This short seminar presentation discusses the basic idea of my dissertation. It uncovers the main ideas of a three players conflict in missile guidance with bounded controls.

1. Missile Guidance
A Game of Two Players
Aircraft Countermeasures
A Game of Three Players
Various Guidance Strategies
Conclusions
A Three Player Pursuit and Evasion Conflict
Doctoral Dissertation
Sergey Rubinsky
under the supervision of
Prof. Shaul Gutman
Department of Mechanical Engineering
Technion – Israel Institute of Technology
March 14, 2015
Sergey Rubinsky under the supervision of Prof. Shaul Gutman A Three Player Pursuit and Evasion Conflict

2. Missile Guidance
A Game of Two Players
Aircraft Countermeasures
A Game of Three Players
Various Guidance Strategies
Conclusions
Outline
1 Missile Guidance
2 A Game of Two Players
Game Definition
Game Model
Game Solution
3 Aircraft Countermeasures
4 A Game of Three Players
Game Definition
Game Model
Game Solution
5 Various Guidance Strategies
1st
Order tgo Based VG
4th
Order tgo Based VG
Transformed VG
Non-ideal Players VG
6 Conclusions
Sergey Rubinsky under the supervision of Prof. Shaul Gutman A Three Player Pursuit and Evasion Conflict

3. Missile Guidance
A Game of Two Players
Aircraft Countermeasures
A Game of Three Players
Various Guidance Strategies
Conclusions
Outline
1 Missile Guidance
2 A Game of Two Players
Game Definition
Game Model
Game Solution
3 Aircraft Countermeasures
4 A Game of Three Players
Game Definition
Game Model
Game Solution
5 Various Guidance Strategies
1st
Order tgo Based VG
4th
Order tgo Based VG
Transformed VG
Non-ideal Players VG
6 Conclusions
Sergey Rubinsky under the supervision of Prof. Shaul Gutman A Three Player Pursuit and Evasion Conflict

4. Missile Guidance
A Game of Two Players
Aircraft Countermeasures
A Game of Three Players
Various Guidance Strategies
Conclusions
What is Missile Guidance?
Missile guidance refers to a variety of methods of guiding a missile
or a guided bomb to its intended target. The classical guidance
problems deal with two players: the missile and the target.
Missile
Target
0 1000 2000 3000 4000 5000 6000
0
200
400
600
x [m]
y[m]
Miss Distance = 0.415 [m]
Sergey Rubinsky under the supervision of Prof. Shaul Gutman A Three Player Pursuit and Evasion Conflict

5. Missile Guidance
A Game of Two Players
Aircraft Countermeasures
A Game of Three Players
Various Guidance Strategies
Conclusions
Example: “Arrow”interception scenario
E.g. Consider the“Arrow”missile interception scenario
Sergey Rubinsky under the supervision of Prof. Shaul Gutman A Three Player Pursuit and Evasion Conflict

6. Missile Guidance
A Game of Two Players
Aircraft Countermeasures
A Game of Three Players
Various Guidance Strategies
Conclusions
Example 2: “Iron Dome”interception scenario
Another example is the“Iron Dome”.
Sergey Rubinsky under the supervision of Prof. Shaul Gutman A Three Player Pursuit and Evasion Conflict

7. Missile Guidance
A Game of Two Players
Aircraft Countermeasures
A Game of Three Players
Various Guidance Strategies
Conclusions
Example 3: “Air-to-Air”missile
An air-to-air missile is launched from an airborne vehicle, and
intercepts another airborne vehicle.
Sergey Rubinsky under the supervision of Prof. Shaul Gutman A Three Player Pursuit and Evasion Conflict

8. Missile Guidance
A Game of Two Players
Aircraft Countermeasures
A Game of Three Players
Various Guidance Strategies
Conclusions
Game Definition
Game Model
Game Solution
Outline
1 Missile Guidance
2 A Game of Two Players
Game Definition
Game Model
Game Solution
3 Aircraft Countermeasures
4 A Game of Three Players
Game Definition
Game Model
Game Solution
5 Various Guidance Strategies
1st
Order tgo Based VG
4th
Order tgo Based VG
Transformed VG
Non-ideal Players VG
6 Conclusions
Sergey Rubinsky under the supervision of Prof. Shaul Gutman A Three Player Pursuit and Evasion Conflict

9. Missile Guidance
A Game of Two Players
Aircraft Countermeasures
A Game of Three Players
Various Guidance Strategies
Conclusions
Game Definition
Game Model
Game Solution
Outline
1 Missile Guidance
2 A Game of Two Players
Game Definition
Game Model
Game Solution
3 Aircraft Countermeasures
4 A Game of Three Players
Game Definition
Game Model
Game Solution
5 Various Guidance Strategies
1st
Order tgo Based VG
4th
Order tgo Based VG
Transformed VG
Non-ideal Players VG
6 Conclusions
Sergey Rubinsky under the supervision of Prof. Shaul Gutman A Three Player Pursuit and Evasion Conflict

10. Missile Guidance
A Game of Two Players
Aircraft Countermeasures
A Game of Three Players
Various Guidance Strategies
Conclusions
Game Definition
Game Model
Game Solution
A Two Players Interception Scenario
Given a two players interception scenario
aM
VM
M
aT
VT
T
γM λ
γT
rMT
where M and T stand for the missile and the target.
Sergey Rubinsky under the supervision of Prof. Shaul Gutman A Three Player Pursuit and Evasion Conflict

11. Missile Guidance
A Game of Two Players
Aircraft Countermeasures
A Game of Three Players
Various Guidance Strategies
Conclusions
Game Definition
Game Model
Game Solution
Simple Differential Game with Bounded Controls
aM
VM
M
aT
VT
T
γM λ
γT
rMT
The players can apply any bounded accelerations, aM = u and
aT = v, such that u ≤ ρu and v ≤ ρv. Given the initial
conditions, and the capabilities, ρu and ρv, find the optimal
strategies, u∗ and v∗, with respect to the saddle point inequality
JMT u∗
, v ≤ JMT u∗
, v∗
≤ JMT u, v∗
)
where JMT = rMT (tf ) is the miss-distance between the missile
and the target, and tf is the final time. Namely, the missile is the
minimizer and the target is the maximizer of the miss-distance.
Sergey Rubinsky under the supervision of Prof. Shaul Gutman A Three Player Pursuit and Evasion Conflict

12. Missile Guidance
A Game of Two Players
Aircraft Countermeasures
A Game of Three Players
Various Guidance Strategies
Conclusions
Game Definition
Game Model
Game Solution
Outline
1 Missile Guidance
2 A Game of Two Players
Game Definition
Game Model
Game Solution
3 Aircraft Countermeasures
4 A Game of Three Players
Game Definition
Game Model
Game Solution
5 Various Guidance Strategies
1st
Order tgo Based VG
4th
Order tgo Based VG
Transformed VG
Non-ideal Players VG
6 Conclusions
Sergey Rubinsky under the supervision of Prof. Shaul Gutman A Three Player Pursuit and Evasion Conflict

13. Missile Guidance
A Game of Two Players
Aircraft Countermeasures
A Game of Three Players
Various Guidance Strategies
Conclusions
Game Definition
Game Model
Game Solution
Equations of Motion
For simplicity, assume that each player can instantly produce any
acceleration of a limited magnitude, u ≤ ρu and v ≤ ρv.
Denote rMT = rT − rM as the relative distance between the
missile and the target, and VMT = VT − VM as the relative
velocity between them. Thus, we obtain the kinematic equation,
¨rMT (t) = v(t) − u(t) (1)
Hence, the open loop block diagram is
1
s
· I3
1
s
· I3
u
v
˙rMT rMT
Sergey Rubinsky under the supervision of Prof. Shaul Gutman A Three Player Pursuit and Evasion Conflict

14. Missile Guidance
A Game of Two Players
Aircraft Countermeasures
A Game of Three Players
Various Guidance Strategies
Conclusions
Game Definition
Game Model
Game Solution
State Space Realization
1
s
· I3
1
s
· I3
aM
aT
˙rMT rMT
By order reduction,
˙rMT (t) = VMT (t) (2)
˙VMT (t) = v(t) − u(t) (3)
the following state space realization is obtained
˙rMT (t)
˙VMT (t)
˙x(t)
=
03 I3
03 03
A
rMT (t)
VMT (t)
x(t)
+
0
−I3
B
u(t) +
0
I3
C
v(t) (4)
Sergey Rubinsky under the supervision of Prof. Shaul Gutman A Three Player Pursuit and Evasion Conflict

