Three-Degree-of-Freedom (DOF) Missile Trajectory ...cradpdf.drdc-rddc.gc.ca/PDFS/unc18/p520640.pdf · Three-Degree-of-Freedom (DOF) Missile Trajectory Simulation Model and Comparative

Three-Degree-of-Freedom (DOF) Missile

Trajectory Simulation Model and

Comparative Study with a High Fidelity

6DOF ModelR. BrochuSummer Research Assistant

R. LestageDRDC Valcartier

Defence R&D Canada – ValcartierTechnical Memorandum

DRDC Valcartier TM 2003-056December 2003

Three-Degree-of-Freedom (DOF) Missile Trajectory Simulation Model and Comparative Study with a High Fidelity 6DOF Model

Raphaël Brochu Summer Research Assistant, DRDC Valcartier

Richard Lestage DRDC Valcartier

Defence R & D Canada - Valcartier Technical Memorandum DRDC Valcartier TM 2003-056 2003-12-19

Authors

Raphaël Brochu and Richard Lestage

Approved by

François Lesage

Acting Head / Precision Weapons

Approved for release by

François Lesage

Acting Head / Precision Weapons

© Her Majesty the Queen as represented by the Minister of National Defence, 2003

© Sa majesté la reine, représentée par le ministre de la Défense nationale, 2003

UNCLASSIFIED

Abstract

A 6 degree-of-freedom (DOF) missile trajectory simulation model has been reduced to a 3DOF model in order to minimise its calculation time. However, this had to be done without compromising the realism of the simulation. The modeling environment is Matlab/Simulin. Starting from a simple punctual mass in the 3-D space (Referenced as model 3DOF-0), three major improvements have brought this model to its final version (3DOF-3): namely, the missile orientation ( )αβ , with missile speed, the trim condition and the autopilot and airframe dynamics (or the lateral acceleration time response). Each time the 3DOF model was upgraded, its accuracy was compared with a high fidelity 6DOF model, and new strategies were deployed to improve the prediction. The final 3DOF model version is simply the insertion of all these improvements into the preliminary version of the 3DOF model. Its level of accuracy and fidelity achieved is satisfactory. Indeed, the relative error between the 3DOF and 6DOF models for maximum range envelope calculations is 3% and 12% at altitudes of 1 and 15 km, respectively. Globally, the 3DOF model is approximately twice as fast compared with the 6DOF one. This can lead to interesting future developments.

Résumé

Un modèle de simulation de trajectoires de missile à 6 degrés de liberté (DDL) a été réduit en un modèle à 3DDL de façon à minimiser le temps de calcul tout en essayant, autant que possible, de maintenir un niveau de précision et de fidélité acceptable. La plate-forme de travail est l’environnement Matlab/Simulink. Réduisant le missile à une simple masse ponctuelle dans l’espace tridimensionnel, trois grandes améliorations apportées à la version préliminaire du modèle à 3DDL (3DOF-0) ont permis d’en arriver à sa version finale (3DOF-3): l’orientation du missile ( )αβ , par rapport au vecteur vitesse, la condition d'équilibre des moments au centre de gravité et la dynamique de l’autopilote (ou temps de réponse aux accélérations latérales commandées). La version finale du modèle à 3DDL est simplement la combinaison des améliorations énumérées précédemment à la version préliminaire du modèle. En somme, le degré de précision et de fidélité atteint est satisfaisant. En effet, en comparant les enveloppes maximales des modèles à 6DDL et 3DDL, l’erreur relative n’est que de 3 et 12 % pour des altitudes de 1 et 15 km, respectivement. De surcroît, le modèle à 3DDL s’avère être jusqu’à 2 fois plus rapide en temps de calcul que son opposant. Ces résultats pourraient amener à d’intéressants développements futurs.

DRDC Valcartier TM 2003-056 i

UNCLASSIFIED

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ii DRDC Valcartier TM 2003-056

Executive summary

Six degree-of-freedom (DOF) missile trajectory simulation models have already been developed and employed in many military applications. Its capability to predict with high accuracy and fidelity the missile speed and trajectory evolution with time is by far its best quality. However, another requirement has to be achieved. In order to see this tool being used for real-time training military applications, batch or Monte-Carlo simulations or for theatre level operation simulations, the missile trajectory simulation model must have a low-cost calculation time.

A 6 degree-of-freedom missile trajectory simulation model has been reduced to a 3DOF model in order to minimise its calculation time. However, this had to be done without compromising the realism of the simulation. The modeling environment used is Matlab/Simulink . Starting from a simple punctual mass in the 3-D space (Referenced as model 3DOF-0), three major improvements have brought this model to its final version (3DOF-3): namely, the missile orientation ( )αβ , with missile speed, the trim condition and the autopilot and airframe dynamics (or the lateral acceleration time response). Each time the 3DOF model was upgraded, its accuracy was compared with a high fidelity 6DOF model, and new strategies were deployed to improve the prediction.

The final 3DOF model version is simply the insertion of all these improvements into the preliminary version of the 3DOF model. Its level of accuracy and fidelity achieved is satisfactory. Indeed, the relative error between the 3DOF and 6DOF models for maximum range envelope calculations is 3% and 12% at altitudes of 1 and 15 km, respectively. Globally, the 3DOF model is approximately twice as fast compared with the 6DOF one. This can lead to interesting future developments.

Brochu, R. and Lestage, R., 2003. "Three-degrees-of-freedom (DOF) missile trajectory simulation model and comparative study with a high fidelity 6DOF model". TM 2003-056 Defence R & D Canada - Valcartier.

DRDC Valcartier TM 2003-056 iii

Sommaire

Des modèles de simulation de trajectoires de missiles à six degrés de liberté (DDL) ont déjà été développés et utilisés dans des applications militaires. Leurs capacités prédictives des positions, vitesses et accélérations du missile sur l’échelle du temps sont leur meilleur attribut. Cependant, d'autres spécifications doivent être satisfaites. Afin de voir cet outil utilisé pour l'entraînement militaire en temps réel, les simulations en lots ou de type Monte-Carlo ou pour les simulations de champs de bataille complets, les simulations de trajectoires de missile doivent avoir un coût et un temps de calcul réduit.

Un modèle de simulation de trajectoires de missile à six degrés de liberté (DDL) a été réduit en un modèle à 3DDL de façon à minimiser le temps de calcul tout en essayant, autant que possible, de maintenir un niveau de précision et de fidélité acceptable. La plate-forme de travail est l’environnement Matlab/Simulink. Réduisant le missile à une simple masse ponctuelle dans l’espace tridimensionnel, trois grandes améliorations apportées à la version préliminaire du modèle à 3DDL (3DOF-0) ont permis d’en arriver à sa version finale (3DOF-3): l’orientation du missile ( )αβ , par rapport au vecteur vitesse, la condition d'équilibre des moments au centre de gravité et la dynamique de l’autopilote (ou temps de réponse aux accélérations latérales commandées).

La version finale du modèle à 3DDL est simplement la combinaison des améliorations énumérées précédemment à la version préliminaire du modèle. En somme, le degré de précision et de fidélité atteint est satisfaisant. En effet, en comparant les enveloppes maximales des modèles à 6DDL et 3DDL, l’erreur relative n’est que de 3 et 12 % pour des altitudes de 1 et 15 km, respectivement. De surcroît, le modèle à 3DDL s’avère être jusqu’à 2 fois plus rapide en temps de calcul que son opposant. Ce résultat pourrait amener à d’intéressants développements futurs.

Brochu, R. and Lestage, R., 2003. "Three degrees of freedom (DOF) missile trajectory simulation model and comparative study with a high fidelity 6DOF model". TM 2003-056 Defence R & D Canada - Valcartier.

iv DRDC Valcartier T

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Table of contents

Abstract........................................................................................................................................ i

Résumé ........................................................................................................................................ i

Executive summary ................................................................................................................... iii

Sommaire................................................................................................................................... iv

Table of contents ........................................................................................................................ v

List of figures ........................................................................................................................... vii

List of tables ............................................................................................................................ viii

Acknowledgements ................................................................................................................... ix

1. Introduction ................................................................................................................... 1

2. Engagement Simulation Modeling ................................................................................ 3 2.1 Global engagement simulation model architecture .......................................... 4 2.2 Homing missile model architecture.................................................................. 5 2.3 Applications of in-service missile simulation models ...................................... 6 2.4 Matlab/Simulink: Our modeling environment .................................................. 7 2.5 Summary .......................................................................................................... 7

3. 3DOF Model Evolution................................................................................................. 8 3.1 Preliminary version (3DOF-0): a punctual mass oriented with missile speed . 8

3.1.1 Characteristics of the 3DOF-0 model.................................................. 9 3.1.2 Limitation .......................................................................................... 10

