Dissertation - A Three Players Pursuit and Evasion Conflict
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This document is a research thesis submitted by Sergey Rubinsky to the Technion – Israel Institute of Technology in partial fulfillment of a Doctor of Philosophy degree. It examines a three player pursuit and evasion conflict using differential game theory. The thesis contains three main parts that analyze the conflict using: 1) a linear kinematic model, 2) a vector guidance approach, and 3) incorporating state estimation. It presents the mathematical formulations of the games, derives the optimal strategies for the players, and evaluates the strategies through simulations. The research makes contributions to understanding pursuit-evasion problems involving multiple agents maneuvering in three dimensions.
![13.9 Linear simulations for kv = −ρv and kv = ρv . . . . . . . . . . . . . . . . . . . . . . 59
13.10Function dv(t∗
go, kv) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
13.11Function dv (kv) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
13.12Linear simulation for kv = 0, te = 1 [sec] . . . . . . . . . . . . . . . . . . . . . . . . 62
13.13Nonlinear simulations for kv = −ρv and kv = ρv . . . . . . . . . . . . . . . . . . . . 63
13.14Nonlinear simulation for kv = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
16.1 Planar Interception Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
17.1 Optimal ZEM P-E Trajectories for amax
P > amax
E (left) and amax
P < amax
E (right) . . . 69
19.1 Optimal Missile-Defender ZEM Trajectories . . . . . . . . . . . . . . . . . . . . . . 72
19.2 Optimal Missile-Target ZEM Trajectories . . . . . . . . . . . . . . . . . . . . . . . . 73
19.3 Bound Functions A and B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
19.4 fail-safe Function C in addition toA and B . . . . . . . . . . . . . . . . . . . . . . . 75
20.1 VG1 Planar Simulation 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
20.2 VG1 Vs. LMG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
20.3 Planar Simulation and ZEM Trajectories . . . . . . . . . . . . . . . . . . . . . . . . 79
20.4 VG1 Planar Simulation 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
20.5 Relative Distance rMT (t) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
20.6 VG1 3D Simulation 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
20.7 VG1 Planar Simulation 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
20.8 Relative Distance rMT (t) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
20.9 VG1 3D Simulation 2 and ZEM Trajectories . . . . . . . . . . . . . . . . . . . . . . 82
22.1 Missile-Target Game Optimal ZEM Trajectory . . . . . . . . . . . . . . . . . . . . . 85
22.2 Missile-Defender Optimal ZEM Trajectories . . . . . . . . . . . . . . . . . . . . . . 90
22.3 Function g tMD
go For Different Values of q . . . . . . . . . . . . . . . . . . . . . . . 92
22.4 Missile-Defender Relative Distance . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
22.5 Function ˙g tMD
go . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
22.6 Function g tMD
go . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
22.7 Evaluation of tMD
go for VG1 and VG4 . . . . . . . . . . . . . . . . . . . . . . . . . . 96
22.8 Functions A, C, yMT and yMD . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
23.1 Functions yMT , yMD , and C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
24.1 VG4 Planar Simulation 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
24.2 VG1 Vs. VG4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
24.3 Relative M-T Distances, rMT (t) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
24.4 VG14 Vs. VG4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
24.5 Demonstration of the Target using VG1 . . . . . . . . . . . . . . . . . . . . . . . . 107
24.6 Acceleration Angle, χ(t) vs. Planar Simulation . . . . . . . . . . . . . . . . . . . . . 108
24.7 VG4 3D Simulation 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
24.8 VG4 Planar Simulation 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
24.9 VG4 3D Simulation 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
24.10VG4 vs. VG4∗
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
24.11VG4 vs. VG4∗
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
25.1 PVG4 vs. LMG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
25.2 PVG4 Planar Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
25.3 PVG4 3D Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
25.4 Elliptical Bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
25.5 TVG4 Planar Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
25.6 TVG4 3D Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
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![1 Introduction
1.1 Motivation
THE protection of an airborne vehicle against a homing missile has become a significant issue,
since a modern interceptor carries a substantial threat to such a vehicle. As interceptor missiles
become more sophisticated, the current passive countermeasure systems are not sufficient. There-
fore, a more advanced countermeasure system is needed. Such possible countermeasure is a short
range homing missile (Defender), aimed at the interception of the interceptor. In such a scenario,
the protected aircraft (Target) can use both its own evasive maneuver and the defender, in order
to evade the missile. In generating guidance strategies, a common practice is a linearization with
respect to a collision course, which implies simplified linear kinematics. However, in a game of
three players, linearization assumptions can be unrealistic. As a result, generated guidance strate-
gies can be inaccurate. Thus, this research provides an alternative approach which is not based on
linearization.
This research is based on Differential Game (DG) theory [1, 2], as a natural way to describe
conflicts. In formulating a DG, there are two main approaches. In the first, the Linear Quadratic
Differential Games (LQDG) approach, the cost is formed of a terminal quadratic state to account
for the miss distance, and a quadratic control integral to account for the control effort, [3, 4, 5].
As a result, the optimal strategies are linear. This approach suffers several drawbacks. First, it
violates the saturation limit every actuator has. Second, it does not guarantee a miss-distance
value. Third, in game theory, the players must “agree” on the cost. However, the linear strategies
generated by LQDG imply that on a collision course (except at the terminal time), both strategies
are identically zero. While for the pursuer this is acceptable, no rational evader can agree to use
such a cost. Indeed, close to termination, the evader has in many cases the potential to increase
the miss-distance. In the second approach [6–7], called Differential Game Guidance Law (DGL),
hard bounds are imposed on the controls and the cost is purely terminal to account for the miss-
distance. As a result, the optimal strategies are nonlinear. Moreover, the saddle-point property
implies a guaranteed miss-distance to each player. In classical terms, the navigation gain increases
with time, and at a certain time before termination the guidance law becomes pure bang-bang.
1.2 Noticeable Contributions
In the field of active aircraft defense against an attacking missile, some noticeable contributions
have been made. In [8], a closed form relation was derived for the initial missile-target range ratio
as well as at interception for the missile-defender conflict, under the assumption of a constant
collision course. Later, [9] finds the requirements on the defender firing angle and the distance
it will run to intercept the attacker as a function of the game geometry and the point at which
the target launches the defender. In that paper, the author derives the location of intercept point
in the target-centered coordinates. This work assumes a constant collision course and therefore
suffers many drawbacks, as in a real battle situation the vehicles do maneuver. In [10], a discretized
and linearized solution to the three player differential game is presented, under the assumptions
that the target is fixed or slowly moving (a battleship for example), the defender is launched from
the target to intercept the missile, while the missile’s objective is to intercept the target. However,
in this scenario, the missile has no knowledge about the defender and therefore will not revise
its collision course with respect to the defender. This study suggests that the missile should use
a random pursuit strategy; otherwise, its trajectory is predictable and can be easily intercepted
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![by the defender (assuming the defender has a greater maneuvering capability). Recently, [11]
has presented a solution to the three player problem, using a linearized model. In his research,
the author has defined a quadratic cost function that represents the player’s objectives and is
formed of a terminal quadratic state to account for the miss-distance, and a quadratic integral
to account for the control effort. That work presents a solution to the full-knowledge differential
game, however the LQDG solution suffers the mentioned drawbacks. More recently, [12] has
presented a cooperative target-defender guidance strategy against a pursuing missile. That article
is based on a two team LQDG and provides an optimal analytic solution for the target-defender
pair. Moreover, a parametric analysis has been done to study the conditions for existence of a
saddle point. The authors have provided numerical simulations to prove their theoretical analysis.
That article implies that all optimal strategies are linear, and therefore, suffers the drawbacks
mentioned above. Using a different approach, [13–14] have presented a multiple model adaptive
guidance strategy to defend the target from the missile. That work applies a multiple model
adaptive estimator with measurement fusion, where each model represents a possible guidance
law and guidance parameters of the incoming homing missile. Thus, under the assumption that
the homing missile uses one of the known guidance strategies, the defender may anticipate the
missile’s maneuver, as the target maneuver is known. That article provides a very interesting
insight into the three player differential game strategy but cannot guarantee any result if the
homing missile doesn’t use any of the known linear strategies. Moreover it cannot guarantee a
miss-distance value. Articles [15–17] have also made some noticeable contributions on this problem.
However, the obtained guidance laws in these articles are still linear, and suffer the same drawbacks
mentioned above. Other noticeable contributions can be found at [20–41].
