multi-missiles guidance and allocation571732/fulltext01.pdf · this thesis deals with the...

Multi-missiles Guidance and Allocation

JEAN HASSE

Masters’ Degree ProjectStockholm, Sweden Oct 2009

XR-EE-RT 2009:021


Jean Hasse

Master of Science Thesis

Royal Institute of Technology, Sweden Department of Electrical Engineering

Grenoble INP, France Ecole Nationale Supérieur d’Ingénieur

Electriciens de Grenoble

Tutor (MBDA France)

Stephane Le Menec

Advance concept department, MBDA France

Examiner (ENSIEG)

Gildas Besançon Département Automatique, GIPSA-lab

Examiner (KTH)

Elling W. Jacobsen

Department of Electrical Engineering, Kungliga Tekniska Högskola

I

Acknowledgement

This thesis has been going on for 25 weeks during spring 2008. It takes place at MBDA France in the Advance Concept department in Le Plessis near Paris. It is the conclusive project in the double degree at the Department of Electrical Engineering at Kungliga Tekniska Högskolan (Stockholm, Sweden) and at the Ecole Nationale Supérieure d’Ingénieurs Electriciens (Grenoble, France). I would like to express my deepest thanks to the following people for their support during this thesis: My supervisor, Stephane Le Menec, for his precious help and support during this project and for commenting my report My examiners, at the Royal Institute of Technology and at the ENSIEG for answering my questions and commenting my report The chief of the Advance Concept department for accepting me in his department during this thesis All the other people of the department, I will not quote all of them in order to avoid forgetting someone, for their sympathy that makes my master thesis an interesting and rewarding time.


II

Abstract The purpose of the project is the development and study of guidance algorithms of missiles in the particular configuration of the simultaneous use of two pursuers to intercept one evader. The profits of this configuration are applied to allocation in a three missiles – two targets scenario. The implemented techniques deal with optimal control theory, more particularly with differential game theory, mainly the linear and linear quadratics, which are commonly used in guidance law. Differential game theory allows determining the optimal commands of the players of one pursuer - one evader game with the minimization of the miss distance for the pursuer and maximization of the same miss distance for the evader as criteria of this optimal problem. A new optimal command law for the evader is developed in the two pursuers against one evader game (called 2x1 games) with the same time-to-go as well as with different time-to-go. These new optimal commands are implemented and validates in two 2D simulations: a linear one, developed in Matlab script, and a non linear one, based on a Simulink model. An extension of the capture zones of each missile is demonstrated and tested in simulations. The 2x1 game and the induced no-escape-zone extension are finally integrated and verified in realistic simulation with a 6 d.o.f missile Simulink model. The 2x1 configuration’s use is applied to allocation on 3 pursuers - 2 evaders scenarios based on the same 3D missile model and the benefits are analysed. Le but de ce rapport et le développement et l'étude d'un algorithme de guidage missile dans le cas particulier de l'utilisation simultanée de deux missiles pour intercepter une cible. L'avantage de cette configuration est utilisé pour l'allocation des missiles dans un scenario de trois missiles contre deux cibles. Les techniques mises en œuvre sont issues de la théorie du control optimal et plus particulièrement de la théorie des jeux différentiels linéaire et quadratique linéaire, qui sont couramment utilisés dans les lois de guidage. La théorie des jeux différentiels permet de déterminer les commandes optimales des joueurs d'un jeu un missile-une cible avec comme critère optimal la minimisation de la distance de passage finale pour le missile et la maximisation de cette même distance pour la cible. Une nouvelle loi de commande optimale pour la cible est établie pour le jeu de deux missiles – une cible (appelé jeu 2x1) avec aussi bien des time-to-go identiques que des time-to-go différents. Ces lois de commande optimales sont implémentées et validées dans deux simulations en 2D: une linéaire, développé en script Matlab, et une non-linéaire, développé sous Simulink. Une extension des zones de captures (no-escape-zone) de chacun des missiles est démontrée et testée avec les simulations. Le jeu 2x1 et l'extension des no-escape-zone qui en découle est finalement intégré et vérifié dans une simulation 3D sous Simulink plus réaliste, basé sur un modèle de missile à six degrés de libertés. L'utilisation d'une configuration 2x1 pour de l'allocation est appliqué à un scenario de trois missiles contre deux cibles basé sur les même modèle de missile à six degrés de libertés.


III

Table of content

1. INTRODUCTION ..................................................................................................... 1

2. THE 1X1 GAME: ONE PURSUER, ONE EVADER............................................ 2

2.1. PROBLEM STATEMENT ........................................................................................... 2 2.2. GUIDANCE LAWS MODELS...................................................................................... 3

2.2.1. The Linear Differential Quadratic Game guidance law (LQDG)..................... 3 2.2.2. The Linear Differential Game guidance laws (DGL)........................................ 3

2.3. OPTIMAL SOLUTION OF THE 1X1 GAME .................................................................. 6 2.4. MAXIMUM LAUNCH TIME....................................................................................... 9

3. THE 2X1 GAME: TWO PURSUERS, ONE EVADER....................................... 11

3.1. PROBLEM STATEMENT ......................................................................................... 11 3.2. THE OPTIMAL EVADER COMMAND IN 2X1 GAME .................................................. 13

3.2.1. Validation in linear simulation........................................................................ 14 3.2.2. Validation in nonlinear simulation.................................................................. 17

3.3. EXTENSION OF THE NO-ESCAPE-ZONES ................................................................ 19 3.3.1. Validation in linear simulation........................................................................ 22 3.3.2. Validation in nonlinear simulation.................................................................. 23 3.3.3. Three axes representation 21,, PPZEMgo of the 2x1 game....................... 25

3.4. THE OPTIMAL 2X1 COMMAND WITH DIFFERENT TIME-TO-GO ............................... 28 3.4.1. New expression of the command ..................................................................... 29 3.4.2. Validation in linear simulation........................................................................ 30 3.4.3. Extension of the no-escape-zones .................................................................... 32 3.4.4. Three axes representation 21,, PPZEMgo ................................................. 33

4. THE 2X1 GAME IN THREE DIMENSION ENVIRONMENT ........................ 36

4.1. PROBLEM STATEMENT ......................................................................................... 36 4.1.1. Geometry ......................................................................................................... 36 4.1.2. Simulation’s model .......................................................................................... 36

4.2. SIMULATION IN THE SIMULINK MODEL ................................................................ 38

5. APPLICATION IN THREE PURSUERS - TWO EVADERS SCENARIO ..... 43

5.1. PROBLEM STATEMENT ......................................................................................... 43 5.1.1. Geometry ......................................................................................................... 43 5.1.2. Simulation’s model .......................................................................................... 43

5.2. ALLOCATION ....................................................................................................... 44 5.3. SIMULATION SCENARIOS ...................................................................................... 46

5.3.1. Scenario 1........................................................................................................ 46 5.3.2. Scenario 2........................................................................................................ 48

6. CONCLUSION ........................................................................................................ 52


IV

Table of figures

Figure 1 : Game geometry. .................................................................................................... 2 Figure 2 : Game space regions and three different game solutions....................................... 7 Figure 3 : Different singular region shape in DGL/1. ........................................................... 8 Figure 4 : Illustration of maximum launch time.................................................................... 9 Figure 5: Trajectories from instant tir =7. .......................................................................... 10

Figure 6 : 2x1 Game geometry. ........................................................................................... 11 Figure 7 : No escape zone in the 2x1 game......................................................................... 11 Figure 8 : Non-optimality of *

1/11 Pxv in 2x1 game. .............................................................. 12

Figure 9: Trajectories when E plays *1/11 Pxv ......................................................................... 12

Figure 10 : Calculation of *12xv ............................................................................................ 13

Figure 11 : Comparison of strategies: *1/11 Pxv , *

2/11 Pxv and *12xv . .......................................... 15

Figure 12: Trajectories when E plays *1/11 Pxv ....................................................................... 16

Figure 13 : Trajectories when E plays *2/11 Pxv ..................................................................... 16

Figure 14: Trajectories when E plays *12xv .......................................................................... 16

Figure 15: *12xv in configuration 1 ....................................................................................... 17

Figure 16 : *12xv in configuration 2 ...................................................................................... 17

Figure 17: *12xv in configuration 1. ...................................................................................... 18

Figure 18 : *12xv in configuration 2. ..................................................................................... 18

Figure 19: Evader’s escape without *12xv saturation. ........................................................... 19

Figure 20: Evader’s escape with *12xv saturation. ............................................................... 19

Figure 21: Evader’s capture................................................................................................. 19 Figure 22: Trajectories in XY space of figures 19-21. ........................................................ 20 Figure 23 : No-escape-zone extension in 2x1 game in configuration 1 and 2. ................... 21 Figure 24: 2x1 no-escape-zone in configuration 1 (Table 1). ............................................. 22 Figure 25: 2x1 no-escape-zone in configuration 2 (Table 2). ............................................. 23 Figure 26: 2x1 no-escape-zone in configuration 1 (Table 1). ............................................. 24 Figure 27: 2x1 no-escape-zone in configuration 2 (Table 2). ............................................. 24 Figure 28: 2x1 game plot in ZEMgo , . ............................................................................ 25

Figure 29: 2x1 game plot in 21,, PPZEMgo . ................................................................. 26

Figure 30: 2x1 game plot in 21,, PPZEMgo with X-Y view ......................................... 26

Figure 31: Section in ZEMgo , of Figure 29 at initial and final time ............................... 27

Figure 32: Trajectory evolution with initial condition on the 2x1 limit. ............................. 27 Figure 33: Section ZEMgo , of Figure 32 at six different fixed times. .......................... 28

Figure 34: Calculation of *12xv with different time-to-go ( 12 ) ...................................... 29

Figure 35: 6000,9, 00 y in configuration 1................................................................. 31


V






Figure 41: 2x1 no-escape-zone in configuration 1 with =1........................................... 32 Figure 42: 2x1 no-escape-zone in configuration 2 with =1........................................... 33 Figure 43: 2x1 game in configuration 2 plot in ZEMgo , ................................................. 34

Figure 44: Trajectory evolution in 21,, PPZEMgo with initial condition on the 2x1 limit.