3.2 First improvement (3DOF-1): missile orientation.......................................... 10 3.2.1 Linearization of the equations of motion........................................... 11 3.2.2 Procedure implemented ..................................................................... 12 3.2.3 Consequences of the 3DOF-1 model missile orientation .................. 13 3.2.4 Case study: Lateral intercept scenario for 6DOF and 3DOF models comparison ..................................................................................................... 13

DRDC Valcartier TM 2003-056 v

3.2.5 Gain and limitation of the 3DOF-1 model ........................................ 14 3.3 Second improvement (3DOF-2): Trim condition........................................... 15

3.3.1 Characteristics of the 3DOF-2 model and procedure implemented .. 15 3.3.2 Gain and limitation of the 3DOF-2 model ........................................ 16

3.4 Third improvement (3DOF-3): autopilot and airframe dynamics .................. 18 3.4.1 Autopilot and airframe dynamic transfer function ............................ 18 3.4.2 State-space implementation............................................................... 20 3.4.3 Gain and limitation of the 3DOF-3 model ........................................ 21

3.5 The final 3DOF model: calculation time reduction and conclusion............... 22

4. Launch Acceptability Region (LAR) for Varying Azimuth........................................ 24 4.1 LAR termination criteria ................................................................................ 25 4.2 LAR comparison between the 3DOF-3 and 6DOF models for both low and high altitudes ............................................................................................................... 27 4.3 X-range limits dependency............................................................................. 31 4.4 Calculation-time comparison and conclusion................................................. 32

5. Conclusion................................................................................................................... 33

References ................................................................................................................................ 34

List of symbols and abbreviations ............................................................................................ 35

Distribution list ......................................................................................................................... 38

vi DRDC Valcartier TM 2003-056

List of figures

Figure 1. Engagement simulation modeling global architecture. ............................................... 4

Figure 2. 1962 US Standard Atmosphere properties (Ref. 3)..................................................... 5

Figure 3. Homing missile model architecture............................................................................. 6

Figure 4. The 3DOF-0 model (preliminary version): a punctual mass....................................... 8

Figure 5. Missile Mach number as a function of the x-distance for a tail chase pursuit (a) without gravity, and (b) with gravity (1g of lateral acceleration commanded in the pitch plane during all flight time). Altitude: 5km. ..................................................................... 10

Figure 6. The 3DOF-1 model: missile orientation (pitch plane only, body-fixed coordinate system). ............................................................................................................................. 13

Figure 7. Scenario «Lateral Intercept» at an altitude of 1 km (no gravity). (a) XY missile trajectory; (b) Missile Mach number as a function of the X-distance. Note: the maximum angle of attack observed for the 3DOF-1 model is 21.50. ................................................. 14

Figure 8. Scenario «Lateral Intercept» at an altitude of 14 km (no gravity). (a) XY missile trajectory; (b) Missile Mach number as a function of the X distance. Note: the maximum angle of attack observed for the 3DOF-1 model is 1060! .................................................. 15

Figure 9. Scenario «Lateral Intercept» at an altitude of 14 km (no gravity). (a) XY missile trajectory; (b) Missile Mach number as a function of the X distance. Note: the maximum angle of attack observed for the 3DOF-1 model is 1060, the 3DOF-2 model, 300 and the 6DOF model, 220............................................................................................................... 17

Figure 10. Lateral acceleration time response and corresponding missile Mach number (ζ = 1.4). (a) At sea level; (b) At the altitude of 15 km. ........................................................... 19

Figure 11. State-space scheme implemented in the 3DOF-3 model......................................... 20

Figure 12. Scenario «Lateral Intercept» at an altitude of 14 km (no gravity). (a) XY missile trajectory; (b) Missile Mach number as a function of the X distance. Note: For the 3DOF-3 model, .4.1=ζ ............................................................................................................. 21

Figure 13. LAR comparison between 3DOF and 6DOF models, altitude 1km. (a) Maximum range envelopes; (b) Minimum range envelopes; (c) 3DOF minimum range envelopes varying minimum flight time termination criterion. ......................................................... 29

Figure 14. LAR comparison between 3DOF and 6DOF models, altitude 15km. (a) Maximum range envelope; (b) Minimum range envelope.................................................................. 30

DRDC Valcartier TM 2003-056 vii

Figure 15. Missile 6DOF miss distance function of the initial x-range limit, altitude 15km, aspect angle °= 150ψ . .................................................................................................... 31

List of tables

Table 1. Final end-game results for the 3DOF missile models in comparison with the 6DOF model, scenario «Lateral intercept», altitude 1 km. .......................................................... 22

Table 2. Final end-game results for the 3DOF missile models in comparison with the 6DOF model, scenario «Lateral intercept», altitude 14 km. ........................................................ 22

Table 3. Calculation time reduction of the 3DOF missile model in comparison with the 6DOF model, scenario «Lateral intercept», altitude 14 km. ........................................................ 23

Table 4. Simulation control subsystem..................................................................................... 27

Table 5. Calculation-time of the 3DOF missile model in comparison with the 6DOF model, LAR simulation, altitude 1 km, without initialization....................................................... 32

viii DRDC Valcartier TM 2003-056

Acknowledgements

I would sincerely like to thank Dr. Richard Lestage, my supervisor during this summer job, as well as Alfred Jeffrey and Marc Lauzon for their supports, their clear explanations, their encouragement and most of all, for their availability, each one of them at a different period during the summer. Being supported by such a team made the working relations so easy and pleasant. Thank you!

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1. Introduction

Six-degree-of-freedom (DOF) missile trajectory simulation models have already been developed and employed in many military applications. Its capability to predict with high accuracy and fidelity the missile speed and trajectory evolution with time is by far its best quality. However, another requirement has to be achieved. In order to see this tool being used for real-time training military applications, batch or Monte-Carlo simulations or for theatre level operation simulations, the missile trajectory simulation model must have a low-cost calculation time. Examples that support this criterion are numerous and two of them will be presented here.

• The Javelin Detachment Trainer (JDT) gives the opportunity to the military to perform weapon system training and rehearsal in a virtual environment. The JDT training unit includes a full-size mock-up of the Javelin launcher system, a surrounding trench environment and a wide-screen display showing the target motion and missile trajectory following a launch command. The need for near real-time computation of the missile-target simulated flight dictated the use of a 3DOF versus a more complex 6DOF model. Consequently, for this type of simulation, the calculation time must be shorter than the missile flight time.

• Military aircraft pilots would ideally like to know at any time if the target that they have to intercept is located in its missile Launch Acceptability Region (LAR), as a function of the target flight path and the specific atmospheric conditions. Of course, since the target position, speed and acceleration are likely to evolve as a function of time, information update time must be adequate. This would increase pilot survivability and the probability chance to successfully complete the mission.

With the objective of reducing the 6DOF model calculation time, the Weapon Engagement Simulation has decided to reduce the latter to a 3DOF missile trajectory simulation model, trying to maintain comparable accuracy and fidelity. Undoubtedly, the 3DOF model will not be as performing as the 6DOF model for all possible scenarios. The problematic lies in the meaning of «degrees of freedom». A 3DOF model is nothing less than a punctual mass in a 3-D space, where the degrees of freedom are the x, y, and z coordinates alone. At the opposite, the 6DOF model calculates the position (x, y, z) and orientation, given by the Eulerian angles

, of a true rigid body. Hence, the basic 3DOF model has no missile orientation and thus strategies will be deployed to predict it. Hopefully, when the new 3DOF model is created, calculation time will be reduced by a satisfactory factor. Finally, since the flight dynamics of the 3DOF model are less complex compared with the 6DOF model, it will be much easier to implement it in the Matlab-Simulink environment.

( ΦΘΨ ,, )

This report presents the complete 3DOF missile trajectory simulation model and more precisely, the modifications brought to the missile aerodynamic, airframe and control subsystems. A quantification of the calculation time gains and accuracy losses is also given.

DRDC Valcartier TM 2003-056 1

The first part of the report introduces the reader to the global architecture and characteristics of the engagement simulation model. Then, the second part describes the 3DOF missile trajectory simulation model in its preliminary version followed by three improvements, which will lead to the complete 3DOF model. Comparative study with a high fidelity 6DOF model will illustrate the relative performance of the 3DOF model throughout the improvements. Qualitative and quantitative measurements of the accuracy, fidelity and calculation time will be assesses using various mission critical scenarios. Finally, in a third part, LAR simulations, for both minimum and maximum range envelopes, will be computed and compared with the 6DOF model. The limitations and difficulties encountered will also be discussed.

2 DRDC Valcartier TM 2003-056

2. Engagement Simulation Modeling

Engagement simulation models predict weapon trajectory, aerodynamic behaviour, speed and performance by the resolution of the equations of the flight dynamics. Its main goal is to improve the operational effectiveness of weapon systems through modeling and simulation. The main systems are:

• Aerodynamics;

• Guidance;

• Control (wings, fins, thrust vector control);

• Combustion;

• Fuzing and warhead.

