Differential Game Based Air Combat Maneuver Generation Using...

Copyright ⓒ The Korean Society for Aeronautical & Space SciencesReceived: June 18, 2015 Revised: April 28, 2016 Accepted: June 19, 2016

204 http://ijass.org pISSN: 2093-274x eISSN: 2093-2480

PaperInt’l J. of Aeronautical & Space Sci. 17(2), 204–213 (2016)DOI: http://dx.doi.org/10.5139/IJASS.2016.17.2.204

Differential Game Based Air Combat Maneuver Generation Using Scoring Function Matrix

Hyunju Park*, Byung-Yoon Lee** and Min-Jea Tahk*** Department of Aerospace Engineering, KAIST, Daejeon 48235, Republic of Korea

Dong-Wan Yoo**** Agency for Defense Development, Daejeon 34060, Republic of Korea

Abstract

A differential game theory based approach is used to develop an automated maneuver generation algorithm for Within Visual

Range (WVR) air-to-air combat of unmanned combat aerial vehicles (UCAVs). The algorithm follows hierarchical decision-

making structure and performs scoring function matrix calculation based on differential game theory to find the optimal

maneuvers against dynamic and challenging combat situation. The score, implying how much air superiority the UCAV has,

is computed from the predicted relative geometry, relative distance and velocity of two aircrafts. Security strategy is applied

at the decision-making step. Additionally, a barrier function is implemented to keep the airplanes above the altitude lower

bound. To shorten the simulation time to make the algorithm more real-time, a moving horizon method is implemented. An

F-16 pseudo 6-DOF model is used for realistic simulation. The combat maneuver generation algorithm is verified through

three dimensional simulations.

Key words: Air-to-Air Combat, Differential Game, Scoring Function Matrix, Pursuit-Evasion Game

1. Introduction

On account of the recent development in guidance, control

and navigation technology, the Unmanned Combat Aerial

Vehicles (UCAVs) have begun to replace manned aircraft in

not only for reconnaissance missions [1]. Instead of well-

trained human pilots, UCAV must be implemented with an

algorithm to successfully carry out air-to-air combat. To deal

with the dynamic and challenging nature of air combat, the

pilot algorithm should be able to predict the maneuver of the

enemy and decide its own optimal maneuver.

Previous studies have treated the air-to-air combat in

several ways. Rule-based method has been studied by Burgin

and Sidor in [2]. The rule-based adaptive maneuvering logic

combat program was successful in combat against human

pilot; yet requires hardcoding and frequent validation to

apply experiences of skilled pilots. As a variation of the rule-

based method, Jang [3] uses an artificial neural network as a

pilot maneuver decision method. Fast computation time and

re-learning are the prime advantages of this method. Existing

combat simulation programs such as BRAWLER [4] uses a

heuristic value-driven system suggested in [5] to avoid time

and effort consuming characteristic of rule-based method.

The paper combines value-driven system with hierarchical

decision-making structure to follow natural straight-forward

decision-making paradigm of human pilots.

As an optimal control theory approach, [6] and [7]

introduced a nonlinear model predictive tracking controller

for a fixed wing UAV that performs three dimensional evasive

maneuvers. The control problem is formulated as a cost

optimization problem and solved using a gradient-descent

method.

Dynamic programming method has been studied for a long

time to solve the general pursuit-evade game. The optimal

This is an Open Access article distributed under the terms of the Creative Com-mons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/3.0/) which permits unrestricted non-commercial use, distribution, and reproduc-tion in any medium, provided the original work is properly cited.

* MS Student ** Ph. D Student *** Professor, Corresponding author: [email protected] **** Research Engineer

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Hyunju Park Differential Game Based Air Combat Maneuver Generation Using Scoring Function Matrix

http://ijass.org

solution of the game is first defined in [8]. To overcome

the complexity of calculation and difficulty in real-time

simulation, variations of dynamic programming have been

suggested by researchers. McGrew et al. [9-10] have suggested

an approximate dynamic programming method to find the

air combat strategy. Approximate dynamic programming

is different from classical dynamic programming in that it

formulates a continuous function to approximately represent

the future reward. Approximate dynamic programming does

not have to perform the future reward calculation for every

discrete state thus results in much shorter computing time.

Austin et al. [11-12] have additionally combined a game

theoretic approach to discrete dynamic game. The authors

performed recursive decision-making procedures with a

fixed planning horizon by establishing scoring function

matrix.

In this paper, a three-dimensional pseudo 6-DOF model

is used to construct the air combat model for two airplanes.

The proposed combat maneuver generation algorithm

mainly utilizes a differential game theory based scoring

function matrix which is built by grafting the game matrix on

a value-driven based decision model. The decision-making

procedure follows the hierarchical structure and adopts

security strategy.

The organization of this paper is as follows: in section 2,

a three dimensional air combat simulation model used in

this study is described. In section 3, the combat generation

algorithm is addressed. In section 4, several cases of air-to-

air combat are investigated by computer simulation to verify

the combat model. Conclusions are given in section 5.

2. 3-Dimensional Air Combat Modeling

2.1 F-16 Pseudo 6DOF Model

A three-dimensional F-16 pseudo 6-DOF model is used

for a realistic simulation in this paper. Two fighters are

simulated simultaneously in this model. The equation

of motions and the kinematics for model is derived by

assuming each aircraft as a point mass. The 3-DOF equations

of motion of this model are as (1), where Vt implies the total

velocity at body axis and Tt is thrust, D is drag, and γt means

the elevation angle where ψt means the azimuth angle.

3

reward calculation for every discrete state thus results in much shorter computing time. Austin et al.

[11-12] have additionally combined a game theoretic approach to discrete dynamic game. The authors

performed recursive decision-making procedures with a fixed planning horizon by establishing

scoring function matrix.

In this paper, a three-dimensional pseudo 6-DOF model is used to construct the air combat model

for two airplanes. The proposed combat maneuver generation algorithm mainly utilizes a differential

game theory based scoring function matrix which is built by grafting the game matrix on a value-

driven based decision model. The decision-making procedure follows the hierarchical structure and

adopts security strategy.

The organization of this paper is as follows: in section 2, a three dimensional air combat simulation

model used in this study is described. In section 3, the combat generation algorithm is addressed. In

section 4, several cases of air-to-air combat are investigated by computer simulation to verify the

combat model. Conclusions are given in section 5.

2. 3-Dimensional Air Combat Modeling

2.1 F-16 Pseudo 6DOF Model

A three-dimensional F-16 pseudo 6-DOF model is used for a realistic simulation in this paper. Two

fighters are simulated simultaneously in this model. The equation of motions and the kinematics for

model is derived by assuming each aircraft as a point mass. The 3-DOF equations of motion of this

model are as (1), where tV implies the total velocity at body axis and tT is thrust, D is drag, and

t means the elevation angle where t means the azimuth angle.