............................................................................................................................................. 34 Figure 45: Section ZEMgo , of Figure 44 at six different fixed times. ........................... 35

Figure 46: Problem’s geometry - (XY) plan. ...................................................................... 36 Figure 47: Problem’s geometry - (XZ) plan........................................................................ 36 Figure 48: No-escape-zones in (XY) plan........................................................................... 37 Figure 49: Missiles trajectories in (XYZ) with capture....................................................... 38 Figure 50: Missiles trajectories in (XY) with capture. ........................................................ 39 Figure 51: Missiles trajectories in (XZ) with capture.......................................................... 39 Figure 52: Missiles accelerations. ....................................................................................... 39 Figure 53: ZEM of the game in (XY).................................................................................. 39 Figure 54: Missiles trajectories in (XYZ) without *

12xv saturation....................................... 40

Figure 55: Missiles trajectories in (XY) without *12xv saturation. ........................................ 40

Figure 56: Missiles trajectories in (XZ) without *12xv saturation.......................................... 40

Figure 57: Missiles accelerations. ....................................................................................... 41 Figure 58: ZEM of the game in (XY).................................................................................. 41 Figure 59: Missiles trajectories in (XY) with *

12xv saturation. ............................................. 42

Figure 60: Missiles accelerations. ....................................................................................... 42 Figure 61: Problem’s geometry - (XY) plan. ...................................................................... 43 Figure 62: Problem’s geometry - (XZ) plan........................................................................ 43 Figure 63: Determination of 11t for A1 and A2................................................................... 45 Figure 64: Trajectories estimation for max time calculation............................................... 46 Figure 65: Allocation table in scenario 1............................................................................. 47 Figure 66: Trajectories for scenario 1.................................................................................. 47 Figure 67: Trajectories in (XY) for scenario 1. ................................................................... 48 Figure 68: ZEM in (XY)...................................................................................................... 48 Figure 69: Trajectories estimation for max time calculation............................................... 49 Figure 70: Allocation table in scenario 2............................................................................. 50 Figure 71: Trajectories for scenario 2.................................................................................. 50 Figure 72: Trajectories in (XY) for scenario 2. ................................................................... 51 Figure 73: ZEM in (XY)...................................................................................................... 51

Introduction

Page 1

1. Introduction

This thesis deals with the utilization of two anti-missile missiles against a hostile one in the scope of a ground defence. The goal of this report is to determine and investigate the interest of the use of such configuration. This report presents, in a first time, the state of art of the different guidance laws descended from the differential game theory. The couple pursuer - evader, modelised with state vectors of dimension 2, 3 or 4, is linearized around the collision trajectory. These laws are sorted in two groups: the LQDG laws which are based on linear differential quadratics games with unbounded commands and the DGL laws which rely on linear differential games with bounded commands. For DGL laws, the optimisation criterion is the final miss distance. The DGL/1 law takes into account the saturation of the missile’s command which is very important against very manoeuvring evaders and particularly adapted for anti-missiles use. The report focuses on the DGL laws and particularly on DGL/1 which one are used later one for missiles in the simulations. As a matter of fact, this law allows guaranteeing a minimal miss distance or even, under conditions, a zero miss distance and gives maximal capture zones (no-escape-zone). The solution of one pursuer - one evader game (1x1 game) playing with DGL/1 law is presented as well as the corresponding no-escape-zone definition. The notion of maximum launch time is introduced. This concept is useful for missile allocation in multi-missiles and multi-target scenarios and is used on the last part of the report in a 3 missiles against 2 targets scenarios. In a second part, the report focuses on two pursuers - one evader game. A new optimal command law for the evader is introduced and checked in linear Matlab simulation and then in a nonlinear simulation. From the simulation results, the influence of such game on the missiles capture zones will be investigated. The following part takes an interest in the integration of this new optimal command law and in confirmation of the previous results through a 3D simulation based on a six degree of freedom missile model. Finally, a three pursuers - two evaders game is considered and the missiles allocation principle introduced. The results found on the 2x1 games are used to build the allocation matrix. The designed allocation method is tested on two different scenarios based on the same six degree of freedom missile model as before. The results are analysed to investigate the improvement due to the use of the 2x1 game.

The 1x1 game: one pursuer, one evader

Page 2

2. The 1x1 game: one pursuer, one evader

2.1. Problem statement

The game analysis is based on the following set of simplifying assumptions:

The engagement between the interceptor (pursuer, P) and the missile (evader, E) takes place in a plane. Both missiles have constant speeds, respectively PV and EV , and bounded lateral

accelerations, EPiaa ii ,,max .

The trajectories of both missiles can be linearized along the initial line of sight which is the line passing through P and E.

Figure 1 : Game geometry.

In Figure 1, a schematic view of the game geometry is shown. Note that the respective velocity vectors of the missiles are generally not aligned with the reference line of sight. The aspect angles E and P are, however, small. Thus, the approximations 1cos

and PE,i 0,s in , are uniformly valid and coherent with assumption 3. Moreover, based on assumptions 2 and 3, the final time of interception can be derived:

EPf VV

rt

Allowing to define the time to go by ttt fgo

The variable that we are interested in is the relative position, normal to the reference line, between the players:


Page 3

PE yyy Which have to be minimized by the pursuer and maximized by the evader. The equations of motion depend on the choice of the dynamic model for the players. It will be developed in 2.2.2. Different guidance laws models can be used to determine the optimal missile strategy according to the control model:

If the controls are unbounded, we use linear differential quadratic game laws If the controls are bounded then we use linear differential game laws

2.2. Guidance laws models

2.2.1. The Linear Differential Quadratic Game guidance law (LQDG)

The LQDG laws are continuous and the cost function to minimize is the energy and the miss distance. In this way, the final miss distance is generally not null because the optimal solution gives a compromise between the final miss distance and the energy consumption. Moreover, as the optimal controls given by the LQDG law are unbounded, we can easily reach the maximum real controls capacity of the missile. Due to this limitation, DGL laws will be used instead in this paper. Because LQDG laws are out of scope of this paper, no more details will be introduced but further information can be found in references1.

2.2.2. The Linear Differential Game guidance laws (DGL)

There are three DGL law models according to the dynamic considered for the players:

DGL/00 - Evader and Pursuer are ideals (order 0) DGL/0 - Evader is ideal and Pursuer is modelised as by first order DGL/1 – Both players are modelised by a first order

2.2.2.1. DGL/00

This is the simplest dynamic model where the players are considered as ideal i.e. as a zero order dynamic.

EcPc aay

Where Eca and Pca are the commended lateral accelerations of E and P, respectively:

tVaatUaa EEcPPc maxmax


Page 4

In this case the state vector is of dimension 2,

TyytX

And the corresponding state space form, obtained from the set of equations, is:

tVa

tUa

tXtXEP

maxmax

00

00

10

The corresponding unbounded linear quadratic game formulation, with an ideal dynamic model, leads to proportional navigation2, one of the most used nowadays, as an optimal guidance law. However, a more realistic approach should take into account that the controls are bounded and thus, that interceptor missile dynamics should be represented at least by a first-order transfer function3.

2.2.2.2. DGL/0

In this model, the evader remains ideal but the pursuer dynamic is modelised by a first order transfer function with time constant P

P

PcP

PP

EcP

ayy

ayy

1

The state vector is thus extend of one dimension,

TPyyytX


tVatU

a

tXtX E

PPP

0

0

0

0

100

100

010

max

max

This differential game guidance law takes into account the limited interceptor manoeuvrability, and the assumption of ideal target manoeuvre dynamics (the worst case for the interceptor) eliminated the need of knowing the actual target manoeuvre. DGL/0 provides robustness with respect to the type of target manoeuvre but cannot guarantee a zero miss distance.


Page 5

If a perfect information on players is available (particularly on the target), a first order (non-ideal) manoeuvre dynamics can be assumed for both players and then an improved guidance law can be used4: DGL/1.

2.2.2.3. DGL/1

In this model, both players dynamic are modelised by a first order transfer function with time constant P and E

E

EcE

EE

P

PcP

PP

EP

ayy

ayy

yyy

1

1

The state vector is thus extend of one more dimension,

TEP yyyytX


tV

a

tUa

tXtX

EE

PP

E

P

max

max 0

0

0

0

0

0

1000

0100

1100

0010

The problem involves two non-dimensional parameters of physical significance. One is the pursuer-evader maximum manoeuvrability ratio, μ, and the other is the ratio of the evader-pursuer time constants, ε.

