software [3].
\clearpage
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% CONCLUSIONS %%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{CONCLUSIONS}
In testing we found that our grand mean velocity was calculated as 0.67 m/s for our quad rotor. Non-comparable to Vijay Kumar's quad rotor experimentally reaching a trajectory velocity of 3.6 m/s [1], [128]. Thus, our experiments show that we have a much slower velocity and cannot handle the carrying load capacity of that of University of Pennsylvania's quad rotor. There vehicle demonstrates how they can predict with a quadratic air drag model to fly through a circular hoop. In their case they conclude they may be able to trade off speed for accuracy enabling them to carry a higher payload [1]. The faster trajectories have higher lead times thus contributing to the overall payload capacity. in addition, further research needs to be handled with care in the control algorithm for following trajectories and automated approach to increase the speed and lead time; therefore, a higher contribution to the carrying capacity for the MAV quad rotor. As increased research in this area the capabilities for furthering the pursuit to the applications of the MAV quad rotor to be able to assemble a building, regulate police operations, regulate traffic control, control crowd management, dispose of waste, surveillance neighborhoods with attached cameras, search and rescue and many more. Future work will encompass adding a camera to the MAV quad rotor to utilize the application for surveillance and intelligence gathering [113], [123], [135], [74], [75].
\clearpage
\section{FUTURE WORKS}
Future work will be aimed at developing an autonomous algorithm to navigate the quad copter micro air vehicle. A controller for maneuvering the vehicle will be used as can be seen in figure 26, [126], [127], [129], [134].
\begin{figure}[h!]
\centering
\includegraphics[width=01\textwidth]{systemconfiguration.jpg}
\caption{Overall System Configuration, \cite{c113}.}
\end{figure}
Future studies should include a three dimensional (3D) Large Eddy Simulation (LES) of the possible flow through the wings of the MAV in ANSYS-FLUENT.
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MAV Research Team Summer 2012.
Since the current model has not been made in a CAD system we should desire to render engineering drawings of the quad copter MAV to model the flow over the flow over the thing, and look at the drag on the associated wings. In reality, everything is 3D and we could appreciate a 3D model that would be much more complicated and would show what is really happening when the wings rotate. This would be possible by creating the model in ANSYS 13 to simulate the motion of flight for the quad rotors application on intelligence gathering. Another aspect of research will be to embed the 2D cavities along the 3D models and gather drag data and flow visualization data to be able to study trends of flight performance with computational fluid dynamic technology.
\clearpage
\section{RELATION TO GRADUATE STUDIES}
This semesters activities included experimental methods for various payloads and looking for documents that have been published that relate to our research topic. This type of research is what is typically envisioned for a graduate student, considering their topic might be something that few others have looked into. Having the dedication to push through the papers and ultimately come to an understanding, show early signs to a successful graduate degree process. When regurgitating the information learned from the texts and trying to create a simple, idealized model was another activity conducted during the semester. This activity demonstrates the level of critical and analytical thinking that are characteristics worthy of graduate students at Arizona State University research program.
\begin{figure}[h!]
\centering
\includegraphics[width=1\textwidth]{teamtwo.jpg}
\caption{ MAV Research Team (From left to right: Michael Thompson, Ivan Ramirez, Mariela Robledo)}
\end{figure}
\begin{figure}[h!]
\centering
\includegraphics[width=1\textwidth]{teamone.jpg}
\caption{ MAV Research Team Spring 2012}
\end{figure}
\clearpage
\section{Team Management}
\begin{enumerate}
\item \bf{Michael Thompson ANSYS-FLUENT Mechanical Engineer.}
\begin{itemize}
\item From a young age, I have been fascinated with flying machines and miniaturization. While at Arizona State University (ASU), I have had many research experiences that have significantly shaped my career goals.
\item After earning my MS and PhD, I plan to become a professor of Mechanical Engineering at a Research I university in the area of nano/micro air vehicles (NAVs/MAVs). This would allow me to teach, conduct world-class research, mentor students, supervise their work, and contribute significantly to the technological development of the nation. I will contribute to the STEM discipline by exposing the research to younger engineers.
\item My website will be a continuation of the ongoing research for further work in the field of nano and micro air vehicles found here:\url{https://sites.google.com/a/asu.edu/michael-thompson/}
\end{itemize}
\item \bf{Ivan Ramirez Civil Engineer.}
\begin{itemize}
\item My role on this team is to analyze the fluid and work closely with the CFD personnel to ensure the correct properties are being used for flow parameters. I will gather parameters of 3D wings for micro air vehicles. Keeping record of the meetings and post them to the website:
\end{itemize}
\item \bf{Mariela Robledo Chemical Engineer.}
\end{enumerate}
\begin{itemize}
\item My role on this team is to analyze the fluid and work closely with the CFD personnel to ensure the correct properties are being used for flow parameters. I will gather parameters of 3D wings for micro air vehicles. Keeping record of the meetings and post them to the website:
\end{itemize}
\subsection{TEAM MEETINGS}
The teams meeting times were typically on Mondays from 7-9pm, Tuesday: 6-8 pm. Our reasons for meeting were to work 4 hours per week, every week, so we can achieve our goal.
\clearpage
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[132] Fix me please
%James F. Roberts, Timothy S. Stirling, Jean Christophe Zufferey and Dario Floreano, "Quadrotor Using Minimal Sensing For Autonomous Indoor Flight”, Ecole Polytechnique Federale de Lausanne (EPFL), Lausanne, 1015, Switzerland, (2007)
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Haoxiang Luo, Rajat Mittal and Steven A. Bielamowicz, "Analysis of flow-structure interaction in the larynx during phonation using an immersed-boundary method", Department of Mechanical Engineering, Vanderbilt University, Department of Mechanical and Aerospace Engineering, George Washington University and Division of Otolaryngology, George Washington University Acoustical Society of America, DOI: 10.1121/1.3158942, (2009) \url{https://docs.google.com/a/asu.edu/viewer?a=v&pid=sites&srcid=YXN1LmVkdXxtaWNoYWVsLXRob21wc29ufGd4OjI1ZTBlNzFlYzc3MWQ1NDA}