The aerodynamic characteristics of munitions and missiles in terms of precision, speed and manoeuvrability are studied and computational tools are developed in order to accomplish these objectives. Numerical simulations with a semi-empirical 6DOF trajectory model are carried out to investigate missile aerodynamic behaviour.

For accurate predictions, the engagement simulation modeling is based on the resolution of the rigid body (6DOF) equations of motions (EOM). They are:

∑ = amF rr (1)

[ ]∑ = ωr&

rIM (2)

Neglecting Coriolis and gravitational forces, the forces and moments in Eqs. 1 and 2 are of two kinds: thrust and aerodynamic. The latter force or moment is generated by aerodynamic coefficients, which, in the 6DOF model case, depends on 7 variables: missile Mach number, angle of attack TOTα , roll angle Mφ , and fin deflections (4). They are found by linear interpolation in 7-D look-up table, and it is why the model is semi-empirical in nature. This type of simulation differs from Computational Fluid Dynamics (CFD) models, where more precise investigations about missile aerodynamic behaviour are realised, using Finite Volume Method, boundary layers analysis and atmospheric turbulence models, etc.

The next section illustrates the global engagement simulation modeling architecture. Secondly, a description of the homing missile architecture will permit to outline modifications of the systems to reduce this 6DOF model to a 3DOF model. Thirdly, studies made with the simulation engagement models will be briefly discussed, and finally the reader will be introduced to the benefits of using the Matlab/Simulink environment.


2.1 Global engagement simulation model architecture

Figure 1 illustrates the global architecture of a missile trajectory simulation model (Ref. 1). The weapon modelled is a generic air-to-air medium range missile representative of the DND AIM-7 (Ref. 2). The engagement depicted is between an aerial fighter and a target, where typical engagement ranges are usually below 55 km.

Figure 1. Engagement simulation modeling global architecture.

The model is globally divided into different components, namely, the environment, missile, launcher, target, relative kinematics and simulation control. The connectivity between the various simulation model components results in the closed-loop nature of the simulation. The environment includes a description of the atmospheric properties (density, pressure, temperature and speed of sound) as a function of altitude (Figure 2). The missile contains a large number of sub-components which are critical to its effectiveness and must therefore be appropriately described in the high-fidelity 6DOF model. The missile includes the dynamics, seeker, guidance, control and kinematics. The operator (or launcher) launches the missile through specific firing commands, and therefore sets the initial conditions of the missile simulation. The target can manoeuvre according to a pre-specified profile. The simulation control determines the success of the engagement based on a pre-defined set of termination criteria. The kinematics model computes the relative position and velocity between the missile, target and launch platform.

The principal termination criteria employed in the model are:

• Maximum guided flight time t ; max

• Minimum distance occurring at closest approach( minV ) ( )0=CV ;


• Ground impact ( ) ; 0=Mz

• Minimum flight time (security distance between the missile and the fighter after launch before the warhead is armed).

mint

The crucial component of this global engagement simulation model architecture is the homing missile itself, and this will be discussed in more detail in the next section.

Figure 2. 1962 US Standard Atmosphere properties (Ref. 3).

2.2 Homing missile model architecture

A homing missile has, by definition, a guidance system to command lateral accelerations (Ref. 1). A typical homing missile architecture is shown in Figure 3. Homing missiles use either infrared or radio frequency seekers to track the target, a guidance law to generate the appropriate missile acceleration commands and a flight control system to achieve the desired acceleration response to the guidance commands. In a 6DOF missile trajectory simulation model, the flight control system utilises an autopilot that converts guidance commands into respective control surface deflection commands. The actuators apply the aerodynamic control surface deflections that cause the missile to manoeuvre. The resultant dynamics are measured by rate gyros and accelerometers to form a feedback control system that stabilizes the airframe and produces the flight control transient required for accurate homing. The reader will be informed in the subsequent chapter that the 3DOF model has no flight control system but instead, tries to reproduce the airframe and autopilot dynamics together by introducing lateral acceleration time response to the commanded lateral accelerations.


For the 6DOF model, the fin deflections are sent to the 7-D table look-up as well as the missile Mach number, angle of attack and roll angle Mφ to extract, by linear interpolation, the approximate aerodynamic coefficients required to calculate the corresponding aerodynamic forces and moments over the missile airframe body. With the addition of gravitational and thrust forces, the missile 6DOF EOMs can be integrated and solved for position, velocities and accelerations. This process gives at each time step the missile trajectory, which can be referenced to the missile body-fixed or the earth-fixed coordinates by quaternion or eulerian transformations. The reader will note in the next chapter that, in the 3DOF model, aerodynamic coefficients are linearized and represented as a function of the Mach number only. Also note that, to reduce the 6DOF to a 3DOF model, emphasis will be put on missile aerodynamic and airframe dynamics with an integrated autopilot system, representing flight control dynamics.

Figure 3. Homing missile model architecture

2.3 Applications of in-service missile simulation models

In-service missile simulation models can be employed for many applications. They can evaluate:

• Missile performance;

• Missile behaviour for various engagement conditions;

• Firing zone calculations.

• Missile lethality assessment support;

• Tolerance and miss distance studies, etc.


Furthermore, they can be helpful for test and evaluation support, definition of flight trials and also, for weapon subsystems analysis, like in the present case, for study of the 3DOF missile aerodynamic, airframe and autopilot subsystems.

2.4 Matlab/Simulink: Our modeling environment

Within the Precision Weapons Section at DRDC Valcartier, missile trajectory simulation models are mainly created in the Matlab/Simulink environment. The model hierarchy is based on a multi-layered structure of systems and subsystems, with their corresponding inputs and outputs. This structure accelerates model implementation, verification and validation, and it facilitates team development. A library that contains different types of missile subsystems was built and allows for quick prototyping.

2.5 Summary

Throughout this chapter, the reader was introduced to simulation engagement modeling with a semi-empirical 6DOF simulation model. Each time it was possible, differences between this model and the 3DOF model were highlighted. These modifications will be explained and analysed more thoroughly in the next section.


3. 3DOF Model Evolution

This section follows the evolution of the 3DOF missile trajectory simulation model, from the preliminary to the final version. Three major improvements were implemented in the models. Each improved model will be described and the results compared with the previous to show the gain made by this modification. The results will then be compared with the high fidelity 6DOF model to show its limitations and determine which adjustments could be employed to further improve the 3DOF model.

3.1 Preliminary version (3DOF-0): a punctual mass oriented with missile speed

As stated earlier in the introduction, the 3DOF model problematic comes from its orientation in a 3-D space. The missile is no longer considered as a rigid body but simply as a punctual mass. Figure 4 illustrates this version of the model with its principal characteristics. The elevation angles are defined ( )rollpitchyaw γγγ ,, as a reference transformation matrix from the earth-fixed (EF) to the missile-velocity-fixed coordinate systems. It is interesting to note that, for this model in particular, since the body-fixed (BF) coordinate system is oriented towards missile velocity vector

without any orientation ( M )φαβ ,, , the elevation angles and Eulerian angles

Figure 4. The 3DOF-0 model (preliminary version): a punctual mass


( ΦΘΨ ,, ) defined in the 6DOF model are identical. However, it should be noted that while the Eulerian angles take into account the missile orientation, the elevation angles do not. Mathematically, the relationship between the Eulerian and elevation angles for the 3DOF model can be expressed as:

( )αβ ,

MV =

βγ +=Ψ yaw (3)

αγ +=Θ pitch (4)

Mroll φγ +=Φ (5)

Of course, for the 3DOF model, 0== Mroll φγ . Also, it should be noted that gravity acts positively along the z-axis EF(Defined positive downwars).

3.1.1 Characteristics of the 3DOF-0 model

Firstly, since the 3DOF-0 model represents a punctual mass, there is no active moment on the missile, and thus, no angular velocities. Consequently, equation 2 disappears, which eliminates 6 integration processes per time step and significantly reduces the calculation time. Secondly, since there is no missile attitude in the yaw and pitch plane

, the velocity vector acts always toward the x-axis of the BF coordinate system (i.e., ). As a consequence, the aerodynamic axial force represents the total drag of the missile irrespective of the angle of attack, hence,

MUD

XF

XFD = (6)

In other words, since the aerodynamic lateral forces and are always perpendicular to the missile velocity, they do not induce any drag on the missile and the latter can manoeuvre in the yaw and/or pitch plane without speed loss. This is, of course, a huge simplification that should be remedied in the first model improvement. Finally, this preliminary version of the 3DOF model directly generates the lateral accelerations commanded by the guidance section:

YF ZF

( ) ( )CMDpitchyawpitchyaw aaaa ,, = (7)

This instantaneous response is, undoubtedly, far from reality and future improvements should consider this aspect.