( ) / sin( sin ) cos cos

( sin )sincos

t t t

t tt

t t

tt

t t

V T D m gL T W

mVL T

mV

, (1)

The kinematic equations are as in (2) where X, Y and h means position at inertia axis. There are

three input commands in this model as real fighters: thrust, roll rate, and load factor. All of the input

(1)

The kinematic equations are as in (2) where X, Y and

h means position at inertia axis. There are three input

commands in this model as real fighters: thrust, roll rate,

and load factor. All of the input commands include lag

compensators.

4

commands include lag compensators.

cos cos

cos sin

sint

t t t

t t t

t

X V

Y V

h V

, (2)

To avoid a singularity which occurs when pitch angle is 90 , this model utilized direction cosine

matrix updated method instead of Euler angle update method. More detailed figure and explanation of

the pseudo 6-DOF model of F-16 fighter is explained in Ref. [16].

2.2 Terminal Condition

A terminal condition and gun score are defined in this combat model. The situation when red

aircraft is in blue aircraft’s gun-range is defined as shown in Fig.1: blue’s bearing angle is smaller

than 60 and red’s aspect angle is smaller than 30 and also the distance between two aircrafts is

smaller than 500m. If the above condition is held for more than 5 seconds, algorithm regards the red is

shot down by the blue. Even though the terminal condition has not been satisfied, the algorithm

calculates how long each aircraft has been in the shooting mode and plot the cumulated time as “gun

score” of each after the algorithm finished.

30BA

60AA

Fig. 1. Definition of Gun Range

(2)

To avoid a singularity which occurs when pitch angle is

±90o, this model utilized direction cosine matrix updated

method instead of Euler angle update method. More detailed

figure and explanation of the pseudo 6-DOF model of F-16

fighter is explained in Ref. [16].


A terminal condition and gun score are defined in this

combat model. The situation when red aircraft is in blue

aircraft’s gun-range is defined as shown in Fig.1: blue’s

bearing angle is smaller than 60o and red’s aspect angle is

smaller than 30o and also the distance between two aircrafts

is smaller than 500m. If the above condition is held for more

than 5 seconds, algorithm regards the red is shot down by

the blue. Even though the terminal condition has not been

satisfied, the algorithm calculates how long each aircraft has

been in the shooting mode and plot the cumulated time as

“gun score” of each after the algorithm finished.

3. Combat Maneuver Generation Algorithm

3.1 Combat Algorithm Outline

The overall combat maneuver decision process is

decomposed into a hierarchical structure as shown in Fig.2.

The most influencing decision is made in the top level, pilot

4

commands include lag compensators.

cos cos

cos sin

sint

t t t

t t t

t

X V

Y V

h V

, (2)

To avoid a singularity which occurs when pitch angle is 90 , this model utilized direction cosine

matrix updated method instead of Euler angle update method. More detailed figure and explanation of

the pseudo 6-DOF model of F-16 fighter is explained in Ref. [16].


A terminal condition and gun score are defined in this combat model. The situation when red

aircraft is in blue aircraft’s gun-range is defined as shown in Fig.1: blue’s bearing angle is smaller

than 60 and red’s aspect angle is smaller than 30 and also the distance between two aircrafts is

smaller than 500m. If the above condition is held for more than 5 seconds, algorithm regards the red is

shot down by the blue. Even though the terminal condition has not been satisfied, the algorithm

calculates how long each aircraft has been in the shooting mode and plot the cumulated time as “gun

score” of each after the algorithm finished.

30BA

60AA

Fig. 1. Definition of Gun Range Fig. 1. Definition of Gun Range

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DOI: http://dx.doi.org/10.5139/IJASS.2016.17.2.204 206

Int’l J. of Aeronautical & Space Sci. 17(2), 204–213 (2016)

behavior decision. In this block, the algorithm analyzes

current combat situation to decide whether to pursuit or

evade. In this paper, different strategies for each behavior are

defined. In the lower decision level, the algorithm generates

maneuver according to selected pilot behavior and finally

determines whether to fire or not based on the gun range

mentioned before.

In this paper, the pilot algorithm generates the maneuvers

based only on the combat situations of current state and

predicted next few seconds to grasp the constant changes in

the air combat. The generated maneuver is therefore a short-

term solution that is not available until the end of the game

according to the nature of differential-game theory. However

this method can effectively reduce the simulation time and

also increase the adaptability of this algorithm.

3.2 Scoring Function Matrix

To determine which maneuver is the “optimal maneuver”

that guarantees the advantageous position after few seconds,

every possible consequence of performing various maneuver

choices should be evaluated. The number of maneuver

choices of each aircraft is limited to 7. As shown in Fig.3,

the 7 choices are steady level flight, pull up, push down, left

pull up, left push down and right pull up and push down,

respectively. These maneuvers are selected to resemble the

real motion of fighters in air combats as much as possible.

Since blue and red UCAVs have 7 maneuver choices each,

there are total 49 possible maneuver combinations for both

blue and red as shown in Fig.4. The scoring function matrix

is used to systematically conduct the comparisons between

each end states. The end states are calculated by numerically

integrating the equations of motion with application of the

corresponding maneuver command. This is the “prediction”

step. The prediction is done for 1~2 seconds. The end states

are then scored by the scoring function, which will be

addressed in the next section. The scores imply how much

advantage corresponding aircraft has.

The “decision-making” procedure is carried out based

on the predicted scores. Security strategy based on the

game theorem is applied in this step to decide the optimal

maneuver commands. First, select the worst score of each

column: to figure out the magnitude of severity caused by

the enemy’s performance. Second, select the row including

the best case among the selected worst. By using this min-

max algorithm, the airplane is able to avoid the worst case

no matter which maneuver the enemy has chosen. In this

paper, both blue and red airplanes are assigned with the

same strategy.

3.3 Scoring Function

3.3.1 Orientation Score

The evaluation criterion of the end states mentioned in

previous section is the scoring function. The scoring function

is derived from the meaning of gaining air superiority.

During WVR air-to-air combat, the necessary conditions for

blue’s air superiority are as follows: (1) Blue has to be located

at the rear of the red, (2) Blue should be heading for the red.