1.3 Main Results and Contribution
This research is divided into three main parts. In Part I, one finds the Linear Model Guidance
(LMG) approach for the three players conflict, in which the kinematics is linear, the controls are
bounded, and the cost is the miss distance. The LMG approach suggests that in certain regions of
the state space, the missile can perform an evasive maneuver with respect to the defender, without
losing its pursuit capabilities. Moreover, sufficient conditions under which a missile can hit a
target while evading a defender launched by the target, are derived. Moreover, the guaranteed
cost strategies are optimized with respect to a robustness measure. However, Part I is based on
the linearized model; as a result, the obtained guidance strategies do not always accurately reflect
the actual situation. A detailed analysis of the linearization problem is provided in Part II. There,
one finds the contradiction of the optimal guidance strategies in the linear kinematics scenario,
and the real, nonlinear world. The reason of such contradiction is described in Part II, as well as
a partial solution. In addition, Part II provides a deep analysis of all parameters relevant to the
problem, and an optimality proof for the target and the defender. Part III continues the study
presented in Part I. While it relies on similar principles, Part III is based on a three dimensional
Vector Guidance (VG) instead of the Linearized Model Guidance (LMG) provided in Part I. A
detailed discussion about the VG in a two player scenario can be found in [18]. As a result, the
obtained strategies are much better than in Part I, as they reflect the actual situation instead
of the linearized one. Planar and three dimensional simulations are provided in order to confirm
the results. In order to account for endo-atmospheric interception conflict, where the aerodynamic
forces are dominant, a Transformed Vector Guidance approach is derived in Part III. This approach
suggests that by using a transformation matrix, one can account for the difference between the
lateral and axial acceleration capabilities of the players. In addition, in order to account for noisy
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![5 Differential Game Definition
The Target maximizes ˙VMT (t) = d
dt
|yMT (t)| with its controller v(t). Therefore, from (4.6), its
optimal strategy is1
v∗
= ρvsign(yMT ) (5.1)
The Defender, minimizes ˙VMD(t) = d
dt
|yMD(t)| with its controller w(t). Analogically, from (4.7),
its optimal guidance law is
w∗
= −ρwsign(yMD) (5.2)
The Missile has two objectives: Defender evasion and Target pursuit. To derive the game bounds,
two separate game situations are analyzed.
1. Missile-Defender Game − The Missile evades the Defender by maximizing ˙VMD(t). In such
case, by (4.7), its optimal guidance law is
u∗
e = −ρusign(yMD) (5.3)
Substituting u∗
e and w∗
into (4.7) gives,
˙V∗
MD(t) = tMD
go (ρu − ρw) (5.4)
Integration yields
|y∗
MD(t)| = |y∗
MD(t = 0)| +
ˆ t
0
tMD
f (ρu − ρw) dξ −
ˆ t
0
ξ (ρu − ρw) dξ
= |y∗
MD(t = 0)| + tMD
f (ρu − ρw) ξ|t
0 −
1
2
(ρu − ρw) ξ2
t
0
(5.5)
= |y∗
MD(t = 0)| + tMD
f t (ρu − ρw) −
1
2
(ρu − ρw) t2
Define
y∗
MD t = tMD
f = (5.6)
where is the minimal desired M-D miss distance. Consequently,
y∗
MD t = tMD
f = = |y∗
MD(t = 0)| +
1
2
(ρu − ρw) tMD
f
2
(5.7)
|y∗
MD(t = 0)| = −
1
2
(ρu − ρw) tMD
f
2
(5.8)
thus
|y∗
MD(t)| = −
1
2
(ρu − ρw) tMD
f
2
+ tMD
f t (ρu − ρw) −
1
2
(ρu − ρw) t2
= −
1
2
(ρu − ρw) tMD
f − t
2
(5.9)
From here, we have the final form of the first bound.
y∗
MD tMD
go = −
1
2
(ρu − ρw) tMD
go
2
(5.10)
1
For a complete derivation of DGL refer to [6]
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![9 Nonlinear Simulations
Example 9.1. Simulation results for ku = ρu and the following parameters is shown on Fig. 9.1.
ρu = 120
m
sec2
, ρv = 60
m
sec2
, ρw = 70
m
sec2
, m = 0.5 [m] , = 150 [m]
Missile
Target
Defender
tf
MD
t*
0 1000 2000 3000 4000 5000 6000 7000
0
1000
2000
3000
4000
x [m]
y[m] Miss MD = 206 , tf
MD
= 5.66
Miss MT = 0.17 , tf
MT
= 15.35
Figure 9.1: Nonlinear Simulation 1 (ku = ρu)
The actual M-D miss distance is greater than the required. This happens because the actual time-
to-go isn’t linear since the Missile evades the Defender and “breaks” the collision triangle which is
the base for our linearization assumptions. Fig. 9.2 shows the measured tgo as a function of the
simulation time t.
0 1 2 3 4 5
0
1
2
3
4
5
Time, t
Estimatedtgo
MD
t*
Figure 9.2: Measured tMD
go as a function of time
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![Clearly, the time-to-go is nonlinear until the switch point.
Example 9.2. One can use the proposed guidance law for ku,1 in order to reduce Missile’s maneuver
so that the collision triangle would suffer less distortion. To obtain ku,1 it is necessary to know
the final times tMD
f , tMT
f . Since these values are unknown, it is possible compute them online by
substituting tMD
go , tMT
go instead of tMD
f , tMT
f and updating it every time step. In such a case, (Fig.
9.3) a much closer result is obtained.
Missile
Target
Defender
tf
MD
t*
0 1000 2000 3000 4000 5000 6000 7000
0
1000
2000
3000
4000
x [m]
y[m]
Miss MD = 156.5 , tf
MD
= 5.6
Miss MT = 0.05 , tf
MT
= 15.3
Figure 9.3: Nonlinear Simulation 2 (ku = ku,1)
and the time-to-go is closer to linear as shown in Fig. 9.4.
0 1 2 3 4 5
0
1
2
3
4
5
Time, t
Estimatedtgo
MD
Figure 9.4: Estimated tMD
go as a function of time
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![One must understand that the greater ∆ρuw = ρu − ρw is, the more distortion suffers the M-D
collision triangle; therefore, linearization assumptions become less valid.
Example 9.3. Consider the parameters
ρu = 170
m
sec2
, ρv = 30
m
sec2
, ρw = 60
m
sec2
, m = 0.5 [m] , = 150 [m]
The result, shown in Fig. 9.5, is the outcome of the nonlinear simulation, using the stated above
parameters.
Missile
Target
Defender
tf
MD
t*
0 1000 2000 3000 4000 5000 6000
0
1000
2000
3000
4000
x [m]
y[m]
Miss MD = 325 , tf
MD
= 5.74
Miss MT = 0.1 , tf
MT
= 13.37
Figure 9.5: Nonlinear Simulation 3 (ku = ρu)
Denote t∗
as the switch time (refer to Fig. 9.5). It is readily seen that t∗
< tMD
f ; therefore, the
Missile switches to pursuit strategy before it passes by the Defender. In fact, this is a big advantage
of the proposed guidance strategy, as it allows the Missile to pursue the Target while it still plays
against the Defender. One can also see a big difference between the requested M-D miss distance
and the actual one.
Example 9.4. It is possible to use ku = ku,1 in order to reduce the Missile’s evasive maneuver
and cause less distortion to the collision triangle. The outcome of such simulation is shown in Fig.
9.6.
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![Missile
Target
Defender
tf
MD
t*
0 1000 2000 3000 4000 5000 6000
0
1000
2000
3000
4000
x [m]
y[m]
Miss MD = 154.5 , tf
MD
= 5.57
Miss MT = 0.18 , tf
MT
= 14.57
Figure 9.6: Nonlinear Simulation 4 (ku = ku,1)
As expected, the Missile – Defender miss distance is much closer to the linear kinematics simulation.
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![Part II
LMG Analysis
This part has two main purposes:
• It provides deep parametric analysis of the results obtained in Part I. Also, it proves opti-
mality for the Target’s and Defender’s maneuvers.
• Analyzes the problem caused by linearization. This analysis emphasizes the need for a
different approach discussed in Part III.
12 Parametric Analysis
Recall the inequality derived in Section 8 of Part I.
− (ρv + ρw) t3
f − 2∆t (ρv + ρw) t2
f
+ (ρu − ρv) ∆t2
+ 2 m − + yMD
0 − yMT
0 tf + 4∆t yMD
0 −
(ρv + ρw) t2
f − 2∆t (ρu − ρv) tf − (ρu − ρv) ∆t2 − 2 (m − + |yMD
0 | − |yMT
0 |)
≤
t2
f (ρu − ρw) − 2 + 2 |yMD
0 |
2ρu
(12.1)
Provided (12.1) holds, a M-D miss distance of , and a M-T miss distance of m can be guaranteed.
From (12.1), one obtains the solution for ρumin
, the minimal maneuver capability required by the
Missile to complete its task. However, analytic solution for ρumin
is too long to be written here;
therefore, qualitative and quantitative properties of ρumin
yMT
0 , yMD
0 , ρv, ρw, m, , tf , ∆t and
its dependence on the various parameters is explored.