3.1.2 Limitation

As just stated in the last paragraph, any lateral acceleration generated by the missile does not reduce its speed. Consequently, the only possible scenarios where this preliminary model would give the same missile Mach and trajectory as the realistic 6DOF model is on a tail chase pursuit or head-on trajectory without the action of gravity. In that case, the 3DOF and 6DOF missiles would follow a perfect straight line with the same amount of drag loss. Thus, missile Mach number evolution would be the same for both models. This is shown in Figure 5(a). By adding just 1g of lateral acceleration in the elevation plane during the flight time (i.e., the gravity), both missiles would manoeuvre and due to the specific formulation of each model, they would no longer follow the same path. Indeed, Figure 5(b) shows that the 3DOF missile does not lose speed sufficiently: hence, it will intercept the target sooner compared to the reference model. Moreover, since its speed is greater than the 6DOF model, the 3DOF model has to manoeuvre faster, and consequently, the missile trajectories differ. This difference is relevant with the only addition of gravity. Larger differences are expected when high-g manoeuvres are performed by the missile.

Figure 5. Missile Mach number as a function of the x-distance for a tail chase pursuit (a) without gravity, and (b) with gravity (1g of lateral acceleration commanded in the pitch plane during all flight time).

Altitude: 5km.

As part of the next stage, it will be crucial to implement a methodology that will allow the missile orientation to move the missile velocity from its BF x-axis and hence increase the total drag. This will be the scope of the next section.

3.2 First improvement (3DOF-1): missile orientation

In this model, the missile is not represented any longer by a punctual mass but instead, by a rigid body with orientations ( )αβ , in the yaw and pitch planes, respectively. For the 6DOF model, the attitude angles ( )Mφαβ ,, can be expressed as only two components ( )MTOT φα , by first rotating the BF coordinate system around the x-axis


by an angle Mφ to keep the missile velocity toward in the elevation plane and then, by finding the total angle of attack that orients the body. The resulting equations are simply:

= −

MTOT V

u1cosα (8)

= −

wv

M1tanφ (9)

where u, v, w are the velocity components in the x, y, z directions of the BF coordinate system. The 6DOF model orientation angles are therefore given by the BF velocity components, which are integrated with time. Since the 3DOF model cannot represent these velocity components without the Eulerian angles1 (and not the elevation angles), a different strategy has to be taken in order to find the missile orientation.

3.2.1 Linearization of the equations of motion

The strategy employed to extract the missile orientation values from the aerodynamic coefficients is to proceed to the linearization of the equations of motion (EOMs). These equations are first linearized using small perturbations around a given trim or equilibrium point (typically, °=∂=∂ 1pitchδα ) to solve in closed-form the EOMs

and to facilitate the extraction of the given angles of attack ( )αβ , . For a symmetric missile body, the trim point is usually 0,0 == pitchδα , and the linearized axial, normal and pitch moment aerodynamic coefficients become:

( ) ( ) ( ) TOTXXTOTX MCMCMC αα α+= 0, (10)

( ) ( ) ( ) pitchNNpitchN MCMCMC δαδα δα +=,, (11)

( ) ( ) ( ) pitchmmpitchm MCMCMC δαδα δα +=,, (12)

)(),,( MCMC mqpitchmq =δα (13)

where the new set of linear coefficients is only a function of Mach number. The coefficients must be extracted from tables or polynomial equations by evaluating the first order derivatives at the trim point ( )0, =pitchδα . In the present case, the aerodynamic coefficients are extracted from the 6DOF 7-D look-up tables. By re-

1 Note that for the 6DOF model, the missile Eulerian angles ( )ΦΘΨ ,, are determined by integrating

the Eulerian angular rates ( )ΦΘΨ &&& ,, , which are given by the missile angular velocities p, q, r.


arranging the above equations, the required expressions of the polynomial aerodynamic coefficients are:

( )

( ) ( )( )0

0XTOTXX

0TOTX0X

)M(C,MCMC

),M(CMC

=αα

=α

α∂−α∂

=

α=

(14)

( ) ( )

( ) ( )0

0

,0

,0

,,

,,

=

=

=

=

∂

∂=

∂

∂=

pitch

pitch

pitch

pitchNN

pitchNN

MCMC

MCMC

δα

δ

δα

α

δδα

αδα

(15)

( ) ( )

( ) ( )0

0

,0

,0

,,

,,

=

=

=

=

∂

∂=

∂

∂=

pitch

pitch

pitch

pitchmm

pitchmm

MCMC

MCMC

δα

δ

δα

α

δδα

αδα

(16)

0,0

),,()(==

=pitch

pitchmqmq MCMCδα

δα (17)

Equations 14 to 17 clearly indicate that these coefficients are now only a function of the Mach number.

3.2.2 Procedure implemented

Once the polynomial aerodynamic coefficients are computed for a specific Mach number, the procedure to implement is quite simple. The procedure is identical for the yaw and pitch planes. When the 3DOF airframe subsystem defined earlier receives its commanded lateral accelerations, lateral forces and aerodynamic coefficients can be calculated and used to find the angles of attack. With these attitude angles, the 3DOF missile can be fully oriented by calculations of the Eulerian angles (Eqs. 3, 4 and 5). Mathematically, the algorithm implemented in the 3DOF-1 model, for the pitch aerodynamic force alone, can be expressed as follow:

( ) αγαα

+=Θ→→→→ ∞ pitchN

NrefZCMDpitch MC

CSqFa

)( (18)

Where all the coefficients assume a trim condition ( )0, =pitchδα .


3.2.3 Consequences of the 3DOF-1 model missile orientation

Figure 6 illustrates the consequence on missile orientation over the total drag of the 3DOF missile for the pitch plane alone. Knowing the angles of attack ( )αβ , , the total 3-D drag becomes:

ααβαβ sincossincoscos ZYX FFFD ++= (19)

which is quite different if compared with the previous model (Eq 6). Indeed, results will show that, for small angles of attack, typically below 300, missile trajectory and Mach number comparison is greatly improved since the drag is more accurately computed. This will be the subject of the next sub-sections.

Figure 6. The 3DOF-1 model: missile orientation (pitch plane only, body-fixed coordinate system).

3.2.4 Case study: Lateral intercept scenario for 6DOF and 3DOF models comparison

To compare the 3DOF-0 and subsequent improved versions with the accurate 6DOF model, a scenario that requires high missile manoeuvres is required. However, since the implemented algorithm is the same for both yaw and pitch planes, there is no need to make the missile manoeuvre in both planes. Keeping this in mind, a scenario called «Lateral Intercept» was created. In this scenario, the fighter and missile are at the origin, where the initial target position is at 3km (x-axis). The gravity is set to zero, since we only want the missile to manoeuvre in the yaw plane. At t , the missile is launched with the fighter initial speed of 100 m/s (360 km/h) along the x-axis. At the same time, the target is moving along the positive y-axis at a constant speed of 100m/s, the pitch plane of the target BF coordinate system being superimposed to the yaw plane EF coordinate system. At

0=launch

3=flightt s, a pitch centripetal acceleration of 6g is given to the target until the end of the simulation. During the manoeuvre, the target is making a circular trajectory with a diameter of


approximately 340m. The scenario «Lateral intercept» will be the reference scenario and allow for testing at different altitudes and also to test the effect of the new model improvements.

3.2.5 Gain and limitation of the 3DOF-1 model

Figure 7 presents the scenario «Lateral Intercept» at the altitude of 1km for the 3DOF-0, 3DOF-1 and 6DOF models. Compared with the preliminary version of the 3DOF model, the 3DOF-1 missile trajectory and Mach number are noticeably improved for the simulation that requires small angles of attack (as we mentioned earlier, typically

( ) °≤ 30,αβ ). This fact means that the 3DOF-1 model is accurate for low altitude and high speed simulations where smaller angles of attack are required to produce the desired manoeuvrability. This can be understood by considering the expression for the aerodynamic force given below:

Figure 7. Scenario «Lateral Intercept» at an altitude of 1 km (no gravity). (a) XY missile trajectory; (b) Missile Mach number as a function of the X-distance. Note: the maximum angle of attack observed for

the 3DOF-1 model is 21.50.

refaeroMrefaeroaero SCVSCqF 2

21

∞∞ == ρ (20)

At low altitude, the air density is greater (which also means higher aerodynamic coefficients). Moreover, aerodynamic forces are proportional to the square of the missile speed, so as the speed increases, so does the manoeuvrability. Hence, at a low altitude and high missile speed, the angles of attack are kept low and missile trajectory is more accurate.

However, this effect becomes worse for high altitude simulations (see Figure 8). Since the missile speed at launch time is very low (100m/s) and air density as well, the guidance section commands a very high lateral acceleration, which is reflected back as a very high angle of attack, around 1060 at the beginning of the flight! Again, this is far from reality and missiles could never sustain such an angle without being unstable.