Above conditions can be expressed using the relative

geometry of two aircrafts (see Fig.5). By definition, the sum

of blue’s aspect angle (AAB) and red’s bearing angle (BAR)

5

3. Combat Maneuver Generation Algorithm

3.1 Combat Algorithm Outline

The overall combat maneuver decision process is decomposed into a hierarchical structure as

shown in Fig.2. The most influencing decision is made in the top level, pilot behavior decision. In this

block, the algorithm analyzes current combat situation to decide whether to pursuit or evade. In this

paper, different strategies for each behavior are defined. In the lower decision level, the algorithm

generates maneuver according to selected pilot behavior and finally determines whether to fire or not

based on the gun range mentioned before.

Fig. 2 Hierarchical Structure of Decision-Making

In this paper, the pilot algorithm generates the maneuvers based only on the combat situations of

current state and predicted next few seconds to grasp the constant changes in the air combat. The

generated maneuver is therefore a short-term solution that is not available until the end of the game

according to the nature of differential-game theory. However this method can effectively reduce the

simulation time and also increase the adaptability of this algorithm.

3.2 Scoring Function Matrix

To determine which maneuver is the “optimal maneuver” that guarantees the advantageous position

after few seconds, every possible consequence of performing various maneuver choices should be

evaluated. The number of maneuver choices of each aircraft is limited to 7. As shown in Fig.3, the 7

choices are steady level flight, pull up, push down, left pull up, left push down and right pull up and

push down, respectively. These maneuvers are selected to resemble the real motion of fighters in air

Fig. 2. Hierarchical Structure of Decision-Making

6

combats as much as possible.

① Left Up② Left Down③ Pull Up④ Level FlightManeuvers are operated in Body axis

⑤ Push Down⑥ Right Up⑦ Right Down

Fig. 3 Maneuver Choices

Since blue and red UCAVs have 7 maneuver choices each, there are total 49 possible maneuver

combinations for both blue and red as shown in Fig.4. The scoring function matrix is used to

systematically conduct the comparisons between each end states. The end states are calculated by

numerically integrating the equations of motion with application of the corresponding maneuver

command. This is the “prediction” step. The prediction is done for 1~2 seconds. The end states are

then scored by the scoring function, which will be addressed in the next section. The scores imply

how much advantage corresponding aircraft has.

Fig. 3. Maneuver Choices

7

RED MNVR Choices

1 2 3 4 5 6 7

BLUE MNVRChoices

1 0.50096 0.4345 0.49813

2 0.56668 0.50075 0.56437

3 0.50221 0.4355 …

4 0.55787 0.47789 …

5 0.49687 0.44113 …

6 0.53704 0.47971 …

7 0.51065 0.42849 …

BLUE Maneuver Choices

1 2 3 4 5 6 7

REDMNVRChoices

1 0.49904 0.43332 0.49779 0.44213 0.50303 0.46296 0.48935

2 0.5655 0.49925 0.5645 0.52211 0.55887

3 0.50187 …

4 0.55744 …

5 0.49616 …

6 0.53647 …

7 0.50906 …

Fig. 4 Scoring Function Matrix The “decision-making” procedure is carried out based on the predicted scores. Security strategy

based on the game theorem is applied in this step to decide the optimal maneuver commands. First,

select the worst score of each column: to figure out the magnitude of severity caused by the enemy’s

performance. Second, select the row including the best case among the selected worst. By using this

min-max algorithm, the airplane is able to avoid the worst case no matter which maneuver the enemy

has chosen. In this paper, both blue and red airplanes are assigned with the same strategy.

3.3 Scoring Function

3.3.1 Orientation Score

The evaluation criterion of the end states mentioned in previous section is the scoring function. The

scoring function is derived from the meaning of gaining air superiority. During WVR air-to-air

combat, the necessary conditions for blue’s air superiority are as follows: (1) Blue has to be located at

the rear of the red, (2) Blue should be heading for the red.

Fig. 4. Scoring Function Matrix

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http://ijass.org

equals to π and vice versa.

8

rBA

bAA

rV

bV

bBA

rAA

Fig. 5 Combat Geometry and Relative Angles between Two Fighters

Above conditions can be expressed using the relative geometry of two aircrafts (see Fig.5). By

definition, the sum of blue’s aspect angle ( BAA ) and red’s bearing angle ( RBA ) equals to and

vice versa.

R B B RAA BA AA BA , (3)

Using this relationship, the scoring function can be derived as follows:

,

,

2 ( )

2 ( )

B BB Ori

R RR Ori

BA AAS

BA AAS

, (4)

Consequently, the scores would have values between 0 and 2. Blue’s score will be 2 when it is on

red’s tail; and 0 when red is on blue’s tail.

3.3.2 Range-Error Score

To maintain the gun range while keeping the safety distance between red and blue, a scoring

function is derived based on the range error. The range error is E safetyR R R , where R means the

distance between blue and red and safetyR represents the safety distance. Following equation is used

to calculate the range error score and the shape is shown in Fig.6.

2 23

2( ) /1 0

1

1 E dR a aRE

E

a aS eR a

(5)

(3)

Using this relationship, the scoring function can be

derived as follows:

8

rBA

bAA

rV

bV

bBA

rAA




vice versa.



,

,

2 ( )

2 ( )

B BB Ori

R RR Ori

BA AAS

BA AAS

, (4)








2 23

2( ) /1 0

1

1 E dR a aRE

E

a aS eR a

(5)

(4)

Consequently, the scores would have values between

0 and 2. Blue’s score will be 2 when it is on red’s tail; and 0

when red is on blue’s tail.


To maintain the gun range while keeping the safety

distance between red and blue, a scoring function is derived

based on the range error. The range error is RE=|R|-Rsafety,

where R means the distance between blue and red and Rsafety

represents the safety distance. Following equation is used

to calculate the range error score and the shape is shown in

Fig.6.

8

rBA

bAA

rV

bV

bBA

rAA




vice versa.



,

,

2 ( )

2 ( )

B BB Ori

R RR Ori

BA AAS

BA AAS

, (4)








2 23

2( ) /1 0

1

1 E dR a aRE

E

a aS eR a

(5) (5)

In the above equation, ad is designed as the most desirable

value of RE where a1 is the vertical asymptote and a0 is the

intercept with x-axis and a3 controls the decay for RE>ad.

The range error score has value between 0 and 1, and 1 is

regarded as the best.

3.3.3 Velocity-Error Score

The desired velocity of aircrafts is maintained at the corner

speed to achieve the maximum turn-rate at every moment.

Using the aerodynamic data of given aircraft model, a V-n

diagram of F-16 pseudo 6-DOF model has plotted (see Fig.7)

in the following condition: altitude of 5000 meters and 10G

of load factor limitation. Since the F-16 model used in this

study has aerodynamic data for only positive angle of attack,

only the upper half of V-n diagram is plotted. The accelerated

stall line in Fig.7 represents the maximum load factor that

an airplane can produce based on airspeed; the desirable

velocity to maintain the maximum turn-rate is derived from

this accelerated stall line.

