Integrated Flight Dynamics Modelling for UnmannedAerial Vehicles

Qing Ou, XiaoQi Chen, Senior Member, IEEE, David Park, Aaron Marburg, James Pinchin

Abstract- The primary motivation to build a flight building a model from scratch is the best way to understanddynamics model was for dead reckoning of Unmanned Aerial the internal mechanism of the model, and hence effectivelyVehicle, a process of estimating the aircraft's motions from the customize the model to meet specific applications.last known state during the interval of losing GPS signals. To simplify the flight dynamics model, non-linearWhen a wind model was added to the flight simulation, it had . . . .

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the ability to either load wind data externally, or to produce equatisofpmotio inarfight model linearised Evenrandom turbulence internally, from light to severe. This though this approach scarifies the modelling accuracy, it isadditional feature gave the simulation a more realistic flight still acceptable for many applications where ease ofenvironment. As a result, it can be used to predict the stability developing control strategies is required [7]. Linearisedand flying characteristics of UAVs. Likewise, autopilot model is also widely use in an embedded system of microcontrollers can be tested in extreme conditions before they are UAVs with a wing span of less than 50 cm to compromiseimplemented. The modified generic flight model permits the limited processing power of the onboard computer [8].modelling of completely arbitrary vehicle configurations, fromminiature Unmanned Aerial Vehicles (UAVs) to transonic It has been recognized that the significant improvementsfighters. Building the model in Matlab Simulink allowed rapid of dynamic performance of current and new generation ofdevelopment of new aircraft models, and has provided a very advanced airplanes is possible if flight systems designflexible platform for users to tailor their flight models. integrates nonlinear analysis, control, and identification [9].

There were several UAV flight control system designI. INTRODUCTION projects applying non-linear flight models to simulate the

N umerical modelling of flight dynamics has been a long dynamic behaviour of their vehicles [5, 10].history in the aerospace industry. Any of the successful The other aspect of flight modelling is representation of

modern aerospace vehicle development projects must be orientation. The most common way to represent the attitudecredited to their advanced flight dynamics models. A flight of an aircraft is a set of three Euler angles. These are populardynamics model is mathematical representation of the steady because they are easy to understand and easy to use.state performance and dynamic response that is expected of However, the main disadvantages of Euler angles are: i) thatthe proposed vehicle[1]. The usage of flight dynamics certain important functions of Euler angles havemodels is diverse. Commercial, military, government singularities, and ii) that they are less accurate than unitorganisations and academic sectors employ flight models to quaternions when used to integrate incremental changes inachieve their specific tasks[2]. Example applications are attitude over time [11]. Nowadays, quaternion has beencontrol strategies and/or algorithms test bed, stability and fly increasingly adapted in flight models because it is better ablecharacteristics evaluation for preliminary designs, onboard to avoid singularities and high data rates associated in Eulerembedded autopilot system, and inertial navigation system angle representation [12].

Research in flight dynamics modelling has covered a wide Recent works on UAV simulations are undertaken torange of area. In most projects, researchers favour to build enable multiple UAVs to collaborate. Military applicationstheir own flight models in Matlab, or using other high level are the major motivator for research in this area. In thecomputer languages like C++, C or Java [3-6]. Apparently Office of the Secretary of Defense perspective, a swarm of

tactical UAVs aim to cooperate to accomplish complicated

This work was supported in part by Geospatial Research Centre (NZ)missions [13]. A multi-UAV model has been built in MatlabLimited. Engineers and scientist at GRC are acknowledged for their Simulink and made available to the public [3]. Such a modelparticipation in technical discussions, andinputs provides a platform for researchers to evaluate their

Qing Ou and XiaoQi Chen are with the Department of Mechanical cooperative control algorithms.Engineering Department, the University of Canterbury, Private Bag 4800, This paper first gives an overview of tools and the processChristchurch 8140, New Zealand (phone: +64 3 364 2987 Ext.: 7221; fax.r±64-3-364-2087; e-mails: qoulO@student.canterbury.ac.nz, used in flight dynamics modelling. After representing thexiaoqi.chen@canterbury.ac.nz mathematical model using Ordinary Differential Equations