Furthermore, because of the excessive manoeuvres of the 3DOF-1 missile model, its Mach number tends to decrease significantly (increased drag).

Figure 8. Scenario «Lateral Intercept» at an altitude of 14 km (no gravity). (a) XY missile trajectory; (b) Missile Mach number as a function of the X distance. Note: the maximum angle of attack observed for

the 3DOF-1 model is 1060!

Because the 3DOF-1 missile model does not include a sum of the moments around its centre of gravity (CG), orientation angles cannot be saturated to a maximum value. In the next section, the reader will notice that each time a missile become unbalanced (i.e. the sum of moments around the missile centre of gravity is non-zero), there is a maximum angle of attack for which the missile can reach another state of equilibrium (or «trimmed» state). This condition is called the «trim condition» and will be the topic of the next model improvement.

3.3 Second improvement (3DOF-2): Trim condition

As stated at the end of the last section, the trim condition allows to the 3DOF missile the ability to manoeuvre without being unbalanced (Ref. 4). There is a maximum angle of attack limit of which a missile remains stable and can hence return to equilibrium state.

3.3.1 Characteristics of the 3DOF-2 model and procedure implemented

When a moment acts on a missile, the latter changes its orientation until the sum of moments around its CG becomes zero again. The trim condition for the pitch plane can be definite by:

∑ =+=⇔= 0)()(),,(0 δαδα δα MCMCMCm mmpitchmCG CG

r (21)


which is very similar to Eq. 12. Again, the same definition applies for the yaw plane also. This equation shows that maximum angle-of-attack is obtained at maximum fin deflection. The maximum angle of attack value can be determined by:

( ) maxmaxmax , δαβα

δ

−=

m

m

CC

(22)

For the model, the value used for the maximum fin deflection is °== 20maxδδ as in the 6DOF model. Equation 22 has been generalized for the yaw and pitch planes since the same coefficient mC is used for both calculations. Note that maxβ and maxα must be the saturation values for both the negative and positive angle of attack values. This condition can be achieved by applying the following algorithm (pitch plane):

( )( )maxmax ,,minmax ααα −⇒Saturation (23)

With the trim condition, Eq 21, and Eq 11, we have now two unknowns instead of three, i.e. ( )yawδβ , in the yaw plane, or ( )pitchδα , in the pitch plane. These unsaturated values can now be solved using these relations:

( ) ( )δαδα

δ

δαδα

δ αβNmmN

mZ

NmmN

mY

CCCCCC

CCCCCC

−=

−= , (24)

( ) ( )αδαδ

α

αδαδ

α δδNmmN

mZpitch

NmmN

mYyaw CCCC

CCCCCC

CC−

=−

= , (25)

Notice that the deflection angles must also pass through the saturation algorithm by replacing maxα with maxδ in Eq 23. Here is the procedure implemented in the 3DOF-2 model for the pitch aerodynamic force alone :

( ) ( ) ( ) ( )

effpitch

pitch

effZeffpitchpitchZZCMDpitch

a

FSaturationCFa

αγ

δαδα

+=Θ→

→

→→→→→ ,,

(26)

Finally, now that the attitude angles ( )αβ , have a ceiling and floor values, theaerodynamic axial forces can be evaluated by using both terms on the right side of Eq 10.


Figure 9 shows the results for the problematic case study of the previous section: «Lateral intercept» at an altitude of 14km. For scenarios that involve large commanded


lateral accelerations, the 3DOF-2 missile trajectory and Mach number have improved in comparison with the previous 3DOF versions. However, the 3DOF-2 model indicates a successful intercept with a miss distance of 4 meters ( )WHR< whereas the accurate 6DOF model indicates a miss of 73 m. This is due to the higher 3DOF-2 achieved missile speed (Figure 9(b)). For the last half of the flight, the 3DOF-2 missile maintains a 25m/s (90km/h) velocity advantage over the 6DOF missile. Since increased velocity translate into increased manoeuvrability, it can be seen in Figure 9(a) that, at the x-distance of 2500 m, the 6DOF missile has difficulty in generating the required lateral accelerations to reach the target.

Figure 9. Scenario «Lateral Intercept» at an altitude of 14 km (no gravity). (a) XY missile trajectory; (b) Missile Mach number as a function of the X distance. Note: the maximum angle of attack observed for

the 3DOF-1 model is 1060, the 3DOF-2 model, 300 and the 6DOF model, 220.

For the same scenario but at an altitude of 1 km (low angles of attack, results not shown), one sees little difference between the 3DOF-2 missile trajectory and the Mach number, in comparison with the 3DOF-1 model. At low altitude, this difference is far from being of the same importance as that at the altitude of 14km. At low altitude, both model versions indicate a successful intercept as the 6DOF model.

There is one more idealistic assumption to consider in the modeling of our 3DOF missile and it has been mentioned in the section of the preliminary version model. The 3DOF-2 model may have lateral force saturation, instantaneous response of the airframe with a precise static gain. Unfortunately, the dynamic response of the airframe can not be that fast or precise: there is always a certain time constant (lag) before achieving the commanded acceleration value and even when this value is reached, the missile always oscillates around its new equilibrium position. A solution to that problem is to implement a dynamic time response to the lateral commanded accelerations that would be a function of both the Mach number and the altitude. In reality, this would try to mimic the dynamics of the 6DOF autopilot.


3.4 Third improvement (3DOF-3): autopilot and airframe dynamics

The autopilot provides a mechanism to transition from one trim condition to the next in a dynamically stable manner. It receives commands from the guidance section, applies a correction (or a tuning) and returns a dynamic time response to the missile airframe. For the 6DOF model, the tuning is applied to the commanded lateral acceleration with a gain scheduling proportional-integral (PI) autopilot as a function of missile Mach number and altitude. The outputs are the four fin deflection values. For the 3DOF model, the correction applied to the commanded lateral accelerations and the outputs are the effective lateral accelerations. However, this time, the autopilot chosen is a second order transfer function, with gain scheduling also a function of missile Mach number and altitude. The reason for this choice is to save some calculation time by adding only two more integrators instead of three in the 6DOF model autopilot.

The autopilots presented above, being linear, require a linear airframe model for tuning. Fortunately, the linearization of the EOMs has already been undertaken for both 3DOF and 6DOF models. The reader may refer to section 3.2.1 and/or Ref. 5 for more information.

3.4.1 Autopilot and airframe dynamic transfer function

The autopilot and airframe dynamic transfer function has the same form for both yaw and pitch accelerations. In the pitch plane, it is equal to:

( ) 2

2

2² NN

N

CMDpitch

pitch

ssK

aa

ωζωω

++= (27)

where K is the static gain and is usually set to one in order to have a unitary gain, ζ is the lateral damping factor and Nω is the natural frequency that best represents the natural dynamic of both the autopilot and airframe. The value of ζ is usually set to its critical damping value, or 7.0=ζ . However, since a second order transfer function always have a first rising time constant lower than a third order transfer function, the damping value is too low and must be increased. For this reason, experiments between 6DOF and 3DOF models evaluating both lateral acceleration time responses at different missile Mach numbers and altitudes have been undertaken (see Figure 10). For these experiments, the autopilot is stimulated by successful step acceleration commands of 100 m/s² starting at the speed of Mach 2 for both altitudes. This set up covers different flight speeds because the missile progressively loses speed as a result of the drag and the absence of propulsion. Each successive step is thus at a lower speed than the previous one. Because of the speed variation, the autopilot has to continuously modify its gains according to the values found in the lookup table (6DOF model) or calculated (3DOF-3 model). After several experiments, results showed that the best overall curve-fit value is 4.1=ζ . Notice that, for the 3DOF model in Figure


10(b), the effective yaw acceleration does not oscillate around its equilibrium upper point because of the lateral acceleration saturation imposition.

Figure 10. Lateral acceleration time response and corresponding missile Mach number (ζ = 1.4). (a) At sea level; (b) At the altitude of 15 km.

On the other hand, the expression of the natural frequency is:

ααω MMZ qN −= (28)

where and are the dimensional aerodynamic coefficients. These coefficients depend on missile airframe properties (mass, moment of inertia…), Mach number and altitude. There respective definitions are:

αα MZ , qM

)(),(

),( MCmV

SMhqMhZ N

m

refαα

∞= (29)

)(),(

)(),(

),( MCmV

SMhqMC

IdSMhq

MhM Nm

refCGm

yy

refααα

∞∞ ∆−= (30)

)(2

²),(),( MC

VIdSMhq

MhM mqmyy

refq

∞= (31)

where designates the slope of the normal force per angle of attack and M and are the slope of pitch moment per angle of attack and per pitch angle rate,

respectively. These dimensional aerodynamic coefficients come from the linearization of the EOMs in yaw and pitch plane separately. Again, additionnal information can be found in Ref. 5.