\bibitem{c164}
H. Dong, M. Bozkurttas, R. Mittal, P. Madden and G. V. Lauder, "Computational modelling and analysis of the hydrodynamics of a highly deformable fish pectoral fin", Department of Mechanical and Aerospace Engineering, The George Washington University and The Museum of Comparative Zoology, Harvard University, doi:10.1017/S0022112009992941, (2010) \url{https://docs.google.com/a/asu.edu/viewer?a=v&pid=sites&srcid=YXN1LmVkdXxtaWNoYWVsLXRob21wc29ufGd4OjY0MmI4MWZlOWEwY2Y0OTg}
\bibitem{c165}
Ehsan Aram, Rajat Mittal, John Griffin and Louis Cattafesta, "Towards Effective ZNMF Jet Based Control of a Canonical Separated Flow", Department of Mechanical Engineering, Johns Hopkins University and Florida Center for Advanced Aero-Propulsion (FCAAP) Interdisciplinary Microsystems Group, AIAA 2010-4705, (2010) \url{https://docs.google.com/a/asu.edu/viewer?a=v&pid=sites&srcid=YXN1LmVkdXxtaWNoYWVsLXRob21wc29ufGd4OjNlYTY5NWIzYjc5ZDcyN2U}
\bibitem{c166}
Lingxiao Zheng, Rajat Mittal and Tyson L. Hedrick, "A Search for Optimal Wing Strokes in Flapping Flight: Can Engineers Improve Upon Nature? ", Department of Mechanical Engineering, Johns Hopkins University , Department of Biology, University of North-Carolina , AIAA 2010-4944, (2010) \url{https://docs.google.com/a/asu.edu/viewer?a=v&pid=sites&srcid=YXN1LmVkdXxtaWNoYWVsLXRob21wc29ufGd4OjI2MDZmNTljMTViMTVlNTA}
\bibitem{c167}
Rupesh B. Kotapati, Rajat Mittal, Olaf Marxen, Frank Ham, Donghyun You and Louis N. Cattafesta I I I, "Nonlinear dynamics and synthetic-jet-based control of a canonical separated flow", Department of Mechanical and Aerospace Engineering, The George Washington University, Centre for Turbulence Research, Stanford University, Department of Mechanical Engineering, Carnegie Mellon University and Department of Mechanical and Aerospace Engineering, University of Florida, doi:10.1017/S002211201000042X, (2010) \url{https://docs.google.com/a/asu.edu/viewer?a=v&pid=sites&srcid=YXN1LmVkdXxtaWNoYWVsLXRob21wc29ufGd4OjYyZDg0N2JjMzEzMGRiZmM}
\bibitem{c168}
J. H. Seo and R. Mittal, "A New Immersed Boundary Method for Aeroacoustic Sound Prediction around Complex Geometries", Department of Mechanical Engineering, Johns Hopkins University, AIAA 2010-4434, (2010) \url{https://docs.google.com/a/asu.edu/viewer?a=v&pid=sites&srcid=YXN1LmVkdXxtaWNoYWVsLXRob21wc29ufGd4OmY5ZWJmZWVmMTUyZTNkNg}
\bibitem{c169}
Shawn Aram and Rajat Mittal, "Computational Study of the Effect of Slot Orientation on Synthetic Jet-Based Separation Control", Department of Mechanical Engineering, Johns Hopkins University, (2011) \url{https://docs.google.com/a/asu.edu/viewer?a=v&pid=sites&srcid=YXN1LmVkdXxtaWNoYWVsLXRob21wc29ufGd4Ojc0NTJkNGM5ZmNlNjc2MDQ}
\bibitem{c170}
X. Zheng, R. Mittal and S. Bielamowicz, "A computational study of asymmetric glottal jet deflection during phonation", Department of Mechanical Engineering, Johns Hopkins University and Division of Otolaryngology, The George Washington University, (2010) \url{https://docs.google.com/a/asu.edu/viewer?a=v&pid=sites&srcid=YXN1LmVkdXxtaWNoYWVsLXRob21wc29ufGd4OjQyMDA5Y2NhOGJiMzM5Yg}
\bibitem{c171}
Xudong Zheng, Steve Bielamowicz, Haoxinag Luo and Rajat Mittal, "A Computational Study of the Effect of False Vocal Folds on Glottal Flow and Vocal Fold Vibration During Phonation", Department of Mechanical and Aerospace Engineering, Division of Otolaryngology, The George Washington University and Department of Mechanical Engineering, Vanderbilt University, DOI: 10.1007/s10439-008-9630-9, (2009) \url{https://docs.google.com/a/asu.edu/viewer?a=v&pid=sites&srcid=YXN1LmVkdXxtaWNoYWVsLXRob21wc29ufGd4OjExMGMyYzY3ZDE0M2Y5NGQ}
\bibitem{c172}
B. R. Ravi, R. Mittal and F.M. Najjar, "Study of Three-Dimensional Synthetic Jet Flow Fields Using Direct Numerical Simulation", Department of Mechanical and Aerospace Engineering, The George Washington University and Center for Simulation of Advanced Rockets, University of Illinois, AIAA 2004-0091 (2004) \url{https://docs.google.com/a/asu.edu/viewer?a=v&pid=sites&srcid=YXN1LmVkdXxtaWNoYWVsLXRob21wc29ufGd4OjcxYWRmOThlOWU4MzkyMmE}
\bibitem{c173}
M. Bozkurttas, H. Dong., V. Seshadri, R. Mittal and F. Najjar, "Towards Numerical Simulation of Flapping Foils on Fixed Cartesian Grids", Department of Mechanical and Aerospace Engineering, The George Washington University and Center for Simulation of Advanced Rockets, University of Illinois, AIAA 2005-0079, (2005) \url{https://docs.google.com/a/asu.edu/viewer?a=v&pid=sites&srcid=YXN1LmVkdXxtaWNoYWVsLXRob21wc29ufGd4OjUxZDcxMmVkZDY2MGM3NTU}
\bibitem{c174}
H. Dong, R. Mittal, M. Bozkurttas and F. Najjar, "Wake Structure and Performance of Finite Aspect-Ratio Flapping Foils", Department of Mechanical and Aerospace Engineering, The George Washington University and Center for Simulation of Advanced Rockets, University of Illinois, AIAA 2005-0081(2005) \url{https://docs.google.com/a/asu.edu/viewer?a=v&pid=sites&srcid=YXN1LmVkdXxtaWNoYWVsLXRob21wc29ufGd4OjFjYjk1Mzc1MTljNGM3MmE}
\bibitem{c175}
K. Mohseni, R. Mittal, F. E. Fish, "Special issue featuring selected papersfrom the Mini-Symposium on Biomimetic and Bio-Inspired Propulsion (Boulder, CO, USA, 26 June 2006)", Department of Aerospace Engineering, University of Colorado, Department of Mechanical and Aerospace Engineering, George Washington University and Department of Biology, West Chester University doi:10.1088/1748-3182/1/4/E01, (2006) \url{https://docs.google.com/a/asu.edu/viewer?a=v&pid=sites&srcid=YXN1LmVkdXxtaWNoYWVsLXRob21wc29ufGd4OjZiZmU4ODcwYjNhODNhOGI}
\bibitem{c176}
Alfred von Loebbecke, Rajat Mittal, "Comparative Analysis of Thrust Production for Distinct Arm-Pull Styles in Competitive Swimming", Mechanical and Aerospace Engineering, The George Washington University and Mechanical Engineering, Johns Hopkins University, (2012) \url{https://docs.google.com/a/asu.edu/viewer?a=v&pid=sites&srcid=YXN1LmVkdXxtaWNoYWVsLXRob21wc29ufGd4OjI5MDUwMTAxMzk3OGI1MzU}
\bibitem{c177}
Louis N. Cattafesta III, Ye Tian and R. Mittal, "Adaptive Control of Post-Stall Separated Flow Application to Heavy Vehicles", Interdisciplinary Microsystems Group Department of Mechanical and Aerospace Engineering and Department of Mechanical and Aerospace Engineering The George Washington University, (2009) \url{https://docs.google.com/a/asu.edu/viewer?a=v&pid=sites&srcid=YXN1LmVkdXxtaWNoYWVsLXRob21wc29ufGd4OjNiMmFjOWZhODUyMzdhNDQ}
\bibitem{c178}
Abel Vargas and Rajat Mittal, "Aerodynamic Performance of Biological Airfoils",The George Washington University, AIAA-2004-2319, (2004) \url{https://docs.google.com/a/asu.edu/viewer?a=v&pid=sites&srcid=YXN1LmVkdXxtaWNoYWVsLXRob21wc29ufGd4OmIzOWNlYjAxNmMxYjUxZQ}