Remark 12.1. For yMD
0 = yMT
0 = m = 0 we have a simpler solution,
ρumin
=
∆t (∆t3
ρv − tf (tf (3∆t + 2tf ) (ρv + ρw) + 4 )) + 3∆t2
−
√
8∆t (∆t + tf ) tf ρv (∆t + tf ) + t2
f ρw + 2
2
− ρv∆t2 t2
f ρw + 2
∆t2 (∆t2 − 4tf (∆t + tf ))
12.1 Initial Conditions
The first topic to explore is the influence of the initial conditions, yMT
0 and yMD
0 , on ρumin
.
Example 12.1. For the following numerical values:
ρv = 30
m
Sec2 , ρw = 50
m
Sec2 , tf = 3 [Sec] , ∆t = 4 [Sec] , m = 0.5 [m] , = 150 [m]
the plot of ρumin
yMT
0 , yMD
0 is shown in Fig. 12.1.
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![Example 12.2. Consider the following values,
ρu = 170
m
sec2
, ρv = 30
m
sec2
, ρw = 50
m
sec2
, tf = 3 [sec] , ∆t = 4 [sec] ,
m = 0.5 [m] , = 150 [m]
The simulations in Fig. 12.3 demonstrate the above analysis.
||yMT|| ||yMD|| ℬ
tgo
*
t*
tf
MD
tf
MT
Time, t
ℓ
||yMT
cr
||
ℬ(t*
)
||ZEM||
(a) yMD
0 = 0, yMT
0 = 0
||yMT|| ||yMD|| ℬ
tgo
*
t*
tf
MD
tf
MT
Time, t
ℓ
||yMT
cr
||
ℬ(t*
)
||ZEM||
(b) yMD
0 = 200, yMT
0 = 0
||yMT|| ||yMD|| ℬ
tgo
*
t*
tf
MD
tf
MT
Time, t
ℓ
||yMT
cr
||
||ZEM||
(c) yMD
0 = 0, yMT
0 = 200
||yMT|| ||yMD|| ℬ
tgo
*
t*
tf
MD
tf
MT
Time, t
ℓ
||yMT
cr
||
ℬ(t*
)
||ZEM||
(d) yMD
0 = 200, yMT
0 = 200
Figure 12.3: Linear Simulations for Different Initial Conditions
Notice that the influence of yMD
0 is greater than of yMT
0 . From Fig. 12.3, we conclude that
yMT
0 = yMD
0 = 200 is better for the Missile than yMT
0 = yMD
0 = 0.
43](https://image.slidesharecdn.com/d0df201a-e3c2-46c0-9eef-17e7d604c5fa-150722113059-lva1-app6892/85/Dissertation-A-Three-Players-Pursuit-and-Evasion-Conflict-59-320.jpg)
![12.2 Target’s and Defender’s Maneuver Capabilities
This subsection explores the influence of ρv and ρw on ρumin
.
Example 12.3. Consider the following numerical values,
yMT
0 = yMD
0 = 0, tf = 3 [Sec] , ∆t = 4 [Sec] , m = 0.5 [m] , = 150 [m]
The plot of ρumin
(ρv, ρw) is shown in Fig. 12.4.
0
20
40
ρv
0
20
40
ρw
50
100
150
ρumin
(a) Plot of ρumin
(ρv, ρw)
0 10 20 30 40 50
0
10
20
30
40
50
ρv
ρw
ρumin
(ρv, ρw)
60
100
140
180
(b) Contour Plot of ρumin
(ρv, ρw)
Figure 12.4: Plot and Contour Plot of ρumin
(ρv, ρw)
and the section plots of ρumin
(ρv) and ρumin
(ρw) are depicted in Fig. 12.5.
ρw = 0 ρw = 25 ρw = 50
10 20 30 40 50
ρv
50
100
150
ρumin
(a) ρumin (ρv)
ρv = 0 ρv = 25 ρv = 50
10 20 30 40 50
ρw
50
100
150
ρumin
(b) ρumin (ρw)
Figure 12.5: Section Plots
44](https://image.slidesharecdn.com/d0df201a-e3c2-46c0-9eef-17e7d604c5fa-150722113059-lva1-app6892/85/Dissertation-A-Three-Players-Pursuit-and-Evasion-Conflict-60-320.jpg)
![Note that, ρumin
(ρv, ρw) behaves almost as a linear function of ρv and ρw. As expected, the grater
ρv and ρw are, the harder it is for the Missile to complete its task.
Example 12.4. Consider the following numerical values,
ρu = 170
m
Sec2 , yMT
0 = yMD
0 = 0, tf = 3 [Sec] , ∆t = 4 [Sec] , m = 0.5 [m] , = 150 [m]
Linear simulation results for different values of ρv and ρw are shown in Fig. 12.6.
||yMT|| ||yMD|| ℬ
tgo
*
t*
tf
MD
tf
MT
Time, t
ℓ
||yMT
cr
||
||ZEM||
(a) ρv = 30, ρw = 50
||yMT|| ||yMD|| ℬ
tgo
*
t*
tf
MD
tf
MT
Time, t
ℓ
||yMT
cr
||
||ZEM||
(b) ρv = 40, ρw = 50
||yMT|| ||yMD|| ℬ
tgo
*
t*
tf
MD
tf
MT
Time, t
ℓ
||yMT
cr
||
||ZEM||
(c) ρv = 30, ρw = 60
||yMT|| ||yMD|| ℬ
tgo
*
t*
tf
MD
tf
MT
Time, t
ℓ
||yMT
cr
||
||ZEM||
(d) ρv = 40, ρw = 60
Figure 12.6: Linear Simulations for Different Values of ρv and ρw
Indeed, the increase of ρv and ρw makes it harder for the Missile to achieve its goal.
45](https://image.slidesharecdn.com/d0df201a-e3c2-46c0-9eef-17e7d604c5fa-150722113059-lva1-app6892/85/Dissertation-A-Three-Players-Pursuit-and-Evasion-Conflict-61-320.jpg)
![12.3 Required M-D and M-T miss distances
While we impose our requirements on the miss distances m and , it is important to understand
their impact on the Missile’s required capability, ρumin
.
Example 12.5. Consider the numerical values,
ρv = 30
m
Sec2 , ρw = 50
m
Sec2 , tf = 3 [Sec] , ∆t = 4 [Sec] , yMT
0 = yMD
0 = 0
Fig. 12.7 depicts the plot of ρumin
(m, ).
0
2
4
50
100
150
200ℓ
120
140
160
ρumin
(a) Plot of ρumin (m, )
0 1 2 3 4 5
50
100
150
200
ℓ
ρumin
(, ℓ)
120
130
140
150
160
(b) Contour Plot of ρumin (m, )
Figure 12.7: Plot and Contour Plot of ρumin
(m, )
Also, the section plots of ρumin
(m) and ρumin
( ) are shown in Fig. 12.8.
ℓ = 10 ℓ = 50 ℓ = 100
2 4 6 8 10
115
120
125
130
135
ρumin
(a) ρumin
(m)
= 0 = 50 = 100
50 100 150 200
ℓ
110
120
130
140
150
160
ρumin
(b) ρumin
( )
Figure 12.8: Section Plots
46](https://image.slidesharecdn.com/d0df201a-e3c2-46c0-9eef-17e7d604c5fa-150722113059-lva1-app6892/85/Dissertation-A-Three-Players-Pursuit-and-Evasion-Conflict-62-320.jpg)
![Again, the dependence of ρumin
on m and is close to linear. However, the required M-T miss
distance, m, has small influence on ρumin
.
Example 12.6. Consider the numerical values
ρu = 170
m
sec2
, ρv = 30
m
sec2
, ρw = 50
m
sec2
, tf = 3 [sec] , yMT
0 = yMD
0 = 0, ∆t = 4 [sec]
Linear simulations depicted in Fig. 12.9 illustrate the above analysis.
||yMT|| ||yMD|| ℬ
tgo
*
t*
tf
MD
tf
MT
Time, t
ℓ
||yMT
cr
||
ℬ(t*
)
||ZEM||
(a) m = 0, = 150
||yMT|| ||yMD|| ℬ
tgo
*
t*
tf
MD
tf
MT
Time, t
ℓ
||yMT
cr
||
ℬ(t*
)
||ZEM||
(b) m = 10, = 150
||yMT|| ||yMD|| ℬ
tgo
*
t*
tf
MD
tf
MT
Time, t
ℓ
||yMT
cr
||
ℬ(t*
)
||ZEM||
(c) m = 0, = 300
||yMT|| ||yMD|| ℬ
tgo
*
t*
tf
MD
tf
MT
Time, t
ℓ
||yMT
cr
||
ℬ(t*
)
||ZEM||
(d) m = 10, = 300
Figure 12.9: Linear Simulations for Different Values of m and
Hence, for any practical use, m = 0 can be chosen, as it simplifies the expressions and has small
effect on the required capability.