15. Missile Guidance
A Game of Two Players
Aircraft Countermeasures
A Game of Three Players
Various Guidance Strategies
Conclusions
Game Definition
Game Model
Game Solution
Zero-Effort-Miss Transformation
The cost function of the game is the miss distance,
JMT = rMT (tf ) = I3 03 x(tf ) = g x(tf ) (5)
where x(t) =
rMT (t)
VMT (t)
is the system’s state vector, and tf is the
final time. Define the Zero-Effort-Miss (ZEM) variable
yMT (t) = gΦ(tf , t)x(t) (6)
where Φ(tf , t) = eA(tf −t) is the transition matrix of A, which
satisfies
˙Φ(tf , t) = −Φ(tf , t)A , Φ(tf , tf ) = I (7)
Sergey Rubinsky under the supervision of Prof. Shaul Gutman A Three Player Pursuit and Evasion Conflict

16. Missile Guidance
A Game of Two Players
Aircraft Countermeasures
A Game of Three Players
Various Guidance Strategies
Conclusions
Game Definition
Game Model
Game Solution
Zero-Effort-Miss Transformation (Contd.)
Differentiate the ZEM and obtain,
˙yMT (t) = XMT (tf , t)u(t) + YMT (tf , t)v(t) (8)
where,
XMT (tf , t) = gΦ(tf , t)B = −tgoI3 (9)
YMT (tf , t) = gΦ(tf , t)C = tgoI3 (10)
where tgo = tf − t, and the explicit form of the ZEM is
yMT (t) = rMT (t) + tgoVMT (t). Therefore, we obtain a reduced
state space model in the ZEM space,
˙yMT (t) = −tgou(t) + tgov(t)
JMT = yMT (tf )
(11)
Sergey Rubinsky under the supervision of Prof. Shaul Gutman A Three Player Pursuit and Evasion Conflict

17. Missile Guidance
A Game of Two Players
Aircraft Countermeasures
A Game of Three Players
Various Guidance Strategies
Conclusions
Game Definition
Game Model
Game Solution
Outline
1 Missile Guidance
2 A Game of Two Players
Game Definition
Game Model
Game Solution
3 Aircraft Countermeasures
4 A Game of Three Players
Game Definition
Game Model
Game Solution
5 Various Guidance Strategies
1st
Order tgo Based VG
4th
Order tgo Based VG
Transformed VG
Non-ideal Players VG
6 Conclusions
Sergey Rubinsky under the supervision of Prof. Shaul Gutman A Three Player Pursuit and Evasion Conflict

18. Missile Guidance
A Game of Two Players
Aircraft Countermeasures
A Game of Three Players
Various Guidance Strategies
Conclusions
Game Definition
Game Model
Game Solution
Simple Differential Game Solution
Define, VMT (t) = yMT (t) . Differentiate VMT (t) with respect to
t, and obtain,
˙VMT (t) = −tgo
yMT (t)
yMT (t)
(u(t) − v(t)) (12)
Thus, the optimal controllers are,
u∗
(t)= ρu
yMT (t)
yMT (t)
= ρu
rMT (t) + tgoVMT (t)
rMT (t) + tgoVMT (t)
(13)
v∗
(t)= ρv
yMT (t)
yMT (t)
= ρv
rMT (t) + tgoVMT (t)
rMT (t) + tgoVMT (t)
(14)
Integration yields the optimal game solution,
y∗
MT (t) = JMT +
1
2
(ρu − ρv) t2
go (15)
Sergey Rubinsky under the supervision of Prof. Shaul Gutman A Three Player Pursuit and Evasion Conflict

19. Missile Guidance
A Game of Two Players
Aircraft Countermeasures
A Game of Three Players
Various Guidance Strategies
Conclusions
Game Definition
Game Model
Game Solution
Optimal Trajectories in the ZEM space
y∗
(t) = J +
1
2
(ρu − ρv) t2
go
If ρu > ρv, we have the following optimal ZEM trajectories,
tgo
||y(tgo)||
Sergey Rubinsky under the supervision of Prof. Shaul Gutman A Three Player Pursuit and Evasion Conflict

20. Missile Guidance
A Game of Two Players
Aircraft Countermeasures
A Game of Three Players
Various Guidance Strategies
Conclusions
Game Definition
Game Model
Game Solution
Optimal Trajectories in the ZEM space (contd.)
y∗
(t) = J +
1
2
(ρu − ρv) t2
go
If ρu < ρv, the following optimal trajectories are obtained,
tgo
||y(tgo)||
Sergey Rubinsky under the supervision of Prof. Shaul Gutman A Three Player Pursuit and Evasion Conflict

21. Missile Guidance
A Game of Two Players
Aircraft Countermeasures
A Game of Three Players
Various Guidance Strategies
Conclusions
Game Definition
Game Model
Game Solution
Game Properties
The obtained strategies are non-linear.
If ρu > ρv, then a zero miss distance can be guaranteed by
the interceptor for every reasonable initial condition.
If ρu < ρv, then a miss distance J > 0 can be guaranteed by
the target.
If ρu > ρv, then a singular area is formed under the trajectory
y∗ = 1
2 (ρu − ρv) t2
go. In that area, the optimal strategies
are arbitrary and the cost is J = 0.
tgo
||y(tgo)||
Sergey Rubinsky under the supervision of Prof. Shaul Gutman A Three Player Pursuit and Evasion Conflict

22. Missile Guidance
A Game of Two Players
Aircraft Countermeasures
A Game of Three Players
Various Guidance Strategies
Conclusions
Game Definition
Game Model
Game Solution
A small problem...
If the missile has a greater maneuver capability, it can always
intercept its target.
Missile
Target
0 1000 2000 3000 4000 5000 6000
0
200
400
600
x [m]
y[m]
Miss Distance = 0.415 [m]
Thus, advanced countermeasures must be used, if one wants to
keep an aircraft safe from an attacking missile.
Sergey Rubinsky under the supervision of Prof. Shaul Gutman A Three Player Pursuit and Evasion Conflict

23. Missile Guidance
A Game of Two Players
Aircraft Countermeasures
A Game of Three Players
Various Guidance Strategies
Conclusions
Outline
1 Missile Guidance
2 A Game of Two Players
Game Definition
Game Model
Game Solution
3 Aircraft Countermeasures
4 A Game of Three Players
Game Definition
Game Model
Game Solution
5 Various Guidance Strategies
1st
Order tgo Based VG
4th
Order tgo Based VG
Transformed VG
Non-ideal Players VG
6 Conclusions
Sergey Rubinsky under the supervision of Prof. Shaul Gutman A Three Player Pursuit and Evasion Conflict

24. Missile Guidance
A Game of Two Players
Aircraft Countermeasures
A Game of Three Players
Various Guidance Strategies
Conclusions
Flare
A (decoy) flare is an aerial infrared countermeasure used by a plane
or helicopter to counter an infrared homing (”heat-seeking”)
surface-to-air missile or air-to-air missile.
Sergey Rubinsky under the supervision of Prof. Shaul Gutman A Three Player Pursuit and Evasion Conflict

25. Missile Guidance
A Game of Two Players
Aircraft Countermeasures
A Game of Three Players
Various Guidance Strategies
Conclusions
Chaff
A Chaff is a radar countermeasure in which aircraft or other targets
spread a cloud of small, thin pieces of aluminum, metalized glass
fiber, or plastic, which either appears as a cluster of primary targets
on radar screens or swamps the screen with multiple returns.
Sergey Rubinsky under the supervision of Prof. Shaul Gutman A Three Player Pursuit and Evasion Conflict

26. Missile Guidance
A Game of Two Players
Aircraft Countermeasures
A Game of Three Players
Various Guidance Strategies
Conclusions
Infrared Countermeasure (ICRM)
IRCM systems are based on modulated source of infrared radiation
with a higher intensity than the target. When this modulated
radiation is seen by a missile seeker, it overwhelms the modulated
signal from the aircraft and provides incorrect steering cues to the
missile.
Sergey Rubinsky under the supervision of Prof. Shaul Gutman A Three Player Pursuit and Evasion Conflict