\bibitem{c179}
R. Mittal, F.M. Najjar, R. Byrganhalli, V. Seshadri and H. Singh, "Simulation of complex biological flows and flow control problems on Cartesian grids", The George Washington University, University of Illinois, Thomas Jefferson School of Science and Technology,( 2004) \url{https://docs.google.com/a/asu.edu/viewer?a=v&pid=sites&srcid=YXN1LmVkdXxtaWNoYWVsLXRob21wc29ufGd4OjJhZmJlYzU5NjU3ZDRmZWY}
\bibitem{c180}
Alfred von Loebbecke, Rajat Mittal, Frank Fish and Russell Mark, "Propulsive Efficiency of the Underwater Dolphin Kick in Humans" Department of Mechanical and Aerospace Engineering, George Washington University, Department of Biology, West Chester University, DOI: 10.1115/1.3116150, (2009) \url{https://docs.google.com/a/asu.edu/viewer?a=v&pid=sites&srcid=YXN1LmVkdXxtaWNoYWVsLXRob21wc29ufGd4OjMxZDk0ODkwODlkNGU3OGU}
\bibitem{c181}
Rajat Mittal and Parviz Moin, "Suitability of Upwind-Biased Finite Difference Schemes for large-Eddy Simulation of Turbulent Flows", Stanford University, AIAA (1997) \url{https://docs.google.com/a/asu.edu/viewer?a=v&pid=sites&srcid=YXN1LmVkdXxtaWNoYWVsLXRob21wc29ufGd4OjFiODJjMDJhNjU3NTZhNjI}
\bibitem{c182}
R. Mittal, J. J. Wilson and F. M. Najjar, Symmetry Properties of the Transitional Sphere Wake, University of Florida and University of Illinois, AIAA0001-1452/02 (2001) \url{https://docs.google.com/a/asu.edu/viewer?a=v&pid=sites&srcid=YXN1LmVkdXxtaWNoYWVsLXRob21wc29ufGd4Ojc1NjkyNTcyOTkyNDM3MDQ}
\bibitem{c183}
Chelakara S. Subramanian,Tahani R. Amer, Donald M. Oglesby and Cecil G. Burkett Jr., "New Self-Referencing Pressure-Sensitive-Paint Measurement", Florida Institute of Technology and NASA Langley Research Center, AIAA 2000-2526, (2000) \url{https://docs.google.com/a/asu.edu/viewer?a=v&pid=sites&srcid=YXN1LmVkdXxtaWNoYWVsLXRob21wc29ufGd4OjZmYWFkZTYyZTNlNmNjNGI}
\bibitem{c184}
J. H. Seo and R. Mittal, "Computation of Aerodynamic Sound around Complex Stationary and Moving Bodies", Department of Mechanical Engineering, Johns Hopkins University, AIAA (2001) \url{https://docs.google.com/a/asu.edu/viewer?a=v&pid=sites&srcid=YXN1LmVkdXxtaWNoYWVsLXRob21wc29ufGd4OjE2NzQyMDI5MWJmOTYzMTA}
\bibitem{c185}
Rajat Mittal, "Planar Symmetry in the Unsteady Wake of a Sphere," University of Florida, AIAA (1998) \url{https://docs.google.com/a/asu.edu/viewer?a=v&pid=sites&srcid=YXN1LmVkdXxtaWNoYWVsLXRob21wc29ufGd4OjU1NGUzOTUyMjU2NmUxZjY}
\bibitem{c186}
R. Mittal, Y. Utturkar, and H.S. Udaykumar "Computational Modeling and Analysis of Biomimetic Flight Mechanisms", Department of Mechanical and Aerospace Engineering The George Washington University, Department of Mechanical Engineering University of Florida and Department of Mechanical Engineering, University of Iowa AIAA 2002-0865 (2002) \url{https://docs.google.com/a/asu.edu/viewer?a=v&pid=sites&srcid=YXN1LmVkdXxtaWNoYWVsLXRob21wc29ufGd4OmMyNmY4NTliNDJkZTdlNw}
\bibitem{c187}
T. Ye, R. Mittal, H. S. Udaykumar and W. Shyy, "A Cartesian Grid Method for Viscous Incompressible Flows with Complex Immersed Boundaries", Department of Mechanical Engineering, University of Florida and Department of Aerospace Engineering, Mechanics and Engineering Science, University of Florida AIAA 99 3312, (1999) \url{https://docs.google.com/a/asu.edu/viewer?a=v&pid=sites&srcid=YXN1LmVkdXxtaWNoYWVsLXRob21wc29ufGd4OmYxZjYxNmQyOGNlNzI2NQ}
\bibitem{c188} \url{https://docs.google.com/a/asu.edu/viewer?a=v&pid=sites&srcid=YXN1LmVkdXxtaWNoYWVsLXRob21wc29ufGd4OjNhMzcwZmIyNWJlNWY5ZjQ}
Etkin, Bernard, and Lloyd Duff Reid. "Dynamics of Flight Stability and Control." Canada. John Wiley and, Inc., 1996. Print.
\bibitem{c189}
R. Mittal, F.M. Najjar, "Vortex Dynamics in the Sphere Wake", Department of Mechanical Engineering, University of Florida and Center for Simulation of Advanced Rockets, University of Illinois, AIAA 99-3806 (1999) \url{https://docs.google.com/a/asu.edu/viewer?a=v&pid=sites&srcid=YXN1LmVkdXxtaWNoYWVsLXRob21wc29ufGd4OjU5NzU2Mzg1MTgwOGNmMw}
\bibitem{c190}
Y. Utturkar1, R. Mittal, P. Rampunggoon and L. Cattafesta, "Sensitivity of Synthetic Jets to the Design of the Jet Cavity", 1Department of Mechanical Engineering, University of Florida, Department of Mechanical and Aerospace Engineering, The George Washington University and 3Department of Aerospace Engineering, Mechanics and Engineering Sciences, University of Florida, AIAA 2002-0124, (2002) \url{https://docs.google.com/a/asu.edu/viewer?a=v&pid=sites&srcid=YXN1LmVkdXxtaWNoYWVsLXRob21wc29ufGd4OjYxZjU1MmIwZGViZWU0YTY}
\bibitem{c191}
Yogen Utturkar, Ryan Holman, Rajat Mittal, Bruce Carroll, Mark Sheplak, and Louis Cattafesta, "A Jet Formation Criterion for Synthetic Jet Actuators", Department of Mechanical and Aerospace Engineering,University of Florida and Department of Mechanical and Aerospace Engineering, The George Washington University, AIAA-2003-0636, (2003) \url{https://docs.google.com/a/asu.edu/viewer?a=v&pid=sites&srcid=YXN1LmVkdXxtaWNoYWVsLXRob21wc29ufGd4OjUyZWZmMzMxMjZkYTE1NzU}
\bibitem{c192}
Donghyun You, Meng Wang, Rajat Mittal, and Parviz Moin, "Study of Rotor Tip-Clearance Flow Using Large-Eddy Simulation", Stanford University and The George Washington University, AIAA 2003-0838, (2003) \url{https://docs.google.com/a/asu.edu/viewer?a=v&pid=sites&srcid=YXN1LmVkdXxtaWNoYWVsLXRob21wc29ufGd4OjIxNzAyZTdmOTUwYTk1ZTM}
\bibitem{c193}
Reza Ghias, Rajat Mittal, Thomas S. Lund, "A Non-Body Conformal Grid Method For Simulation of Compressible Flows With Complex Immersed Boundaries", Department of Mechanical and Aerospace Engineering, The George Washington University and Department of Aerospace Engineering Sciences, The University of Colorado, AIAA 2004-0080, (2004) \url{https://docs.google.com/a/asu.edu/viewer?a=v&pid=sites&srcid=YXN1LmVkdXxtaWNoYWVsLXRob21wc29ufGd4OjI1MmNlNjQwZjZkOGM4Yw}
\bibitem{c194}
Quentin Gallas, Ryan Holman, Reni Raju, Rajat Mittal, Mark Sheplak and Louis Cattafesta, "Low Dimensional Modeling of Zero-Net Mass-Flux Actuators, "University of Florida, The George Washington University, University of Florida, AIAA 2004-2413 (2004) \url{https://docs.google.com/a/asu.edu/viewer?a=v&pid=sites&srcid=YXN1LmVkdXxtaWNoYWVsLXRob21wc29ufGd4OjMwZDczYzc2NzY0YWRlNzc}
\bibitem{c195}
Rupesh B. Kotapati and Rajat Mittal, "Time-Accurate Three-Dimensional Simulations of Synthetic Jets in Quiescent Air, The George Washington University, AIAA 2005-0103 (2005) \url{https://docs.google.com/a/asu.edu/viewer?a=v&pid=sites&srcid=YXN1LmVkdXxtaWNoYWVsLXRob21wc29ufGd4OjYyMzBjYjg4ODIxZmNkY2Y}
\bibitem{c196}
Rajat Mittal, Rupesh B. Kotapati and Louis N. Cattafesta III, "Numerical Study of Resonant Interactions and Flow Control in a Canonical Separated Flow", Department of Mechanical and Aerospace Engineering, The George Washington University and Department of Mechanical and Aerospace Engineering, University of Florida, AIAA 2005-1261, (2005) \url{https://docs.google.com/a/asu.edu/viewer?a=v&pid=sites&srcid=YXN1LmVkdXxtaWNoYWVsLXRob21wc29ufGd4Ojc0NTBlOGViOGQwNGI5NGI}