P

E

E

P

a

a

max

max

Then, we can define the natural cost function of the game as the miss distance which is the relative position between the players at the final time:

ff tXtyJ 1

The state space can be reduced to a scalar one by using the transformation

tXtttZEM f .,

where tt f , is the transition matrix of the original homogenous system:


Page 6

hhttt EPgof ..1, 22

with 1 eh The new state variable tZEM is the zero-effort miss distance, a widely used term in guidance analysis5. It is the miss distance that results if both players do not apply any further acceleration commands. As the calculation of tZEM is based on the solution

of the state space, tZEM is a function of the current state tX . For analysis, non-dimensional variables are defined. The independent variable is the normalized time to go

P

f tt

The non-dimensional state variable is the normalized zero-effort miss distance

max

2EP a

tZEMZ

When the non-dimensional variables are used, equation Z , the normalized game dynamics can be derivated and becomes

vhuhZ

....

The non-dimensional cost function is the normalized miss distance,

0 ZZJ f

to be minimized by the pursuer and maximized by the evader. For the rest of this paper only DGL/1 will be considered (except contrary mention). First-order dynamics for both players will be used in the linear simulation models.

2.3. Optimal solution of the 1x1 game

The game solution provides simultaneously the optimal pursuer strategy (the missile’s guidance law), the optimal evader strategy (the worst target manoeuvre), and the value of the game defined by the cost criteria (the miss distance guaranteed to the pursuer as well as to the manoeuvring target, by using the respective optimal strategies)6. The complete resolution of the DGL/1 game will not be given here but can be found in the references4. The solution of this game provide the optimal control strategies of the players as

Zsignvu **

Note that the resulting controls are of bang-bang type (due to the sign function) and cause high energy consumption.


Page 7

Substituting the optimal commands into the Z

expression yields the dynamics along

candidate optimal trajectories,

Zsignhh

Z...

*

When this expression is integrated, a family of regular optimal trajectories is generated (dashed lines in Figure 2). The game solution, and its trajectory, is a unique function of the integration’s initial condition. The family of trajectories divides the game space ,Z into two regions, a regular one and a singular one. Within the regular region, R (filled with the dashed lines), the optimal strategies are of bang-bang type and are of the form:

...00

**

hhZsignZZ

The borders between R and S (solid line) are the pair of optimal trajectories, denoted

** , ZZ , that lead to a zero miss distance i.e. where 0)0(* Z .

**

22* .

2.

2.

ZZ

hhZ

In the other (singular) region, S, enclose by ** , ZZ , the optimal strategies cannot be uniquely determined and every trajectories in this region can be considered as optimal.

0 2 4 6 8 10 12 14 16 18 20-400

-300

-200

-100

0

100

200

300

400

z

Figure 2 : Game space regions and three different game solutions.


Page 8

In concrete terms, all trajectories starting in S (for example, from the circle in S, Figure 2) must go through the throat in Z(0)=0 whatever if the evader command is optimal (solid line) or not (dashed line). The evader is always captured whatever is its evasive manoeuvre, i.e. a zero miss distance is guarantee, and the region S is commonly called No-escape-zone (NEZ). In other hand, trajectories starting in R (for example, from the circle in R, Figure 2) do not guarantee a zero miss distance but a maximum miss distance (60 m here) that correspond to the worst case for the pursuer, when evader applies its optimal command (and can be less if evader don’t play optimally). One last important aspect of the region shape is the influence due to parameters μ and ε on *Z . Two cases can be distinguished based on the value of μ.ε, which can be interpreted as the pursuer-evader agility ratio:

If and , the singular region is open.

If and , the singular region is close.

Figure 3 shows these different shapes of the NEZ according to the value of and .

0 20 40 60 80-200

-100

0

100

200

(s)

z

0 5 10 15 20-1000

-500

0

500

1000

go (s)

ZE

M (

m)

= 0.8 and = 9

0 5 10-200

-100

0

100

200

(s)

z

0 0.5 1 1.5 2-200

-100

0

100

200

go (s)

ZE

M (

m)

= 3 and = 2

Figure 3 : Different singular region shape in DGL/1.

Note that the boundaries shape of the no-escape-zone depends on the dynamic model chosen for the players and thus are different for each law: DGL/00, DGL/0 and DGL/1.


Page 9

2.4. Maximum launch time

An important data in anti-missile defence is the last launch time ( max ) that guarantees a

zero miss distance. In other words, this is the time-to-go where the missile trajectory goes out of the anti-missile no-escape-zone in the (ZEM,tgo) space. The principle of max determination is to suppose the evader command in the future

( hypV ) and to deduce its trajectory in order to compute the corresponding max . Figure 4

illustrate this. Pursuer is waiting (u=0) and the evader is assume to play a constant command equal to hypV ( 1hypV here) that gives the hypothetic trajectory (blue dashed

line), the no-escape-zone (red lines) and allow to compute max (about 7.9s). The green

line represent a trajectory simulation where evader effectively plays hypV (and pursuer

u=0) until switching to *11xv when pursuer is launched at 7 (and now plays is optimal

command too: *11xu ). The right figure gives, for a given launch time tir , the guarantee

miss distance in worst case. In our example, for tir =7, the predicted miss distance is about 1100m that can be

verified with the real trajectory in green in the left figure. In this example, the pursuer wait too long and thus it is sure to miss the evader unless the evader doesn’t play optimally. The hazardous part of such computation is the trajectory prediction of the evader. In practise the evader command (lateral acceleration) is considered as constant but can also be based from the few first observation or from predictive value of a Kalman filter.

0 2 4 6 8 10-6000

-4000

-2000

0

2000

4000

6000

go [s]

ZE

M [

m]

Miss distance [m] = 1182.66

NEZ

-NEZZEM

hyp

ZEM

0 2 4 6 8 100

1000

2000

3000

4000

5000

6000

go [s]

Mis

s di

stan

ce [

m]

max [s] = 7.8691

tir

Figure 4 : Illustration of maximum launch time.


Page 10

The predicted miss distance can also be checked in Figure 5 which show the pursuer and evader trajectories in the XY space.

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

x 104

-2000

-1000

0

1000

2000

3000

4000

5000

Distance X [m]

Dis

tanc

e Y

[m

]

P

E

Figure 5: Trajectories from instant tir =7.

This maximum launch time is an important data. Indeed, the allocation strategy is very important to maximize the number of target intercepted and can be useful for target allocation in case of multi-missile and multi-target configuration. This data will be use in the allocation strategy of the three missiles against two evaders’ scenario we will focus on in part 5 of this report.

The 2x1 game: Two pursuers, one evader

Page 11

3. The 2x1 game: Two pursuers, one evader


In this new problem configuration we assume two pursuers, called P1 and P2, with P1 above P2 and one evader (E) initially located between them (Figure 6). All parameters notation used in the 1x1 game (Figure 1) hold for both couple P1-E and P2-E.

Figure 6 : 2x1 Game geometry.

Note that the case where E is not between the two pursuer then E can only face one pursuer that amounts to a 1x1 game, respectively P1-E if E is over P1 and P2-E if E is under P2 (Figure 7).

0 1 2 3 4 5 6 7 8 9 10-1

-0.5

0

0.5

1

1.5

2x 10

4

go (s)

Dis

tanc

e (m

)

Figure 7 : No escape zone in the 2x1 game

The 2x1 game only makes a change in the evader optimal command. Indeed, as the presence of pursuer 1 doesn’t change the behaviour of the pursuer 2 and vice versa, the optimal command for each pursuer in the 2x1 game are the same as the ones in their respective 1x1 game. As a matter of fact, in a point of view of each pursuer, they play each like in a 1x1 game to determine their optimal command, respectively P1-E and P2-E, i.e. the command (called *

1u and *2u ) defines in DGL/1 (2.2.2.3).


Page 12

However, the evader gets two commands (denoted *1/11 Pxv and *

2/11 Pxv ) from each 1x1

game with P1 and P2 it is involved in. They are no longer optimal against two pursuers because of their non-unicity and need to be redefined. Indeed, if the evader plays 1v , it risks to enter in the P2’s no-escape-zone (but not necessary see Figure 11) and then risks to be captured by P2 in trying to avoid P1 (Figure 8) that point out the non-optimality of this command on the 2x1 game.

0 1 2 3 4 5 6 7-8000

-6000

-4000

-2000

0

2000

4000

ZEM P1-EMiss distance P1-E [m] = -3948.48

ZE

M

go (s)

0 1 2 3 4 5 6 7-4000

-2000

0

2000

4000

6000


ZE

M

go (s)

Figure 8 : Non-optimality of *

1/11 Pxv in 2x1 game.

0 0.5 1 1.5 2 2.5 3 3.5

x 104

-6000

-4000

-2000

0

2000

4000

6000

Distance X [m]

Dis

tanc

e Y

[m

]

P1

P2E

Figure 9: Trajectories when E plays *

1/11 Pxv .


Page 13

3.2. The optimal evader command in 2x1 game

The optimal command for the evader is the command that satisfy the same criteria as in 1x1 game, namely maximization of the final distance, for both pursuers. In other words, the optimal evader command, called *

12xv , should lead to the equality of the final

distances to satisfy the criteria:

ff tZtZ 21

Note that the minus sign is due to the y-axes orientation like in Figure 6. In normalized variable, we get:

222

211 PfPf zz

Looking at Figure 10, the normalized zero-effort-miss at final time can be written:

01max111 zzzz f

02max222 zzzz f

In order to be clearer, the 2x1 game can be plotted on a single figure like in Figure 10. It’s done by superimposition of the two plots of each couple pursuer-evader (as in Figure 8) with the game initial condition as point of coincidence. However some disadvantages remain and will be discussed in (3.3.3).