David Park and Aaron Marburg are with Geospatial Research Centre(NZ) Ltd, Private Bag 4800, Christchurch 8140, New Zealand (phone: ±64 (ODE), it discussed an integrative generic flight dynamics3 364 3830; fax: ±64-3-364-3880; e-mail: {david.park, aaron.marburg} modelling system. Finally simulation results were compared@grcnz.com. with the outputs from the software JSBSim, and the dead

James Pinchin is with both the Department of Mechanical Engineering, rcoigapctonfthFDwspooedthe University of Canterbury; and Geospatial Research Centre (NZ) Ltd. E- rco1gaplalno hDa rpsdmail: jtp24@student.canterbury.ac.nz, or james.pinchin@grcnz.com.

1-4244-2368-2/08/$20.00 ©2008 IEEE 570

II. TOOLS AND PROCESS OF FLIGHT DYNAMICS MODELLING vast number of different flavours.

A. Modelling Tools B. Steps ofModellingThe most common implementation of flight dynamics Regardless of implementation methods, developing a

model is a full nonlinear six-degree of freedom rigid-body flight dynamics model generally need to go through theflight dynamics model. Expanding its role in assisting the following processes.development of UAVs, a flight model not only constructs (1) Identify methods and software platform. At the firstand solves equations of motion, but also interface with and step, a number of decisions need to be made in terms ofprocess all the necessary flight parameters in the real world, model implementation method, software platform andas shown in Fig. 1. computer language. To take the advantage of the existing

aircraft 3D graphical packages, the ability for the flightController Environment model to interface with a data visualization tool such as

model SimGraph[14] and FlightGear need to be considered. Thedecision for a development platform is applicationdependent, and often depends on the experience the

Aerodynamics Flight Signal Onboard sensorestimation Dynamics processing & actuators developers have in the field of computer programming andsoftware Model & filtering model flight simulations.

(2) Aerodynamics data. Aerodynamic properties areData required by the mathematical model. These data are eithervisualization obtained from wind tunnel tests or analytical means, all of

Fig. 1. Flight dynamics model as an integration system of flight simulation. which are subject to a number of assumptions andlimitations. These coefficient data have reasonable accuracy

Representative existing modelling tools available to the for steady state lift and drag estimates but are relativelypublic domain are X-plane (NASA), JSBSim (Berndt J.) and inaccurate for dynamic characteristics due to limitations inFlight Simulator X (Microsoft). Among the three tools, only wind tunnel measurement accuracy and assumptions basedJSBSim is open source software, whereas the other two on small linear perturbations [15].simulation programs are like a 'black box' where users have (3) Flight characteristics data. To achieve accurate flightno access to the internal of the models. The open source simulation, the flight characteristics data set is essential. Itfeature of JSBSim has gained a lot of attention from describes the response of the vehicle for certain controlresearchers, because the significant cost saving in using inputs. These data are normally obtained with high qualityJSBSim. A full six-degree of freedom simulator for flight instrument onboard.simulation and pilot training was constructed at the (4) Validation. As the last and most essential phase ofUniversity of Naples used JSBSim as its physics engine flight model development, the model needs to be validated[16]. against flight results in the real world. Real flight data areThe fundamental goal of flight dynamics modelling is to always contained some degrees of uncertainties and noise

represent the flight motion numerically, as close to the flight because of limitations of sensors. As a result, signalmotion in the real world as the applications need. To processing and data acquisition techniques are usuallyaccommodate a wide range of applications, various required for flight model validation.implementations of flight models in terms of assumptions C. Coordinate Systemsand algorithms therefore exist. Nevertheless, all flight