A velocity scoring function is derived to follow this corner

speed during the combat.

10

Fig. 7. V-n Diagram for F-16 Pseudo 6DOF Model

A velocity scoring function is derived to follow this corner speed during the combat.

des refV

ref

V VS

V

(6)

In above equation, desV represents the desired corner speed at corresponding altitude and refV is

the current velocity. That is, the difference between desired velocity and current velocity becomes the

score. Therefore, unlike other scores, smaller value of VS is more favorable.

The implementation of velocity-error score enables energy conserving maneuvers such as

high-yo-yo or low-yo-yo. Following Fig.8 and Fig.9 show the difference before and after application

of the velocity-error score. Both figures are the resultant trajectory of two simulation cases with same

initial condition and are applied with different scoring functions. In both simulation cases, the initial

velocity of blue is 270 /m s and the initial altitude is 7000m while the calculated corner speed at

7000m is 250 /m s . Two aircrafts are slanted head-on and the red performs level flight. It is obvious

that to achieve air superiority over red, blue should turn around at first.

When only orientation score and range-error score are applied, blue makes turn in lower altitude as

0 50 100 150 200 250 3000

2

4

6

8

10

12

14

16

18V-n Diagram for F-16 pseudo 6-DOF dynamics

Velocity (m/s)

Load

fact

or (G

)

Accelerated Stall LineLimit LoadAOA 15(deg)AOA 20(deg)AOA 30(deg)

(6)

In above equation, Vdes represents the desired corner

speed at corresponding altitude and Vref is the current

velocity. That is, the difference between desired velocity and

current velocity becomes the score. Therefore, unlike other

scores, smaller value of SV is more favorable.

The implementation of velocity-error score enables

energy conserving maneuvers such as high-yo-yo or low-

yo-yo. Following Fig.8 and Fig.9 show the difference before

8

rBA

bAA

rV

bV

bBA

rAA




vice versa.



,

,

2 ( )

2 ( )

B BB Ori

R RR Ori

BA AAS

BA AAS

, (4)








2 23

2( ) /1 0

1

1 E dR a aRE

E

a aS eR a

(5)

Fig. 5. Combat Geometry and Relative Angles between Two Fighters

9

In the above equation, da is designed as the most desirable value of ER where 1a is the

vertical asymptote and 0a is the intercept with x-axis and 3a controls the decay for E dR a . The

range error score has value between 0 and 1, and 1 is regarded as the best.

Fig. 6. Range Error Scoring Function 3.3.3 Velocity-Error Score

The desired velocity of aircrafts is maintained at the corner speed to achieve the maximum turn-rate

at every moment. Using the aerodynamic data of given aircraft model, a V-n diagram of F-16 pseudo

6-DOF model has plotted (see Fig.7) in the following condition: altitude of 5000 meters and 10G of

load factor limitation. Since the F-16 model used in this study has aerodynamic data for only positive

angle of attack, only the upper half of V-n diagram is plotted. The accelerated stall line in Fig.7

represents the maximum load factor that an airplane can produce based on airspeed; the desirable

velocity to maintain the maximum turn-rate is derived from this accelerated stall line.

Fig. 6. Range Error Scoring Function

10


A velocity scoring function is derived to follow this corner speed during the combat.

des refV

ref

V VS

V

(6)

In above equation, desV represents the desired corner speed at corresponding altitude and refV is

the current velocity. That is, the difference between desired velocity and current velocity becomes the

score. Therefore, unlike other scores, smaller value of VS is more favorable.

The implementation of velocity-error score enables energy conserving maneuvers such as

high-yo-yo or low-yo-yo. Following Fig.8 and Fig.9 show the difference before and after application

of the velocity-error score. Both figures are the resultant trajectory of two simulation cases with same

initial condition and are applied with different scoring functions. In both simulation cases, the initial

velocity of blue is 270 /m s and the initial altitude is 7000m while the calculated corner speed at

7000m is 250 /m s . Two aircrafts are slanted head-on and the red performs level flight. It is obvious

that to achieve air superiority over red, blue should turn around at first.

When only orientation score and range-error score are applied, blue makes turn in lower altitude as

0 50 100 150 200 250 3000

2

4

6

8

10

12

14

16

18V-n Diagram for F-16 pseudo 6-DOF dynamics

Velocity (m/s)

Load

fact

or (G

)

Accelerated Stall LineLimit LoadAOA 15(deg)AOA 20(deg)AOA 30(deg)


(204~213)15-100.indd 207 2016-07-05 오후 7:54:37



and after application of the velocity-error score. Both figures

are the resultant trajectory of two simulation cases with

same initial condition and are applied with different scoring

functions. In both simulation cases, the initial velocity of

blue is 270m/s and the initial altitude is 7000m while the

calculated corner speed at 7000m is 250m/s . Two aircrafts

are slanted head-on and the red performs level flight. It is

obvious that to achieve air superiority over red, blue should

turn around at first.

When only orientation score and range-error score

are applied, blue makes turn in lower altitude as in Fig.8.

However, to conserve its energy as fighters do in real air

combat, blue ought to turn with higher altitude as shown in

Fig.9.

3.4 Pilot Behavior

In this paper, the combat situation is classified into four

cases by the combat geometry: Offensive, neutral, defensive

and head-on (see Fig.10). The pilot behavior block makes its

decision based on this combat situation category.

The strategic difference according to pilot behaviors is

revealed when constructing the final scoring functions.

When the behavior is determined as neutral, defensive or

head-on, the scoring function is derived as follows.

12

0 90 180

180

90

Fig. 10. Pilot Behavior Classification

The strategic difference according to pilot behaviors is revealed when constructing the final scoring

functions. When the behavior is determined as neutral, defensive or head-on, the scoring function is

derived as follows.

total Ori VS S S (7)

Since the purpose of range error score is to avoid crashes with opponent or pursuit it, it does not

included in the total score. Velocity score is considered to maintain optimal turn rate.

Otherwise, when the aircraft puts itself in the state of offense, the total score is derived as follows.

total Ori RES S S (8)

RES would operate to shorten the distance when the adversary is too far away and keep gun range

when the distance is too close. Additionally, in the offensive maneuver, the aircraft would follow the

enemy velocity instead of its corner speed to avoid overtaking. A P-D controller is implemented to

generate corresponding thrust commands. This velocity control is named “Tracking mode” in this

study. Fig.11 shows the different gun score cumulative plots for two cases: with and without Tracking

mode. In Fig.11 (Left), the cumulative gun score does not have any constant section after once it goes

into the enemy’s gun range.