0 1 2 3 4 5 6 7 8 9 100

2000

4000

6000

8000

10000

12000

go (s)

Dis

tanc

e (m

)

v*2x1

v*1x1

= -1

z

-z1max

Figure 10 : Calculation of *

12xv


Page 14

We know that the no-escape-zone expressions are:

211111max1 HHz

222122max2 HHz

Where:

2,1,,1,1 iHH ii

hdhH 2

2

0

1

and 2,1,,2,2 iHH ii

hdhH2

2

0

2

The third terms that correspond to the changes in ifz due to *

12xv are:

1*

12211*

11*

122111 xxxf vHvvHtz

1*12222

*11

*122222 xxxf vHvvHtz

Substituting these expression leads to:

1*1221121111111 xf vHHHzz

1*1222222212222 xf vHHHzz

From which, using the initial equality, we can isolated *

12xv :

21211

22222

222122

211111*

12

PP

PP

HH

zHzHzv

3.2.1. Validation in linear simulation

In order to validate the evader command *

12xv , it was added to a linear Matlab model

based on DGL/1. The linear model used the state space defined in (2.2.2.3) to simulate the game. Two set of configuration parameters for players was used, in order to see the influence of different μ and so on the shape of each no-escape-zone, and was reported in Table 1 and Table 2 below.


Page 15

Table 1: First players’ configuration parameters

Table 2: Second players’ configuration parameters

The improvement of the evader command thanks to *

12xv compare to the 1x1 optimal

command can be seen in Figure 11 where *12xv is compared to *

1/11 Pxv and *2/11 Pxv . The

game was set up with configuration 2 (Table 2).

Figure 11 : Comparison of strategies: *1/11 Pxv , *

2/11 Pxv and *12xv .

Misssile 1 Missile 2 Evader

1PV 2300 m/s² 2PV 2300 m/s² EV 2700 m/s²

max1Pa 20 g max2Pa 20 g maxEa 10 g

1P 0.3 s 1P 0.7 s E 0.6 s

1 2 2 0.85714

1 2 2 2

Misssile 1 Missile 2 Evader

1PV 2300 m/s² 2PV 2300 m/s² EV 2700 m/s²


1P 0.3 s 1P 0.7 s E 0.6 s

1 2 2 0.85714

1 2 2 3

0 1 2 3 4 5-8000

-6000

-4000

-2000

0

2000


ZE

M

go (s)

0 1 2 3 4 5-2000

-1000

0

1000

2000

3000

4000

5000

ZEM P2-EMiss distance P2-E [m] = 1249.75

ZE

M

go (s)

0 1 2 3 4 5-8000

-6000

-4000

-2000

0

2000


ZE

M

go (s)

0 1 2 3 4 5-2000

-1000

0

1000

2000

3000

4000

5000


ZE

M

go (s)

0 1 2 3 4 5-8000

-6000

-4000

-2000

0

2000


ZE

M

go (s)

0 1 2 3 4 5-2000

-1000

0

1000

2000

3000

4000

5000


ZE

M

go (s)


Page 16

The right figure in Figure 11 confirmed that *12xv is the optimal evader command for a

2x1 game. In the same time, the equality of the miss distances demonstrated the accuracy of the *

12xv expression founded in (3.2).

The corresponding players’ trajectories in the XY space, for each case, were illustrated in Figure 12, Figure 13 and Figure 14.

0 0.5 1 1.5 2 2.5

x 104

-6000

-4000

-2000

0

2000

4000

6000

8000

Distance X [m]

Dis

tanc

e Y

[m

]

P1

P2

E


12xv

The following figures (Figure 15 - Figure 16) present the linear simulation results of a 2x1 game with the geometrical initial condition s7 and my 700002 but with

different players configuration, respectively configuration 1 and 2.

0 0.5 1 1.5 2 2.5

x 104

-6000

-4000

-2000

0

2000

4000

6000

8000

Distance X [m]

Dis

tanc

e Y

[m

]

P1

P2E


1/11 Pxv

0 0.5 1 1.5 2 2.5

x 104

-6000

-4000

-2000

0

2000

4000

6000

8000

Distance X [m]

Dis

tanc

e Y

[m

]

P1

P2

E

Figure 13 : Trajectories when E plays *

2/11 Pxv


Page 17

0 5 10 15-2000

0

2000

4000

6000

8000

10000

12000

go [s]

Dis

tanc

e [m

]

Miss dist P1E [m]= -1324.86 Miss dist P2E [m]= 1324.87

0 0.5 1 1.5 2 2.5 3 3.5

x 104

-8000

-6000

-4000

-2000

0

2000

4000

Distance X [m]

Dis

tanc

e Y

[m

]

P1

P2

E

Figure 15: *

12xv in configuration 1

0 5 10 15-2000

0

2000

4000

6000

8000

10000

12000

go [s]

Dis

tanc

e [m

]


0 0.5 1 1.5 2 2.5 3 3.5

x 104

-8000

-6000

-4000

-2000

0

2000

4000

Distance X [m]

Dis

tanc

e Y

[m

]

P1

P2

E

Figure 16 : *

12xv in configuration 2

As we can see, the final distances are equal in both cases which confirm the accuracy of the *

12xv expression for different game parameter. Note that in configuration 2 the final

distances are smaller than in configuration 1 due to the bigger maximum lateral acceleration of missile 2. It can be seen in Figure 16 where missile 2 turn more to its left and then close more the windows for the evader.

3.2.2. Validation in nonlinear simulation

The purpose here is to test the *

12xv expression in a more realistic simulation than the

linear one. The simulation is based on a Simulink model which integrates nonlinear model of missiles with 2 degree of freedom. This model includes more parameters than in a linear simulation like aerodynamics, propulsion or inertial measurement unit.


Page 18

After integration of the *12xv law in the model, the same initial conditions and

configurations as in the linear simulation are tested in this new model. The Simulink simulation of the 2x1 game with initial condition s7 and

my 700002 gives the following results in the Figure 17 and Figure 18 for respectively

configuration 1 and 2.

0 1 2 3 4 5 6 70

2000

4000

6000

8000

10000

12000

go [s]

Dis

tanc

e [m

]

0 0.5 1 1.5 2 2.5 3 3.5

x 104

-8000

-6000

-4000

-2000

0

2000

4000

Distance X [m]

Dis

tanc

e Y

[m

]

P1

P2

E

Figure 17: *

12xv in configuration 1.

0 1 2 3 4 5 6 70

2000

4000

6000

8000

10000

12000

go [s]

Dis

tanc

e [m

]

0 0.5 1 1.5 2 2.5 3 3.5

x 104

-8000

-6000

-4000

-2000

0

2000

4000

Distance X [m]

Dis

tanc

e Y

[m

]

P1

P2

E

Figure 18 : *

12xv in configuration 2.

The same results as in the linear simulation were obtained in both configuration with miss distances of about respectively 1320m and 340m. The equality of the miss distances demonstrated the accuracy of the *

12xv expression with a more realistic

missiles model. Many simulations have been performed with different initial conditions and different behaviours have been observed.


Page 19

3.3. Extension of the no-escape-zones

From the different simulations that was done, three sets of initial conditions in the 2x1 zone involve three different behaviour of the game which can be distinguished.

0 1 2 3 4 5 6 7 8 9 100

2000

4000

6000

8000

10000

12000

go [s]

Dis

tanc

e [m

]

Miss dist P1E [m]= -0.000347279 Miss dist P2E [m]= -0.000200684

0 1 2 3 4 5 6 7 8 9 10-100

-50

0

50

100

go [s]

Acc

eler

atio

n (m

/s²)

Target lateral acceleration

v*2x1

Figure 21: Evader’s capture.

0 2 4 6 8 100

2000

4000

6000

8000

10000

12000

go [s]

ZE

M [

m]


0 2 4 6 8 100

20

40

60

80

100

go [s]

Acc

eler

atio

n [m

/s²]


v*2x1

Figure 19: Evader’s escape without *

12xv saturation.

0 2 4 6 8 100

2000

4000

6000

8000

10000

12000

go [s]

ZE

M [

m]


0 2 4 6 8 10-100

-80

-60

-40

-20

0

go [s]

Acc

eler

atio

n [m

/s²]


v*2x1

Figure 20: Evader’s escape with *

12xv saturation.


Page 20

Figure 19, Figure 20 and Figure 21 are three examples of 2x1 game played with configuration 1 and initial conditions taken in each set. The first and second case leads to the escape of the evader but with different behaviour. Initial condition in Figure 19 gives equality of the final distance as desired in the *

12xv

design. However, in the case of Figure 20, even if *12xv also guarantees the evader’s

escape, the final distances are not equals. The lateral acceleration profile show that *12xv

saturate, pointing out that the desired command to satisfy the criteria is too large (over maxEa ) and so impossible to be achieved by the missile. As a matter of fact,

max*

12 Ex av causes non equality of the final distances.

The third case causes and guarantees the evader’s capture (Figure 21). Whatever is the command chosen by the evader, it will be captured by both pursuers.