B

dynaics odesar basd o themathmatcs mdel Body, wind and stability coordinate systems are usedderamive fromdNewtonian Physics.eFomaNewatos sconde extensively in the flight model. It is important to understandlaw,vedaarct mwtionian Phsits.s reo feedom scanb their exact definitions in order to interchange from one

dsrb byai sm ofinon-learfr orde differe. nt coordinate system to another. The definitions of these threedescribed by a system of non-linear first order differentialcodnt ytm a ecnuig seilywe

equations. These equations of motion served as the 'core' coordinaersystems can be confusing, especially whenfor most flight dynamics models. In today's computing eslightly different definitions can be found in differentpower, the processing time of solving these equations is references This work adopts these coordinate systems usedtriia coprn to ote sinlpoesn .agrtm eg in Aerospace Blockset. All coordinate systems satisfy thetrivial comparing to other signal processing algorithms (e.g. ih an ue

Kalman filter) that might be implemented as part of the RightHandrule.flight model. A number of popular techniques used to solve 1) Body Coordinates

The body coordinate system is fixed in both origin andthese non-linear systems are ranged from less accurate butfast tohiglyacurte ut ompuatinalintesiv: Eler orientation to the moving craft. The craft is assumed to be

Hen Boak-hmie Rug-ut an'orad rigid. Fig. 2 shows the aircraft flying at body velocities [u vPrince.~~~~~~~~~~~~~~~~Asfih.oelneoe vrmr ohsiae w] and speed V, where u, v, w are velocities along x-, y-,

and z-axis respectively. The x-axis points through the noseand ore ppliatin spcifi, thy ae imlemeted o a of the craft, and the y-axis to the right of the x-axis (facing

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in the pilot's direction of view), perpendicular to the x-axis. III. MATHEMATICAL MODELThe z-axis points down through the bottom of the craft, Essentially, a dynamics flight model can be represented asperpendicular to the xy plane and satisfying the right hand a system of ODE. Solving the system is effectively the samerule. as running the flight model. In other words, all the state

variables are solved with respect to time. Such amathematical model is very useful, because it is independentof any software platform and lnguages It gives a gatewayto implementing a FDM in any languages (VHDL, assembly

v 7 - ~ u or C language) for embedded system applications.Furthermore, transfer functions can be derived from the

v system of ODE for later development of autopilot PIDFig. 2. Body coordinates, controllers.

Aircraft properties include aircraft's mass, moment of2) Stability coordinates inertia and centre of gravity. The current FDM assumed theThey are primarily used for the fores and moments aircraft has constant mass, although variable mass can also

calculated from aerodynamic coefficients. In stability be modelled by changing the equations of motion block incoordinates, the drag, lift and side forces are on the Xstab-, FDM. The entry ofmoment of inertia is a 3-by-3 matrixystab- and Zstab axes respectively. Similarly, rolling, pitchingand yawing moments are about the xstab-, ystab- and zstab-axes I - -Irespectively. The stability coordinates are defined in Fig. x Y

3(a), with the x-axis points in the direction of the projection I:=:K:Yx IY-§yzof relative wind on the x-z plane of the body-axis. The y- -Izx -Izy Iz,axis points to the right of the x-axis (facing in the pilot'sdirection of view), perpendicular to the x-axis (same as body For traditional aircraft, symmetry and uniformly distributedcoordinates). Stability coordinates can be obtained by mass about the x-z plane can be assumed. As a result, therotating the body coordinates about y-axis, for angle c Since I and(AOA), and then make the x- and z axes pointing to the p o I Y 0.ZY xz

opposite directions. IZX are generally very much smaller than Ix, Iy and I , a

further simplification can be made by neglecting them, so

I = 0 [16]. This assumption is a very satisfactory

stab

\xwmdu =rv-qw+~Fx/m (1)

(a) (b) v =pw-ru+ZEFy/m (2)Fig. 3. (a) stability coordinates, and (b) wind coordinates.