(7)

Since the purpose of range error score is to avoid crashes

with opponent or pursuit it, it does not included in the total

score. Velocity score is considered to maintain optimal turn

rate.

Otherwise, when the aircraft puts itself in the state of

offense, the total score is derived as follows.

12

0 90 180

180

90




derived as follows.













(8)

SRE would operate to shorten the distance when the

adversary is too far away and keep gun range when the

distance is too close. Additionally, in the offensive maneuver,

the aircraft would follow the enemy velocity instead of

its corner speed to avoid overtaking. A P-D controller is

implemented to generate corresponding thrust commands.

This velocity control is named “Tracking mode” in this study.

Fig.11 shows the different gun score cumulative plots for two

cases: with and without Tracking mode. In Fig.11 (Left), the

cumulative gun score does not have any constant section

after once it goes into the enemy’s gun range.

3.5 Altitude Barrier Function

To improve the validity of simulation, a barrier function to

11

in Fig.8. However, to conserve its energy as fighters do in real air combat, blue ought to turn with

higher altitude as shown in Fig.9.

Fig. 8. Before implement the Velocity-Error Score

Fig. 9. After implement the Velocity-Error Score

3.4 Pilot Behavior

In this paper, the combat situation is classified into four cases by the combat geometry: Offensive,

neutral, defensive and head-on (see Fig.10). The pilot behavior block makes its decision based on this

combat situation category.

Foe

Friend

Foe

Friend


11

in Fig.8. However, to conserve its energy as fighters do in real air combat, blue ought to turn with

higher altitude as shown in Fig.9.



3.4 Pilot Behavior

In this paper, the combat situation is classified into four cases by the combat geometry: Offensive,

neutral, defensive and head-on (see Fig.10). The pilot behavior block makes its decision based on this

combat situation category.

Foe

Friend

Foe

Friend


12

0 90 180

180

90




derived as follows.














13

Fig. 11. (Left) Tracking Mode Implemented, (Right) Without Tracking Mode


To improve the validity of simulation, a barrier function to protect the aircrafts from ground hitting

is implemented. This algorithm is dependent on two variables, the z-axis velocity ( zV ), and the

altitude of the aircraft ( Z ). The feasible clearance of aircrafts is defined as above an altitude of

500m~1000m and the barrier function is designed to prevent the aircrafts going down lower than the

feasible region. For detailed adjustment of the barrier function, adjustment of the barrier function, two

bounds are defined: named caution bound and the critical bound, respectively. The former is located

on 2000 meters up, and the latter is on around 1000~1200 meters. The operation principle of the

barrier function is as follow: when the aircraft goes down below the caution bound, velocity penalty

vP occurs and the value is defined as the following equation.

/ 1, , 00 ,

z v c zv

V K for Z B VP

otherwise

(9)

In (9), cautionB refers to the caution bound where vK is an arbitrary constant. The value of vK

depends on the maneuverability of aircraft: better maneuverability requires smaller vK . The penalty

vP is valid when not only the caution bound is violated but also the direction of ZV is negative.

That is, the penalty is paid only when the aircraft is heading toward the ground.

If the aircraft continues going down and finally invade the critical bound, another penalty term HP ,

Foe

Friend

Foe

Friend


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209


http://ijass.org

protect the aircrafts from ground hitting is implemented. This

algorithm is dependent on two variables, the z-axis velocity

(Vz), and the altitude of the aircraft (Z). The feasible clearance

of aircrafts is defined as above an altitude of 500m~1000m

and the barrier function is designed to prevent the aircrafts

going down lower than the feasible region. For detailed

adjustment of the barrier function, adjustment of the barrier

function, two bounds are defined: named caution bound

and the critical bound, respectively. The former is located

on 2000 meters up, and the latter is on around 1000~1200

meters. The operation principle of the barrier function is

as follow: when the aircraft goes down below the caution

bound, velocity penalty occurs and the value is defined as

the following equation.

13



To improve the validity of simulation, a barrier function to protect the aircrafts from ground hitting

is implemented. This algorithm is dependent on two variables, the z-axis velocity ( zV ), and the

altitude of the aircraft ( Z ). The feasible clearance of aircrafts is defined as above an altitude of

500m~1000m and the barrier function is designed to prevent the aircrafts going down lower than the

feasible region. For detailed adjustment of the barrier function, adjustment of the barrier function, two

bounds are defined: named caution bound and the critical bound, respectively. The former is located

on 2000 meters up, and the latter is on around 1000~1200 meters. The operation principle of the

barrier function is as follow: when the aircraft goes down below the caution bound, velocity penalty

vP occurs and the value is defined as the following equation.

/ 1, , 00 ,

z v c zv

V K for Z B VP

otherwise

(9)

In (9), cautionB refers to the caution bound where vK is an arbitrary constant. The value of vK

depends on the maneuverability of aircraft: better maneuverability requires smaller vK . The penalty

vP is valid when not only the caution bound is violated but also the direction of ZV is negative.

That is, the penalty is paid only when the aircraft is heading toward the ground.

If the aircraft continues going down and finally invade the critical bound, another penalty term HP ,

Foe

Friend

Foe

Friend

(9)

In (9), Bcaution refers to the caution bound where Kv is

an arbitrary constant. The value of Kv depends on the

maneuverability of aircraft: better maneuverability requires

smaller Kv. The penalty Pv is valid when not only the caution

bound is violated but also the direction of Vz is negative.

That is, the penalty is paid only when the aircraft is heading

toward the ground.

If the aircraft continues going down and finally invade the

critical bound, another penalty term PH, which is dependent

of the height of the aircraft, occurs and defined as follows:

14

which is dependent of the height of the aircraft, occurs and defined as follows:

/ / ,0 ,

H L H LH

Z K B K for Z BP

otherwise

(10)

Here, HK is also a constant and CriticalB means the lower bound. These penalty terms are applied

to the scoring function exponentially right after the relative orientation of two aircrafts is evaluated.

The final scores of the aircrafts are derived as follows:

( )10 v hP Pfinal totalS S (11)

The “decision-making” procedure is conducted using above final scores.

Fig. 9 Application of Altitude Barrier Function

Figure 12 shows how the altitude barrier function works. Only blue aircraft is implemented with the

altitude barrier function in the above figure. Blue and red aircrafts have maneuvered scissors at the

first time; but after they go down below 1500m, the blue aircraft maneuvers to recover the height.