0 0.5 1 1.5 2 2.5 3

x 104

-6000

-4000

-2000

0

2000

4000

6000

8000

Distance X [m]

Dis

tanc

e Y

[m

]

P1

P2

E

0 0.5 1 1.5 2 2.5

x 104

-6000

-4000

-2000

0

2000

4000

Distance X [m]

Dis

tanc

e Y

[m

]

P1

P2

E

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5

x 104

-8000

-6000

-4000

-2000

0

2000

4000

6000

Distance X [m]

Dis

tanc

e Y

[m

]

P1

P2

E

Figure 22: Trajectories in XY space of figures 19-21.

This also can bee seen in the trajectories of each case (Figure 22). When *

12xv saturate,

the evader doesn’t succeed to rich the middle of the windows created by pursuers (top right figure) and when evader is too far, the window is closed before reaching it that leads to the evader capture (bottom figure).


Page 21

This last set, with evader’s capture, is very important because it extend the reunion of the two no-escape-zone of each 1x1 game into a bigger 2x1 no-escape-zone. Initial conditions under this area will guarantee the evader’s capture. If evader plays in the worst way )-sign(zv ii then a new limit can be considered:

211111min1 HHz

222122min2 HHz

For reminder, the two no-escape-zone of each 1x1 game are:

211111max1 HHz

222122max2 HHz

The new limit is then the line between the two intersections that was solution of the equations:

21min1max2 PPzz

21max1min2 PPzz

These equations give an unique solution, 12x , which correspond to the new limit of the

no-escape zone in the 2x1 game. Figure 23 show the boundary of this new 2x1 no-escape-zone (in blue) where

9.712 x s and 2.712 x s for respectively configuration 1 and 2.

0 2 4 6 8 10 120

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

11000

go [s]

Dis

tanc

e [m

]

0 2 4 6 8 10 12

0

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

11000

go [s]

Dis

tanc

e [m

]

Figure 23 : No-escape-zone extension in 2x1 game in configuration 1 and 2. This new limit, as well as the zone of *

12xv saturation, will be investigated in our linear

and non-linear simulation in order to confirm the computation of 12x .


Page 22


Each set of sample can be distinguished by comparison of the final distance: If 01 fz and 02 fz then the initial conditions belongs to the 2x1 no-escape-zone.

If 0ifz and ff zz 21 then the initial conditions belongs to the 2x1 non-saturate zone.

If 0ifz and ff zz 21 then the initial conditions belongs to the 2x1 saturate zone.

The method used to determine the three sets was to run the linear simulation with as many initial conditions as possible and sort it out by their final distances at the end of the simulation as describe previously.

0 1 2 3 4 5 6 7 8 9 100

2000

4000

6000

8000

10000

12000

go [s]

ZE

M [

m]

Figure 24: 2x1 no-escape-zone in configuration 1 (Table 1).

The shape of the saturation zone as well as the 12x limit can be seen for respectively

configuration 1 and 2 in Figure 24 and Figure 25. The red area correspond to the non-saturation zone of *

12xv , i.e. where the final distances are equals.

These two figures, results of the linear simulations for both configuration, confirm the previous computation of 12x . The same results are founded for this new limit:

respectively 9.712 x s and 2.712 x s


Page 23

0 1 2 3 4 5 6 7 8 9 100

2000

4000

6000

8000

10000

12000

go [s]

ZE

M [

m]


As the validation of the *

12xv law, the no-escape-zone extension should also be validated

with the nonlinear simulation.

3.3.2. Validation in nonlinear simulation

The same method has been applied to the Simulink model. The comparison of the final distances, as previously, allows plotting the non-saturation zone and then compares the results obtained with the linear simulation. Note that, due to model limitation, simulation with a time to go less than 2 seconds are not reliable in this case. Indeed, for too small time, the pursuers and evader are too close in x-axis compare to the initial y-distance between them and then the approximation on the aspect angles state in (2.1) are no longer available. Figure 26 and Figure 27 represent, for respectively configuration 1 and 2, the shape of the saturation zone as well as the 12x limit.

The same results as in linear simulation are obtained and particularly for the 12x limit

with respectively 9.712 x s and 2.712 x s.


Page 24




Page 25

3.3.3. Three axes representation 21,, PPZEMgo of the 2x1 game

As see in the previous parts, the 2x1 game has been plotted on a single figure ZEMgo , . This superimposition of the two plots of each couple pursuer-evader has

some disadvantages. The main one is that these figures only show the initial pursuers’ position and no-escape-zone with consequence of dividing artificially in two the evader trajectory. Indeed, during simulation, the distance between pursuers changes and pursuers’ position and no-escape-zone should be plotted again at each step time. The idea is to plot the game simulation in a three dimension plot with 21PP as z-axis. Figure 28 and Figure 29 are the plots of the same 2x1 game in configuration 1 and with initial conditions 7000,6, 00 y respectively in the ZEMgo , coordinates and

21,, PPZEMgo coordinates.

0 1 2 3 4 5 6 7 8 9 100

2000

4000

6000

8000

10000

12000 Miss dist P1E [m]= -1807 Miss dist P2E [m]= 1806

go [s]

ZE

M [

m]

Figure 28: 2x1 game plot in ZEMgo , .


Page 26

0

5

10

0200040006000800010000120003000

4000

5000

6000

7000

8000

9000

10000

11000

go [s]

ZEM [m]

P

1P2

Figure 29: 2x1 game plot in 21,, PPZEMgo .

Figure 30 allows understanding how the representation 21,, PPZEMgo works. The

miss distance of this 2x1 game example, easily measurable in plot section of Figure 31, are about 1810 meters which are the same as the one found in the ZEMgo ,

representation of Figure 28.

0 1 2 3 4 5 6 7 8 9 100

2000

4000

6000

8000

10000

12000

go [s]

ZE

M [

m]

Figure 30: 2x1 game plot in 21,, PPZEMgo with X-Y view


Page 27

0 1 2 3 4 5 6 7 8 9 100

2000

4000

6000

8000

10000

12000

go [s]

ZE

M [

m]

0 1 2 3 4 5 6 7 8 9 10

0

2000

4000

6000

8000

0000

2000

go [s]

Figure 31: Section in ZEMgo , of Figure 29 at initial and final time

The disadvantage of such representation is to not be very suitable for a printed report due to lake of legibility for the reader. A convenient solution to avoid such problem is to extract section plot as in Figure 31 or Figure 33. However, the principal interest of this representation is to follow the evader position in relation to the capture boundaries of the game all along the time.

0

5

10

020004000600080001000012000

0

2000

4000

6000

8000

10000

12000

go [s]

ZEM [m]

P

1P2 [

m]

Figure 32: Trajectory evolution with initial condition on the 2x1 limit.


Page 28

In Figure 32, the initial conditions are set up to be in the 12x limit. The main result is

that the evader trajectory remains in the limit it started from all along the simulation. It is more clear to see this in Figure 33 were several sections are plotted during simulation.

0 2 4 6 8 100

5000

10000

go [s]

ZE

M [

m]

0 2 4 6 8 100

2000

4000

6000

8000

go [s]

ZE

M [

m]

0 2 4 60

1000

2000

3000

4000

go [s]

ZE

M [

m]

0 2 4 60

500

1000

1500

2000

go [s]

ZE

M [

m]

0 1 2 30

50

100

150

200

go [s]

ZE

M [

m]

0 1 2 30

50

100

go [s]

ZE

M [

m]

Figure 33: Section ZEMgo , of Figure 32 at six different fixed times.

3.4. The optimal 2x1 command with different time-to-go

In this configuration, the two pursuers are launched at different times and so different time-to-go. The previous expression of *

12xv (3.2) is no longer available and thus need to

be generalized to take into account the difference in time-to-go. Note that in this new case, the evader is assumed to switch its command to *

11xv when it goes beyond its first

opponent. Indeed, during this time where first opponent is passed, the evader faces only one pursuer and then plays like in a 1x1 game.


Page 29

3.4.1. New expression of the command

The optimal evader command, called *

12xv , should still lead to the equality of the final

distances and thus, the equality set in (3.2) is still available. In time-to-go variable τ:

00 2211 ZZ

Note that the minus sign is due to the y-axes orientation like in Figure 6. In normalized variable, we still get:

222

211 00 PP zz

0 1 2 3 4 5 6 7 8 9 100

2000

4000

6000

8000

10000

12000

z1

-z1max

z2max

z2

v*1x1

= -1

v*2x1

Figure 34: Calculation of *

12xv with different time-to-go ( 12 )

Looking at Figure 34, case where 12 , the normalized zero-effort-miss at final time can be now written:

11max1111 0 zzzz

00 22max2222 zzzz

Since 11 0 zz because evader plays only against P1 ( 1* 11 xv ) during

(see Figure 34). The third terms that corresponds to the new change in fz1 due to *

12xv is:

*

11*

1212111 xx vvHz


Page 30

And, thanks to the properties of integrals,

2121121 1

1

HHdhH

Expression becomes: 1*

1221112111 xvHHz

Substituting these expressions leads to:

10 *12211121112111111111 xvHHHHzz

*1221112112111111111 0 xvHHHHzz

However the expression of fz2 remain the same as before (3.2):

10 *12222222222122222 xvHHHzz

From which, using the initial equality, we can isolated *

12xv , in the case where 12 :

212111211

222222

22222122

21211111111*

12

PP

PP

HHH

zHHzHzv

Due to the problem symmetry, the same calculus can be done switching the role of pursuers 1 and 2. Expression of *

12xv in case where 12 is then:

211211

222222222

22222222122

21111111*

12

PP

PP

HHH

zHHzHzv

Note that the particular case where 0 leads to the expression of *

12xv found in 3.2.