w=qu-pv+~Fz/m (3)3) Wind coordinatesThey are defined by rotating the body-axis so that the x- where r, p, q are angular velocity about body coordinate x-,

axis is aligned with the speed direction V. Taking the y-, and z-axis respectively Fx, F, F are forceix, - nexample showed in Fig. 2, the wind-axis at that moment will z-axis respectively. m is the mas efhveincx,l-,an

approimaton fo UAVmadess.ftevhce

be like the one shown in Fig. 3(b). The y-axis points to the The angular accelerations about x-, y- and z-axis ean beright of the x-axis (pointing to the direction of V), expressed as [17]:perpendicular to the x-axis. The z-axis pointsperpendicularly to the xy plane in whatever way needed to .rvM (II -IJ)rhI (4)satisfy the nrulewithrespectto the x- andy axes.

Note that the wind-axis is unlikely to stay the same with q [ Mv ± (I - Ijpr]/Iy (5)respect to the aircraft body, because the direction of V is r=L[Mz+±(JI - J)pq]/Jz (6)likely to change over time. In other word, the orientation ofthe wind oordinate axes is fixed by the direction of wind Transformation from body velocities to earth-fixedspeed as apposed to the craft body in body coordinates,

The arrefe yrtaigtebd-xs ota h - werenefr,,qare anglaretlocityaboutv boy coordilynathex-

inverse of Direction Cosine Matrix (DCM) [18].

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K;; 1vj i4)±vV2+2 u2 2+W27 1where Xe, Ye, Ze are x-, y-, z-coordinates in the earth-fixed (14)reference frame, and The above system of ODE is a general aircraftDCJVK1= mathematical model. The sum of forces and moments are

!cosqt'cosO cosqt,sinOsino5- sinqt'cosoz cosqt,sin0cos0+sin/sin0 not specified, because for different aircraft, the functions ofsinqt'cosO sinqt,sin0sin0z ±cosqt'cos0 sinVt,sin0coso5-coqsn these terms are different. Since forces and moments are

-sinO cosOsinoz CoOSCOSOz obtained from aerodynamic coefficients, the summationterms are actually functions of flight conditions and aircraft

where 0, 0, y' are rolling, pitching and yawing angle in state. Due to the inherent complexity of aerodynamics, theearth-fixed reference frame respectively, forces and moments are generally defined by lookup tables,

which was the case throughout this project. However, byHence, the body velocities to earth-fixed reference frame applying mathematical approximation techniques and usingvelocities can be obtained: the 'right' basis (e.g. polynomials), the coefficients of the

X~e = u cos yii cos 0 + v(cos q/insin 0sin y -co 7 least square approximation can be determined. Hence,+ w(cos y sin 0 co 0 + sin / sin 0)approximation functions can be substituted to each of the

+~=w(sn/cosw (snysin 0 COSn/5+ co y os0 summation terms in Equation 1 to 6 to complete the= u sn w cs 0 v(si w Sfl 0 in ~5+ ~OW~OS(/~)(8) mathematical model.

+ w/(sin yii sin 0 cos (/ - cos (/ sin 05)Z~e = -u sin0+ vcosO0sin(5+ wcosO cos(/ (9) IV. GENERIc FLIGHT DYNAMICS MODEL

This work focuses on the development of a generic flightThe rates of rolling, pitching and jawing angles are given as dynamics model (FDM). The role of the FDM in the wholefollows [18]: simulation is a physics engine for dead reckoning of an

aerial vehicle. By manipulating input variables=p + (qsino + rcoso)sinO/cosO (10) mathematically, FDM predicts the future states of an

=qcoso - rsino (I11) aircraft. Fig. 4 shows the schematic diagram of flighti~=(qsin0 + rcos 0)/cosO0 (12) dynamics modelling for dead reckoning.