3.6 Moving Horizon Method

Red

Blue

(10)

Here, KH is also a constant and BCritical means the lower

bound. These penalty terms are applied to the scoring

function exponentially right after the relative orientation of

two aircrafts is evaluated. The final scores of the aircrafts are

derived as follows:

14


/ / ,0 ,

H L H LH

Z K B K for Z BP

otherwise

(10)











Red

Blue

(11)

The “decision-making” procedure is conducted using

above final scores.

Figure 12 shows how the altitude barrier function works.

Only blue aircraft is implemented with the altitude barrier

function in the above figure. Blue and red aircrafts have

maneuvered scissors at the first time; but after they go down

below 1500m, the blue aircraft maneuvers to recover the

height.


This method is created based on sliding window

method in order to reduce the simulation time and prevent

unreasonable maneuvering. Since the maneuver choices

defined in Fig.13 include both left turn and right turn, the

algorithm might select radical roll command. Extreme roll

rate command exceeds maneuvering limitation of aircraft

should be rejected for more realistic simulation.

The moving horizon method limits the possible maneuver

choice with regard to the current attitude of the aircraft. The

former maneuver choice history is saved and applied at the

sizing of the window. In this paper, if the aircraft has chosen

left of right direction of maneuver at the very last moment,

the window excludes the opposite side of maneuvers from

choices. Therefore, the size of scoring function matrix is

reduced to 5×7 matrix from 7×7 matrix as circumstance

requires.

The implement of the moving window method also

decreases the calculation time of the algorithm about 71.6%

as predicted.

14


/ / ,0 ,

H L H LH

Z K B K for Z BP

otherwise

(10)











Red

Blue

Fig. 12. Application of Altitude Barrier Function

15

This method is created based on sliding window method in order to reduce the simulation time and

prevent unreasonable maneuvering. Since the maneuver choices defined in Fig.13 include both left

turn and right turn, the algorithm might select radical roll command. Extreme roll rate command

exceeds maneuvering limitation of aircraft should be rejected for more realistic simulation.

The moving horizon method limits the possible maneuver choice with regard to the current attitude

of the aircraft. The former maneuver choice history is saved and applied at the sizing of the window.

In this paper, if the aircraft has chosen left of right direction of maneuver at the very last moment, the

window excludes the opposite side of maneuvers from choices. Therefore, the size of scoring function

matrix is reduced to 5 7 matrix from 7 7 matrix as circumstance requires.

The implement of the moving window method also decreases the calculation time of the algorithm

about 71.6% as predicted.

RED MNVR Choices

1 2 3 4 5 6 7

BLUE MNVRChoices

1 …

2 …

3 ...

4 ...

5 ...

6 ...

7 ...

RED MNVR Choices

1 2 3 4 5 6 7

BLUE MNVRChoices

1 …

2 …

3 ...

4 ...

5 ...

6 ...

7 ...

Fig. 10 Moving Horizon Method

4. Simulations

Fig. 13. Moving Horizon Method

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4. Simulations

4.1 Simulation Parameters

In order to verify the algorithm suggested in this paper,

several simulations are conducted. Simulation parameters

that are used in simulations are listed in Table.1. Detailed

control commands are also listed in Table.4.1. Two different

initial combat conditions are designed in this paper. The

specific conditions are listed in Table.2.

4.2 Simulation Results

First simulation case is the head-on case. Two fighters are

flying in opposite direction and they originally located far

enough to past each other if they do not start chasing. When

the aircrafts approach to each other within the maneuvering

distance, they start chasing each other as shown in Fig.14.

The two star-marks are the start point of each aircraft. The

result is draw as predicted. Two aircrafts are maneuvering

scissors. Fig.15 shows the relative angle history of blue and

red aircrafts. It is shown that two aircrafts has started from

Head-on position and stay in neutral and head on behaviors.

As it is mentioned in the Table.2, the second case is the

tail-chase where blue is advantageous at the beginning. Blue

is not only on the rear of the red aircraft and heading for the

red but also located at the higher altitude than red. In general

WVR air combat, the initial advantage usually has large

influence on the result of combat. Fig.17 represents that the

relative angle history of blue is mostly positioned in offensive

domain while angle history of red is cornered at defensive

domain as predicted.

4.3 Simulation on Different Prediction Time

Prediction time is also a parameter influencing the

performance of this algorithm. The originally set prediction

time is 1 second as written in Table.1. To show the influence

of prediction time on combat result, three more simulations

Table 1. Simulation Parameters and Maneuver Option Sets

16


In order to verify the algorithm suggested in this paper, several simulations are conducted.

Simulation parameters that are used in simulations are listed in Table.1. Detailed control commands

are also listed in Table.4.1. Two different initial combat conditions are designed in this paper. The


Table 1. Simulation Parameters and Maneuver Option Sets Simulation Parameters Maneuver Option Sets #1

Simulation Time 80 sec Max Pull-up Load Factor (G) 10

Prediction Time 1 sec Max Push-down Load Factor (G) -3

Integration Method RK 4th Max Allowed Roll Rate (deg/s) 150

Table 2. Initial Combat Conditions Scenario #1 Scenario #2

Situation Head-on (Fair Game) Tail-Chase (Blue Adv)

NED Pos (m) RED [1000, 50, -7000] [500, 50, -7000]

BLUE [-1000, -50, -7000] [-500, -50, -8000]

Heading (deg) RED 180 0

BLUE 0 0

Velocity (m/s) RED 250 250

BLUE 250 270


First simulation case is the head-on case. Two fighters are flying in opposite direction and they

originally located far enough to past each other if they do not start chasing. When the aircrafts

approach to each other within the maneuvering distance, they start chasing each other as shown in

Fig.14. The two star-marks are the start point of each aircraft. The result is draw as predicted. Two

aircrafts are maneuvering scissors. Fig.15 shows the relative angle history of blue and red aircrafts. It

Table 2. Initial Combat Conditions

16


In order to verify the algorithm suggested in this paper, several simulations are conducted.

Simulation parameters that are used in simulations are listed in Table.1. Detailed control commands

are also listed in Table.4.1. Two different initial combat conditions are designed in this paper. The


Table 1. Simulation Parameters and Maneuver Option Sets Simulation Parameters Maneuver Option Sets #1

Simulation Time 80 sec Max Pull-up Load Factor (G) 10

Prediction Time 1 sec Max Push-down Load Factor (G) -3

Integration Method RK 4th Max Allowed Roll Rate (deg/s) 150

Table 2. Initial Combat Conditions Scenario #1 Scenario #2

Situation Head-on (Fair Game) Tail-Chase (Blue Adv)

NED Pos (m) RED [1000, 50, -7000] [500, 50, -7000]

BLUE [-1000, -50, -7000] [-500, -50, -8000]

Heading (deg) RED 180 0

BLUE 0 0

Velocity (m/s) RED 250 250

BLUE 250 270


First simulation case is the head-on case. Two fighters are flying in opposite direction and they

originally located far enough to past each other if they do not start chasing. When the aircrafts

approach to each other within the maneuvering distance, they start chasing each other as shown in

Fig.14. The two star-marks are the start point of each aircraft. The result is draw as predicted. Two

aircrafts are maneuvering scissors. Fig.15 shows the relative angle history of blue and red aircrafts. It

17

is shown that two aircrafts has started from Head-on position and stay in neutral and head on

behaviors.