\bibitem{c197}
I. Akhtar and R. Mittal, "A Biologically Inspired Computational Study of Flow Past Tandem Flapping Foils", Department of Mechanical and Aerospace Engineering, The George Washington University, AIAA 2005-4760 (2005) \url{https://docs.google.com/a/asu.edu/viewer?a=v&pid=sites&srcid=YXN1LmVkdXxtaWNoYWVsLXRob21wc29ufGd4OmFlMWJkYzRhMTQ3YjE3OA}
\bibitem{c198}
B. R. Ravi and R. Mittal,"Numerical Study of Large Aspect-Ratio Synthetic Jets, "Department of Mechanical and Aerospace Engineering, The George Washington University, AIAA 2006-0315, (2006) \url{https://docs.google.com/a/asu.edu/viewer?a=v&pid=sites&srcid=YXN1LmVkdXxtaWNoYWVsLXRob21wc29ufGd4Ojc1M2IzMzU4Yjk4MThjMDY}
\bibitem{c199}
Rupesh B. Kotapati, Rajat Mittal and Louis N. Cattafesta III, "Numerical Experiments in Synthetic Jet Based Seperation Control", Department of Mechanical and Aerospace Engineering, The George Washington University and Department of Mechanical and Aerospace Engineering, University of Florida, AIAA 2660-0320 (2006) \url{https://docs.google.com/a/asu.edu/viewer?a=v&pid=sites&srcid=YXN1LmVkdXxtaWNoYWVsLXRob21wc29ufGd4Ojc5OWJhMzdjZjRhNDE0ZTQ}
\bibitem{c200}
M. Bozkurttas, H. Dong., R. Mittal, P. Madden and G.V. Lauder, "Hydrodynamic Performance of Deformable Fish Fins and Flapping Foils", Department of Mechanical and Aerospace Engineering, The George Washington University and Museum of Comparative Zoology Harvard University, AIAA 2006-1392 (2006) \url{https://docs.google.com/a/asu.edu/viewer?a=v&pid=sites&srcid=YXN1LmVkdXxtaWNoYWVsLXRob21wc29ufGd4OjYyMGJjNzhmMmY5YTdjZg}
\bibitem{c201}
Ye Tian, Louis N. Cattafesta III and Rajat Mittal, "Adaptive Control of Separated Flow", Department of Mechanical and Aerospace Engineering, University of Florida and Department of Mechanical and Aerospace Engineering, The George Washington University, AIAA 2006-1401 (2006)
\url{https://docs.google.com/viewer?a=v&pid=sites&srcid=YXN1LmVkdXxtaWNoYWVsLXRob21wc29ufGd4OjdhZjdiNWU0MTRjY2Uz}
\bibitem{c202}
R. Mittal, H. Dong, M. Bozkurttas, A. Von Loebbecke and F. Najjar, "Analysis of Flying and Swimming in Nature Using an Immersed Boundary Method", Department of Mechanical and Aerospace Engineering, The George Washington University and Center for Simulation of Advanced Rockets, University of Illinois, AIAA 2006-2867 (2006)
\url{https://docs.google.com/viewer?a=v&pid=sites&srcid=YXN1LmVkdXxtaWNoYWVsLXRob21wc29ufGd4OjZlODU2Y2QzYzg5ZGJjODY}
\bibitem{c203}
M. Bozkurttas, H. Dong, R. Mittal, James Tangorra, Ian Hunter, G.V. Lauder and P. Madden, "CFD based Analysis and Design of Biomimetic Flexible Propulsors for Autonomous Underwater Vehicles", Department of Mechanical and Aerospace Engineering, The George Washington University, AIAA-2007-4213 (2007)
\url{https://docs.google.com/viewer?a=v&pid=sites&srcid=YXN1LmVkdXxtaWNoYWVsLXRob21wc29ufGd4OjdmZWVjN2U1NWUwMzZkN2I}
\bibitem{c204}
Srinivas Ramakrishnan, Rajat Mittal, George V. Lauder and Meliha Bozkurttas, "Analysis of Maneuvering Fish Fin Hydrodynamics Using an Immersed Boundary Method", Department of Mechanical and Aerospace Engineering, George Washington University and The Museum of Comparative Zoology, Harvard University, AIAA 2008-3717, (2008)
\url{https://docs.google.com/viewer?a=v&pid=sites&srcid=YXN1LmVkdXxtaWNoYWVsLXRob21wc29ufGd4OjZhZjk0NjZhYmMwY2Q0OWI}
\bibitem{c205}
Xudong Zheng and Rajat Mittal, "A High Fidelity Computational Method for Flow-Tissue Interaction in Biological Flows", Department of Mechanical and Aerospace Engineering, The George Washington University, AIAA 2008-3954, (2008)
\url{https://docs.google.com/viewer?a=v&pid=sites&srcid=YXN1LmVkdXxtaWNoYWVsLXRob21wc29ufGd4OjNiOGU1MDIyMDg4MmRkMzE}
\bibitem{c206}
Reni Raju, Ehsan Aram, Rajat Mittal and Louis Cattafesta, "Reduced-Order Models of Zero-Net Mass-Flux Jets for Large-Scale Flow Control Simulations", Department of Mechanical and Aerospace Engineering, The George Washington University and Interdisciplinary Microsystems Group, Department of Mechanical and Aerospace Engineering, University of Florida AIAA 2008-6404, (2008)
\url{https://docs.google.com/viewer?a=v&pid=sites&srcid=YXN1LmVkdXxtaWNoYWVsLXRob21wc29ufGd4OjU0N2M4NmI5NWEyYWRjZTg}
\bibitem{c207}
Lingxiao Zheng, Xiaolin Wang, Afzal Khan, R. R. Vallance Rajat Mittal and Tyson L. Hedrick, "A Combined Experimental Numerical Study of the Role of Wing Flexibility in Insect Flight", Department of Mechanical and Aerospace Engineering, George Washington University, Department of Biology, University of North Carolina, AIAA 2009-382, (2009)
\url{https://docs.google.com/viewer?a=v&pid=sites&srcid=YXN1LmVkdXxtaWNoYWVsLXRob21wc29ufGd4OjRkYTZlNzM5ODEyNDRiZTI}
\bibitem{c208}
Jonathan H. Tu, Clarence W. Rowley, Ehsan Aram and Rajat Mittal, " Koopman spectral analysis of separated flow over a finite-thickness flat plate with elliptical leading edge", Princeton University and Johns Hopkins University, AIAA 2011-38, (2011)
\url{https://docs.google.com/viewer?a=v&pid=sites&srcid=YXN1LmVkdXxtaWNoYWVsLXRob21wc29ufGd4OjI5NjM0MTVlN2FkMGM1NmY}
\bibitem{c209}
Donghyun You, Meng Wang, Rajat Mittal and Parviz Moin, "Large-Eddy Simulations of Longitudinal Vortices Embedded in a Turbulent Boundary Layer," Stanford University and George Washington University, DOI: 10.2514/1.22043, (2006)
\url{https://docs.google.com/viewer?a=v&pid=sites&srcid=YXN1LmVkdXxtaWNoYWVsLXRob21wc29ufGd4OjU1M2YyMjlmMmIzYzIxZWQ}
\bibitem{c210}
Reni Raju, Rajat Mittal and Louis Cattafesta, "Dynamics of Airfoil Separation Control Using Zero-Net Mass-Flux Forcing", The George Washington University and University of Florida, DOI: 10.2514/1.37147, (2008)
\url{https://docs.google.com/viewer?a=v&pid=sites&srcid=YXN1LmVkdXxtaWNoYWVsLXRob21wc29ufGd4OjQ0NWIwY2YxZjM5ZTk2MzA}
\bibitem{c211}
Reni Raju, Rajat Mittal, Quentin Gallas and Louis Cattafesta, "Scaling of Vorticity Flux and Entrance Length Effects in Zero-Net Mass-Flux Devices", Department of Mechanical and Aerospace Engineering,The George Washington University and Department of Mechanical and Aerospace Engineering, University of Florida, AIAA 2005-4751, (2005)
\url{https://docs.google.com/viewer?a=v&pid=sites&srcid=YXN1LmVkdXxtaWNoYWVsLXRob21wc29ufGd4OjE2Yjg2MzQ4NDE4OWZlYWQ}
\bibitem{c212}
Rupesh B. Kotapati, Rajat Mittal, Olaf Marxen, Frank Ham and Donghyun You, "Numerical Simulations of Synthetic Jet Based Separation Control in a Canonical Separated Flow", Department of Mechanical and Aerospace Engineering, The George Washington University and Center for Turbulence Research, Stanford University, AIAA 2007-1308, (2007)
\url{https://docs.google.com/viewer?a=v&pid=sites&srcid=YXN1LmVkdXxtaWNoYWVsLXRob21wc29ufGd4OjMzODFmMDgzNDVhNjE0OTY}