InlitialThe angle of attack a is related to the body velocity u (in x-........................................... oidtls

Differentiatingbothsides,~~~~~~~~~~~~~~~~~~~~~~~~~~~~................ titd

Fig.4. Flight.......dynamics.....modelling...fordedgeconn2WUWU~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~.............cxseccx= 2~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~HH..........................................U~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~..................Rearrangetheequtin.t.obainth.rae.o.anleof ttak:.hegenricFDMwa. deeloed.n.atlb.Smulnkusing Aerospace Blockset.................It...models.....the...motion.....of...anyW

WU-WU 2 vehicleconfigurations,fromaball (potentially for~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~...................C056Y '~~~~~~~~~~'~~~'debugging),toatransonic fighter.TheFDM,likeanyother~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~................

Byffdifferentiation: es,...aircraft ....properties.... (inertia...and...gravity),...aerodynamic~ ~~ ~ ~~~~~ ~~~ ~~~ ~~~ ~~~ ~~~ ~~~ ~~~ ~~~ ~~~ ~~~ ~~~ ~~ ~~~ ~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~HHHHHH~...........................d sinfl = v(v2 + + w2V2coefficients,controls and wind conditions. It then outputs~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~...........dtdt/ thedynamicresponseof the aircraftwithrespecttotime.~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~.........................

~~cosg=~~~~~~~(v2+u2+w2)'2 Among the input data, determiningaerodynamic~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~............- 1(2. + 2v~~~~~~~~~~~~~~~~~~~~~~~~) coefficients is one of the keyaspectsinthisproject.Infact,~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~................

2+2wvi2)~~~~~~~v~~+ U2 + W~~~~~~~ )2 it is ultimately responsible for the accuracy ofsimulations.~~~~~~.............Hence the rate of sideslip angle is obtained:Two methods to determine the coefficients are: i)~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~................

573~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~..............

experimental approach, and ii) mathematical approach. The logged into the Matlab Workspace. Hence any relationshipformer is generally very accurate, but time consuming and of interest can be plotted as a graph in Matlab commandexpensive. On the other hand, the mathematical approach is prompt or in a script file. This flexibility provides users withmuch faster, with little cost and very repeatable by the aid of the absolute control of all variables before, during and aftercomputers. However, the inaccuracy in the mathematical a simulation. As an option, the FDM can produce a 3Dapproach is inevitably caused by the complex aerodynamics graphic animation in real time by interfacing it withinvolved in aircraft modelling, and the uncertainty in the real FlightGear. With the Datcom added in the front-end, and theenvironment conditions. Even though the best compromise FlightGear added in the back-end, the simulation hasis to use a combination of these two methods, all the become more users friendly and versatile. The integratedcoefficients were determined through mathematical flight dynamics modelling system is shown in Fig. 5.approach at the early stage of the project.Most commonly, aerodynamic coefficients describe forces hnitial

and moments in the stability coordinates. The six-degree of I.condition.sIfreedoms are drag, side, lift, roll, pitch and yaw. Each PropertllllllWiesllllllllaerodynamic coefficient can be categorised into one of these F1ji Poitosix axes to describe a force or a moment. As more and more AXcrf Dac_ A_oya Vlctecoefficients being built up along each axis, the sum of forces _Mdland moments become more accurate, and hence the.simulation results can be improved. Iiiiiiiiiiiiiii iiiiiiiiiiiiil

This work used the software Datcom [19] to calculate11111111111111aerodynamic coefficients from first principles. By writing a Fig. 5. Block diagram of integrated flight dynamics modelling.Datcom input file, which is a collection of all essentialgeometries of an aircraft, Datcom can produce an output file V. VERIFICATION AND DISCUSSIONSwith aerodynamic coefficients. The coefficients in the six To determine whether or not the model correctlyd.o.f are drag, lift, side, pitching moment, rolling moment, calculates the outputs for the given inputs, a verificationand yawing moment coefficient. process has been carried out. The FDM was verified by