As a matter of fact, this is this more general *12xv expression that is used from now in

simulation models.


To validate the new general expression of *

12xv , the same linear simulation as the one

exploited in (3.2.1) is used but with different initial positions for the two pursuers. Indeed, the initial position in x-axis of P1 and P2 are shift in order to create the wanted time shift between the P1-E’s time to go and the P2-E’s one. In the same time, three different initial condition ( 6000,9, 00 y , 5000,7, 00 y

and 3000,6, 00 y ) have been studied in order to determine the extension of the no-

escape-zones as they have been defined previously in (3.3). These three cases have been simulated for configuration 1 and 2 and the results plotted from Figure 35 to Figure 40.


Page 31

0 2 4 6 8 10 120

2000

4000

6000

8000

10000

12000 Miss dist P1E [m]= -0.00656313 Miss dist P2E [m]= -0.00993991

go [s]

ZE

M [

m]

Figure 35: 6000,9, 00 y in configuration 1.

0 2 4 6 8 10 120

2000

4000

6000

8000

10000

12000 Miss dist P1E [m]= -0.00590742 Miss dist P2E [m]= -0.00728553

go [s]

ZE

M [

m]


0 2 4 6 8 10 120

2000

4000

6000

8000

10000

12000 Miss dist P1E [m]= -1949.78 Miss dist P2E [m]= 1949.72

go [s]

ZE

M [

m]


0 2 4 6 8 10 120

2000

4000

6000

8000

10000


go [s]

ZE

M [

m]


0 2 4 6 8 10 120

2000

4000

6000

8000

10000


go [s]

ZE

M [

m]


0 2 4 6 8 10 120

2000

4000

6000

8000

10000


go [s]

ZE

M [

m]



Page 32

As we can see in Figure 37 and Figure 38, the final distances are equal in both cases which confirm the accuracy of the new *

12 xv expression for the two configurations. In

the same time the determination of the no-escape-zone extension (dashed red lines in the figures) seems to hold as explain in (3.3) because the evader is captured in the top figures or its command saturate in the bottom ones. As in (3.3.1), this new limit computation needs to be validated.

3.4.3. Extension of the no-escape-zones

The same method as in (3.3.1) has been applied to the model. The comparison of the final distances, as previously, allows plotting the shape of the non-saturation zone as well as the 12x limit for respectively configuration 1 and 2 in Figure 41and Figure 42.

The red area corresponds to the non-saturation zone of *12 xv , i.e. where the final

distances are equals. The results obtained confirm, particularly for the 12x limit, the previous simulation and

the determination of 12x made in (3.3). The main result is that the 12x limit remains a

straight line which will be useful for time allocation.

Figure 41: 2x1 no-escape-zone in configuration 1 with =1.


Page 33

0 2 4 6 8 10 120

2000

4000

6000

8000

10000

12000

go [s]

ZE

M [

m]

Figure 42: 2x1 no-escape-zone in configuration 2 with =1.

3.4.4. Three axes representation 21,, PPZEMgo

As see for the first simulations, the problem of legibility of the 2x1 game plotted on a single figure ZEMgo , remains. The 21,, PPZEMgo plot can be used to show the

game simulation in a three dimension plot with 21PP as z-axis. Figure 43 and Figure 44 are the plots of the same 2x1 game in configuration 2 with initial conditions 00 , y on the 12x limit respectively in the ZEMgo , coordinates

and 21,, PPZEMgo coordinates.


Page 34

0 2 4 6 8 10 120

2000

4000

6000

8000

10000

12000

go [s]

ZE

M [

m]

Figure 43: 2x1 game in configuration 2 plot in ZEMgo , .

0

5

10

-20000200040006000800010000120000

2000

4000

6000

8000

10000

12000

goZEM [m]

P

1P2

Figure 44: Trajectory evolution in 21,, PPZEMgo with initial condition on the 2x1 limit.


Page 35

0 5 100

5000

10000

ZE

M [

m]

0 5 100

5000

10000

0 2 4 60

1000

2000

3000

4000

ZE

M [

m]

0 2 4 60

1000

2000

3000

4000

0 1 2 3 4 50

500

1000

go [s]

ZE

M [

m]

0 1 2 3 4 50

500

1000

go [s]

Figure 45: Section ZEMgo , of Figure 44 at six different fixed times.

As said, the initial conditions are set up to be in the 12x limit (Figure 32). The principal

interest of this representation is to follow the evader position in relation to the capture boundaries of the game all along the time. As expected (compared with the simulations in (3.3.3)) the evader trajectory remains in the limit it started from all along the simulation (Figure 44). It is more clear to see this in Figure 45 were several sections are plotted during simulation.

The 2x1 game in three dimension environment

Page 36

4. The 2x1 game in three dimension environment


4.1.1. Geometry

The problem’s geometry is then in three dimension (XYZ). In the (XY) plan, the problem’s geometry remains the same as in two dimension (3.1) i.e. the two pursuers, called P1 and P2, are assumed to be with M1 under M2 and one evader (E) initially located between them (Figure 46). In the altitude plan (XZ), the two pursuers are based on the ground and the hostile missile come from air (Figure 47).

0 1 2 3 4 5 6 7 8

x 104

-6000

-4000

-2000

0

2000

4000

6000

X [m]

Y [

m]

Defending Area

Missile 2Missile 1

data4

Figure 46: Problem’s geometry - (XY) plan.

0 1 2 3 4 5 6 7 8

x 104

0

500

1000

1500

2000

2500

3000

Z [

m]

X [m]

Defending Area

Missile 2Missile 1

data4

Figure 47: Problem’s geometry - (XZ) plan.

The game scenario consists of two missiles defending an area, in green, which is the objective of the hostile missile, in red, as it can be seen in Figure 46 and Figure 47.

4.1.2. Simulation’s model

The simulation model used to simulate the 2x1 game is based on a 6 degree of freedom missile model library. The two missiles are simulated by the same model library and thus are totally identical. Contrary to the previous models where the characteristic time constants for missiles were set in the initialization script, the missiles time constants are inherent to the guidance and control units within the model and need to be estimated. The model studies have permitted to determine the characteristic time constants of the missile and the target and report with other characteristic constants in Table 3.


Page 37

Table 3: Missiles’ configuration on the 6 d.o.f. simulation.

With the knowledge of the players’ characteristic parameters from Table 3, the no-escape-zone and all other boundaries determined from now, like the 12x limit, can be

drawn for the (XY) plan in Figure 48.

0 2 4 6 8 10 12 140

2000

4000

6000

8000

10000

12000

go [s]

ZE

M [

m]

Figure 48: No-escape-zones in (XY) plan.

Thanks to the calculation method of the no-escape-zone extension (see 3.3) and with parameters from Table 3 as well as the knowledge of 21PP = 11000m, the 12x limit can

be determined and, as it can be seen in Figure 48, was 12x = 7.78 s.

As all the previous preparing simulations, the validity of the *12 xv law and the impact of

the no-escape-zone extension ( 12x limit) must be investigated in this new highly

realistic model to confirm the predictions.

Missile 1 Missile 2 Target

1PV 2300 m/s² 2PV 2300 m/s² EV 2700 m/s²


1P 0.3 s 1P 0.3 s E 0.6 s

1 2 2 2

1 2 2 2


Page 38

4.2. Simulation in the Simulink model

The missiles are playing, for both plan (XY) and (XZ), with their optimal command. In other hand, the target was supposed to hit the defending area. To achieve this goal, the target plays proportional navigation1 in the (XZ) plan and plays, in (XY), the *

12xv law

until it pass the missiles. If the target succeeds at going beyond them, it switches as in (XZ) plan to proportional navigation towards its initial goal. The simulation has been run for many different initial conditions but the result of only three particular ones are showed. The initial conditions are 6000,9, 00 y ,

6500,5.6, 00 y and 7500,5.6, 00 y and have been chosen, thanks to Figure

48, to highlight the impact of the no-escape-zone extension. For all the simulations, the initial condition in the (XZ) plan has been fixed to 0z = -2000 m. Note that the minus

sign come from the fact that the Z axis in oriented to the ground in the aeronautics convention. The first case set the initial conditions of the game in the extended no-escape-zone i.e. after 12x = 7.78 s (Figure 53). Some results of this simulation are plotted from Figure 49

to Figure 53.

01

23

45

x 104

-1

-0.5

0

0.5

1

x 104

-2000

-1500

-1000

-500

0

X [m]

Trajectories

Y [m]

Z [

m]

Target

M1

M2

Defending Area

Figure 49: Missiles trajectories in (XYZ) with capture.


Page 39

As it can be seen in the trajectories plots (Figure 49 to Figure 51) the target is captured by the two defending missiles. The interception is easy in the (XZ) plan since the evader plays proportional navigation without trying to avoid the defending missiles (Figure 51). In the (XY) plan, the evader plays its optimal command to go between missiles as indicate the target acceleration profile (Figure 52). However, the target is too far and the window is closed before reaching it that leads to its capture.