αZ α

qM


3.4.2 State-space implementation

Since vectors cannot be manipulated with Laplace transfer functions, and since the natural frequency Nω of the system is evolving with time, Eq. 27 has to be implemented in the state-space form:

BuxAx +=r&r (32)

DuxCy +=r

(33)

where A, B, C, D are matrices, xr is the state vector and and are respectively the input and output of the system. Because Eq. 27 is a second order differential equation, the state vector has two states, namely x and (which have no physical meaning with position and velocity). The input of the system is one of the two commanded lateral accelerations and the output, one of the two effective lateral acceleration responses. Assuming a unitary gain (K = 1) in Eq. 27, Eqs. 32 and 33 take the following form for the pitch plane (Ref. 6):

u y

x&

(34) [ ]CMDpitch

NN

axx

xx

+

−−

=

10

210

2 &&&

&

ζωω

(35) [ ]

=

xx

a Neffpitch &0

2ω

Equations 34 and 35 also applies for the yaw plane. The state-space equations are easily implemented with the scheme presented in Figure 11.

Figure 11. State-space scheme implemented in the 3DOF-3 model.



For high altitude scenarios, Figure 12(a) shows good agreement between the 3DOF-3 and 6DOF missile trajectory. In fact, lateral acceleration time response makes missile direction changes smoother, especially at launch time. However, the time response is still too prompt compared to the 6DOF model. Nonetheless, Figure 12(b) presents a significant improvement between the 3DOF-3 missile Mach number and the 6DOF one, in comparison with the old 3DOF models. Indeed, the magnitude of missile oscillations around its static gain generates higher angles of attack, which in return, increase the overall missile drag. It should be remembered, however, that these oscillations cannot be effective if the lateral acceleration time response is too close from the saturation limit. This is a very encouraging improvement, since we know that the closer the missile Mach curve is to the 6DOF model, a similar manoeuvrability profile will be attained, which in turn will provide better overall agreement (trajectory

and end-game).

Figure 12. Scenario «Lateral Intercept» at an altitude of 14 km (no gravity). (a) XY missile trajectory; (b)

Missile Mach number as a function of the X distance. Note: For the 3DOF-3 model, .4.1=ζ

Table 1 and Table 2 summarize the final end-game results as given by each model for the scenario «Lateral Intercept» at the altitude of 1 and 14 km, respectively. The objective here is to give reader a numerical tool of comparison between the 3DOF and 6DOF models. By analysing data in Table 2, it is possible to appreciate significant improvement with the 3DOF model for high altitude scenarios. Undoubtedly, the 3DOF-3 model is, by far, the most accurate model for this kind of scenario. Unfortunately, for low altitude scenarios, the 3DOF model missile final velocities are at least 17 m/s higher compared to that given by the 6DOF model. This will lead to major differences when comparing 3DOF and 6DOF launch envelopes for this range of altitude.


Table 1. Final end-game results for the 3DOF missile models in comparison with the 6DOF model, scenario «Lateral intercept», altitude 1 km.

MODEL FINAL VERDICT MISS DISTANCE FINAL VELOCITY

[m] [m/s]

6DOF Success 0.0 500

3DOF-3 Success 0.0 524




Table 2. Final end-game results for the 3DOF missile models in comparison with the 6DOF model, scenario «Lateral intercept», altitude 14 km.

MODEL FINAL VERDICT MISS DISTANCE FINAL VELOCITY

[m] [m/s]

6DOF Miss 75 524

3DOF-3 Miss 28 524

3DOF-2 Success 4 550



In summary, the lateral acceleration time response of the 3DOF-3 missile model does bring a significant gain in accuracy if compared with the previous 3DOF models for high altitude scenarios. For low altitude scenarios, no noticeable break away is achieved. However, the addition of four integrators for the implementation of the lateral acceleration time response in each plane has increased the calculation time of the 3DOF-3 model by almost 30% when compared with the 3DOF-2 model.

3.5 The final 3DOF model: calculation time reduction and conclusion

Globally, the 3DOF-3 missile simulation model remains the most complete and realistic version for a 3 degrees-of-freedom missile trajectory simulation model.


Consequently, it becomes the official 3DOF model with a ζ value set to 1.4 for better accuracy.

Now that the final version of the 3DOF model is completed, it is possible to evaluate the calculation time reduction and its performance in comparison with the 6DOF. It is interesting to note that the time integration during a simulation remains the process with the highest cost of calculation time. Hence, by knowing the number of integrations per time step we have for each models, it is possible to evaluate the time calculation gain of the 3DOF model. The 6DOF model has 2 integrators for the 3 components of acceleration and angular acceleration, which make a total of 12 integrations. Moreover, the autopilot PI uses 3 integrators for the two components of commanded lateral accelerations, for a total of 6 integrations. Overall, the 6DOF model makes 18 integrations per iteration. In comparison to this, the final 3DOF model makes only 10 integrations (6 for the aerodynamics and 4 for the autopilot integrations). Therefore, it can be stated that the 3DOF time calculation gain is approximately 18/10 = 1.8. The results are tabulated in Table 1 for a typical simulation and the gain in time of the 3DOF model, without initialization, is very close to what has been estimated. Furthermore, if we calculate the time reduction taking into consideration the initialization process (in the 6DOF model, the autopilot gains are computed only when missile properties and/or geometries are changing), the gain increased to 3.4.

Table 3. Calculation time reduction of the 3DOF missile model in comparison with the 6DOF model, scenario «Lateral intercept», altitude 14 km.

MODEL SIMULATION TIME WITH

INITIALIZATION

SIMULATION TIME WITHOUT

INITIALIZATION

FLIGHT TIME

[s] [s] [s]

6DOF 13.9 5.7 8.1

3DOF-3 4.1 3.0 8.1

GAIN 3.4 1.9 -

In conclusion, without initialization, the final 3DOF model is approximately 2 times faster than the 6DOF model, and even higher with the initialization. However, further improvements are still possible to reduce the 3DOF model calculation time.


4. Launch Acceptability Region (LAR) for Varying Azimuth

The Launch Acceptability Region (LAR) for varying azimuth is an envelope in the horizontal plane inside which a missile will successfully intercept a target. It is also called a Launch Envelope (LE). Simulations required to determine this region depend on many parameters, such as target and fighter initial conditions and termination criteria. In the current document, the 3DOF-3 and 6DOF model Launch Envelopes have been plotted and compared for both low and high altitude simulations with constant target and fighter velocities. The fighter is initially positioned at the origin and is given an initial velocity of 100 m/s along the x-axis. The missile launch time is at s and the gravity force is enabled, since we want the LAR simulation to be as realistic as possible. The target has a constant velocity magnitude of 200 m/s but the vector direction will change with the aspect angle

0=launcht

ψ in the following manner:

0sincos

===

T

TT

TT

wVvVu

ψψ

(36)

The goal of the LAR simulation is to find the initial x-position of the target where a successful missile intercept will occur for each prescribed target velocity vector. The computed target initial x-position for a given ψ is also called the Initial X-range Limit (IXL). It should be noted that the target always starts on the x-axis, the only changing parameter is the velocity direction (or the aspect angle ψ ). However, the LAR plot gives the IXL value as a function of the aspect angle ψ on a polar graph, which gives a circular envelope decentred from the origin (and the fighter). This is illustrated in Figures 13 and 14.

An important precision has to be made in regard to the computation of the LAR. If target x position is too far from the fighter, the missile will fail to intercept the target due to the stated termination criteria. However, if the target is too close to the fighter, there exists a second IXL value inside of which the missile will fail to intercept the target. The first type of simulation is called a maximum range envelope (Rmax), and the second type is a minimum range envelope (Rmin). In the present case, Rmin and Rmax launch envelopes for varying azimuth will be calculated for both low and high altitudes using the 6DOF and 3DOF models. It should be noted, however, that Rmin calculations are highly dependent on termination criteria (see section 4.1), and results should be analysed with great care, as shown in Figure 13(c).

Another important aspect of launch envelopes in general is that they will be symmetric about the 0-180 degrees axis if the fighter and target velocity inputs are constants (no target manoeuvres);


LAR simulations are very useful for aircraft pilots, where they can determine if the target is within the performance limits of their weapons. For our purpose, the main objective of these simulations is to obtain a better understanding of the missile aerodynamic behaviour and to compare the accuracy between 3DOF and 6DOF models.

The next section describes the other parameters that have to be set before executing LAR simulations. The termination criteria will determine what is defined as a successful target interception and what is not.

4.1 LAR termination criteria

The simulation control subsystem of a typical 3DOF or 6DOF model enables to stop the simulation when some precise criteria are met and to output if the intercept was a success or a failure. Table 4 lists all the criteria for which a simulation stops and all those for which a verdict of successful or miss intercept is given. The objective here is not to give an exhaustive presentation of the termination criteria but primarily, to point out which and how a termination criterion can influence LAR simulations.