27. Missile Guidance
A Game of Two Players
Aircraft Countermeasures
A Game of Three Players
Various Guidance Strategies
Conclusions
Laser Countermeasure
This countermeasure is designed to fire a laser beam in order to
corrupt the guidance system of the attacking missile.
Sergey Rubinsky under the supervision of Prof. Shaul Gutman A Three Player Pursuit and Evasion Conflict

28. Missile Guidance
A Game of Two Players
Aircraft Countermeasures
A Game of Three Players
Various Guidance Strategies
Conclusions
Defending Missile Countermeasure
By using this countermeasure, the target releases a short range
defending missile (Defender) in order to intercept the attacking
missile.
Sergey Rubinsky under the supervision of Prof. Shaul Gutman A Three Player Pursuit and Evasion Conflict

29. Missile Guidance
A Game of Two Players
Aircraft Countermeasures
A Game of Three Players
Various Guidance Strategies
Conclusions
Defending Missile Countermeasure (contd.)
When using a defender, the target has two countermeasures
against the attacking missile:
Available Countermeasures
The target can maneuver in order to evade the attacking
missile or lead it to collision with the defender.
The target can use the defender in order to intercept the
attacking missile, or not to let it get to a collision course with
the target.
As a result, we obtain a game of three players: The missile, the
target, and the defender.
Sergey Rubinsky under the supervision of Prof. Shaul Gutman A Three Player Pursuit and Evasion Conflict

30. Missile Guidance
A Game of Two Players
Aircraft Countermeasures
A Game of Three Players
Various Guidance Strategies
Conclusions
Game Definition
Game Model
Game Solution
Outline
1 Missile Guidance
2 A Game of Two Players
Game Definition
Game Model
Game Solution
3 Aircraft Countermeasures
4 A Game of Three Players
Game Definition
Game Model
Game Solution
5 Various Guidance Strategies
1st
Order tgo Based VG
4th
Order tgo Based VG
Transformed VG
Non-ideal Players VG
6 Conclusions
Sergey Rubinsky under the supervision of Prof. Shaul Gutman A Three Player Pursuit and Evasion Conflict

31. Missile Guidance
A Game of Two Players
Aircraft Countermeasures
A Game of Three Players
Various Guidance Strategies
Conclusions
Game Definition
Game Model
Game Solution
Outline
1 Missile Guidance
2 A Game of Two Players
Game Definition
Game Model
Game Solution
3 Aircraft Countermeasures
4 A Game of Three Players
Game Definition
Game Model
Game Solution
5 Various Guidance Strategies
1st
Order tgo Based VG
4th
Order tgo Based VG
Transformed VG
Non-ideal Players VG
6 Conclusions
Sergey Rubinsky under the supervision of Prof. Shaul Gutman A Three Player Pursuit and Evasion Conflict

32. Missile Guidance
A Game of Two Players
Aircraft Countermeasures
A Game of Three Players
Various Guidance Strategies
Conclusions
Game Definition
Game Model
Game Solution
Three Players Interception Scenario
u
VM
rM
v
VT
rT
w
VD
rD
rMT
rMD
rTD
In this scenario, the Missile (M), the Target (T) and the Defender
(D) can apply a bounded acceleration of u, v and w respectively.
Sergey Rubinsky under the supervision of Prof. Shaul Gutman A Three Player Pursuit and Evasion Conflict

33. Missile Guidance
A Game of Two Players
Aircraft Countermeasures
A Game of Three Players
Various Guidance Strategies
Conclusions
Game Definition
Game Model
Game Solution
A Game of Three Ideal Players
Let all players have bounded magnitude accelerations, such that,
u ≤ ρu, v ≤ ρv, and w ≤ ρw. Also define m and as the
M-T and M-D interception miss distances; namely, the minimal
required distances for interception. The task of this research is to
Find a control strategy u∗ that enables the missile to
guarantee a M-D miss distance rMD tMD
f ≥ , and
guarantee a M-T miss distance rMT tMT
f ≤ m, for
tMD
f < tMT
f ≤ tb, where tb is the missile’s engine burning
time. Also, algebraic conditions under which this scenario is
possible must also be found.
Find the optimal controllers v∗ and w∗, that make the missile
pay the highest price for using u∗.
Sergey Rubinsky under the supervision of Prof. Shaul Gutman A Three Player Pursuit and Evasion Conflict

34. Missile Guidance
A Game of Two Players
Aircraft Countermeasures
A Game of Three Players
Various Guidance Strategies
Conclusions
Game Definition
Game Model
Game Solution
Outline
1 Missile Guidance
2 A Game of Two Players
Game Definition
Game Model
Game Solution
3 Aircraft Countermeasures
4 A Game of Three Players
Game Definition
Game Model
Game Solution
5 Various Guidance Strategies
1st
Order tgo Based VG
4th
Order tgo Based VG
Transformed VG
Non-ideal Players VG
6 Conclusions
Sergey Rubinsky under the supervision of Prof. Shaul Gutman A Three Player Pursuit and Evasion Conflict

35. Missile Guidance
A Game of Two Players
Aircraft Countermeasures
A Game of Three Players
Various Guidance Strategies
Conclusions
Game Definition
Game Model
Game Solution
Equations of Motion
Define rM , rT , rD and VM , VT , VD as the positions and the
velocities of the Missile, the Target and the Defender respectively.
Assuming ideal players, we have the following kinematic equations
˙rMT (t) = VMT (t) (16)
˙rMD(t) = VMD(t) (17)
˙VMT (t) = v(t) − u(t) (18)
˙VMD(t) = w(t) − u(t) (19)
where rMT and rMD are the M-T and M-D relative positions, and
VMT and VMD are the M-T and M-D relative velocities.
Sergey Rubinsky under the supervision of Prof. Shaul Gutman A Three Player Pursuit and Evasion Conflict

36. Missile Guidance
A Game of Two Players
Aircraft Countermeasures
A Game of Three Players
Various Guidance Strategies
Conclusions
Game Definition
Game Model
Game Solution
State Space Realization
Therefore, the state space realization becomes,




˙rMT
˙VMT
˙rMD
˙VMD



 =




0 In 0 0
0 0 0 0
0 0 0 In
0 0 0 0








rMT
VMT
rMD
VMD



+




0
−In
0
−In



 u+




0
In
0
0



 v+




0
0
0
In



 w
and the open loop block diagram is,
1
s
· I3
1
s
· I3
1
s
· I3
1
s
· I3
u
w
v
VMD
VMT
rMD
rMT
Sergey Rubinsky under the supervision of Prof. Shaul Gutman A Three Player Pursuit and Evasion Conflict

37. Missile Guidance
A Game of Two Players
Aircraft Countermeasures
A Game of Three Players
Various Guidance Strategies
Conclusions
Game Definition
Game Model
Game Solution
Zero-Effort-Miss Transformation
Define two terminal cost functions, with fixed final times
JMT = rMT tMT
f = In 0 0 0 x tMT
f = gx tMT
f
JMD = rMD tMD
f = 0 0 In 0 x tMD
f = hx tMD
f
and two ZEM variables,
yMT (t)= gΦ tMT
f , t x(t) (20)
yMD(t)= hΨ tMD
f , t x(t) (21)
where Φ tMT
f , t = eA tMT
f −t
, and Ψ tMD
f , t = eA tMD
f −t
.
Sergey Rubinsky under the supervision of Prof. Shaul Gutman A Three Player Pursuit and Evasion Conflict

38. Missile Guidance
A Game of Two Players
Aircraft Countermeasures
A Game of Three Players
Various Guidance Strategies
Conclusions
Game Definition
Game Model
Game Solution
Zero-Effort-Miss Transformation (Contd.)
For ideal players we obtain the following systems in the ZEM
space. The M-D system,
˙yMD(t) = tMD
go (−u(t) + w(t)) (22)
JMD = yMD tMD
f (23)
and the M-T system
˙yMT (t) = tMT
go (−u(t) + v(t)) (24)
JMT = yMT tMT
f (25)
Sergey Rubinsky under the supervision of Prof. Shaul Gutman A Three Player Pursuit and Evasion Conflict

39. Missile Guidance
A Game of Two Players
Aircraft Countermeasures
A Game of Three Players
Various Guidance Strategies
Conclusions
Game Definition
Game Model
Game Solution
Outline
1 Missile Guidance
2 A Game of Two Players
Game Definition
Game Model
Game Solution
3 Aircraft Countermeasures
4 A Game of Three Players
Game Definition
Game Model
Game Solution
5 Various Guidance Strategies
1st
Order tgo Based VG
4th
Order tgo Based VG
Transformed VG
Non-ideal Players VG
6 Conclusions
Sergey Rubinsky under the supervision of Prof. Shaul Gutman A Three Player Pursuit and Evasion Conflict