Fig. 14. Trajectory: Head-on (Blue ‘-’, Red ‘o’)

Fig. 15. Relative Angle History: Head-on

As it is mentioned in the Table.2, the second case is the tail-chase where blue is advantageous at the

-1500 -1000-500 0 500

1000

-1000-500

0500

10001500

5000

5500

6000

6500

7000

7500

8000

East Y (m)North X (m)

Alti

tude

-Z (m

)

0 20 40 60 80 100 120 140 160 1800

20

40

60

80

100

120

140

160

180

Neutral Defensive

Offensive

Offensive

Head-on

Relative Angle History (Blue)

Aspect Angle (Deg)

Bea

ring

Ang

le (D

eg)

0 20 40 60 80 100 120 140 160 1800

20

40

60

80

100

120

140

160

180

Neutral Defensive

Offensive

Offensive

Head-on

Relative Angle History (Red)

Aspect Angle (Deg)

Bea

ring

Ang

le (D

eg)

Red

Blue


17

is shown that two aircrafts has started from Head-on position and stay in neutral and head on

behaviors.



As it is mentioned in the Table.2, the second case is the tail-chase where blue is advantageous at the

-1500 -1000-500 0 500

1000

-1000-500

0500

10001500

5000

5500

6000

6500

7000

7500

8000


Alti

tude

-Z (m

)

0 20 40 60 80 100 120 140 160 1800

20

40

60

80

100

120

140

160

180

Neutral Defensive

Offensive

Offensive

Head-on


Aspect Angle (Deg)

Bea

ring

Ang

le (D

eg)

0 20 40 60 80 100 120 140 160 1800

20

40

60

80

100

120

140

160

180

Neutral Defensive

Offensive

Offensive

Head-on


Aspect Angle (Deg)

Bea

ring

Ang

le (D

eg)

Red

Blue


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are conducted: 0.5 seconds, 1.5 seconds and 2 seconds of

prediction time for red fighter. The results are shown in Fig.

18 ~ 23. All the simulations are conducted under the Head-

on initial condition.

Not only when the prediction time is shorter(Fig. 18, 19)

but when is longer(Fig. 20 ~ 23), red seems disadvantageous

against blue. From the results, it seems that a certain best

prediction time exists, and it is 1 second here. This occurs

because each fighter holds the decided maneuver for “hold

time”, 0.5 seconds in this paper. The relationship between

hold time and prediction time would be studied in future

research.

5. Conclusion

In this paper, a security strategy algorithm using the scoring

function matrix is developed for automated 3-dimensional

air-to-air combat. Using F-16 pseudo 6-DOF model, scoring

function and scoring matrix are defined and applied to

18

beginning. Blue is not only on the rear of the red aircraft and heading for the red but also located at

the higher altitude than red. In general WVR air combat, the initial advantage usually has large

influence on the result of combat. Fig.17 represents that the relative angle history of blue is mostly

positioned in offensive domain while angle history of red is cornered at defensive domain as predicted.

Fig. 16. Trajectory: Tail-Chase (Blue ‘-’, Red ‘o’)

Fig. 117. Relative Angle History: Tail-Chase 4.3 Simulation on Different Prediction Time

-3000

-2500

-2000

-1500

-1000

-500

0

500

1000-1000 -500 0 500 1000 1500 2000 2500

7000

7500

8000

8500

9000

9500

10000

10500

North X (m)

East Y (m)

Alti

tude

-Z (m

)

0 20 40 60 80 100 120 140 160 1800

20

40

60

80

100

120

140

160

180

Neutral Defensive

Offensive

Offensive

Head-on


Aspect Angle (Deg)

Bea

ring

Ang

le (D

eg)

0 20 40 60 80 100 120 140 160 1800

20

40

60

80

100

120

140

160

180

Neutral Defensive

Offensive

Offensive

Head-on


Aspect Angle (Deg)

Bea

ring

Ang

le (D

eg)

Red

Blue


18

beginning. Blue is not only on the rear of the red aircraft and heading for the red but also located at

the higher altitude than red. In general WVR air combat, the initial advantage usually has large

influence on the result of combat. Fig.17 represents that the relative angle history of blue is mostly

positioned in offensive domain while angle history of red is cornered at defensive domain as predicted.


Fig. 117. Relative Angle History: Tail-Chase 4.3 Simulation on Different Prediction Time

-3000

-2500

-2000

-1500

-1000

-500

0

500

1000-1000 -500 0 500 1000 1500 2000 2500

7000

7500

8000

8500

9000

9500

10000

10500

North X (m)

East Y (m)

Alti

tude

-Z (m

)

0 20 40 60 80 100 120 140 160 1800

20

40

60

80

100

120

140

160

180

Neutral Defensive

Offensive

Offensive

Head-on


Aspect Angle (Deg)

Bea

ring

Ang

le (D

eg)

0 20 40 60 80 100 120 140 160 1800

20

40

60

80

100

120

140

160

180

Neutral Defensive

Offensive

Offensive

Head-on


Aspect Angle (Deg)

Bea

ring

Ang

le (D

eg)

Red

Blue

Fig. 17. Relative Angle History: Tail-Chase

19

Prediction time is also a parameter influencing the performance of this algorithm. The originally set

prediction time is 1 second as written in Table.1. To show the influence of prediction time on combat

result, three more simulations are conducted: 0.5 seconds, 1.5 seconds and 2 seconds of prediction

time for red fighter. The results are shown in Fig. 18 ~ 23. All the simulations are conducted under the

Head-on initial condition.