\bibitem{c213}
Reni Raju, Rajat Mittal and Louis N. Cattafesta III, "Towards Physics Based Strategies for Separation Control over an Airfoil using Synthetic Jets", The George Washington University and University of Florida, AIAA 2007-1421 (2007)
\url{https://docs.google.com/viewer?a=v&pid=sites&srcid=YXN1LmVkdXxtaWNoYWVsLXRob21wc29ufGd4Ojk5OTZjZjg5OTI0YzlkZg}
\bibitem{c214}
Ryan Holman, Yogen Utturkar, Rajat Mittal, Barton L. Smith, Louis Cattafesta, "Formation Criterion for Synthetic Jets", University of Florida, George Washington University and Utah State University, AIAA, 0001-1452/05, (2005)
\url{https://docs.google.com/viewer?a=v&pid=sites&srcid=YXN1LmVkdXxtaWNoYWVsLXRob21wc29ufGd4OjE5MWZjZjk2Yzc0NmNhYWM}
\bibitem{c215}
R. Ghias, R. Mittal and H. Dong, "A sharp interface immersed boundary method for compressible viscous flows", Department of Mechanical and Aerospace Engineering, The George Washington University, doi:10.1016/j.jcp.2006.12.007 (2006)
\url{https://docs.google.com/viewer?a=v&pid=sites&srcid=YXN1LmVkdXxtaWNoYWVsLXRob21wc29ufGd4OmVlYTM5NzcwZmE3Y2I0NQ}
\bibitem{c216}
Isaac I. Kaminer, Oleg A. Yakimenko, Vladimir N. Dobrokhodov and Kevin D Jones, "Rapid Flight Test Prototyping System and the Fleet of UAVs and MAVs at the Naval Postgraduate School", Naval Postgraduate School, AIAA (2005)
\url{https://docs.google.com/viewer?a=v&pid=sites&srcid=YXN1LmVkdXxtaWNoYWVsLXRob21wc29ufGd4OjE4MGFkMzU1NDYxNTQ4ODQ}
\bibitem{c217}
Soon-Jo Chung, Michael Dorothy and Jeremiah R. Stoner, "Neurobiologically Inspired Control of Engineered Flapping Flight", Iowa State University, AIAA, (2009)
\url{https://docs.google.com/viewer?a=v&pid=sites&srcid=YXN1LmVkdXxtaWNoYWVsLXRob21wc29ufGd4OjZiYmZjMmM2MWUzMWIwNTU}
\bibitem{c218}
Alfred Von Loebbecke, Rajat Mittal, Russell Mark and James Hahn, "A Computational Method for analysis of Underwater Dolphin Kick Hydrodynamics in Human Swimming", George Washington University, DOI: 10.1080/1470802629982, (2008)
\url{https://docs.google.com/viewer?a=v&pid=sites&srcid=YXN1LmVkdXxtaWNoYWVsLXRob21wc29ufGd4OjZlNGI1ZWRlZmQ3NmM0N2E}
\bibitem{c219}
Rajat Mittal and S. Balachandar, " On the Inclusion of Three-Dimensional Effects in Simulations of Two-Dimensional Bluff-Body Wake Flows", Stanford University and University of Illinois, (1997)
\url{https://docs.google.com/viewer?a=v&pid=sites&srcid=YXN1LmVkdXxtaWNoYWVsLXRob21wc29ufGd4OjQ5N2M2YmRiYTIxYzU0MzA}
\bibitem{c220}
Frank E. Fish, George V. Lauder, Rajat Mittal, Alexandra H. Techet, Michael S. Triantafyllou, Jeffery A. Walker, and Paul W. Webb, "Conceptual Design for the Construction of a Biorobotic AUV Based on Biological Hydrodynamics", West Chester University, Harvard University, George Washington University, MIT, University of Southern Maine, University of Michigan, (2003)
\url{https://docs.google.com/viewer?a=v&pid=sites&srcid=YXN1LmVkdXxtaWNoYWVsLXRob21wc29ufGd4OjdiZGU1OTU5MGNmMmMxNzk}
\bibitem{c221}
Abel Vargas, Rajat Mittal and Haibo Dong, "A computational study of the aerodynamic performance of a dragonfly wing section in gliding flight", The George Washington University, Bioinspiration and Biomimetics, doi:10.1088/1748-3182/3/2/026004, (2008)
\url{https://docs.google.com/viewer?a=v&pid=sites&srcid=YXN1LmVkdXxtaWNoYWVsLXRob21wc29ufGd4OjE2NWFjMDQ1YjlhZjIzYjQ}
\bibitem{c222}
Rajneesh Bhardwaj and Rajat Mittal, "Benchmarking a Coupled Immersed-Boundary-Finite-Element Solver for Large-Scale Flow-Induced Deformation", Johns Hopkins University, AIAA, (2011)
\url{https://docs.google.com/viewer?a=v&pid=sites&srcid=YXN1LmVkdXxtaWNoYWVsLXRob21wc29ufGd4OjU3YzEzNGM0MTgyNzg4NjI}
\bibitem{c223}
Rajat Mittal, "Computational Modeling in Biohydrodynamics: Trends, Challenges, and Recent Advances", The George Washington University, IEEE, D.O.I.10.1109/JOE.2004.833215, (2004)
\url{https://docs.google.com/viewer?a=v&pid=sites&srcid=YXN1LmVkdXxtaWNoYWVsLXRob21wc29ufGd4OjViNWUxNmQyNjY0NDBlNzg}
\bibitem{c224}
M. Bozkurttas, R. Mittal, H. Dong, G. V. Lauder and P. Madden, "Low-dimensional models and performance scaling of a highly deformable fish pectoral fin", The George Washington University and Harvard University, doi:10.1017/S0022112009007046, (2009)
\url{https://docs.google.com/viewer?a=v&pid=sites&srcid=YXN1LmVkdXxtaWNoYWVsLXRob21wc29ufGd4OjU0YjE4YTkzYzlkOGNhMGU}
\bibitem{c225}
Jung Hee Seo and Rajat Mittal, "A Coupled Flow-Acoustic Computational Study of Bruits from a Modeled Stenosed Artery", Johns Hopkins University, (2011)
\url{https://docs.google.com/viewer?a=v&pid=sites&srcid=YXN1LmVkdXxtaWNoYWVsLXRob21wc29ufGd4OjU0YjE4YTkzYzlkOGNhMGU}
\bibitem{c226}
T. Ye, R. Mittal, H. S. Udaykumar and W. Shyy, "An Accurate Cartesian Grid Method for Viscous Incompressible Flows with Complex Immersed Boundaries", University of Florida, Journal of Computational Physics 156, 209 240 (1999)
\url{https://docs.google.com/viewer?a=v&pid=sites&srcid=YXN1LmVkdXxtaWNoYWVsLXRob21wc29ufGd4OjY4MTdlMTdmYWNiOWYyMWY}
\bibitem{c227}
H. S. Udaykumar, R. Mittal and P. Rampunggoon, "Interface tracking finite volume method for complex solid fluid interactions on fixed meshes", University of Iowa and University of Florida, DOI: 10.1002/cnm.468 (2002)
\url{https://docs.google.com/viewer?a=v&pid=sites&srcid=YXN1LmVkdXxtaWNoYWVsLXRob21wc29ufGd4OjRlY2E0MDZkNTg3MTU1M2E}
\bibitem{c228}
Donghyun You, Rajat Mittal, Meng Wang and Parviz Moin, "Progress in Large-Eddy Simulation of a Rotor Tip-Clearance Flow", Stanford University and The George Washington University, (2002)
\url{https://docs.google.com/viewer?a=v&pid=sites&srcid=YXN1LmVkdXxtaWNoYWVsLXRob21wc29ufGd4OjI1YTZhODU0OGJmYmMzMTM}
\bibitem{c229}
Michelle Kwok and Rajat Mittal, "Experimental Investigation of the Aerodynamics of a Modeled Dragonfly Wing Section", The George Washington University, AIAA (2005)
\url{https://docs.google.com/viewer?a=v&pid=sites&srcid=YXN1LmVkdXxtaWNoYWVsLXRob21wc29ufGd4Ojc2MjNlNmM4MDk2Zjg2NWY}
\bibitem{c230}
X. Zheng, J.H. Seo, V. Vedula, T. Abraham and R.Mittal, "Computational Modeling and Analysis of Intracardiac Flows in Simple Models of the Left Ventricle", Johns Hopkins University, (2012)
\url{https://docs.google.com/viewer?a=v&pid=sites&srcid=YXN1LmVkdXxtaWNoYWVsLXRob21wc29ufGd4OjJlYjE0ODc0MGRjMTIxNGE}
\bibitem{c231}
R. Mittal and S. Balachandar, "Direct Numerical Simulation of Flow Past Elliptic Cylinders", University of Illinois, Journal of Computational Physics 124, 351 367 (1996)
\url{https://docs.google.com/viewer?a=v&pid=sites&srcid=YXN1LmVkdXxtaWNoYWVsLXRob21wc29ufGd4OjE4ZWNmZjdhNzdhZjk3MjA}
\bibitem{c232}
Donghyun You, Rajat Mittal, Meng Wang and Parviz Moin, "A Computational Methodology for Large-Eddy Simulation of Tip-Clearance Flows", Stanford University and The George Washington University, FEDSM2003-45395 (2003)
\url{https://docs.google.com/viewer?a=v&pid=sites&srcid=YXN1LmVkdXxtaWNoYWVsLXRob21wc29ufGd4OjVmN2MxNDY2ZGU4MzdkOGE}