40. Missile Guidance
A Game of Two Players
Aircraft Countermeasures
A Game of Three Players
Various Guidance Strategies
Conclusions
Game Definition
Game Model
Game Solution
Optimal Strategies
Define VMD = yMD , and VMT (t) = yMT . Differentiate, and
obtain
˙VMD(t) = −tMD
go
yMD(t)
yMD(t)
(u(t) − w(t)) (26)
˙VMT (t) = tMT
go
yMT (t)
yMT (t)
(u(t) − v(t)) (27)
The target maximizes yMT ; therefore, its optimal strategy is
v∗
= ρv
yMT
yMT
= ρv
rMT + tMT
go VMT
rMT + tMT
go VMT
(28)
The defender minimizes yMD , thus
w∗
= −ρw
yMD
yMD
= −ρw
rMD + tMD
go VMD
rMD + tMD
go VMD
(29)
Sergey Rubinsky under the supervision of Prof. Shaul Gutman A Three Player Pursuit and Evasion Conflict

41. Missile Guidance
A Game of Two Players
Aircraft Countermeasures
A Game of Three Players
Various Guidance Strategies
Conclusions
Game Definition
Game Model
Game Solution
Optimal Strategies (Contd.)
The missile has two objectives: The optimal evasion law that
maximizes yMD is,
u∗
e = −ρu
yMD
yMD
= −ρu
rMD + tMD
go VMD
rMD + tMD
go VMD
(30)
and the optimal pursuit law that minimizes yMT is,
u∗
p = ρu
yMT
yMT
= ρu
rMT + tMT
go VMT
rMT + tMT
go VMT
(31)
Assuming the worst case (for the missile), where yMT
yMT
= yMD
yMD
,
we have u∗
e = −u∗
p. Hence, The missile’s optimal evasive and
pursuit maneuvers are opposite.
Sergey Rubinsky under the supervision of Prof. Shaul Gutman A Three Player Pursuit and Evasion Conflict

42. Missile Guidance
A Game of Two Players
Aircraft Countermeasures
A Game of Three Players
Various Guidance Strategies
Conclusions
Game Definition
Game Model
Game Solution
Game Bounds
Integration yields the bound functions
A tMD
go = −
1
2
(ρu − ρw) tMD
go
2
(32)
B tMT
go = m +
1
2
(ρu − ρv) tMT
go
2
(33)
where and m are the M-D and M-T interception miss distances.
ℓ m
(t)
ℬ(t)
tf
MD
tf
MT
Time, t
||ZEM||
Sergey Rubinsky under the supervision of Prof. Shaul Gutman A Three Player Pursuit and Evasion Conflict

43. Missile Guidance
A Game of Two Players
Aircraft Countermeasures
A Game of Three Players
Various Guidance Strategies
Conclusions
Game Definition
Game Model
Game Solution
Game Bounds (Contd.)
ℓ m
(t)
ℬ(t)
tf
MD
tf
MT
Time, t
||ZEM||
Namely, in order to evade the defender and intercept the target,
the missile must keep yMD outside the area bounded by A(t),
and yMT inside the area bounded by B(t). Let us keep in mind
that u∗
e = −u∗
p; thus, by increasing yMD the missile also
increases yMT , and vice-versa.
Sergey Rubinsky under the supervision of Prof. Shaul Gutman A Three Player Pursuit and Evasion Conflict

44. Missile Guidance
A Game of Two Players
Aircraft Countermeasures
A Game of Three Players
Various Guidance Strategies
Conclusions
Game Definition
Game Model
Game Solution
Fail-safe function: C(t)
Situation is not hopeless though, as there is a function which
describes the maximal decrease rate of yMD ,
C tMD
go = +
1
2
(ρu + ρw) tMD
go
2
(34)
ℓ m
(t)
ℬ(t)
(t)
tf
MD
tf
MT
Time, t
||ZEM||
Sergey Rubinsky under the supervision of Prof. Shaul Gutman A Three Player Pursuit and Evasion Conflict

45. Missile Guidance
A Game of Two Players
Aircraft Countermeasures
A Game of Three Players
Various Guidance Strategies
Conclusions
Game Definition
Game Model
Game Solution
Fail-safe function: C(t)
ℓ m
(t)
ℬ(t)
(t)
tf
MD
tf
MT
Time, t
||ZEM||
Namely, if yMD reaches C, then the missile can safely switch to
u∗
p, as a M-D miss distance rMD tMD
f ≥ is guaranteed.
Sergey Rubinsky under the supervision of Prof. Shaul Gutman A Three Player Pursuit and Evasion Conflict

46. Missile Guidance
A Game of Two Players
Aircraft Countermeasures
A Game of Three Players
Various Guidance Strategies
Conclusions
Game Definition
Game Model
Game Solution
Optimal Guidance Strategies
The Guidance strategy which maximizes a robustness measure;
namely, keeps yMT as“deep”as possible inside the singular area1
defined by B(t) is,
u∗
=
u∗
e = −ρu
yMD
yMD
, yMD < C tMD
go
u∗
p = ρu
yMT
yMT
, yMD ≥ C tMD
go
(35)
and the optimal strategies for the Target-Defender team are
v∗
= ρv
yMT
yMT
(36)
w∗
= −ρw
yMD
yMD
(37)
1
This is true when assuming constant final times. Otherwise, there is no
singular area, and the proposed strategy provides a local optimization for
minimum time.
Sergey Rubinsky under the supervision of Prof. Shaul Gutman A Three Player Pursuit and Evasion Conflict

47. Missile Guidance
A Game of Two Players
Aircraft Countermeasures
A Game of Three Players
Various Guidance Strategies
Conclusions
Game Definition
Game Model
Game Solution
Optimal Guidance in ZEM Plane
For {u∗, v∗, w∗}, and prescribed tMD
f , and tMT
f , we obtain the
following ZEM trajectories,
||yMT|| ||yMD||  ℬ 
tgo
*
t*
tf
MD
tf
MT
Time, t
ℓ
||yMT
cr
||
ℬ(t*
)
||ZEM||
Sergey Rubinsky under the supervision of Prof. Shaul Gutman A Three Player Pursuit and Evasion Conflict

48. Missile Guidance
A Game of Two Players
Aircraft Countermeasures
A Game of Three Players
Various Guidance Strategies
Conclusions
1st
Order tgo Based VG
4th
Order tgo Based VG
Transformed VG
Non-ideal Players VG
Outline
1 Missile Guidance
2 A Game of Two Players
Game Definition
Game Model
Game Solution
3 Aircraft Countermeasures
4 A Game of Three Players
Game Definition
Game Model
Game Solution
5 Various Guidance Strategies
1st
Order tgo Based VG
4th
Order tgo Based VG
Transformed VG
Non-ideal Players VG
6 Conclusions
Sergey Rubinsky under the supervision of Prof. Shaul Gutman A Three Player Pursuit and Evasion Conflict

49. Missile Guidance
A Game of Two Players
Aircraft Countermeasures
A Game of Three Players
Various Guidance Strategies
Conclusions
1st
Order tgo Based VG
4th
Order tgo Based VG
Transformed VG
Non-ideal Players VG
Outline
1 Missile Guidance
2 A Game of Two Players
Game Definition
Game Model
Game Solution
3 Aircraft Countermeasures
4 A Game of Three Players
Game Definition
Game Model
Game Solution
5 Various Guidance Strategies
1st
Order tgo Based VG
4th
Order tgo Based VG
Transformed VG
Non-ideal Players VG
6 Conclusions
Sergey Rubinsky under the supervision of Prof. Shaul Gutman A Three Player Pursuit and Evasion Conflict

50. Missile Guidance
A Game of Two Players
Aircraft Countermeasures
A Game of Three Players
Various Guidance Strategies
Conclusions
1st
Order tgo Based VG
4th
Order tgo Based VG
Transformed VG
Non-ideal Players VG
1st
Order time-to-go
The 1st order time-to-go is defined by
tPE
go =
rPE
VPE
(38)
This definition implies that
1 The players do not accelerate; therefore, rPE = tPE
go VPE .
This is not necessarily true, as the players do accelerate.
2 The players are close to collision triangle. In such a case, the
closing speed VC = VPE is approximately constant, and
tPE
go is approximately linear.
3 The final time tPE
f is approximately constant.
4 time-to-go may cause the guidance law to enter sliding mode
(chattering) at some point.
Sergey Rubinsky under the supervision of Prof. Shaul Gutman A Three Player Pursuit and Evasion Conflict