Four criteria are used to stop the simulation. The first one, the closest approach, is the most encountered one. In this case, the simulation will stop only if the missile starts by taking some speed over the target but finally runs off, and vice-versa. In other words, the simulation will stop when the missile-target range velocity (or closing velocity) is crossing zero (V ). The closest approach termination criterion is not activated before a minimum flight time ( ) to allow the missile to gain some speed over the target and prevent premature flight termination just after launch. The second termination criterion happens when the missile flight time reaches the maximum time allowed for guided flight ( t ). For our purpose, this criterion has been turned off for Rmax calculations to test missile aerodynamic behaviour. In practice, this criterion is never met for Rmin calculation. The third termination criterion is activated if ground impact occurs ( ). Finally, the fourth termination criterion has been added to the simulation control subsystem in order to reduce calculation time when calculating Rmin envelope. This new criterion arises when the target is inside the missile warhead lethal radius , but the missile flight time is below the minimum flight time allowed for aircraft safety, resulting in an unsuccessful mission. Since closest approach has happened before t , this criterion is not activated and the simulation would normally go on until maximum flight time ( t ) is reached. Fortunately, with this new criterion, calculation time is often reduced. Note also that this criterion is never met for Rmax calculation.

0=C

maxz

WH

mint

min

max

R

max

When a termination criterion is met, the simulation control subsystem outputs the first final verdict to occur. For example, if the simulation stops at closest approach and the first verdict to occur is that the missile reaches its minimum velocity, even if the missile is outside the lethal radius, the final verdict will be a miss intercept caused by minimum missile velocity. In most simulation cases, successful interceptions are


stopped at closest approach. Rarely a missile will reach t or and being inside before the activation of closest approach termination criterion.

max maxz

WHR

As mentioned in the last section, Rmin calculations depend strongly on termination criteria and for that, they make their final verdict highly unpredictable. One example of this is when the missile arrives at his closest approach but the target is outside the

and . In this special case, the simulation cannot be stopped neither by the fourth criterion (because R > ), nor by the first one (because t

WHR mintt <

WHR mint< ). Consequently, a few final verdicts may occur. The first one is called the «second chance» scenario: the missile makes a turn during t mint< , then starts to overtake the target and hits it, resulting in a successful intercept. This type of scenario tends to lower the IXL value (see Figure 13(c)) and should be avoided in order to stay as conservative as possible. However, the extensive study of this particular subject is out of the scope of this report. Fortunately, during its come back, the missile can also hit the ground, reach its minimum velocity, minimum closing velocity or its maximum flight time. This is what we would expect most of the time.

It makes no doubt that all final verdict parameters used to declare a successful or miss intercept have a direct influence on LE results. Thus, LE plots will be annotated with the relevant missile initial conditions and simulation parameters.


Table 4. Simulation control subsystem.

VERDICTS TERMINATION CRITERIA

MISS intercept SUCCESSFUL intercept (Simulation stopped)

• . WHMT RR > • Closest approach: V .0=C

• but minimum flight time:

.

WHMT RR ≤

mintt < 1

• Maximum guided flight time

.maxtt ≥ 3

• Minimum missile speed: V .min−< MM V 2 • Ground impact . minzzM =

• Minimum closing velocity: .min−< CMT VR& 2

• Ground impact . minzzM =

• Maximum guided flight time .maxtt ≥ 3

WHMT RR ≤

• but WHMT RR ≤

mintt < .4

1. is based on the warhead arming delay. Not useful for maximum range calculations. mint

2. The criteria V and V can be in effect simultaneously. V is not useful for minimum range

calculations. min−C min−M min−M

3. Turned off for maximum range calculations to test missile aerodynamic behaviour.

4. Termination criterion when calculating minimum launch envelope.

4.2 LAR comparison between the 3DOF-3 and 6DOF models for both low and high altitudes

Figures 13 and 14 illustrate 6DOF and 3DOF models maximum and minimum launch envelopes for both low and high altitudes, with gravity. Analysis of the 3DOF and 6DOF maximum range envelope at both altitudes leads to the same conclusion: the missile 6DOF maximum range LEs are most of the time more restrictive compared with the 3DOF ones. In fact, the only time where both missiles have exactly the same IXL value is when no lateral acceleration is required, i.e., when there is no gravity and the aspect angle ψ = 0 or 180 . °

The difference between the 6DOF and 3DOF LE can be quantified by the use of a unique measure. The IXL ratio is simply the mean of the relative errors between the 3DOF and 6DOF initial x-range limits, expressed in percentage:


N

IXLIXLIXL

ratioIXL

N

i DOF

DOFDOF∑=

− ×−

= 1 6

633 100 (37)

Here, N is simply the length of the vector aspect angle ψ . Unsurprisingly, the relative error between the 3DOF and 6DOF models grows as the missile has to manoeuvre to hit the target. In addition, the IXL ratio between the two models increases progressively with increasing altitude. For the present cases, the IXL ratio is only 3.3% in Figure 13 and 12.4% in Figure 14. Hence, for the type of scenario mentioned at the beginning of section 4 and for any intermediate altitude between 1 and 15 km, the possible IXL ratios encountered are expected to be between 3.3 and 12.4%.

The results for the 3DOF and 6DOF minimum range envelope calculations are difficult to represent since they are sensitive to many different termination criteria. However, for a wide-angle range, the same conclusion applied for both models, i.e. the 6DOF IXL value is usually equal or more restrictive than the 3DOF value (for Rmin calculations, a more restrictive LE means higher IXL values).

Globally, we can affirm that the 3DOF model accuracy is a function of two parameters. They are:

),(3 altitudeafDOF LATERALTOTaccuracy −= (38)

Since , we can therefore state that the 3DOF accuracy, in comparison with the 6DOF model, is a function of the Mach number and altitude.

),( altitudeMfa LATERALTOT =−


Figure 13. LAR comparison between 3DOF and 6DOF models, altitude 1km. (a) Maximum range

envelopes; (b) Minimum range envelopes; (c) 3DOF minimum range envelopes varying minimum flight time termination criterion.


Figure 14. LAR comparison between 3DOF and 6DOF models, altitude 15km. (a) Maximum range

envelope; (b) Minimum range envelope.


4.3 X-range limits dependency

For calculations requiring high manoeuvres (typically for Rmin calculations), the IXL is determined using the bisection method to search for the transition between a success and a miss. The method can converge to different IXL values dependent on the choice of the input minimum and maximum x values necessary to start the bisection method in the envelope calculation program.

One way to ensure ourselves that the proper IXL value is chosen is to plot the miss distance as a function of the IXL value in the x-range of interest and for a specific aspect angle ψ . This is illustrated in Figure 15. The R_WH line designates the warhead lethal radius limit for a successful or missed intercept. In this example, Miss/Success transitions occur at approximately 1350, 1475 and 2075m. Miss/Success transition occurring at 1350 and 1475 m are not the IXL because engagements at larger distances (between 1600m and 2075m) are unsuccessful. The true IXL value would be at 2075 m.

0

10

20

30

40

50

60

1200 1400 1600 1800 2000 2200 2400

Initial x-range limit [m]

Mis

s di

stan

ce [m

]

Miss distance

R_WH

Figure 15. Missile 6DOF miss distance function of the initial x-range limit, altitude 15km, aspect angle °= 150ψ .


4.4 Calculation-time comparison and conclusion

As expected, the gain in calculation-times for minimum or maximum range envelopes without initialization is similar to the ones encountered for only one simulation scenario. These gains are given in Table 5. It should be noted, however, that these times and the ones given in Table 3 are based on clock time and not CPU time. Thus, it is normal to see different gains, especially for long simulations like Rmax envelope calculations because CPU memory allocation can fluctuate during this period.

Table 5. Calculation-time of the 3DOF missile model in comparison with the 6DOF model, LAR simulation, altitude 1 km, without initialization.

MODEL MAXIMUM RANGE LAUNCH ENVELOPE

MINIMUM RANGE LAUNCH ENVELOPE

[min] [min]

6DOF 52 4.6

3DOF-3 30 2.4

GAIN 1.7 1.9

In conclusion, the calculation of maximum range envelopes and their corresponding IXL ratio gives a good evaluation of the final 3DOF model accuracy in comparison with the high fidelity 6DOF model. Average relative error for low and high altitude maximum launch envelopes of 3 and 12% respectively seem a good approximation of the 3DOF missile aerodynamic behaviour, which is what we wanted to evaluate. It is also in fair agreement with the qualitative analysis realised with the «lateral intercept» scenario. While Rmax launch envelopes give a better understanding of the missile aerodynamic behaviour, Rmin launch envelopes are too dependent on termination criteria and more specifically, on minimum flight time criterion, to consider evaluating any other behaviour than simulation control.