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5

x 104

-6000

-4000

-2000

0

2000

4000

6000 Trajectories

X [m]

Y [

m]

Target

M1M2

Defending Area

Figure 50: Missiles trajectories in (XY) with capture.

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5

x 104

-2000

-1800

-1600

-1400

-1200

-1000

-800

-600

-400

-200

0

Trajectories

X [m]

Z [

m]

Target

M1M2

Defending Area

Figure 51: Missiles trajectories in (XZ) with capture.

0 1 2 3 4 5 6 7 8 9

-200

0

200

Acc. M1

Acc

eler

atio

n [m

/s²]

0 1 2 3 4 5 6 7 8 9

-200

0

200

Acc. M2

ay

az

0 1 2 3 4 5 6 7 8 9-100

0

100

Acc. T

Time [s]

Figure 52: Missiles accelerations.

0 2 4 6 8 10 120

2000

4000

6000

8000

10000

12000

go [s]

ZE

My [

m]

Figure 53: ZEM of the game in (XY).

The simulation confirms, in this scenario, that if the evader is in the extended no-escape-zone (Figure 53) then the evader is captured.


Page 40

The second case set the initial conditions of the game out of the extended no-escape-zone i.e. before 12x = 7.78 s (Figure 58). Figure 54 to Figure 58 illustrated some results

obtained from the simulation.

01

23

4

x 104

-1

-0.5

0

0.5

1

x 104

-2000

-1500

-1000

-500

0

X [m]

Trajectories

Y [m]

Z [

m] Target

M1M2

Defending Area

Figure 54: Missiles trajectories in (XYZ) without *

12xv saturation.

0 0.5 1 1.5 2 2.5 3 3.5

x 104

-6000

-4000

-2000

0

2000

4000

6000 Trajectories

X [m]

Y [

m]

Target

M1

M2

Defending Area

Figure 55: Missiles trajectories in (XY) without *

12xv saturation.

0 0.5 1 1.5 2 2.5 3 3.5

x 104

-2000

-1800

-1600

-1400

-1200

-1000

-800

-600

-400

-200

0

Trajectories

X [m]

Z [

m]

Target

M1M2

Defending Area

Figure 56: Missiles trajectories in (XZ) without

*12xv saturation.


Page 41

As it can be seen in the trajectories plots (Figure 54 to Figure 56), the target escapes from the two defending missiles. In the (XZ) plan nothing changes since last simulation, the evader still plays proportional navigation in this plan and the interception remains easy in (XZ) (Figure 56). However, in the (XY) plan, the target is too close for the defending missiles and even if they play their maximum acceleration (pointing out by saturation in commanded acceleration in Figure 57), the target pass trough the window. The evader plays its optimal command until passing the two missiles as indicate the target acceleration profile at go =6.5 s (Figure 57). At this time the target switch from a negative

acceleration ( *12xv ) to a positive saturate acceleration commanded by the proportional

navigation to go from left to right and hit the ground base.

0 2 4 6 8 10 12

-200

0

200

Acc. M1

Acc

eler

atio

n [m

/s²]

0 2 4 6 8 10 12

-200

0

200

Acc. M2

ay

az

0 2 4 6 8 10 12-100

0

100

Acc. T

Time [s]


0 2 4 6 8 10 120

2000

4000

6000

8000

10000

12000

go [s]

ZE

My [

m]

Figure 58: ZEM of the game in (XY).

The simulation confirms, in this scenario, that if the evader is out of the extended no-escape-zone (Figure 58) then the evader passes the two defending missiles and reaches its goal in hitting the ground base. The last case set the initial conditions of the game still out of the extended no-escape-zone but with a higher y initial position. The result is almost the same as the previous scenario except that the commanded target’s acceleration saturates to pass the defending missiles (Figure 60). The main difference is that, even if the target switch to its proportional navigation command, the target doesn’t reach its goal which is to hit the ground base (Figure 59).


Page 42

0 0.5 1 1.5 2 2.5 3 3.5

x 104

-6000

-4000

-2000

0

2000

4000

6000 Trajectories

X [m]

Y [

m]

Target

M1M2

Defending Area

Figure 59: Missiles trajectories in (XY) with *

12xv saturation.

0 2 4 6 8 10 12

-200

0

200

Acc. M1

Acc

eler

atio

n [m

/s²]

0 2 4 6 8 10 12

-200

0

200

Acc. M2

ay

az

0 2 4 6 8 10 12-100

0

100

Acc. T

Time [s]


Application in three pursuers - two evaders scenario

Page 43

5. Application in three pursuers - two evaders scenario


5.1.1. Geometry

The problem’s geometry is still in three dimensions (XYZ). In the (XY) plan, the three pursuers, called M1, M2 and M3, are assumed to be with M1 under M3 and M2 between them and all aligned in the y-axis, like in Figure 61. The two targets (T1 and T2) could be everywhere in the positive x-axis. In the altitude plan (XZ), the three pursuers are based on the ground and the hostile missiles come from air like in the last chapter (Figure 62).

0 1 2 3 4 5 6 7 8 9 10

x 104

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1x 10

4

X [m]

Y [

m]

data1GB 2

T 1

T 2

M 1

M 2M 3

Figure 61: Problem’s geometry - (XY) plan.

0 1 2 3 4 5 6 7 8 9 10

x 104

0

500

1000

1500

2000

2500

3000[

]

X [m]

GB 1GB 2

T 1

T 2

M 1

M 2M 3

Figure 62: Problem’s geometry - (XZ) plan.

The game scenario consists of three missiles defending two areas, ground bases in green, which are the objectives of the hostile missiles, in dark green, as it can be seen in Figure 61 and Figure 62.

5.1.2. Simulation’s model

The simulation model used to simulate this scenario of three pursuers - two evaders is based on the same missile and target Simulink model library as previously. The missiles and targets are simulated by duplication of the same model library and thus are totally identical and thus the characteristic time constants remain the same as in Table 3.


Page 44

Concerning the command strategies, the missiles and targets plays like in the previous 2x1 game simulation (4.2). In other words, the missiles are playing, for both plan (XY) and (XZ), with their optimal command when the targets plays proportional navigation1 in the (XZ) plan and plays, in (XY), the *

12xv law until it pass the missiles. If the target

succeeds at going beyond them, it switches as in (XZ) plan to proportional navigation towards its initial goal. However, the main problem of such configuration is to decide which missile(s) is allocated on which target before launching. This allocation need to be done in the simulation model.

5.2. Allocation

The decision of allocation is made with help of an allocation matrix which provide, according to a criterion, the launch time and the allocated target for each missile. The allocation matrix is included in the simulation and updated at each simulation step time until launch. Moreover, the chosen criterion for allocation decision is to maximize the launch time of all missiles. The allocation matrix is built by listing all the possible allocation of the 3 missiles on T1 and T2, which is of 12 possibilities in this case (see first row of Table 4), and by filling it with the corresponding max times ( Table 4). The max times are the corresponding times defined in (2.4), max , and in (3.3),

12x . In other words, jit , is the time where the supposed trajectory of the target j

intersects the no-escape zone of missile i and kjit ,, is the time where the supposed

trajectory of the target k intersects the extended no-escape zone of missile i and j. In our simulation, for the max times computation, the target trajectory is supposed to be a straight line between its position and a ground base. M1

/ M2

M1 / M3

M2 / M3

M2 / M1

M3 / M1

M3 / M2

M1 / M23

M2 / M13

M3 / M12

M23 / M1

M13 / M2

M12 / M3

T1 11t 11t 21t 21t 31t 31t 11t 21t 31t 231t 131t 121t

T2 22t 32t 32t 12t 12t 22t 232t 132t 122t 12t 22t 32t

Table 4: Allocation matrix in 3x2 game.

However, the target trajectory estimation and thus max time computation depends on which ground bases the target should hit and we can’t know about it. As we get 2 ground bases in the simulation, for each matrix term, two max times can be computed: one if Ti goes to ground base 1 and another if it goes to ground base 2. As a matter of fact, two matrixes for each ground base like in Table 4 should be made. An example is given in Figure 63 on determination of 11t for A1 and A2.


Page 45

These two matrixes, A1 and A2, should be merged into the final allocation matrix A. The method used to merge them is to take, for a given term of A, the minimum time between the same term of A1 and A2 in order to guarantee the capture without knowledge on the target objective:

),(2);,(1min),( nmAnmAnmA In the example of Figure 63, the chosen values for the allocation matrix A that guarantee capture by M1 whatever is the real trajectory of T1 is 11t for A2.

0 2 4 6 8 10 12 14 16 18 20 22

-1

-0.5

0

0.5

1

x 104

go [s]

ZE

M [

m]

T1 trajectory

T1 trajectory

NEZ M3NEZ M2

-NEZ M2

NEZ M1

GB2

GB1

T1

t11

for A2

t11

for A1

Figure 63: Determination of 11t for A1 and A2.

Note that the figure is plot in time-to-go whereas allocation is based on time. If necessary, the conversion can be done by taking into account that, before missiles’ launch, the closing velocity is only the target velocity, TV whereas after, the closing

velocity is the sum of missile and target velocity cV . The formula is then:

TcgoT VVXt .0

Finally, the criterion is applied to the allocation matrix A to establish the allocation plane given by the column of A:

),2(),1(max_ jAjAcaseAllocation j


Page 46

The launch times of each missile correspond to the adequate terms A(1,Allocation_case) and A(2,Allocation_case).