51. Missile Guidance
A Game of Two Players
Aircraft Countermeasures
A Game of Three Players
Various Guidance Strategies
Conclusions
1st
Order tgo Based VG
4th
Order tgo Based VG
Transformed VG
Non-ideal Players VG
VG1 Planar Simulation 1
For ρu = 17 [g], ρv = 3 [g], and ρw = 6 [g] we have
Missile
Target
Defender
tf
MD
t*
-4000 -2000 0 2000 4000 6000
0
2000
4000
6000
8000
x [m]
y[m] Miss MD = 150.4 , tf
MD
= 5.92
Miss MT = 0.5 , tf
MT
= 26.14
Sergey Rubinsky under the supervision of Prof. Shaul Gutman A Three Player Pursuit and Evasion Conflict

52. Missile Guidance
A Game of Two Players
Aircraft Countermeasures
A Game of Three Players
Various Guidance Strategies
Conclusions
1st
Order tgo Based VG
4th
Order tgo Based VG
Transformed VG
Non-ideal Players VG
VG1 3D Simulation 1
And a three dimensional version looks like...
Missile Target Defender tf
MD t*
0
5000
x [m]
0
2000
4000
6000
8000y [m]
0
1000
2000
3000
z [m]
Sergey Rubinsky under the supervision of Prof. Shaul Gutman A Three Player Pursuit and Evasion Conflict

53. Missile Guidance
A Game of Two Players
Aircraft Countermeasures
A Game of Three Players
Various Guidance Strategies
Conclusions
1st
Order tgo Based VG
4th
Order tgo Based VG
Transformed VG
Non-ideal Players VG
A Small Problem...
Observe the M-T range along the simulation
0 5 10 15 20 25
0
1000
2000
3000
4000
5000
6000
Time, t
M-TRange,||rMT(t)||
The presence of local minimum tells us that the missile misses the
target, but turns around and chases it.
Sergey Rubinsky under the supervision of Prof. Shaul Gutman A Three Player Pursuit and Evasion Conflict

54. Missile Guidance
A Game of Two Players
Aircraft Countermeasures
A Game of Three Players
Various Guidance Strategies
Conclusions
1st
Order tgo Based VG
4th
Order tgo Based VG
Transformed VG
Non-ideal Players VG
VG1 Planar Simulation 2
In order to improve the situation, the missile needs more capability.
For example, setting ρu = 27 [g], yields
Missile
Target
Defender
tf
MD
t*
0 1000 2000 3000 4000 5000 6000
0
1000
2000
3000
x [m]
y[m]
Miss MD = 151.4 , tf
MD
= 6.13
Miss MT = 0.5 , tf
MT
= 10.26
Sergey Rubinsky under the supervision of Prof. Shaul Gutman A Three Player Pursuit and Evasion Conflict

55. Missile Guidance
A Game of Two Players
Aircraft Countermeasures
A Game of Three Players
Various Guidance Strategies
Conclusions
1st
Order tgo Based VG
4th
Order tgo Based VG
Transformed VG
Non-ideal Players VG
VG1 3D Simulation 2
And in three dimensions we have,
Missile Target Defender tf
MD t*
0
2000
4000
6000x [m]
0
1000
2000
3000y [m]
0
500
1000
z [m]
Sergey Rubinsky under the supervision of Prof. Shaul Gutman A Three Player Pursuit and Evasion Conflict

56. Missile Guidance
A Game of Two Players
Aircraft Countermeasures
A Game of Three Players
Various Guidance Strategies
Conclusions
1st
Order tgo Based VG
4th
Order tgo Based VG
Transformed VG
Non-ideal Players VG
The M-T Range becomes...
0 2 4 6 8 10
0
1000
2000
3000
4000
5000
6000
Time, t
M-TRange,||rMT(t)||
Note that now, rMT (t) has no local minima; however, the
missile’s capability advantage is enormous.
Sergey Rubinsky under the supervision of Prof. Shaul Gutman A Three Player Pursuit and Evasion Conflict

57. Missile Guidance
A Game of Two Players
Aircraft Countermeasures
A Game of Three Players
Various Guidance Strategies
Conclusions
1st
Order tgo Based VG
4th
Order tgo Based VG
Transformed VG
Non-ideal Players VG
Outline
1 Missile Guidance
2 A Game of Two Players
Game Definition
Game Model
Game Solution
3 Aircraft Countermeasures
4 A Game of Three Players
Game Definition
Game Model
Game Solution
5 Various Guidance Strategies
1st
Order tgo Based VG
4th
Order tgo Based VG
Transformed VG
Non-ideal Players VG
6 Conclusions
Sergey Rubinsky under the supervision of Prof. Shaul Gutman A Three Player Pursuit and Evasion Conflict

58. Missile Guidance
A Game of Two Players
Aircraft Countermeasures
A Game of Three Players
Various Guidance Strategies
Conclusions
1st
Order tgo Based VG
4th
Order tgo Based VG
Transformed VG
Non-ideal Players VG
Optimal Time-to-go
Previously we had that if both hypothetical players (P and E) play
optimal, then the ZEM trajectory is
rPE + tPE
go VPE = JPE +
1
2
(amax
P − amax
E ) tPE
go
2
(39)
Previously (VG1), the final time was assumed to be fixed. Now,
this assumption is lifted. For given relative position and velocity
vectors, rPE and VPE, substitute JPE as the desired miss-distance
and obtain an equation in time-to-go. Therefore, the optimal
time-to-go is the solution of (39) for a given set of
rPE, VPE, JPE, amax
P , and amax
E . This definition has the
following advantages:
1 It does not assume constant speed.
2 It has nothing to do with collision triangle.
3 It does not assume constant tf .
Sergey Rubinsky under the supervision of Prof. Shaul Gutman A Three Player Pursuit and Evasion Conflict

59. Missile Guidance
A Game of Two Players
Aircraft Countermeasures
A Game of Three Players
Various Guidance Strategies
Conclusions
1st
Order tgo Based VG
4th
Order tgo Based VG
Transformed VG
Non-ideal Players VG
M-T Game: Optimal Time-to-go
Define m as the desired M-T miss distance (usually, m = 0).
Recall that VG4 assumes optimal maneuver; therefore, the
time-to-go is the solution of
rMT + tMT
go VMT = m +
1
2
(ρu − ρv) tMT
go
2
(40)
Square both sides of the equation, simplify and obtain that tMT
go is
the positive real root of the 4th order polynomial equation
1
4
(ρu − ρv)2
tMT
go
4
+ m (ρu − ρv) − VMT
2
tMT
go
2
(41)
− 2rMT VMT · tMT
go + m2
− rMT
2
= 0
Note that solving numerically (40) is easier and more accurate
than solving (41).
Sergey Rubinsky under the supervision of Prof. Shaul Gutman A Three Player Pursuit and Evasion Conflict

60. Missile Guidance
A Game of Two Players
Aircraft Countermeasures
A Game of Three Players
Various Guidance Strategies
Conclusions
1st
Order tgo Based VG
4th
Order tgo Based VG
Transformed VG
Non-ideal Players VG
M-T Game: Optimal Time-to-go (Contd.)
Therefore, the ZEM norm, yMT is always placed on the function
B tMT
go =
1
2
(ρu − ρv) tMT
go
2
(42)
tgo
MT
||yMT||
Sergey Rubinsky under the supervision of Prof. Shaul Gutman A Three Player Pursuit and Evasion Conflict