\bibitem{c233}
Q. Xue, X. Zheng, S. Bielamowicz and R. Mittal, "Sensitivity of vocal fold vibratory modes to their three-layer structure: Implications for computational modeling of phonation", Johns Hopkins University and The George Washington University, Acoustical Society of America DOI: 10.1121/1.3605529, (2011)
\url{https://docs.google.com/viewer?a=v&pid=sites&srcid=YXN1LmVkdXxtaWNoYWVsLXRob21wc29ufGd4OjEzODAyMzEzNzdjN2M4Yw}
\bibitem{c234}
Ge Jin, Nakhoon baek, james k. Hahn, Steven Bielamowicz, Rajat Mittal and Raymond Walsh, John Wiley and Sons, "Image Guided Mediaization Laryngoplasty", Computer Animation and Vitrtual Worlds, 19: 1-10, DOI: 10.1002/cav, (2008)
\url{https://docs.google.com/viewer?a=v&pid=sites&srcid=YXN1LmVkdXxtaWNoYWVsLXRob21wc29ufGd4OjVkNjYyNzIxODQ3OGEyMDc}
\bibitem{c235}
Raman S. Dusaj, Katherine C. Michelis, Megan Terek, Reza Sanai, Rajat Mittal, Jannet F. Lewis, Robert K. Zeman, Brian G. Choi, George Washington University and Johns Hopkins University, "Estimation of right atrial and ventricular hemodynamics by CT coronary angiography" ,Journal of Cardiovascular Computed Tomography, doi:10.1016/j.jcct.2010.10.005, (2011)
\url{https://docs.google.com/viewer?a=v&pid=sites&srcid=YXN1LmVkdXxtaWNoYWVsLXRob21wc29ufGd4OjE0N2U1MmQ5Y2JhNTJkNGU}
\bibitem{c236}
Fady M. Najjar and Rajat Mittal, University of Illinois and The George Washington University, "Simulations of Complex Flows and Fluid-Structure Interaction Problems on Fixed Cartesian Grids", FEDSM2003-45577, ASME, (2003)
\url{https://docs.google.com/viewer?a=v&pid=sites&srcid=YXN1LmVkdXxtaWNoYWVsLXRob21wc29ufGd4Ojc0MmVkZjNlMThhNjkzNzU}
\bibitem{c237}
Donghyun You, Meng Wang, Parviz Moin and Rajat Mittal, Stanford University and The George Washington University, " Vortex Dynamics and Mechanisms for Viscous Losses in the Tip-Clerance Flow", FEDSM2005-77175, ASME, (2005)
\url{https://docs.google.com/viewer?a=v&pid=sites&srcid=YXN1LmVkdXxtaWNoYWVsLXRob21wc29ufGd4Ojc5ZjJkOGI2YjgzYzcxMDE}
\bibitem{c238}
Rajat Mittal, Xudong Zheng, Rajneesh Bhardwaj, Jung Hee Seo, Qian Xue and Steven Bielamowicz, Johns Hopkins University and George Washington University, " Toward a simulation-based tool for the treatment of vocal fold paralysis", doi: 10.3389/fphys.2011.00019, (2011)
\bibitem{c239}
James L. Tangorra, George V. Lauder, Peter G. Madden, Rajat Mittal, Meliha Bozkurttas and Ian W. Hunter, Drexel University, Harvard University, George Washington University and Massachusetts Institute of Technology, "A Biorobotic Flapping Fin for Propulsion and Maneuvering", (2007)
\url{https://docs.google.com/viewer?a=v&pid=sites&srcid=YXN1LmVkdXxtaWNoYWVsLXRob21wc29ufGd4OjFiODFkZWFlOGI2ZmE1ZTU}
\bibitem{c240}
[ Ehsan Aram, Rajat Mittal and Louis Cattafesta, Johns Hopkins University and University of Florida, "Simple Representations of Zero-Net Mass-Flux Jets in Grazing Flow for Flow-Control Simulations", International Journal of Flow Control, ISSN 1756-8250 (2010)
\url{https://docs.google.com/viewer?a=v&pid=sites&srcid=YXN1LmVkdXxtaWNoYWVsLXRob21wc29ufGd4OjE3OGU2MDlkNzFmYjM5ZTE}
\bibitem{c241}
Donghyun You, Meng Wang and Rajat Mittal, Stanford University, University of Notre Dame and George Washington University, "A methodology for high performance computation of fully inhomogeneous turbulent flows", International Journal for Numerical Methods in Fluids, 2007; 53:947 968, DOI: 10.1002/fld (2006)
\url{https://docs.google.com/viewer?a=v&pid=sites&srcid=YXN1LmVkdXxtaWNoYWVsLXRob21wc29ufGd4OmI5ZjFkNzJkYTliOWY3OQ}
\bibitem{c242}
Rajat Mittal and Rupesh B. Kotapati, George Washington University, "Resonant Mode Interaction in A Canonical Seperated Flow", IUTAM Symposium, (2004)
\url{https://docs.google.com/viewer?a=v&pid=sites&srcid=YXN1LmVkdXxtaWNoYWVsLXRob21wc29ufGd4OjQ0MjNiZjVmYzlkYTVlMDM}
\bibitem{c243}
James Louis Tangorra, S. Naomi Davidson, Ian W. Hunter, Peter G. A. Madden, George V. Lauder, Haibo Dong, Meliha Bozkurttas, and Rajat Mittal, "The Development of a Biologically Inspired Propulsor for Unmanned Underwater Vehicles, IEEE Journal of Oceanic Engineering, (2007)
\url{https://docs.google.com/viewer?a=v&pid=sites&srcid=YXN1LmVkdXxtaWNoYWVsLXRob21wc29ufGd4OjY3N2U2OTFmY2RiYTFkYTE}
\bibitem{c244}
Jung Hee Seo and Rajat Mittal, Johns Hopkins University, "A high-order immersed boundary method for acoustic wave scattering and low-Mach number flow-induced sound in complex geometries", Journal of Computational Physics, doi:10.1016/j.jcp.2010.10.017, (2010)
\url{https://docs.google.com/viewer?a=v&pid=sites&srcid=YXN1LmVkdXxtaWNoYWVsLXRob21wc29ufGd4OjYzZDBlN2ZmZDU5NTI1MWQ}
\bibitem{c245}
Jung Hee Seo and Rajat Mittal, Johns Hopkins University, "A sharp-interface immersed boundary method with improved mass conservation and reduced spurious pressure oscillations", Journal of Computational Physics, doi:10.1016/j.jcp.2011.06.003, (2011)
\url{https://docs.google.com/viewer?a=v&pid=sites&srcid=YXN1LmVkdXxtaWNoYWVsLXRob21wc29ufGd4OmRiNzZlN2JlN2I4MGEyZg}
\bibitem{c246}
H. S. Udaykumar, R. Mittal, P. Rampunggoon and A. Khanna, University of Iowa, George Washington University and University of Florida, "A Sharp Interface Cartesian Grid Method for Simulating Flows with Complex Moving Boundaries Journal of Computational Physics, doi:10.1006/jcph.2001.6916, (2001)
\url{https://docs.google.com/viewer?a=v&pid=sites&srcid=YXN1LmVkdXxtaWNoYWVsLXRob21wc29ufGd4OjIxNTBmNjY4M2NkYjVmY2M}
\bibitem{c247}
Donghyun You, Meng Wang, Parviz Moin, and Rajat Mittal, Stanford University, University of Notre Dame and George Washington University, "Vortex Dynamics and Low-Pressure Fluctuations in the Tip-Clearance Flow", Journal of Fluids Engineering,
\url{https://docs.google.com/viewer?a=v&pid=sites&srcid=YXN1LmVkdXxtaWNoYWVsLXRob21wc29ufGd4OjVlOTQ3ZGQ2OTRmNTEwNmY}
\bibitem{c248}
[248] Donghyun You, Meng Wang, Rajat Mittal and Parviz Moin, Stanford University and George Washington University, A Quasi-Generalized-Coordinate Approach for Numerical Simulation of Complex Flows", ASME, DOI: 10.1115/1.2354533, (2006)
\url{https://docs.google.com/viewer?a=v&pid=sites&srcid=YXN1LmVkdXxtaWNoYWVsLXRob21wc29ufGd4OjZmM2YxOWViMDI0M2Y2ODc}
\bibitem{c249}
[249] H. Dong, R. Mittal and F. M. Najjar, George Washington University and University of Illinois, "Wake topology and hydrodynamic performance of low-aspect-ratio flapping foils", doi:10.1017/S002211200600190, (2006)
\url{https://docs.google.com/viewer?a=v&pid=sites&srcid=YXN1LmVkdXxtaWNoYWVsLXRob21wc29ufGd4OjQ0NjIzY2JhMGI0OWE4OGI}
\bibitem{c250}
Mukund Narasimhan, Haibo Dong, Rajat Mittal and Sahjendra N. Singh, University of Nevada and George Washington University, "Optimal Yaw Regulation and Trajectory Control of Biorobotic AUV Using Mechanical Fins Based on CFD Parametrization1", Journal of Fluids Engineering, DOI: 10.1115/1.2201634, (2006).