5.3. Simulation scenarios

5.3.1. Scenario 1

The initial conditions chosen for the first scenario are: 8000,20, 00 y for T1 and

3000,20, 00 y for T2. Figure 64 shows the initial targets positions and the four

possible trajectories to the ground bases. The intersections with all possible no-escape-zones are solved and used to build, as describe in (5.2), the allocation matrix detailed in Table 5.

0 2 4 6 8 10 12 14 16 18 20

-1

-0.5

0

0.5

1

x 104

go [s]

ZE

M [

m]

GB2

GB1

T2

T1

Figure 64: Trajectories estimation for max time calculation.

M1 / M2

M1 / M3

M2 / M3

M2 / M1

M3 / M1

M3 / M2

M1 / M23

M2 / M13

M3 / M12

M23 / M1

M13 / M2

M12 / M3

T1 14,9 14,9 15,1 15,1 1,4 1,4 14,9 15,1 1,4 23,3 17,7 23,3 T2 20,1 11,4 11,4 6,4 6,4 20,1 23,3 17,7 23,3 6,4 20,1 11,4

Table 5: Allocation matrix in scenario 1.

The allocation case obtain with the criterion is the allocation n°7, surrounded in red in Table 5. Practically, the chosen allocation is M1 lock on T1 and M2 - M3 both lock on T2.


Page 47

The allocation matrix can be plot in an allocation table that represent, for each possible allocation, the corresponding max times for T1 and T2. Figure 65 shows this allocation table for scenario 1. We can see that the selected allocation n°7 (black squares) is the best choice to maximize all allocation times. The main reason is that the max time of M2 – M3 on T2 is larger than M2 on T2 (blue circle in line 2) and even more than M3 on T2 (blue circle in line 3) which permit to increase the launch time against T2.

0 5 10 15 20 25 30 350

1

2

3

4

5

6

7

M1

M2

M3

M1-M2

M2-M3

M1-M3

time [s]

T1

T2

Figure 65: Allocation table in scenario 1.

The simulation results can be seen from Figure 66 to Figure 68. Figure 67 shows that both targets are intercepted and that missiles allocation are respected and efficient.

02

46

810

x 104

-1

-0.5

0

0.5

1

x 104

0

500

1000

1500

2000

[m]

Trajectories

[m]

[m]

Target 1Target 2

M1

M2

M3

Defending Area1Defending Area2

Figure 66: Trajectories for scenario 1.


Page 48

We can see in Figure 68 that the ZEM of target 1 start far from missile 1 no-escape-zone. It can be explained by the method used to merge the matrixes (see 5.2). In this case the kept max time to launch M1 is 11t from A2 (with GB2) and not the one from A1 (GB1) which would give a starting point on the missile 1 no-escape-zone. However, the objective of guarantee the capture without knowledge on the target goal is fulfilled since both targets are intercepted.

0 2 4 6 8 10

x 104

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1x 10

4

Trajectories

X [m]

Y [

m]

Figure 67: Trajectories in (XY) for scenario 1.

0 2 4 6 8 10 12 14 16 18 20

-1

-0.5

0

0.5

1

x 104

go [s]

ZE

M [

m]

Figure 68: ZEM in (XY).

Finally, the simulation shows that allocation is efficient with the capture of both targets and that the use of 2 missiles against 1 target, thanks to the no-escape-zone extension, allows increasing the launch time compared to a single missile allocation for the same target.

5.3.2. Scenario 2

The initial conditions chosen for the second scenario are: 6000,20, 00 y for T1 and

4000,20, 00 y for T2. Figure 69 shows all the possible targets trajectories to the

ground bases. The intersections with all possible no-escape-zones are solved and used to build, as describe in (5.2), the allocation matrix detailed in Table 6.


Page 49

0 2 4 6 8 10 12 14 16 18 20

-1

-0.5

0

0.5

1

x 104

go [s]

ZE

M [

m]

GB2

GB1

T1

T2

Figure 69: Trajectories estimation for max time calculation.

M1 / M2

M1/ M3

M2 / M3

M2 / M1

M3 / M1

M3 / M2

M1 / M23

M2 / M13

M3 / M12

M23/ M1

M13 / M2

M12 / M3

T1 3,5 3,5 17,2 17,2 13,9 13,9 3,5 17,2 13,9 23,1 17,8 23,1 T2 19,3 5,7 5,7 12,6 12,6 19,3 23,1 17,8 23,1 12,6 19,3 5,7

Table 6: Allocation matrix in scenario 2.

The allocation case obtain with this scenario is the allocation n°11, surrounded in red in Table 6. Practically, the chosen allocation is M2 lock on T2 and M1 – M2 both lock on T1. The allocation matrix can be plot in an allocation table that represent, for each possible allocation, the corresponding max times for T1 and T2. Figure 70 shows this allocation table for scenario 2. We can see that the selected allocation n°11 (black squares) is the best choice to maximize all allocation times. Indeed, even if the choice of M2-M3 or M1-M2 seems to be better, the best choice remains M1 – M3 because the choice of M2 for T2 is better than M1 or M3 alone lock on T2.


Page 50

0 5 10 15 20 25 30 350

1

2

3

4

5

6

7

M1

M2

M3

M1-M2

M2-M3

M1-M3

time [s]

T1

T2

Figure 70: Allocation table in scenario 2.

The simulation results can be seen from Figure 71 to Figure 73. Figure 72 shows that both targets are intercepted and that missiles allocation are respected and efficient.

0

5

10

x 104

-1

-0.5

0

0.5

1

x 104

0

500

1000

1500

2000

X [m]

Trajectories

Y [m]

Z [

m]

Target 1Target 2

M1

M2

M3

Defending Area1Defending Area2

Figure 71: Trajectories for scenario 2.


Page 51

0 2 4 6 8 10

x 104

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1x 10

4

Trajectories

X [m]

Y [

m]

Figure 72: Trajectories in (XY) for scenario 2.

0 2 4 6 8 10 12 14 16 18 20

-1

-0.5

0

0.5

1

x 104

go [s]

ZE

M [

m]

Figure 73: ZEM in (XY).

This scenario, more complexes than the previous one on account of the crossing of target trajectories, shows that allocation remains very efficient with the capture of both targets. The choice of the allocation criteria (maximization of launch time) proves to be successful because the waiting increases the probability to make the best allocation. It also confirms that the use of 2 missiles against 1 target allows increasing the launch time compared to a single missile allocation for the same target.

Conclusion

Page 52

6. Conclusion

This report demonstrates a new optimal command for the evader as solution of the 2x1 game. Moreover, it points out that the use of two missiles instead of a single one against a target (even if the target plays its new optimal command) increases the capture zone of the two missiles. Indeed, the new capture zone is more than the reunion of the two no-escape-zones of each missile and a new area where the evader’s capture is guaranteed exists. The limit of this no-escape-zone extension has been characterized to be a straight line and validate on three different simulations. Finally, the use of the 2x1 configuration has been studied in allocation strategy for three missiles – two targets scenarios. The chosen allocation strategy is to delay as much as possible all missiles launch time because of two main reasons: Increase the confidence in target estimation process and take into account, in reality, the no unlimited flight time of the defending missiles. So, the allocation criterion has been designed to perform this strategy. The first scenario’s simulation shows that allocation is efficient, with the capture of both targets. The second scenario, more complexes than the first one due to the crossing trajectories of the targets, confirms that allocation remains very efficient too. In both scenarios, the choice of the allocation strategy allows disposing as much as possible time to make the right allocation's decision and proves to be successful from the point of view of strategy's respect. The continuation of this thesis can concern different points for complementary researches:

Use other zone calculation methods with more realistic models over approximation (level sets…) and extend to Nx1 games. Use, if less information is known about the evader, DGL/0 law instead of DGL/1. Implement in-flight allocation and optimize mid-course guidance law (which is the law before allocation) like guidance on the targets’ barycentre position. Improve the target trajectory estimation to increase the allocation matrix accuracy. Take into account all data sharing problems between missiles like data link latency, limited bandwidth in transmission or data fusion.

References 1 Joseph Z. Ben-Asher and Isaas Yaesh, Advances in Missile Guidance Theory, Vol.180, Progress in Astronautics and Aeronautics, AIAA, Reston, VA 1998. 2 Ho Y. C., Brysson A. E., and Baron S., “Differential Games and Optimal Pursuit-Evasion Strategies”, IEEE Transactions on Optimal Control, Vol. AC-10, No 4, 1965, pp. 385-389. 3 Gutman S., “On Optimal Guidance for Homing Missiles”, Journal of Guidance and Control, Vol.3, No.4, 1979, pp296-300. 4 Shinar J., “Solution Techniques for Realistic Pursuit-Evasion Games”, Advances in Control and Dynamic Systems, edited by C.T. Leondes, Vol.17, Academic Press, New York, 1981, pp. 63-124. 5 Zarchan P., Tactical and Strategic Missile Guidance, Vol.176, Progress in Astronautics and Aeronautics, AIAA, Reston, VA 1997, pp. 143-161. 6 Isaacs R., Differential Games, Wiley, New York, 1965.

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