61. Missile Guidance
A Game of Two Players
Aircraft Countermeasures
A Game of Three Players
Various Guidance Strategies
Conclusions
1st
Order tgo Based VG
4th
Order tgo Based VG
Transformed VG
Non-ideal Players VG
M-T Game: Optimal Time-to-go (Contd. 2)
This definition is remarkable because:
1 There is no singular area, and B is not a bound; rather, it is
the only possible optimal ZEM trajectory.
2 This law never chatters as the denominator never vanishes.
3 If ρu > ρv, a zero miss distance achievable from any I.C.
4 The ZEM variable has absolutely different meaning now. The
VG4 ZEM means that if both players play optimal, then
yMT lays on B, and the final time, tMT
f , remains constant.
5 The final time, tMT
f , does not have to be constant, and it is
always the time at which the M-T miss distance is zero.
6 Since the achievable M-T miss distance is always zero, the
M-T conflict becomes about the final time, tMT
f , instead of
the miss distance. VG4 satisfies the saddle point inequality
tMT
f u∗
p, v ≤ tMT
f u∗
p, v∗
≤ tMT
f u, v∗
(43)
Sergey Rubinsky under the supervision of Prof. Shaul Gutman A Three Player Pursuit and Evasion Conflict

62. Missile Guidance
A Game of Two Players
Aircraft Countermeasures
A Game of Three Players
Various Guidance Strategies
Conclusions
1st
Order tgo Based VG
4th
Order tgo Based VG
Transformed VG
Non-ideal Players VG
M-D Game: Optimal Time-to-go
Analogically to M-T game, we expect the M-D time-to-go variable,
tMD
go , to be a solution of,
rMD + tMD
go VMD = q −
1
2
(ρu − ρw) tMD
go
2
(44)
where q is the M-D miss distance (assuming both players play
optimal in the entire time interval t ∈ 0, tMD
f ), for some q ≥ ,
and is the minimal allowed M-D miss distance. Therefore,
yMD is placed on the function
Z tMD
go = q −
1
2
(ρu − ρw) tMD
go
2
(45)
Sergey Rubinsky under the supervision of Prof. Shaul Gutman A Three Player Pursuit and Evasion Conflict

63. Missile Guidance
A Game of Two Players
Aircraft Countermeasures
A Game of Three Players
Various Guidance Strategies
Conclusions
1st
Order tgo Based VG
4th
Order tgo Based VG
Transformed VG
Non-ideal Players VG
M-D Game: Optimal Time-to-go (Contd.)
where Z tMD
go is parallel to
A tMD
go = −
1
2
(ρu − ρw) tMD
go
2
(46)
(tgo
MD
) (tgo
MD
)
tgo
MD
ℓ
q
||yMD||
Sergey Rubinsky under the supervision of Prof. Shaul Gutman A Three Player Pursuit and Evasion Conflict

64. Missile Guidance
A Game of Two Players
Aircraft Countermeasures
A Game of Three Players
Various Guidance Strategies
Conclusions
1st
Order tgo Based VG
4th
Order tgo Based VG
Transformed VG
Non-ideal Players VG
M-D Game: Optimal Time-to-go (Contd. 2)
This research proves that the solution is given by
tMD
go = arg min
tMD
go
rMD + tMD
go VMD +
1
2
(ρu − ρw) tMD
go
2
Sergey Rubinsky under the supervision of Prof. Shaul Gutman A Three Player Pursuit and Evasion Conflict

65. Missile Guidance
A Game of Two Players
Aircraft Countermeasures
A Game of Three Players
Various Guidance Strategies
Conclusions
1st
Order tgo Based VG
4th
Order tgo Based VG
Transformed VG
Non-ideal Players VG
VG4 Planar Simulation 1
For ρu = 17 [g], ρv = 3 [g], and ρw = 6 [g] we have
Missile
Target
Defender
tf
MD
t*
0 1000 2000 3000 4000 5000 6000
0
1000
2000
3000
x [m]
y[m] Miss MD = 155.1 , tf
MD
= 6.76
Miss MT = 0.5 , tf
MT
= 11.68
Sergey Rubinsky under the supervision of Prof. Shaul Gutman A Three Player Pursuit and Evasion Conflict

66. Missile Guidance
A Game of Two Players
Aircraft Countermeasures
A Game of Three Players
Various Guidance Strategies
Conclusions
1st
Order tgo Based VG
4th
Order tgo Based VG
Transformed VG
Non-ideal Players VG
VG1 vs. VG4
Missile
Target
Defender
tf
MD
t*
-4000 -2000 0 2000 4000 6000
0
2000
4000
6000
8000
x [m]
y[m]
Miss MD = 150.4 , tf
MD
= 5.92
Miss MT = 0.5 , tf
MT
= 26.14
Missile
Target
Defender
tf
MD
t*
0 1000 2000 3000 4000 5000 6000
0
1000
2000
3000
x [m]
y[m]
Miss MD = 155.1 , tf
MD
= 6.76
Miss MT = 0.5 , tf
MT
= 11.68
VG4 performs better.
Sergey Rubinsky under the supervision of Prof. Shaul Gutman A Three Player Pursuit and Evasion Conflict

67. Missile Guidance
A Game of Two Players
Aircraft Countermeasures
A Game of Three Players
Various Guidance Strategies
Conclusions
1st
Order tgo Based VG
4th
Order tgo Based VG
Transformed VG
Non-ideal Players VG
VG1 vs. VG4 (Contd.)
0 5 10 15 20 25
0
1000
2000
3000
4000
5000
6000
Time, t
M-TRange,||rMT(t)||
0 2 4 6 8 10
0
1000
2000
3000
4000
5000
6000
Time, t
M-TRange,||rMT(t)||
Unlike VG1, the range in VG4 scenario has no local minima.
Sergey Rubinsky under the supervision of Prof. Shaul Gutman A Three Player Pursuit and Evasion Conflict

68. Missile Guidance
A Game of Two Players
Aircraft Countermeasures
A Game of Three Players
Various Guidance Strategies
Conclusions
1st
Order tgo Based VG
4th
Order tgo Based VG
Transformed VG
Non-ideal Players VG
VG4 3D Simulation 1
In three dimensions we have,
Missile Target Defender tf
MD t*
0
2000
4000
6000
x [m]
0
1000
2000
3000
y [m]
0
500
1000
1500
z [m]
Sergey Rubinsky under the supervision of Prof. Shaul Gutman A Three Player Pursuit and Evasion Conflict

69. Missile Guidance
A Game of Two Players
Aircraft Countermeasures
A Game of Three Players
Various Guidance Strategies
Conclusions
1st
Order tgo Based VG
4th
Order tgo Based VG
Transformed VG
Non-ideal Players VG
VG4 Planar Simulation 2
Since VG4 Performs better, we can even decrease the missile’s
capability to ρu = 12 [g] and obtain
Missile
Target
Defender
tf
MD
t*
0 1000 2000 3000 4000 5000 6000
0
1000
2000
3000
4000
5000
6000
x [m]
y[m]
Miss MD = 150 , tf
MD
= 6.47
Miss MT = 0.5 , tf
MT
= 17.3
Sergey Rubinsky under the supervision of Prof. Shaul Gutman A Three Player Pursuit and Evasion Conflict

70. Missile Guidance
A Game of Two Players
Aircraft Countermeasures
A Game of Three Players
Various Guidance Strategies
Conclusions
1st
Order tgo Based VG
4th
Order tgo Based VG
Transformed VG
Non-ideal Players VG
VG4 3D Simulation 2
A three dimensional simulation yields
Missile Target Defender tf
MD t*
0
2000
4000
6000x [m]
0
2000
4000
6000
y [m]
0
500
1000
1500
2000
z [m]
Sergey Rubinsky under the supervision of Prof. Shaul Gutman A Three Player Pursuit and Evasion Conflict

71. Missile Guidance
A Game of Two Players
Aircraft Countermeasures
A Game of Three Players
Various Guidance Strategies
Conclusions
1st
Order tgo Based VG
4th
Order tgo Based VG
Transformed VG
Non-ideal Players VG
Outline
1 Missile Guidance
2 A Game of Two Players
Game Definition
Game Model
Game Solution
3 Aircraft Countermeasures
4 A Game of Three Players
Game Definition
Game Model
Game Solution
5 Various Guidance Strategies
1st
Order tgo Based VG
4th
Order tgo Based VG
Transformed VG
Non-ideal Players VG
6 Conclusions
Sergey Rubinsky under the supervision of Prof. Shaul Gutman A Three Player Pursuit and Evasion Conflict