47](https://image.slidesharecdn.com/d0df201a-e3c2-46c0-9eef-17e7d604c5fa-150722113059-lva1-app6892/85/Dissertation-A-Three-Players-Pursuit-and-Evasion-Conflict-63-320.jpg)
![12.4 The final times tMD
f and tMT
f
When we talk about the final times, we refer to tMD
f and tMT
f . However, in Subsection 7.3 of Part
I, the following parameters were defined.
tMD
f = tf (12.2)
tMT
f = tf + ∆t (12.3)
Therefore, we explore the influence of the final times in terms of tf and ∆t.
Example 12.7. Consider the numerical values
ρv = 30
m
Sec2 , ρw = 50
m
Sec2 , tf = 3 [Sec] , ∆t = 4 [Sec] , m = 0.5 [m] , = 150 [m]
The plot of ρumin
(tf , ∆t) is shown in Fig. 12.10,
1 2
3
4
5
Δt
2
3
4
5
tf
200
300
400
500
600
ρumin
(a) Plot of ρumin (tf , ∆t)
1 2 3 4 5
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
Δt
tf
ρumin
(Δt, tf )
150
250
350
450
550
(b) Contour Plot of ρumin (tf , ∆t)
Figure 12.10: Plot and Contour Plot of ρumin
(tf , ∆t)
and the section plots are depicted in Fig. 12.11.
48](https://image.slidesharecdn.com/d0df201a-e3c2-46c0-9eef-17e7d604c5fa-150722113059-lva1-app6892/85/Dissertation-A-Three-Players-Pursuit-and-Evasion-Conflict-64-320.jpg)
![Example 12.8. Linear simulations in Fig. 12.12 Illustrate this analysis. Consider the numerical
values:
ρu = 170
m
Sec2 , ρv = 30
m
Sec2 , ρw = 50
m
Sec2 , ∆t = 4 [Sec] ,
m = 0.5 [m] , = 150 [m]
From Fig. 12.11 we have that for ∆t = 4 [sec], the optimal value of tf is tOpt
f ≈ 2.5 [sec].
||yMT|| ||yMD|| ℬ
t*
=tf
MD
tf
MT
Time, t
ℓ
||yMT
cr
||
ℬ(t*
)
||ZEM||
(a) tf = 1.5
||yMT|| ||yMD|| ℬ
tgo
*
t*
tf
MD
tf
MT
Time, t
ℓ
||yMT
cr
||
ℬ(t*
)
||ZEM||
(b) tf = 2.5
||yMT|| ||yMD|| ℬ
tgo
*
t*
tf
MD
tf
MT
Time, t
ℓ
||yMT
cr
||
ℬ(t*
)
||ZEM||
(c) tf = 3.5
||yMT|| ||yMD|| ℬ
tgo
*
t*
tf
MD
tf
MT
Time, t
||yMT
cr
||
||ZEM||
(d) tf = 4.5
Figure 12.12: Linear Simulations for Different Values of tf
At t∗
go, the ZEM |yMT | is most far from its bound B, at tf = 2.5 [sec].
50](https://image.slidesharecdn.com/d0df201a-e3c2-46c0-9eef-17e7d604c5fa-150722113059-lva1-app6892/85/Dissertation-A-Three-Players-Pursuit-and-Evasion-Conflict-66-320.jpg)
![where dt = ti − ti−1 ∀i = 1, 2, . . . , N. Therefore, maximizing (13.10) means
max
kv(t)
ˆ t∗
0
kv(ξ)dξ = lim
N→∞
N
i=1
max
kv(ti)
{kv(ti)} dt (13.11)
where −ρv ≤ kv(t) ≤ ρv. Hence; maximizing the Riemann’s integral means maximizing the
function kv(t) at each time point, ti. The maximizing value for kv(t) at each time point ti is
kOpt
v (ti) = ρv ∀i = 1, 2, . . . , N. The same conclusion can be made for
˜ t∗
0
kv(ξ)dξ. Consequently,
the optimal value of the Target’s maneuver gain is kv(t)=ρv.
As for optimality of kw(t), recall (13.9),
d(·)
1
2
(ρu − ρv) t∗
go + ∆t
2
−
¨ t∗
0
kv(ξ)dξdξ − t∗
go + ∆t
ˆ t∗
0
kv(ξ)dξ − f (t∗
) (13.12)
One can see that d(·) doesn’t depend on kw(t) directly, rather it depends on t∗
which is the
intersection time of
|yMD(t)| = −
¨ t
0
kw(ξ)dξdξ − tgo
ˆ t
0
kw(ξ)dξ + g(t) , tgo = tf − t (13.13)
with the fail-safe function C(tgo) = + 1
2
(ρu + ρw) t2
go. Thus, d(·) is not affected by the shape of
the function kw(t), rather it is only affected by t∗
. However, since −ρw ≤ kw(t) ≤ ρw , the function
|yMD(t)| is bounded,
yMIN
MD (kw = ρw) ≤ yMD kw(t) ≤ yMAX
MD (kw = −ρw) (13.14)
Denote t∗
MIN as the intersection of yMAX
MD with C, and t∗
MAX as the intersection of yMIN
MD with C.
Assuming continuity, the entire range of t∗
∈ [t∗
MIN , t∗
MAX] is reachable by a constant maneuver
gain kw ∈ [−ρw, ρw]. Hence, there always exists a constant maneuver kw that yields the same
intersection time t∗
; thus, the same function d(·) (Fig. 13.3). However, from (13.4) we know that
if kw(t) = kw = const. then the optimal solution is: kw = ρw. Consequently, the optimal maneuver
gain of the Defender is kw = ρw.
|yMD
MAX
| |yMD
MIN
| |yMD| |yMD
Equivalent
(kw=Const)|
tMIN
*
t*
tMAX
*
Time,t
|yMD(t*
)|
|ZEM|
Figure 13.3: Bounds and Different Possibilities of |yMD(t)|
53](https://image.slidesharecdn.com/d0df201a-e3c2-46c0-9eef-17e7d604c5fa-150722113059-lva1-app6892/85/Dissertation-A-Three-Players-Pursuit-and-Evasion-Conflict-69-320.jpg)
![13.2 Optimality in the nonlinear kinematics scenario
For linear kinematics, the optimal maneuvers regarding the “measure of success” d(·), are
u∗
=
−ρusign(yMD) |yMD| < C
ρusign(yMT ) |yMD| ≥ C
(13.15)
v∗
= ρvsign(yMT ) (13.16)
w∗
= −ρwsign(yMD) (13.17)
Example 13.1. Let the Target use v(t) = kvsign(yMT ). Consider the parameters,
ρu = 170
m
Sec2 , ρv = 30
m
Sec2 , ρw = 60
m
Sec2 , m = 0 [m] , = 120 [m], yMT
0 = yMD
0 = 0
Missile
Target
Defender
tf
MD
t*
0 1000 2000 3000 4000 5000 6000
0
500
1000
1500
2000
x [m]
y[m]
Miss MD = 215 , tf
MD
= 5.81
Miss MT = 0.2 , tf
MT
= 8.3
(a) kv = −ρv
Missile
Target
Defender
tf
MD
t*
0 1000 2000 3000 4000 5000 6000
0
500
1000
1500
2000
2500
x [m]
y[m]
Miss MD = 215 , tf
MD
= 5.81
Miss MT = 0.2 , tf
MT
= 9.24
(b) kv = −0.5ρv
Missile
Target
Defender
tf
MD
t*
0 1000 2000 3000 4000 5000 6000
0
1000
2000
3000
4000
x [m]
y[m]
Miss MD = 215 , tf
MD
= 5.81
Miss MT = 0.2 , tf
MT
= 12
(c) kv = 0.5ρv
Missile
Target
Defender
tf
MD
t*
0 1000 2000 3000 4000 5000 6000
0
1000
2000
3000
4000
x [m]
y[m]
Miss MD = 215 , tf
MD
= 5.81
Miss MT = 0.2 , tf
MT
= 13.5
(d) kv = ρv
Figure 13.4: Nonlinear simulation for kv = −ρv, −0.5ρv, 0.5ρv and ρv
54](https://image.slidesharecdn.com/d0df201a-e3c2-46c0-9eef-17e7d604c5fa-150722113059-lva1-app6892/85/Dissertation-A-Three-Players-Pursuit-and-Evasion-Conflict-70-320.jpg)
![From simulation (Fig. 13.4), it is readily seen that regardless of the Target’s strategy, it gets
intercepted by the Missile. Therefore, one might think that optimal strategies for linear kinematics
are indeed optimal in the real (nonlinear) scenario. Generally, since the M-D game takes place at
the first phase of guidance, the collision triangle between them suffers relatively small distortion
(assuming players are close to collision triangle at the beginning, and evasion doesn’t take too
much time), the time-to-go is close to linear, and u∗
and w∗
are arguably justified (although the
actual M-D miss distance considerably bigger than required). However, by evading the Defender,
the Missile also “evades” the Target (recall that u∗
e = −u∗
p), while the Target evades the Missile
(applies v∗
). Consequently, the M-T collision triangle breaks and linearization assumptions fail to
hold.