The contribution from each aerodynamic coefficient is comparing its results with another FDM with exactly thedifferent. While some main coefficients (e.g. lift, drag and same input information. The open source FDM JSBSim waspitching moment coefficients) have huge impact on the chosen to make the comparison, because it is generallyforces and moments, some other coefficients can have considered a very accurate FDM. Verifications were focusedunnoticeable effects for subsonic small aeroplanes like on body velocities, altitude and pitch angle. Simulationunmanned aerial vehicles. Hence, to improve the accuracy results from the two FDMs are shown in Fig. 6.Of a simulation, main coefficients should be determinedexperimentally if possible.. iitiirlril -gr%.Sr r.The complexity caused by aerodynamic coefficients lies k

in the fact that almost every coefficient is not a constant, but _4i 7. = .,.,=,,=,,,.a function of some of the following independent variables: t:altitude, Mach number, angle of attack, sideslip angle and gcontrol surfaces deflection angles. Output file from Datcom ' _fi' = | <<g..a<.is in report format provides aerodynamic coefficientscorresponding to discrete flight conditions and angles, X}},/WSgwhich can be easily used to construct lookup tables. By (a) Body velocity u (b) Body velocity wapplying interpolation technique, continuous aerodynamic -= -_;_coefficients in any flight conditions and aircraft states can be ZZD. Ef aWI t <XTdetermined. t

For low speed and low altitude UAV applications, the t M ..--aerodynamic differences in different Mach numbers and "Z 2altitudes are subtle. Consequently, in this project the effect ,-.= -' =on different Mach numbers and altitudes on the r,'.=i.y ym2i5faerodynamics was ignored. This was a very practical (c) Altitude (d) Pitch angleapproach because it reduced two dimensions on all look up Fig. 6. Verification of simulation results against JBSSimtables. As a result, this marginal accuracy loss has led togreat performance gain in the simulation. In the comparison, propulsion and control surface

The FMD was built in Simulink with all variables being deflection angles inputs were set to be zero in both flightvisible, that is, any variable can be made as an output and models. This simplified the XML aircraft specification file

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required in JSBSim. The simplification will not affect the results. With accurate aerodynamic coefficients, thecomparison results as long as inputs in both FDMs are kept application of stability evaluation seems promising.identical. Fig. 6 plots the comparison results of body Dead reckoning application relies on other sensorsvelocity u & w, altitude and pitch angle versus time in an providing accurate information in real time. Future workarbitrary flight simulation scenario. Body velocity v was not will include the development of onboard sensing such asshown because it was zero at all time from both FDMs. The wind speed sensing so that real-time wind parameters can becomparison shows the two flight dynamics models produced input to the FDM. Experimental methods will be used toalmost identical results for the same inputs. determine important aerodynamic coefficients accurately.By combining with other navigation aids (GPS) and on The success of various modules and their integration will

board sensors, the FDM can be used for dead reckoning. At move a step closer to developing the low-cost autopilotthe early development stage, dead reckoning can be UAV systems for commercial operations.achieved by the use of a ground station as illustrate in Fig. 7.After thoroughly testing, the FDM can be implemented in REFERENCESthe onboard processor, with the integration of sensors to [1] Introduction of Modeling of Aerospace Vehicles for Flight Dynamicform an Inertial Navigation System (INS). The advantage of Studies http://dcb.larc.nasa.gov/Introduction/models.html.INS is that once initial conditions are provided, the system [2] Chavez, F.R., et al., "Advancing the State of the Art in FlightSimulation via the Use of Synthetic Environments".can estimate the aircraft's motion without the need for [3] Rasmussen, S.J. and Chandler, P.R., "Unmanned aerial vehicles:external reference. When the current state of the aircraft is MultiUAV: a multiple UAV simulation for investigation ofknown, the model is able to predict the tendency of the cooperative control", Proceedings of the 34th Conference on Winteraircraft'smotion and give correct command to the aircraft Simulation: exploring new frontiers. 2002, Winter Simulationaircraft's motion and give a correct command to the aircraft Conference: San Diego, California.

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