72. Missile Guidance
A Game of Two Players
Aircraft Countermeasures
A Game of Three Players
Various Guidance Strategies
Conclusions
1st
Order tgo Based VG
4th
Order tgo Based VG
Transformed VG
Non-ideal Players VG
Preface
Previously, we have discussed a scenario where each player could
apply a bounded maneuver in any direction in 3D space. Although
this definition is applicable in exo-atmospheric scenario where the
thrust is the only dominant force, it is hardly applicable to an
atmospheric engagement, where aerodynamic forces are dominant.
Therefore, in an atmospheric conflict, each player’s maneuver
capability is mostly perpendicular to its velocity vector.
Sergey Rubinsky under the supervision of Prof. Shaul Gutman A Three Player Pursuit and Evasion Conflict

73. Missile Guidance
A Game of Two Players
Aircraft Countermeasures
A Game of Three Players
Various Guidance Strategies
Conclusions
1st
Order tgo Based VG
4th
Order tgo Based VG
Transformed VG
Non-ideal Players VG
Elliptical Transformation
In atmospheric engagement, a more realistic bound is an ellipsoid.
Such a bound reflects the difference between axial and lateral
acceleration capabilities. Consider the unitary matrix,
Ui =





vi1√
v2
i1+v2
i2+v2
i3
− vi2√
v2
i1+v2
i2
− vi1vi3√
v2
i1+v2
i2+v2
i3
√
v2
i1+v2
i2
vi2√
v2
i1+v2
i2+v2
i3
vi1√
v2
i1+v2
i2
− vi2vi3√
v2
i1+v2
i2+v2
i3
√
v2
i1+v2
i2
vi3√
v2
i1+v2
i2+v2
i3
0
√
v2
i1+v2
i2√
v2
i1+v2
i2+v2
i3





where i = u, v, w. The first column of Ui is each player’s
normalized velocity vector, Vi, and the other two columns are
orthonormal to Vi and each other. This matrix satisfies Ui = U−1
i .
Sergey Rubinsky under the supervision of Prof. Shaul Gutman A Three Player Pursuit and Evasion Conflict

74. Missile Guidance
A Game of Two Players
Aircraft Countermeasures
A Game of Three Players
Various Guidance Strategies
Conclusions
1st
Order tgo Based VG
4th
Order tgo Based VG
Transformed VG
Non-ideal Players VG
Elliptical Transformation (Contd.)
Also consider a scaling matrix
Σ =


σa 0 0
0 σ⊥1 0
0 0 σ⊥2

 (47)
where σa scales each player’s axial acceleration capability and
σ⊥1, σ⊥2 scale their lateral acceleration capability. Now, consider
each player’s transformation matrix
Ti = UiΣUi (48)
As a result, the standard · 2 ball bound, transforms into a
velocity vector framed ellipsoid.
Sergey Rubinsky under the supervision of Prof. Shaul Gutman A Three Player Pursuit and Evasion Conflict

75. Missile Guidance
A Game of Two Players
Aircraft Countermeasures
A Game of Three Players
Various Guidance Strategies
Conclusions
1st
Order tgo Based VG
4th
Order tgo Based VG
Transformed VG
Non-ideal Players VG
Elliptical Transformation Example
σa = 1 σa = 0.5
σa = 0.25 σa = 0
Sergey Rubinsky under the supervision of Prof. Shaul Gutman A Three Player Pursuit and Evasion Conflict

76. Missile Guidance
A Game of Two Players
Aircraft Countermeasures
A Game of Three Players
Various Guidance Strategies
Conclusions
1st
Order tgo Based VG
4th
Order tgo Based VG
Transformed VG
Non-ideal Players VG
TVG Optimal Strategies
We obtain the following optimal strategies
{includegraphics{Figures/VG1 Planar Simulation 2.pdf}}{Figures/VG1
u∗
=



−ρu
TuyMD
TuyMD
, yMD < C tMD
go
ρu
TuyMT
TuyMT
, yMD ≥ C tMD
go
(49)
v∗
= ρv
TvyMT
TvyMT
(50)
w∗
= −ρw
TwyMD
TwyMD
(51)
Sergey Rubinsky under the supervision of Prof. Shaul Gutman A Three Player Pursuit and Evasion Conflict

77. Missile Guidance
A Game of Two Players
Aircraft Countermeasures
A Game of Three Players
Various Guidance Strategies
Conclusions
1st
Order tgo Based VG
4th
Order tgo Based VG
Transformed VG
Non-ideal Players VG
TVG4 Simulation Parameters
Consider the following maneuver capabilities
ρu = 17 [g] , ρv = 3 [g] , and ρw = 6 [g]
intercept miss distances,
m = 0.5 [m], = 150 [m]
and lateral maneuver scaling factors
σ⊥1 = σ⊥2 = 1
The following simulations demonstrate an engagement for
σa = {1, 0.5, 0.25, 0}.
Sergey Rubinsky under the supervision of Prof. Shaul Gutman A Three Player Pursuit and Evasion Conflict

78. Missile Guidance
A Game of Two Players
Aircraft Countermeasures
A Game of Three Players
Various Guidance Strategies
Conclusions
1st
Order tgo Based VG
4th
Order tgo Based VG
Transformed VG
Non-ideal Players VG
TVG4 Planar Simulations
Missile
Target
Defender
tf
MD
t*
0 1000 2000 3000 4000 5000 6000
0
1000
2000
3000
x [m]
y[m] Miss MD = 155.1 , tf
MD
= 6.76
Miss MT = 0.5 , tf
MT
= 11.68
σa = 1
Missile
Target
Defender
tf
MD
t*
0 1000 2000 3000 4000 5000 6000
0
500
1000
1500
2000
2500
3000
3500
x [m]
y[m]
Miss MD = 150 , tf
MD
= 5.73
Miss MT = 0.5 , tf
MT
= 16.2
σa = 0.5
Missile
Target
Def .
tf
MD
t*
0 1000 2000 3000 4000 5000 6000
0
500
1000
1500
2000
2500
x [m]
y[m]
Miss MD = 150 , tf
MD
= 5.9
Miss MT = 0.5 , tf
MT
= 17.5
σa = 0.25
Missile
Target
Defender
tf
MD
t*
0 1000 2000 3000 4000 5000 6000
0
500
1000
1500
2000
x [m]
y[m]
Miss MD = 150 , tf
MD
= 5.87
Miss MT = 950 , tf
MT
= 9.59
σa = 0
Sergey Rubinsky under the supervision of Prof. Shaul Gutman A Three Player Pursuit and Evasion Conflict

79. Missile Guidance
A Game of Two Players
Aircraft Countermeasures
A Game of Three Players
Various Guidance Strategies
Conclusions
1st
Order tgo Based VG
4th
Order tgo Based VG
Transformed VG
Non-ideal Players VG
TVG4 3D Simulations
Missile Target Defender tf
MD t*
0
2000
4000
6000
x [m]
0
1000
2000
3000
y [m]
0
1000
2000
z [m]
σa = 1
Missile Target Defender tf
MD t*
0
2000
4000
6000
x [m]
0
500
1000
1500
y [m]
0
1000
2000
3000
z [m]
σa = 0.5
Missile Target Defender tf
MD t*
0
2000
4000
6000
x [m]
0
500
1000
1500
y [m]
0
1000
2000
z [m]
σa = 0.25
Missile Target Defender tf
MD t*
0
2000
4000
6000
x [m]
0
500
1000
1500
y [m]
-500
0
500
1000
1500
z [m]
σa = 0
Sergey Rubinsky under the supervision of Prof. Shaul Gutman A Three Player Pursuit and Evasion Conflict

80. Missile Guidance
A Game of Two Players
Aircraft Countermeasures
A Game of Three Players
Various Guidance Strategies
Conclusions
1st
Order tgo Based VG
4th
Order tgo Based VG
Transformed VG
Non-ideal Players VG
Outline
1 Missile Guidance
2 A Game of Two Players
Game Definition
Game Model
Game Solution
3 Aircraft Countermeasures
4 A Game of Three Players
Game Definition
Game Model
Game Solution
5 Various Guidance Strategies
1st
Order tgo Based VG
4th
Order tgo Based VG
Transformed VG
Non-ideal Players VG
6 Conclusions
Sergey Rubinsky under the supervision of Prof. Shaul Gutman A Three Player Pursuit and Evasion Conflict