Example 13.2. Now, consider the same parameters, except: = 150 [m], and a slightly different
geometry. Nonlinear simulations for kv = −ρv and kv = ρv are depicted in Fig. 13.5.
Missile
Target
Defender
tf
MD
t*
0 1000 2000 3000 4000 5000 6000
-400
-200
0
200
400
600
800
x [m]
y[m]
Miss MD = 325 , tf
MD
= 5.74
Miss MT = 380 , tf
MT
= 7.37
(a) kv = −ρv
Missile
Target
Defender
tf
MD
t*
0 1000 2000 3000 4000 5000 6000
0
1000
2000
3000
4000
x [m]
y[m]
Miss MD = 325 , tf
MD
= 5.74
Miss MT = 0.1 , tf
MT
= 13.37
(b) kv = ρv
Figure 13.5: Nonlinear simulations for kv = −ρv and kv = ρv
Clearly, the Target gets intercepted when maneuvering optimally, but manages to escape by apply-
ing the opposite guidance strategy, which by our analysis is the worst for it to choose. This refutes
our optimality analysis for the Target. What went wrong? In order to explain this, observe again
the nonlinear simulations in Fig. 13.5. Notice, that while the final time tMT
f for kv = −ρv is about
7.4 [sec], it is about 13.4 [sec] for kv = ρv. Indeed, by applying kv = −ρv the Target “pursues”
the Missile. Therefore, the M-T collision triangle suffers relatively small distortion and the final
time tMT
f suffers small change during the game. However, by applying kv = ρv the Missile and the
Target maneuver at opposite directions, resulting the collision triangle to break. As a result, the
value of tMT
f is dramatically different from tMT
go at t = 0. Recall that tMT
f = tf + ∆t; thus, loosely
speaking, Target’s evasive maneuver has “increased” ∆t. This is the main idea of this analysis: the
harder the Target evades the Missile, the more it “increases” ∆t.
55](https://image.slidesharecdn.com/d0df201a-e3c2-46c0-9eef-17e7d604c5fa-150722113059-lva1-app6892/85/Dissertation-A-Three-Players-Pursuit-and-Evasion-Conflict-71-320.jpg)
![Example 13.3. Approximate the nonlinear simulations of Example 13.2 with linear simulations.
Consider
ρu = 170
m
Sec2 , ρv = 30
m
Sec2 , ρw = 60
m
Sec2 , m = 0 [m] , = 325 [m] ,
tf = 5.74 [Sec] , yMD
0 = yMT
0 = 0
For kv = −ρv we set ∆t = 1.7 [sec], while for kv = ρv we set ∆t = 7.7 [sec]. Fig. 13.6 shows the
results.
||yMT|| ||yMD|| ℬ
tgo
*
t*
tf
MD
tf
MT
Time, t
ℓ
||yMT
cr
||
ℬ(t*
)
||ZEM||
(a) kv = −ρv, ∆t = 1.7 [sec]
||yMT|| ||yMD|| ℬ
tgo
*
t*
tf
MD
tf
MT
Time, t
||yMT
cr
||
ℬ(t*
)
||ZEM||
(b) kv = ρv, ∆t = 7.7 [sec]
Figure 13.6: Linear simulations for kv = −ρv and kv = ρv
Linear Simulations in Fig. 13.6 justify the results of the nonlinear simulations in Example 13.2.
By using kv = −ρv, the Target minimizes |yMT | instead of maximizing it (which appears to be
optimal); however, ∆t remains almost unchanged. As a result, small ∆t enables it to evade the
Missile despite the opposite maneuver. On the other hand, by applying kv = ρv, the Target
maximizes |yMT |; however, it also adds about 6 [sec] to ∆t. As a result the Target “increases” the
bound B t∗
go by
∆B t∗
go = te (ρu − ρv)
te
2
+ t∗
go (13.18)
where te is the addition to ∆t (in this example te = 6 [sec]). Consequently, the Target has let the
Missile to intercept it, despite the maximization of |yMT | which appears to be optimal. To clarify
even more, Fig. 13.7 presents the results of Example 13.3 on the same plot.
56](https://image.slidesharecdn.com/d0df201a-e3c2-46c0-9eef-17e7d604c5fa-150722113059-lva1-app6892/85/Dissertation-A-Three-Players-Pursuit-and-Evasion-Conflict-72-320.jpg)
![|yMT(kv=-ρv)| |yMT(kv=ρv)| ℬ(kv=-ρv) ℬ(kv=ρv)
Δℬ(t*
)Δ|yMT
cr
|
t*
tf
MT
tf
MT
Time, t
|yMT
cr
|
ℬ(t*
)
|yMT
cr
|
ℬ(t*
)
|ZEM|
Figure 13.7: Results of Fig. 13.6, presented on the same plot
Clearly, ∆ |ycr
MT | is smaller than ∆B(t∗
go). Hence, by performing an evasive maneuver, the Target
has lost in general more than it gained from maximizing its ZEM.
Remark 13.1. In order to intercept the Target in such a scenario (kv = −ρv), the Missile must have
more maneuvering capability, or alternatively, the required M-D miss distance, has to be reduced.
Fig. 13.8 demonstrates the idea.
Missile
Target
Defender
tf
MD
t*
0 1000 2000 3000 4000 5000 6000
-600
-400
-200
0
200
400
600
800
x [m]
y[m]
Miss MD = 241.8 , tf
MD
= 5.84
Miss MT = 0.4 , tf
MT
= 7.47
(a) ρu = 220
Missile
Target
Defender
tf
MD
t*
0 1000 2000 3000 4000 5000 6000
-200
0
200
400
600
800
x [m]
y[m]
Miss MD = 28.5 , tf
MD
= 5.63
Miss MT = 0.3 , tf
MT
= 7.13
(b) ρu = 170
Figure 13.8: Nonlinear Simulations
57](https://image.slidesharecdn.com/d0df201a-e3c2-46c0-9eef-17e7d604c5fa-150722113059-lva1-app6892/85/Dissertation-A-Three-Players-Pursuit-and-Evasion-Conflict-73-320.jpg)
![13.3 Intermediate conclusions
Rewrite the inequality derived in Section 8 of Part I for some Target’s maneuver v = kvsign(yMT ),
where −ρv ≤ kv ≤ ρv.
− (kv + ρw) t3
f − 2∆t (kv + ρw) t2
f
+ (ρu − kv) ∆t2
+ 2 m − + yMD
0 − yMT
0 tf + 4∆t yMD
0 −
(kv + ρw) t2
f − 2∆t (ρu − kv) tf − (ρu − kv) ∆t2 − 2 (m − + |yMD
0 | − |yMT
0 |)
≤
t2
f (ρu − ρw) − 2 + 2 |yMD
0 |
2ρu
(13.19)
1. If (13.19) holds for3
kv = ρv, the Missile can guarantee its success for any Target’s maneuver.
Namely, even if ∆t doesn’t suffer an increase due to the non-linearity caused by Target’s
evasive maneuver, the Missile is still able to intercept it. Moreover, if ∆t grows, or the
Target applies a suboptimal maneuver gain, kv < ρv, it is even easier for the Missile to
intercept it. Such a case is described in Example 13.1, where the Target is being intercepted
regardless of its maneuver.
2. If (13.19) does not hold for4
kv = −ρv, the Target can evade the Missile using any kv ∈
[−ρv, ρv] if ∆t remains constant. However, we know that ∆t remains approximately constant
only if kv = −ρv (again, assuming evasion doesn’t take too much time). Therefore, the Target
can guarantee its safety by performing an opposite maneuver towards the Missile; namely,
by applying kv = −ρv. It is important to understand that even in this case, kv = −ρv is not
the optimal5
maneuver. However, this maneuver guarantees Target’s evasion, while other
strategies have the chance to increase ∆t and enable interception. We can observe this case
in Example 13.2.
3. If (13.19) does not hold for kv = ρv but holds for kv = −ρv, a further analysis (provided
in Subsection 13.4) is required. In this case, the Target cannot apply neither kv = −ρv nor
kv = ρv, because kv = −ρv leads to capture (as (13.19) holds), and kv = ρv makes ∆t grow
and again, (usually) leads to capture. This case is called: The Uncertainty Area.
3
This statement implies that (13.19) also holds for any other kv ∈ [−ρv, ρv]
4
This statement implies that (13.19) doesn’t hold for any kv ∈ [−ρv, ρv]
5
A maneuver which maximizes the M-T miss distance.