5. Conclusion

A 6DOF missile trajectory simulation model has been reduced to a 3DOF model in order to minimise its calculation time. However, this had to be done without compromising the realism of the simulation. The modeling environment employed is Matlab/Simulink.

Starting from a simple punctual mass in the 3-D space, three major improvements have brought the first 3DOF model (3DOF-0) to its final version (3DOF-3): the missile orientation ( )αβ , with the missile velocity vector, the trim condition and the autopilot and airframe dynamics (or the lateral acceleration time response). With the orientation of the 3DOF missile, the lateral forces play an essential role in the total drag force calculation and enable the 3DOF missile to reduce its speed. The trim condition leads to a saturation of the missile angles of attack in order to limit its maximum lateral acceleration. Finally, the lateral acceleration time response subsystem gives smoother direction changes to the 3DOF missile trajectory and also reduces substantially the missile Mach number. Each time the 3DOF model was upgraded, its accuracy was compared with a high fidelity 6DOF model, and new strategies were deployed in order to improve the prediction.

The final 3DOF model version is simply the insertion of all these improvements into the preliminary version of the 3DOF model. Its achieved level of accuracy and fidelity is satisfactory. Indeed, the average relative error between the 3DOF and 6DOF models for maximum range envelope calculations is of 3% and 12% at altitudes of 1 and 15 km, respectively. Therefore, we can state that for intermediate altitudes, the global error stays between 3 and 12%. Globally, the 3DOF model is approximately twice as fast compared with the 6DOF one.


References

1. Felio, D.A. and Duggan D.S. (1999). Autonomous Vehicle Guidance, Control and Simulation. 14-18 June 1999 short course notes. The University of Kansas. 532 p.

2. Missile Simulation Computer Program for AIM-7F, Volume III Analyst Manual (U), Joint Technical Coordination Group for Munitions Effectiveness, 61 JTCG/ME-75-15-3, March 1979, CONFIDENTIAL.

3. (1962) U.S. Standard Atmosphere, 1962. U.S. Government Printing Office, Washington 25, D.C.

4. Etkin, Bernard (1967). Dynamics of flight, Stability and Control. 6th edition. John Wiley & Sons, Inc., New-York. 519 p.

5. Lestage, R., Lauzon, M., and Jeffrey, A. (2002). Automatic Tuning of Gain-Scheduled Autopilot for Computer Simulations. TR 2001-230. Defence R&D Canada—Valcartier, Canada. 47 p.

6. Gills, J.-C., Decaulme, P. and Pélegrin, M. Dynamique de la commande linéaire (1991). 8th edition. Dunod, Paris. Chapitre 14.

7. Gibeau, D. (1993). Missile design PC TRAP: An improved PC TRAP for tactical missile design. Engineer’s Thesis, Naval Postgraduate School, Monterey, CA. 323 p.


List of symbols and abbreviations

6DOF Six-degrees-of-freedom

3DOF Three-degrees-of-freedom

pitchyaw a,a Yaw and pitch lateral accelerations

BF Body-Fixed

CFD Computational Fluid Dynamics

CG Centre of gravity

CMD Commanded

aeroC Aerodynamic coefficient

XC Non-dimensional linearized axial force coefficient

NC Non-dimensional linearized normal force coefficient

mC Non-dimensional linearized pitch moment coefficient

mqC Non-dimensional linearized pitch damping moment coefficient

ZY CC , Non-dimensional yaw and pitch force coefficient

D Drag force

d Aerodynamic chord length

DND Department of National Defence

DRDC Defence R & D Canada

EF Earth-Fixed

EOM Equation of motion

eff Effective (as opposed to commanded)

ZYX FFF ,, Axial, yaw and pitch forces (body-fixed)


ThrustF Thrust force

h Altitude

IXL Initial x-range limit at a specific angle ψ

Iyy Missile polar moment of inertia against y

[I] Inertia matrix

K Static gain

LAR Launch Acceptability Region

m Missile mass

M Missile Mach number

Mq Slope of pitch moment per pitch angle rate

Mα Slope of pitch moment per angle-of-attack

PI Proportional-Integral

p Roll rate

∞q Dynamic pressure

q Pitch rate

r Yaw rate

MTR Missile-target range

MTR& Closing velocity

Rmin Minimum range

Rmax Maximum range

RDDC R & D pour la défense Canada

WHR Missile warhead lethal radius

refS Aerodynamic reference area


s Laplace differential operator

t Time

u Missile velocity component in x, body referenced

v Missile velocity component in y, body referenced

CV Closing velocity

VM Magnitude of the missile velocity

w Missile velocity component in z, body referenced

Zα Slope of normal force per angle-of-attach

∞ρ Free-stream air density

αβ , Missile angles of attack in yaw and pitch planes, respectively

pitchyaw ,δδ Commanded fin deflection in yaw and pitch planes, respectively

rollpitchyaw ,, γγγ

Orientation angles in yaw, pitch and roll from the EF to the BF coordinate system, the BF coordinate system being oriented toward missile speed

ΦΘΨ ,, Missile Eulerian angles in yaw, pitch and roll planes, respectively

Mφ Missile roll angle

∆CG Position of centre of gravity with respect to aerodynamic coefficient reference point

TOTα Total angle of attack

Nω Autopilot and airframe natural frequency

ζ Closed-loop lateral damping factor

ω& Angular acceleration

ψ Azimuth angle for LAR calculation


Distribution list

INTERNAL

DRDC Valcartier

1- Director General

1- Deputy Director General

1- H/Precision Weapons

3- Document Library

1- Dr. R. Lestage (author)

1- Mr. M. Lauzon

1- Mr. A. Jeffrey

1- Ms. N. Harrison

1- Mr. B. Gilbert

EXTERNAL

1- Director Research and Development Knowledge & Information Management (unbound copy)

1- Directorate of Aerospace Equipment Program Management – Fighters & Trainers 3-4

1- Directorate of Technical Airworthiness

1- Directorate of Strategic Intelligence

1- Directorate of Aerospace Equipment Program Management – Radar, Communications & Systems 6-4-2 (attn: Capt Marc Comeau)

1- Director Science and Technology (Air)

1- Directorate of Aerospace System Operational Research

1- Raphäel Brochu (author)

Numerica Technlogies Inc. 3420, Lacoste Québec, Qc, G2E 4P8


dcd03e rev.(10-1999)

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3. TITLE (Its classification should be indicated by the appropriate abbreviation (S, C, R or U) (U) Three-degree-of-freedom (DOF) missile trajectory simulation model and comparative study with a high fidelity 6DOF model

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dcd03e rev.(10-1999)

UNCLASSIFIED SECURITY CLASSIFICATION OF FORM

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13. ABSTRACT (a brief and factual summary of the document. It may also appear elsewhere in the body of the document itself. It is highly desirable that the abstract of classified documents be unclassified. Each paragraph of the abstract shall begin with an indication of the security classification of the information in the paragraph (unless the document itself is unclassified) represented as (S), (C), (R), or (U). It is not necessary to include here abstracts in both official languages unless the text is bilingual).

(U) A 6 degrees-of-freedom (DOF) missile trajectory simulation model has been reduced to a 3DOF model in order to minimise its calculation time. However, this had to be done without compromising the realism of the simulation. The modeling environment is Matlab/Simulin. Starting from a simple punctual mass in the 3-D space (Referenced as model 3DOF-0), three major improvements have brought this model to its final version (3DOF-3): namely, the missile orientation with missile speed, the trim condition and the autopilot and airframe dynamics (or the lateral acceleration time response). Each time the 3DOF model was upgraded, its accuracy was compared with a high fidelity 6DOF model, and new strategies were deployed to improve the prediction. The final 3DOF model version is simply the insertion of all these improvements into the preliminary version of the 3DOF model. Its level of accuracy and fidelity achieved is satisfactory. Indeed, the relative error between the 3DOF and 6DOF models for maximum range envelope calculations is 3% and 12% at altitudes of 1 and 15 km, respectively. Globally, the 3DOF model is approximately 2 times faster compared with the 6DOF one. This can lead to interesting future developments.

14. KEYWORDS, DESCRIPTORS or IDENTIFIERS (technically meaningful terms or short phrases that characterize a document and could be helpful in cataloguing the document. They should be selected so that no security classification is required. Identifiers, such as equipment model designation, trade name, military project code name, geographic location may also be included. If possible keywords should be selected from a published thesaurus, e.g. Thesaurus of Engineering and Scientific Terms (TEST) and that thesaurus-identified. If it is not possible to select indexing terms which are Unclassified, the classification of each should be indicated as with the title.)

Missile model Missile simulation six-degrees-of-freedom simulation maximum range envelope missile engagement autopilot airframe trim

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