81. Missile Guidance
A Game of Two Players
Aircraft Countermeasures
A Game of Three Players
Various Guidance Strategies
Conclusions
1st
Order tgo Based VG
4th
Order tgo Based VG
Transformed VG
Non-ideal Players VG
Preface
Until now all players were ideal; namely, each player could instantly
apply any bounded acceleration in any direction in 3D space.
However, in order to describe a more realistic engagement, we have
to account for each player’s dynamics. In this work, the Missile has
a first order isotropic dynamics, while other players are Ideal.
Therefore, if the Missile is able to evade the Defender and
intercept the Target in this scenario, it will be also able to do this
in a real engagement, where other players are also not ideal.
Sergey Rubinsky under the supervision of Prof. Shaul Gutman A Three Player Pursuit and Evasion Conflict

82. Missile Guidance
A Game of Two Players
Aircraft Countermeasures
A Game of Three Players
Various Guidance Strategies
Conclusions
1st
Order tgo Based VG
4th
Order tgo Based VG
Transformed VG
Non-ideal Players VG
Game Model
Consider a three player engagement,
u
VM
rM
v
VT
rT
w
VD
rD
rMT
rMD
rTD
Sergey Rubinsky under the supervision of Prof. Shaul Gutman A Three Player Pursuit and Evasion Conflict

83. Missile Guidance
A Game of Two Players
Aircraft Countermeasures
A Game of Three Players
Various Guidance Strategies
Conclusions
1st
Order tgo Based VG
4th
Order tgo Based VG
Transformed VG
Non-ideal Players VG
Game Model (Contd.)
However, here the Missile’s dynamics is not ideal, and described by
the transfer matrix, GM : uC → u,
GM (s) =
AM BM
CM DM
(52)
where uC, u ∈ R3, and GM (s) ∈ RH∞
. In order to simplify
matters, we assume isotropic dynamics,
GM (s) = I3Gs(s) (53)
where Gs(s) is a first order transfer function,
Gs(s) =
1
τM s + 1
=
− 1
τM
1
τM
1 0
(54)
Sergey Rubinsky under the supervision of Prof. Shaul Gutman A Three Player Pursuit and Evasion Conflict

84. Missile Guidance
A Game of Two Players
Aircraft Countermeasures
A Game of Three Players
Various Guidance Strategies
Conclusions
1st
Order tgo Based VG
4th
Order tgo Based VG
Transformed VG
Non-ideal Players VG
Game Model (Contd. 2)
The state space realization is now






˙rMT (t)
¨rMT (t)
˙rMD(t)
¨rMD(t)
˙u(t)






=






0 I3 0 0 0
0 0 0 0 −I3
0 0 0 I3 0
0 0 0 0 −I3
0 0 0 0 − I3
τM












rMT (t)
˙rMT (t)
rMD(t)
˙rMD(t)
u(t)






+






0
0
0
0
I3
τM






uC(t) +






0
I3
0
0
0






v(t) +






0
0
0
I3
0






w(t)
Sergey Rubinsky under the supervision of Prof. Shaul Gutman A Three Player Pursuit and Evasion Conflict

85. Missile Guidance
A Game of Two Players
Aircraft Countermeasures
A Game of Three Players
Various Guidance Strategies
Conclusions
1st
Order tgo Based VG
4th
Order tgo Based VG
Transformed VG
Non-ideal Players VG
Game Model (Contd. 3)
The bound functions become,
A tMD
go = −
1
2
(ρu − ρw) tMD
go
2
+ ρuτ2
M ψ tMD
go (55)
B tMT
go = m +
1
2
(ρu − ρv) tMT
go
2
− ρuτ2
M ψ tMT
go (56)
and the fail-safe is
C tMD
go = +
1
2
(ρu + ρw) tMD
go
2
− ρuτ2
M ψ tMD
go (57)
where ψ(tgo) =
tgo
τM
− 1 + e−tgo/τM .
Sergey Rubinsky under the supervision of Prof. Shaul Gutman A Three Player Pursuit and Evasion Conflict

86. Missile Guidance
A Game of Two Players
Aircraft Countermeasures
A Game of Three Players
Various Guidance Strategies
Conclusions
1st
Order tgo Based VG
4th
Order tgo Based VG
Transformed VG
Non-ideal Players VG
Game Model (Contd. 4)
(t) ℬ(t) (t)
ℓ
m
tf
MD
tf
MT
Time, t
||ZEM||
Similar to the ideal players game, In order to evade the Defender
and intercept the Target, the Missile evades the Defender until
yMD reaches C, and then switches to pursuit strategy.
Sergey Rubinsky under the supervision of Prof. Shaul Gutman A Three Player Pursuit and Evasion Conflict

87. Missile Guidance
A Game of Two Players
Aircraft Countermeasures
A Game of Three Players
Various Guidance Strategies
Conclusions
1st
Order tgo Based VG
4th
Order tgo Based VG
Transformed VG
Non-ideal Players VG
Game Solution
Thus, the missile’s strategy is
uC =



−ρu
rMD+tMD
go ˙rMD−τ2
M ψ tMD
go u
rMD+tMD
go ˙rMD−τ2
M ψ tMD
go u
, yMD < C
ρu
rMT +tMT
go ˙rMT −τ2
M ψ tMT
go u
rMT +tMT
go ˙rMT −τ2
M ψ tMT
go u
, yMD ≥ C
(58)
and, the Target’s and Defender’s guidance laws are
v∗
= ρv
rMT + tMT
go ˙rMT − τ2
M ψ tMT
go u
rMT + tMT
go ˙rMT − τ2
M ψ tMT
go u
(59)
w∗
= −ρw
rMD + tMD
go ˙rMD − τ2
M ψ tMD
go u
rMD + tMD
go ˙rMD − τ2
M ψ tMD
go u
(60)
Sergey Rubinsky under the supervision of Prof. Shaul Gutman A Three Player Pursuit and Evasion Conflict

88. Missile Guidance
A Game of Two Players
Aircraft Countermeasures
A Game of Three Players
Various Guidance Strategies
Conclusions
1st
Order tgo Based VG
4th
Order tgo Based VG
Transformed VG
Non-ideal Players VG
Game Simulation
For ρu = 17 [g], ρv = 3 [g], ρw = 6 [g], and τM = {0.1, 0} [sec],
we have
Missile
Target
Defender
tf
MD
t*
0 1000 2000 3000 4000 5000 6000
0
500
1000
1500
2000
2500
3000
3500
x [m]
y[m]
Miss MD = 150.2 , tf
MD
= 6.04
Miss MT = 0.5 , tf
MT
= 11.89
Missile
Target
Defender
tf
MD
t*
0 1000 2000 3000 4000 5000 6000
0
1000
2000
3000
x [m]
y[m]
Miss MD = 155.1 , tf
MD
= 6.76
Miss MT = 0.5 , tf
MT
= 11.68
Sergey Rubinsky under the supervision of Prof. Shaul Gutman A Three Player Pursuit and Evasion Conflict

89. Missile Guidance
A Game of Two Players
Aircraft Countermeasures
A Game of Three Players
Various Guidance Strategies
Conclusions
Outline
1 Missile Guidance
2 A Game of Two Players
Game Definition
Game Model
Game Solution
3 Aircraft Countermeasures
4 A Game of Three Players
Game Definition
Game Model
Game Solution
5 Various Guidance Strategies
1st
Order tgo Based VG
4th
Order tgo Based VG
Transformed VG
Non-ideal Players VG
6 Conclusions
Sergey Rubinsky under the supervision of Prof. Shaul Gutman A Three Player Pursuit and Evasion Conflict

90. Missile Guidance
A Game of Two Players
Aircraft Countermeasures
A Game of Three Players
Various Guidance Strategies
Conclusions
Conclusions
This research provides a solution which enables an attacking
Missile to evade the Defender and intercept its Target.
The provided solution is optimal with respect to a robustness
measure.
The switch time from evasion to pursuit is substantially
smaller than the M-D final time. Therefore, the missile can
start pursuing the Target before it passes by the Defender.
The provided solution separates the game into two phases:
evasion and pursuit. As a result, cooperation between the
target and the defender is impossible. Indeed, this research
proves that the optimal strategies for the Target–Defender
team are not cooperative.
A recursive algorithm for estimation of time-to-go introduces
substantially improved results over the classic, first order
time-to-go.
Sergey Rubinsky under the supervision of Prof. Shaul Gutman A Three Player Pursuit and Evasion Conflict

91. Missile Guidance
A Game of Two Players
Aircraft Countermeasures
A Game of Three Players
Various Guidance Strategies
Conclusions
Questions
Thank You!
Sergey Rubinsky under the supervision of Prof. Shaul Gutman A Three Player Pursuit and Evasion Conflict