58](https://image.slidesharecdn.com/d0df201a-e3c2-46c0-9eef-17e7d604c5fa-150722113059-lva1-app6892/85/Dissertation-A-Three-Players-Pursuit-and-Evasion-Conflict-74-320.jpg)
![13.4 The Uncertainty Area Analysis
13.4.1 The M-T bound function revised
As we know, the function
B(tgo) = m +
1
2
(ρu − ρv) (tgo + ∆t)2
(13.20)
describes the bound of the M-T singular area. However, we also know, that it is not always wise
for the Target to use its maximal evasive maneuver; thus, let us modify (13.20). Consider a Target
maneuvering with v = kvsign(yMT ), where −ρv ≤ kv ≤ ρv. In such a case (13.20) becomes,
B(tgo) = m +
1
2
(ρu − kv) (tgo + ∆t)2
(13.21)
In order to account for the non-linearity of time-to-go, define the M-T pseudo-singular area,
Bv(tgo) = m +
1
2
(ρu − kv) (tgo + ∆t + te(kv))2
(13.22)
where te(kv) is an approximated addition factor to ∆t resulted by the Target’s evasive maneuver.
Although te(kv) cannot be determined analytically, as it would require knowing all players’ strate-
gies during the entire game period, one can approximate it from simulations. Note that te has
to be a monotonically increasing function of kv, since the bigger kv is, the bigger is the addition
to ∆t. The function Bv(tgo) defines the M-T pseudo-singular area; namely, an area in which the
Missile’s strategy is arbitrary, and the M-T miss distance is smaller than m, if the Target uses
v = kvsign(yMT ).
Example 13.4. Consider the following numerical values
ρu = 170
m
Sec2 , ρv = 30
m
Sec2 , ρw = 60
m
Sec2 , m = 0 [m] , = 30 [m] ,
tf = 5.5 [Sec] , yMD
0 = yMT
0 = 0
Linear simulations for kv = ρv and kv = −ρv are depicted in Fig. 13.9. In these simulations
∆t = 1.5 [sec]; however, for kv = ρv we set te = 4 [sec], and for kv = −ρv we set te = 0 [sec]
(These parameters approximate nonlinear simulations which are discussed later).
||yMT|| ||yMD|| ℬv
tgo
*
t*
tf
MD
tf
MT
Time, t
||yMT
cr
||
||ZEM||
(a) kv = −ρv, te = 0 [sec]
||yMT|| ||yMD|| ℬv
tgo
*
t*
tf
MD
tf
MT
Time, t
||yMT
cr
||
||ZEM||
(b) kv = ρv, te = 4 [sec]
Figure 13.9: Linear simulations for kv = −ρv and kv = ρv
59](https://image.slidesharecdn.com/d0df201a-e3c2-46c0-9eef-17e7d604c5fa-150722113059-lva1-app6892/85/Dissertation-A-Three-Players-Pursuit-and-Evasion-Conflict-75-320.jpg)
![For kv = ρv we have a normal singular area (with an addition of te)
Bv(tgo) =
1
2
(ρu − ρv) (tgo + ∆t + te(ρv))2
(13.23)
However, for kv = −ρv we have
Bv(tgo) =
1
2
(ρu + ρv) (tgo + ∆t + te(−ρv))2
(13.24)
Although Bv does not define the actual singular area (unless kv = ρv), |yMT | behaves inside it just
like inside a singular area. Thus, if at the critical time tgo = t∗
go the ZEM yMT (t∗
go) > Bv(t∗
go),
then the Target can guarantee a miss distance greater than m. It is wise for the Target to look for
such a maneuver gain kv, that ensures yMT (t∗
go) > Bv(t∗
go). In this example, for both kv = ρv and
kv = −ρv the Target gets intercepted by the Missile. This is exactly the uncertainty case when
(13.19) does not hold for kv = ρv but holds for kv = −ρv. Let us try to find such a maneuver gain
kv that keeps the Target safe.
13.4.2 Function d(·) Revised
Let the Target apply some maneuver gain kv ∈ [−ρv, ρv]. Hence, (7.35) becomes
|ycr
MT | =
1
2
tf + 2∆t + t∗
go t∗
go
2
(ρu − kv) + t2
f (ρw + kv) + 2 − 2 yMD
0
tf + t∗
go
+ yMT
0 (13.25)
In order to account for the addition to ∆t, rewrite (13.25) as follows
ycr
MTv
=
1
2
tf + 2 (∆t + te(kv)) + t∗
go t∗
go
2
(ρu − kv) + t2
f (ρw + kv) + 2 − 2 yMD
0
tf + t∗
go
+ yMT
0 (13.26)
Consider a revised “measure of success” function. Define
dv(·) Bv(t∗
go) − ycr
MTv
= m +
1
2
(ρu − kv) t∗
go + ∆t + te(kv)
2
(13.27)
−
1
2
tf + 2 (∆t + te(kv)) + t∗
go t∗
go
2
(ρu − kv) + t2
f (ρw + kv) + 2 − 2 yMD
0
tf + t∗
go
− yMT
0
This function has the same meaning as d(·), only now it accounts for the addition to ∆t. Hence,
if te(kv) can be approximated, then dv(·) can be minimized by kv and the optimal solution for the
Target can be found.
Example 13.5. Consider the same numerical values as in Example 13.4. In that example we had,
yMD
0 = yMT
0 = m = 0; thus, (13.27) reduces to
dv(·) =
1
2
(ρu − kv) t∗
go + ∆t + te(kv)
2
(13.28)
−
1
2
tf + 2 (∆t + te(kv)) + t∗
go t∗
go
2
(ρu − kv) + t2
f (ρw + kv) + 2
tf + t∗
go
60](https://image.slidesharecdn.com/d0df201a-e3c2-46c0-9eef-17e7d604c5fa-150722113059-lva1-app6892/85/Dissertation-A-Three-Players-Pursuit-and-Evasion-Conflict-76-320.jpg)
![-ρv ρv
kv
dv
Figure 13.11: Function dv (kv)
It is readily seen that in order to minimize dv(·) it is best for the Target to apply a small maneuver,
or approximately not to maneuver at all; namely, apply kv = 0. Also, for both kv = ρv and kv = −ρv
we have dv(·) > 0 resulting Bv(t∗
go) > yMT (t∗
go) , meaning the Target gets intercepted. Indeed, our
linear simulations in Example 13.4 approve these results.
Example 13.6. Consider the same numerical values as in Example 13.4. Now, examine the linear
simulation for kv = 0, resulting te = 1 [sec] (Fig. 13.12).
||yMT|| ||yMD|| ℬv
tgo
*
t*
tf
MD
tf
MT
Time, t
||yMT
cr
||
||ZEM||
Figure 13.12: Linear simulation for kv = 0, te = 1 [sec]
Clearly, yMT t = tMT
f > m = 0; hence, the Target evades the Missile.
Example 13.7. Consider the following numerical values
ρu = 170
m
Sec2 , ρv = 30
m
Sec2 , ρw = 60
m
Sec2 , m = 0 [m] , = 10 [m]
In addition, all players are on appropriate collision courses; hence, yMD
0 = yMT
0 = 0. Fig.
13.13 depicts nonlinear simulations for kv = ρv and kv = −ρv, while Fig. 13.14 shows a nonlinear
simulation for kv = 0. In these simulations ∆t = 5.5 [sec] for kv ≈ ρv, ∆t ≈ 1.5 [sec] for kv = −ρv,
62](https://image.slidesharecdn.com/d0df201a-e3c2-46c0-9eef-17e7d604c5fa-150722113059-lva1-app6892/85/Dissertation-A-Three-Players-Pursuit-and-Evasion-Conflict-78-320.jpg)
![and ∆t ≈ 2.5 [sec] for kv = 0. Linear simulations in Example 13.4, approximate the nonlinear
simulations presented in Fig. 13.13, while Example 13.6 approximates the nonlinear simulation in
Fig. 13.14.
Missile
Target
Defender
tf
MD
t*
0 1000 2000 3000 4000 5000 6000
-200
0
200
400
600
800
x [m]
y[m]
Miss MD = 28.5 , tf
MD
= 5.63
Miss MT = 0.3 , tf
MT
= 7.13
(a) kv = −ρv
Missile
Target
Defender
tf
MD
t*
0 1000 2000 3000 4000 5000 6000
0
500
1000
1500
2000
2500
3000
x [m]
y[m]
Miss MD = 28.5 , tf
MD
= 5.63
Miss MT = 0.1 , tf
MT
= 11.54
(b) kv = ρv
Figure 13.13: Nonlinear simulations for kv = −ρv and kv = ρv
Missile
Target
Defender
tf
MD
t*
0 1000 2000 3000 4000 5000 6000
0
500
1000
1500
x [m]
y[m]
Miss MD = 28.5 , tf
MD
= 5.63
Miss MT = 461 , tf
MT
= 8.07
Figure 13.14: Nonlinear simulation for kv = 0
These results justify our analysis, as for both kv = −ρv and kv = ρv the Target gets intercepted,
while for kv = 0 it escapes.