\url{https://docs.google.com/viewer?a=v&pid=sites&srcid=YXN1LmVkdXxtaWNoYWVsLXRob21wc29ufGd4OjU2NjZkZDU0NzE1NzZhNzI&pli=1}
\bibitem{c251}
R. Mittal, S. P. Simmons and F. Najjar, George Washington University, University of Florida and University of Illinois, "Numerical study of pulsatile flow in a constricted channel", Journal of Fluid Mechanics, DOI: 10.1017/S002211200300449X, (2003)
\url{https://docs.google.com/a/asu.edu/viewer?a=v&pid=sites&srcid=YXN1LmVkdXxtaWNoYWVsLXRob21wc29ufGd4OjQ4ODM4YjM3MTZhOWU1NjM}
\bibitem{c252}
Dongyun You, Meng Wang, Parviz Moin and Rajat Mittal, Stanford University, University of Notre Dame and George Washington University, "Large-eddy simulation analysis of mechanisms for viscous losses in a turbomachinery tip-clearance flow", Journal of Fluid Mechanics, doi:10.1017/S0022112007006842, (2007)
\url{https://docs.google.com/a/asu.edu/viewer?a=v&pid=sites&srcid=YXN1LmVkdXxtaWNoYWVsLXRob21wc29ufGd4OmMwOGJmNzBiMTk2ODg}
\bibitem{c253}
Ge Jin, Sang-Joon Lee, James K. Hahn, Steven Bielamowicz, Rajat Mittal, and Raymond Walsh, George Washington University, "3D Surface Reconstruction and Registration for Image Guided Medialization Laryngoplasty", (2003)
\url{https://docs.google.com/a/asu.edu/viewer?a=v&pid=sites&srcid=YXN1LmVkdXxtaWNoYWVsLXRob21wc29ufGd4OjdhNmY0NzgwN2I4NDQzZWQ}
\bibitem{c254}
George V Lauder, Peter G A Madden, Rajat Mittal, Haibo Dong and Meliha Bozkurttas, Harvard University and George Washington University, "Locomotion with flexible propulsors: I. Experimental analysis of pectoral fin swimming in sunfish", doi:10.1088/1748-3182/1/4/S04, (2006)
\url{https://docs.google.com/a/asu.edu/viewer?a=v&pid=sites&srcid=YXN1LmVkdXxtaWNoYWVsLXRob21wc29ufGd4OjY5OTUxMTNjYjJiYjVmOTA}
\bibitem{c255}
Reza Ghias, Rajat Mittal, Haibo Dong and Thomas S. Lund, George Washington University and University of Colorado, "Large-Eddy Simulation of the Tip Flow of a Rotor in Hover", AIAA-2004-2432, (2004)
\url{https://docs.google.com/a/asu.edu/viewer?a=v&pid=sites&srcid=YXN1LmVkdXxtaWNoYWVsLXRob21wc29ufGd4OmY5MzIyNTlmOThlODQzNQ}
\bibitem{c256}
Alfred von Loebbecke, Rajat Mittal, Frank Fish and Russell Mark, George Washington University and West Chester University, "A comparison of the kinematics of the dolphin kick in humans and cetaceans", doi:10.1016/j.humov.2008.07.005, (2008)
\url{https://docs.google.com/a/asu.edu/viewer?a=v&pid=sites&srcid=YXN1LmVkdXxtaWNoYWVsLXRob21wc29ufGd4OjYxNjkxOGQ3MmI2ZWI4NWI}
\bibitem{c257}
R. Mittal, S.Venkatasubramanian and F.M. Najjar, University of Florida and University of Illinois, "Large Eddy Simulation of Flow Through a Low Pressure Turbine Cascade", AIAA 2001 2560, (2001)
\url{https://docs.google.com/a/asu.edu/viewer?a=v&pid=sites&srcid=YXN1LmVkdXxtaWNoYWVsLXRob21wc29ufGd4Ojc0N2U5OGZjNGFhMTMzMzQ}
\bibitem{c258}
T. E. Mengesha, R. R. Vallance and R. Mittal, George Washington University and Johns Hopkins University, "Stiffness of desiccating insect wings", doi:10.1088/1748-3182/6/1/014001, (2010)
\url{https://docs.google.com/a/asu.edu/viewer?a=v&pid=sites&srcid=YXN1LmVkdXxtaWNoYWVsLXRob21wc29ufGd4OjMwYjQ0YmI2OWEzNjE5NDI}
\bibitem{c259}
Donghyun You, Rajat Mittal, MengWang and Parviz Moin, Stanford University and George Washington University, "Computational Methodology for Large-Eddy Simulation of Tip- Clearance Flows", AIAA Journal, (2004)
\url{https://docs.google.com/a/asu.edu/viewer?a=v&pid=sites&srcid=YXN1LmVkdXxtaWNoYWVsLXRob21wc29ufGd4OjZhMjg1MTc4YzQ5Y2NlOWI}
\bibitem{c260}
Rajat Mittal, Veeraraghavan Seshadri and Holavanahalli S. Udaykumar, The George Washington University and University of Iowa, "Flutter, Tumble and Vortex Induced Autorotation", Theoretical and Computational Fluid Dynamics, DOI10.1007/s00162-003-0101-5, (2004)
\url{https://docs.google.com/a/asu.edu/viewer?a=v&pid=sites&srcid=YXN1LmVkdXxtaWNoYWVsLXRob21wc29ufGd4Ojc5OTA0ZjA2YThhYjgyMTM}
\bibitem{c261}
Imran Akhtar, Rajat Mittal, George V. Lauder and Elliot Drucker, George Washington University, Virginia Tech and Harvard University, "Hydrodynamics of a biologically inspired tandem flapping foil configuration, Theoretical and Computational Fluid Dynamics, DOI 10.1007/s00162-007-0045-2, (2007)
\url{https://docs.google.com/a/asu.edu/viewer?a=v&pid=sites&srcid=YXN1LmVkdXxtaWNoYWVsLXRob21wc29ufGd4OjM4YThkNjE4YTA3ODBiYzQ}
\bibitem{c262}
Rajat Mittal, Haibo Dong, Meliha Bozkurttas, George V. Lauder and Peter Madden, The George Washington University and Harvard University, "Locomotion with flexible propulsors: II. Computational modeling of pectoral fin swimming in sunfish", doi:10.1088/1748-3182/1/4/S05, (2006)
\url{https://docs.google.com/a/asu.edu/viewer?a=v&pid=sites&srcid=YXN1LmVkdXxtaWNoYWVsLXRob21wc29ufGd4OjczNmY5MTNkZDVmYWM1OTY}
\bibitem{c263}
H. S. Udaykumar and L. Mao and R. Mittal, University of Iowa and George Washington University, "A Finite-Volume Sharp Interface scheme for Dendritic Growth Simulations: Comparison with Microscopic Sovability Theory", (2002)
\url{https://docs.google.com/a/asu.edu/viewer?a=v&pid=sites&srcid=YXN1LmVkdXxtaWNoYWVsLXRob21wc29ufGd4OjczNTQ0NWFiODJlYjFlMDU}
\bibitem{c264}
Wei Shyy and Rajat Mittal, University of Florida, "The Handbook of Fluid Dynamics, Ch. 31 solution Methods for the Incompressible navier-Stokes Equations", (1998)
\url{https://docs.google.com/a/asu.edu/viewer?a=v&pid=sites&srcid=YXN1LmVkdXxtaWNoYWVsLXRob21wc29ufGd4OjMxOGI5ODA4MzZjYzY0MTE}
\bibitem{c265}
W. Shyy, M.-H. Chen, R. Mittal and H. S. Udaykumar, University of Florida, "On the Suppression of Numerical Oscillations Using a Non-Linear Filter", Journal of Computational physics, (1991)
\url{https://docs.google.com/a/asu.edu/viewer?a=v&pid=sites&srcid=YXN1LmVkdXxtaWNoYWVsLXRob21wc29ufGd4OjM0NDUzNzQ1MDliMjQ2OTY}
\bibitem{c266}
S. Balachandar, R. Mittal and F. M. Najjar, University of Illinois, "Properties of the Mean Recirculation Region in the Wakes of Two-Dimensional Bluff Bodies", Journal of Fluid Mechanics, (1997)
\url{https://docs.google.com/a/asu.edu/viewer?a=v&pid=sites&srcid=YXN1LmVkdXxtaWNoYWVsLXRob21wc29ufGd4OjEwY2NiYWMwZGI1ZmRmZDk}
\bibitem{c267}
H. -J. Kaltenbach, M. Fatica, R. Mittal, T. S. Lund and P. Moin, Stanford University, "Study of Flow in a Planar Asymmetric Diffuser Using Large-Eddy Simulation", Journal of Fluid Mechanics, (1999)
\url{https://docs.google.com/a/asu.edu/viewer?a=v&pid=sites&srcid=YXN1LmVkdXxtaWNoYWVsLXRob21wc29ufGd4OmJkMDg0MjhkMjVmNTA1ZA}
\bibitem{c268}
R. Mittal, V. Seshadri, S.E. Sarma and H.S. Udaykumar, George Washington University, Massachusetts Institute of Technology and University of Iowa, "Computational Modeling of Fluidic Micro-Handling Processes, (2002)
\url{https://docs.google.com/a/asu.edu/viewer?a=v&pid=sites&srcid=YXN1LmVkdXxtaWNoYWVsLXRob21wc29ufGd4OjI4YTZmOWRkNGVkMTA1ZGY}
\bibitem{c269}
Meliha Bozkurttas, James Tangorra, George Lauder and Rajat Mittal, Drexel University, Harvard University and The George Washington University, "Understanding the Hydrodynamics of Swimming: From Fish Fins to Flexible Propulsors for Autonomous Underwater Vehicles", Advances in Science and Technology Vol. 58 (2008)