Fig. 18. Trajectory: Prediction time 0.5 sec (Blue ‘-’, Red ‘o’)

Fig. 19. Relative Angle History: Prediction time 0.5 sec

-10000

10002000

30004000

5000

-1000

0

1000

5000

6000

7000

8000

9000

10000


Alti

tude

-Z (m

)

0 20 40 60 80 100 120 140 160 1800

20

40

60

80

100

120

140

160

180

Neutral Defensive

Offensive

Offensive

Head-on


Aspect Angle (Deg)

Bea

ring

Ang

le (D

eg)

0 20 40 60 80 100 120 140 160 1800

20

40

60

80

100

120

140

160

180

Neutral Defensive

Offensive

Offensive

Head-on


Aspect Angle (Deg)

Bea

ring

Ang

le (D

eg)

Red

Blue


19

Prediction time is also a parameter influencing the performance of this algorithm. The originally set

prediction time is 1 second as written in Table.1. To show the influence of prediction time on combat

result, three more simulations are conducted: 0.5 seconds, 1.5 seconds and 2 seconds of prediction

time for red fighter. The results are shown in Fig. 18 ~ 23. All the simulations are conducted under the

Head-on initial condition.



-10000

10002000

30004000

5000

-1000

0

1000

5000

6000

7000

8000

9000

10000


Alti

tude

-Z (m

)

0 20 40 60 80 100 120 140 160 1800

20

40

60

80

100

120

140

160

180

Neutral Defensive

Offensive

Offensive

Head-on


Aspect Angle (Deg)

Bea

ring

Ang

le (D

eg)

0 20 40 60 80 100 120 140 160 1800

20

40

60

80

100

120

140

160

180

Neutral Defensive

Offensive

Offensive

Head-on


Aspect Angle (Deg)

Bea

ring

Ang

le (D

eg)

Red

Blue


(204~213)15-100.indd 211 2016-07-05 오후 7:54:43



generate the maneuver command to conduct WVR combat

successfully. Altitude barrier function and tracking mode is

implemented to improve the validity and performance of the

combat model. Moving horizon method is also introduced to

refine the algorithm.

Two engagement scenarios of head-on, tail-chasing

of blue are investigated to check the performance of the

developed model and proposed algorithms. The verification

of combat model has done under the controlled computation

environment and the combat results and maneuvers are

similar to those of real air combat.

Studies on air-to-air combat between different maneuver

algorithms would be helpful for algorithm improvement.

Acknowledgement

This research was supported by Agency for Defense

Development as part of the research project ‘A study on UAV

Maneuver Generation Algorithm for Air-to-Air Combat’

20



-2000

-1500

-1000

-500

0

-1000

-500

0

500

1000

6000

6500

7000

7500

8000


Alti

tude

-Z (m

)

0 20 40 60 80 100 120 140 160 1800

20

40

60

80

100

120

140

160

180

Neutral Defensive

Offensive

Offensive

Head-on


Aspect Angle (Deg)

Bea

ring

Ang

le (D

eg)

0 20 40 60 80 100 120 140 160 1800

20

40

60

80

100

120

140

160

180

Neutral Defensive

Offensive

Offensive

Head-on


Aspect Angle (Deg)

Bea

ring

Ang

le (D

eg)

Red

Blue


21

Fig. 22. Trajectory: Prediction time 2 sec (Blue ‘-’, Red ‘o’)

Fig. 23. Relative Angle History: Prediction time 2 sec

Not only when the prediction time is shorter(Fig. 18, 19) but when is longer(Fig. 20 ~ 23), red

seems disadvantageous against blue. From the results, it seems that a certain best prediction time

exists, and it is 1 second here. This occurs because each fighter holds the decided maneuver for “hold

time”, 0.5 seconds in this paper. The relationship between hold time and prediction time would be

studied in future research.

-1500

-1000

-500

0

-1000

-500

0

500

1000

1500

5600

5800

6000

6200

6400

6600

6800

7000

7200

East Y (m)

North X (m)

Alti

tude

-Z (m

)

0 20 40 60 80 100 120 140 160 1800

20

40

60

80

100

120

140

160

180

Neutral Defensive

Offensive

Offensive

Head-on


Aspect Angle (Deg)

Bea

ring

Ang

le (D

eg)

0 20 40 60 80 100 120 140 160 1800

20

40

60

80

100

120

140

160

180

Neutral Defensive

Offensive

Offensive

Head-on


Aspect Angle (Deg)

Bea

ring

Ang

le (D

eg)

Red

Blue


21



Not only when the prediction time is shorter(Fig. 18, 19) but when is longer(Fig. 20 ~ 23), red

seems disadvantageous against blue. From the results, it seems that a certain best prediction time

exists, and it is 1 second here. This occurs because each fighter holds the decided maneuver for “hold

time”, 0.5 seconds in this paper. The relationship between hold time and prediction time would be

studied in future research.

-1500

-1000

-500

0

-1000

-500

0

500

1000

1500

5600

5800

6000

6200

6400

6600

6800

7000

7200

East Y (m)

North X (m)

Alti

tude

-Z (m

)

0 20 40 60 80 100 120 140 160 1800

20

40

60

80

100

120

140

160

180

Neutral Defensive

Offensive

Offensive

Head-on


Aspect Angle (Deg)

Bea

ring

Ang

le (D

eg)

0 20 40 60 80 100 120 140 160 1800

20

40

60

80

100

120

140

160

180

Neutral Defensive

Offensive

Offensive

Head-on


Aspect Angle (Deg)

Bea

ring

Ang

le (D

eg)

Red

Blue


20



-2000

-1500

-1000

-500

0

-1000

-500

0

500

1000

6000

6500

7000

7500

8000


Alti

tude

-Z (m

)

0 20 40 60 80 100 120 140 160 1800

20

40

60

80

100

120

140

160

180

Neutral Defensive

Offensive

Offensive

Head-on


Aspect Angle (Deg)

Bea

ring

Ang

le (D

eg)

0 20 40 60 80 100 120 140 160 1800

20

40

60

80

100

120

140

160

180

Neutral Defensive

Offensive

Offensive

Head-on


Aspect Angle (Deg)

Bea

ring

Ang

le (D

eg)

Red

Blue


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under contract UD130041JD.

References

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[3] Jang, S., “Study of Intelligent Pilot Model Based on

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brawler

[5] Lazarus, E., “The Application of Value-Driven Decision-

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Piscataway, NJ, Vol. 3, 1997, pp. 2302-2307.

[6] Sprinkle, J., Eklund, M., Kim, H. and Sastry, S., “Encoding

Aerial Pursuit/Evasion Games with Fixed Wing Aircraft into a

Nonlinear Model Predictive Tracking Controller”, 43rd IEEE

Conference on Decision and Control, Bahamas, 2004.

[7] Sprinkle, J., Eklund, M. and Sastry, S., “Implementing

and Testing a Nonlinear Model Predictive Tracking

Controller for Aerial Pursuit/Evasion Games on a Fixed Wing

Aircraft”, 2005 American Control Conference, Portland, OR,

2005.

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