63](https://image.slidesharecdn.com/d0df201a-e3c2-46c0-9eef-17e7d604c5fa-150722113059-lva1-app6892/85/Dissertation-A-Three-Players-Pursuit-and-Evasion-Conflict-79-320.jpg)
![14 Conclusions
In the current part, a parametric analysis was performed. From that analysis we conclude that
the influence of most parameters on the Missile’s minimal maneuvering capability, ρumin
, is very
rational and not surprising. However, the influence of tf suggests a presence of an optimal final
time, tOpt
f , which minimizes ρumin
. This parameter can be numerically evaluated, and serve as a
tactical consideration for both, the Missile, and the Target.
The present part also provides an optimality analysis with respect to d(·), which is a robust-
ness measure (or “success measure”). Although equations (13.15), (13.16), and (13.17) perfectly
describe the players’ optimal strategies in the linear kinematics scenario, the situation is much
more complicated when the kinematics become nonlinear. Since the M-D collision triangle suf-
fers relatively small distortion (assuming the players start close to the M-D collision triangle, and
evasion doesn’t take much time), the Missile’s and the Defender’s optimal strategies, described by
(13.15) and (13.17), are arguably justified for the nonlinear kinematics. However, by applying an
evasive maneuver, the Target also “increases ∆t”, which is assumed to be constant by linearization.
As a result, the Missile obtains more time to intercept it, and the advantage from the extra inter-
cept time can overcome the disadvantage from the Target’s evasive maneuver. Hence, the optimal
Target’s maneuver, described by (13.16), is not necessarily optimal in a real, nonlinear, conflict.
By performing the analysis described earlier in this part, the following conclusions can be made.
1. If (13.19) holds for kv = ρv (also holds for any other kv ∈ [−ρv, ρv]), the Missile can guarantee
its success for any Target’s maneuver. Namely, even if ∆t doesn’t suffer an increase due to
the non-linearity caused by Target’s evasive maneuver, the Missile is still able to intercept
it. Moreover, if ∆t grows, or the Target applies a suboptimal maneuver gain, kv < ρv, it is
even easier for the Missile to intercept it. Such a case is described in Example 13.1, where
the Target is being intercepted regardless of its maneuver.
2. If (13.19) does not hold for kv = −ρv (doesn’t hold for any kv ∈ [−ρv, ρv]), the Target can
evade the Missile using any kv ∈ [−ρv, ρv] if ∆t remains constant. However, we know that
∆t remains approximately constant only if kv = −ρv (again, assuming evasion doesn’t take
too much time). Therefore, the Target can guarantee its safety by performing an opposite
maneuver towards the Missile; namely, by applying kv = −ρv. It is important to understand
that even in this case, kv = −ρv is not the optimal maneuver. However, this maneuver
guarantees Target’s evasion, while other strategies have the chance to increase ∆t and enable
interception. We can observe this case in Example 13.2.
3. If (13.19) does not hold for kv = ρv but holds for kv = −ρv, the approximated addition factor,
te(kv), must be obtained. Once te(kv) is found, the new robustness measure, dv(·), serves
as the optimization criterion instead of d(·). Of course te(kv) is only a rough approximation
and can be obtained only from simulations; therefore, the uncertainty in this case is very
significant. This case is shown in Example 13.7.
A very important conclusion we can make from the above is that LMG cannot provide accurate
results in a nonlinear conflict. This is because LMG relies on linearization, which fails to hold in
a conflict of three players. In addition, the improvements designed to account for the nonlinear
nature of the game rely on rough approximations and are very hard for implementation. This leads
us to the need of developing a new method, one that does not rely on linearization and provides
accurate results in a real, nonlinear conflict. Such approach is discussed in Part III.
64](https://image.slidesharecdn.com/d0df201a-e3c2-46c0-9eef-17e7d604c5fa-150722113059-lva1-app6892/85/Dissertation-A-Three-Players-Pursuit-and-Evasion-Conflict-80-320.jpg)
![20 VG1 Simulations
One must understand that tMT
f , and therefore ∆t, can change due to the player’s strategies. As
a result, (19.38) provides only a sufficient condition regarding the initial value of ∆t. Thus, if
(19.38) holds for the initial ∆t, then the Missile is able to intercept the Target in the worst case
when ∆t remains unchanged. However, if (19.38) does not hold, the Missile might still be able to
intercept the Target with a different tMT
f . This analysis is similar to the one made for the linear
model; however, the big difference is that all players can now account for the true nature of game
kinematics, and therefore, adopt the optimal strategy regarding the actual ZEM.
Example 20.1. Consider the following parameters,
ρu = 170
m
sec2
, ρv = 30
m
sec2
, ρw = 60
m
sec2
m = 0.5 [m] , = 150 [m]
The M-T-D trajectories are depicted in Fig. 20.1.
Missile
Target
Defender
tf
MD
t*
0 1000 2000 3000 4000 5000 6000
0
500
1000
1500
2000
2500
x [m]
y[m]
Miss MD = 154.5 , tf
MD
= 5.57
Miss MT = 0.18 , tf
MT
= 14.57
Figure 20.1: VG1 Planar Simulation 1
Clearly, the Missile misses the Target. Actually, the simulation stops after d rMT
dt
changes sign. If
not, the Missile is fully capable of intercepting the Target. This result shows that
1. VG1 is more reliable than LMG. As discussed in Part II, a considerable disadvantage of
LMG is its dependance on linearization. Hence, by performing the optimal evasive maneuver
regarding the linear model, the Target increases tMT
f , which is assumed to be constant in
LMG, and increases the bound B. As a result, the advantage gained from maximizing yMT
is smaller then the disadvantage gained from the increase in tMT
f . However, VG accounts
for full kinematics; therefore, the Target’s evasive maneuver is more reliable. Note that the
direction of the Target’s trajectory is similar to the one in Fig. 13.14 of Part II. However,
now this result is obtained without any approximations.
78](https://image.slidesharecdn.com/d0df201a-e3c2-46c0-9eef-17e7d604c5fa-150722113059-lva1-app6892/85/Dissertation-A-Three-Players-Pursuit-and-Evasion-Conflict-94-320.jpg)
![2. The actual M-D miss distance is almost identical to the required one, unlike LMG, where the
actual M-D miss distance is much greater than required due to the nonlinear effects, which
cannot be foreseen in the linearized model.
In Fig. 20.2, one finds the nonlinear simulation of LMG (Fig. 9.5), and the vector guidance (VG1).
Missile
Target
Defender
tf
MD
t*
0 1000 2000 3000 4000 5000 6000
0
500
1000
1500
2000
2500
x [m]
y[m]
Miss MD = 154.5 , tf
MD
= 5.57
Miss MT = 0.18 , tf
MT
= 14.57
(a) VG1
Missile
Target
Defender
tf
MD
t*
0 1000 2000 3000 4000 5000 6000
0
1000
2000
3000
4000
x [m]
y[m]
Miss MD = 325 , tf
MD
= 5.74
Miss MT = 0.1 , tf
MT
= 15.37
(b) LMG
Figure 20.2: VG1 Vs. LMG
Indeed, VG1 shows better results than LMG. In Fig. 20.3, one finds the trajectories plot vs. the
ZEM plot.
Missile
Target
Defender
tf
MD
t*
0 1000 2000 3000 4000 5000 6000
0
500
1000
1500
2000
2500
x [m]
y[m]
Miss MD = 154.5 , tf
MD
= 5.57
Miss MT = 0.18 , tf
MT
= 14.57
(a) Planar Simulation
||yMT|| ||yMD|| ℬ
tgo
*
t*
tf
MD
tf
MT
Time, t
||yMT
cr
||
ℬ(t*
)
||ZEM||
(b) ZEM Trajectories
Figure 20.3: Planar Simulation and ZEM Trajectories
79](https://image.slidesharecdn.com/d0df201a-e3c2-46c0-9eef-17e7d604c5fa-150722113059-lva1-app6892/85/Dissertation-A-Three-Players-Pursuit-and-Evasion-Conflict-95-320.jpg)
![Note that tMD
f and tMT
f are relatively close to each other. However, this picture is not yet complete,
as by using VG1, the Missile has the capability to pursue the Target even if rMT changes its
sign.
Example 20.2. Consider the same parameters as in Example 20.1; however, the simulation does
not stop when the sign of d
dt
rMT changes (Fig. 20.4).
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
Figure 20.4: VG1 Planar Simulation 2
The Missile eventually intercepts the Target. In Fig. 20.5, one finds a local minimum in rMT (t) .