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Nonthipat Thaweewat, Delft University of Technology, "Influence of wing kinematics on vortex dynamics and performance in insect forward flight", MS, (2007)
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Fluent Inc, "Tutorial: Store Separation from a 3D Delta Wing", (2008)
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Frank Martijn, Delft University of Technology "Numerical simulations of flapping foil and wing aerodynamics", PhD Thesis, (2009)
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Yih-Ferng Peng, Rajat Mittal, Amalendu Sau, Robert R. Hwang, National Chi Nan University, George Washington University, Gyeongsang National University and Institute of Physics, "Nested Cartesian grid method in incompressible viscous fluid flow", Journal of Computational Physics 229, doi:10.1016/j.jcp.2010.05.041, (2010)
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Donghyun You, Meng Wang, Parviz Moin, Rajat Mittal, Stanford University, University of Notre Dame, and George Washington University, "Effects of tip-gap size on the tip-leakage flow in a turbomachinery cascade", Physics of Fluids 18, 105102, DOI: 10.1063/1.2354544, (2006)
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R. Mittal and S. Balachandar, University of Illinois, "Generation of Streamwise Vortical Structures in Bluff body Wakes", Physical Review letters, (1995)
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R. Mittal and S. Balachandar, University of Illinois, "Effect of three-dimensionality on the lift and drag of nominally two-dimensional cylinders", American Institute of Physics, (1995)
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R. Mittal, P. Rampunggoon, The George Washington University and University of Florida, "On the virtual aeroshaping effect of synthetic jets", Physics of Fluids, DOI: 10.1063/1.1453470, (2002)
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R. Mittal, University of Florida, "Response of the Sphere Wake to Freestream Fluctuations", Theoretical and Computational Fluid Dynamics, (2000)
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Reni Raju, Quentin Gallasa, Rajat Mittal and Louis Cattafesta, The George Washington University and University of Florida, "Scaling of pressure drop for oscillatory flow through a slot", DOI: 10.1063/1.2749814, (2007)
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Srinivas Ramakrishnan, Meliha Bozkurttas, Rajat Mittal and George V. Lauder, ANSYS, Inc, College of Engineering, Johns Hopkins University and Harvard University, "Thrust Production in Highly Flexible Pectoral Fins: A Computational Dissection", (2007)
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Reza Ghias, Rajat Mittal, Haibo Dong and Thomas S. Lund, The George Washington University and The University of Colorado, "Study of Tip-Vortex Formation Using Large-Eddy Simulation", AIAA-2005-1280, (2005)
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H. S. Udaykumar, R. Mittal and Wei Shyy, University of Florida, "Computation of Solid Liquid Phase Fronts in the Sharp Interface Limit on Fixed Grids," Journal of Computational Physics 153, 535 574 (1999)
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Donghyun You, Rajat Mittal, Meng Wang and Parviz Moin, Stanford University and George Washington University, "Analysis of stability and accuracy of finite-difference schemes on a skewed mesh", Journal of Computational Physics 213, doi:10.1016/j.jcp.2005.08.007, (2006)
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Rajat Mittal, University of Florida, "A Fourier Chebyshev Spectral Collocation Method For Simulating Flow Past Spheres and Spheroids", International Journal For Numerical Methods in Fluids 30: 921 937, (1999)
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R. Mittal, S. P. Simmons, H. S. Udaykumar, University of Florida, University of Iowa, "Application of Large-Eddy Simulation to the Study of Pulsatile Flow in a Modeled Arterial Stenosis", Journal of Biomechanical Engineering, DOI: 10.1115/1.1385840, (2001)
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R. Mittal, P. Rampunggoon and H. S. Udaykumar, University of Florida, University of Iowa, "Interaction of a Synthetic Jet with a Flat Plate Boundary Layer", AIAA 2001 2773, (2001)
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James L. Tangorra, George V. Lauder, Ian W. Hunter, Rajat Mittal, Peter G. A. Madden and Meliha Bozkurttas, Drexel University, MIT, Johns Hopkins University and The George Washington University, "The effect of fin ray flexural rigidity on the propulsive forces generated by a biorobotic fish pectoral fin", The Journal of Experimental Biology 213, 4043-4054, doi:10.1242/jeb.048017, (2010)
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\bibitem{c300}
Donghyun You, Meng Wang, Parviz Moin and Rajat Mittal, Stanford University and George Washington University, Syudy of Tip-Clearance Flow in Turbomachines Using Large-Eddy Simulation", Computing in Science and Engineering, (2004)
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\bibitem{c301}
Rajat Mittal, The George Washington University, "Computational Modeling in Bio-Hydrodynamics: Trends, Challenges and Recent Advances", (2002)
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Rajat Mittal, Imran Akhtar, Meliha Bozkurttas and Fady. M. Najjar, The George Washington University and University of Illinois, "Towards a Conceptual Model of a Bio-Robotic AUV: Pectoral Fin Hydrodynamics", (2003)
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\bibitem{c303}
Veeraraghavan Seshadri, Rajat Mittal and H.S. Udaykumar, The George Washington University, University of Iowa, "Vortex Induced Auto-Rotation of a Hinged Plate: A Computational Study", F E DSM2003-45512, (2003)
\url{https://docs.google.com/viewer?a=v&pid=sites&srcid=YXN1LmVkdXxtaWNoYWVsLXRob21wc29ufGd4OjMyOWRlZTgxZGQxZTg3Yjg}
\bibitem{c304}
Srinivas Ramakrishnan, Lingxiao Zheng, Rajat Mittal, Fady Najjar, George V. Lauder and Tyson L. Hedrick, George Washington University, Harvard University and University of North Carolina, "Large Eddy Simulation of Flows with Complex Moving Boundaries: Application to Flying and Swimming in Animals", AIAA-2009-3976, (2009)
\url{https://docs.google.com/viewer?a=v&pid=sites&srcid=YXN1LmVkdXxtaWNoYWVsLXRob21wc29ufGd4OjRhNDViYmZmZDcwNzAzNjc}
\bibitem{c305}
X. Zheng, Q. Xue, R. Mittal, S. Beilamowicz, Johns Hopkins University and George Washington University, "A Coupled Sharp-Interface Immersed Boundary-Finite-Element Method for Flow-Structure Interaction With Application to Human Phonation, Journal of Biomechanical Engineering, ASME, DOI: 10.1115/1.4002587, (2010)
\url{https://docs.google.com/viewer?a=v&pid=sites&srcid=YXN1LmVkdXxtaWNoYWVsLXRob21wc29ufGd4OjExOWRmMGU4MzI1ZWM2YzQ}
\end{thebibliography}
\end{document}
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