0 5 10 15 20 25
0
1000
2000
3000
4000
5000
6000
Time, t
M-TRange,||rMT(t)||
Figure 20.5: Relative Distance rMT (t)
80](https://image.slidesharecdn.com/d0df201a-e3c2-46c0-9eef-17e7d604c5fa-150722113059-lva1-app6892/85/Dissertation-A-Three-Players-Pursuit-and-Evasion-Conflict-96-320.jpg)
![Example 20.3. A three dimensional version of such a conflict is shown in Fig. 20.6.
Missile Target Defender tf
MD t*
0
5000
x [m]
0
2000
4000
6000
8000y [m]
0
1000
2000
3000
z [m]
Figure 20.6: VG1 3D Simulation 1
Since ρu = 170 is not enough to capture the Target, without having d
dt
rMT change its sign, in
tMT
f ≈ 8.5 [sec] as in Fig. 20.1, the Missile needs more maneuvering capability.
Example 20.4. A simulation for the following parameters is depicted in Fig. 20.7.
ρu = 270
m
sec2
, ρv = 30
m
sec2
, ρw = 60
m
sec2
, m = 0.5 [m] , = 150 [m]
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
Figure 20.7: VG1 Planar Simulation 3
81](https://image.slidesharecdn.com/d0df201a-e3c2-46c0-9eef-17e7d604c5fa-150722113059-lva1-app6892/85/Dissertation-A-Three-Players-Pursuit-and-Evasion-Conflict-97-320.jpg)
![Also, rMT (t) becomes (Fig. 20.8).
0 2 4 6 8 10
0
1000
2000
3000
4000
5000
6000
Time, t
M-TRange,||rMT(t)||
Figure 20.8: Relative Distance rMT (t)
Clearly, rMT (t) has no local minima, and the Missile intercepts the Target without turning
around and chasing it.
Example 20.5. A 3D version of such a conflict is described in Fig. 20.9 (3D trajectories (left),
ZEM trajectories (right)).
Missile Target Defender tf
MD t*
0
2000
4000
6000x [m]
0
1000
2000
3000y [m]
0
500
1000
z [m]
||yMT|| ||yMD|| ℬ
tgo
*
t*
tf
MD
tf
MT
Time, t
||yMT
cr
||
ℬ(t*
)
||ZEM||
Figure 20.9: VG1 3D Simulation 2 and ZEM Trajectories
Although yMD and yMT are distorted due to nonlinearities in tMD
go and tMT
go , yMT stays inside
the bound B, and the Target is intercepted.
82](https://image.slidesharecdn.com/d0df201a-e3c2-46c0-9eef-17e7d604c5fa-150722113059-lva1-app6892/85/Dissertation-A-Three-Players-Pursuit-and-Evasion-Conflict-98-320.jpg)
![Example 22.1. For some rMD and VMD, and different values of q , the function g tMD
go is shown
in Fig. 22.3.
q = 100 [m] q = 1050 [m] q = 2100 [m]
2 4 6 8 10
tgo
MD
2000
4000
6000
8000
g(tgo
MD)
Figure 22.3: Function g tMD
go For Different Values of q
As a result, the following conclusions are to be made (keep in mind that in this analysis both
players are assumed to play optimal):
1. If g tMD
go = 0 has no positive real solutions, then q is too small, and such a miss distance
is unachievable. Refer to the Example 22.2. Here the required q is smaller than qmin; thus,
there is no value of the time, t, for which q < qmin.
2. If g tMD
go = 0 has exactly one positive real solution, then q = qmin is the minimal achievable
M-D miss distance. Clearly, this occurs when rMD reaches its minimum (again, refer to
Example 22.2).
3. If g tMD
go = 0 has more than one positive real solution (in fact, exactly 2 solutions), then
q is too big, and by choosing the bigger of the two solutions for tMD
go , we are aiming for
a higher trajectory than required (while the smaller solution is not physical, as rMD still
decreases). This would require more time for evasion than necessary and produce bigger M-D
miss distance than desired. In Example 22.2 we see that the first solution of g tMD
go = 0
is not physical as it occurs when the relative distance is still decreasing, while the second
solution introduces a bigger evasion time than the required minimum, and a bigger distance
than the actual miss-distance.
92](https://image.slidesharecdn.com/d0df201a-e3c2-46c0-9eef-17e7d604c5fa-150722113059-lva1-app6892/85/Dissertation-A-Three-Players-Pursuit-and-Evasion-Conflict-108-320.jpg)
![Example 22.4. For some numerical values, Fig. 22.7 shows a comparison between 1st
and 4th
order time-to-go.
VG4
VG1
0 1 2 3 4 5 6
0
1
2
3
4
5
6
Time, t
Estimatedtgo
MD
t*
Figure 22.7: Evaluation of tMD
go for VG1 and VG4
It is readily seen that when tMD
go ∈ [0, ts], both VG1 and VG4 produce similar results.
Theorem 22.7. The optimal acceleration direction for both Missile and Defender in fixed coordi-
nates is constant.
Proof. The proof is similar to that of M-T game. The kinematic equation is,
¨rMD(t) = w(t) − u(t) (22.70)
Let both players use optimal maneuvers
u∗
e(t) = −ρu
rMD(t) + tMD
go ˙rMD(t)
rMD(t) + tMD
go ˙rMD(t)
(22.71)
w∗
(t) = −ρw
rMD(t) + tMD
go ˙rMD(t)
rMD(t) + tMD
go ˙rMD(t)
(22.72)
Therefore,
¨rMD(t) = (ρu − ρw)
rMD(t) + tMD
go ˙rMD(t)
rMD(t) + tMD
go ˙rMD(t)
(22.73)
Also let both players use VG4. Hence,
rMD(t) + tMD
go ˙rMD(t) = q −
1
2
(ρu − ρw) tMD
go
2
(22.74)
Assuming that both players play optimal, we have tMD
f = const. Substitute, and obtain
¨rMD(t) = (ρu − ρw)
rMD(t) + tMD
go ˙rMD(t)
q − 1
2
(ρu − ρw) tMD
go
2 =
2 rMD(t) + tMD
go ˙rMD(t)
2q
ρu−ρw
− tMT
go
2 (22.75)
96](https://image.slidesharecdn.com/d0df201a-e3c2-46c0-9eef-17e7d604c5fa-150722113059-lva1-app6892/85/Dissertation-A-Three-Players-Pursuit-and-Evasion-Conflict-112-320.jpg)
![Simplify and obtain
tMD
f − t
2
− T 2
¨rMD(t) + 2tMD
go ˙rMD(t) + 2rMD(t) = 0
rMD(0) = rMD
0
˙rMD(0) = V MD
0
(22.76)
where T = 2q
ρu−ρw
[sec]. The solution of (22.76) is
rMD(t) =
t2
− tMD
f
2
+ T 2
rMD
0 + t T 2
− tMD
f − t · tMD
f V MD
0
T 2 − tMD
f
2 (22.77)
Differentiate with respect to t,
˙rMD(t) =
2t · rMD
0 + 2t · tMD
f − tMD
f
2
+ T 2
V MD
0
T 2 − tMD
f
2 (22.78)
Therefore, the ZEM is
yMD(t) = rMD(t) + tMD
go ˙rMD(t) =
T 2
− tMD
go
2
T 2 − tMD
f
2 · yMD
0 (22.79)
Now, substitute into the optimal guidance laws,
u∗
e(t) = −ρu
rMD(t) + tMD
go ˙rMD(t)
rMD(t) + tMD
go ˙rMD(t)
= −ρu
T 2−(tMD
go )
2
T 2−(tMD
f )
2 · yMD
0
q − 1
2
(ρu − ρw) tMD
go
2 = −ρu
yMD
0
yMD
0
(22.80)
w∗
(t) = −ρw
rMD(t) + tMD
go ˙rMD(t)
rMD(t) + tMD
go ˙rMD(t)
= −ρw
T 2−(tMD
go )
2
T 2−(tMD
f )
2 · yMD
0
q − 1
2
(ρu − ρw) tMD
go
2 = −ρw
yMD
0
yMD
0
(22.81)
Hence, both optimal laws are constant and pointed in the direction of yMD
0 = rMD
0 +tMD
f V MD
0 .
Theorem 22.8. When both players play optimal and start on a collision course, so that rMD
0 and
V MD
0 are collinear, the LOS direction is constant.
Proof. Assuming collision course, we have that rMD
0 and V MD
0 are collinear.
V MD
0 = α · rMD
0 , α ∈ R (22.82)
Substitute into (22.77) and obtain,
rMD(t) =
(1 + αt) T 2
− tMD
go t + (1 + αt) tMD
f
T 2 − tMD
f
2 · rMD
0 = β(t) · rMD
0 , β : R → R (22.83)
Hence the LOS direction is always the direction of LOS0.
Theorem 22.9. When both players play optimal and start on collision course, such that rMD
0 and
V MD
0 are collinear, Missile’s relative to LOS acceleration angle δMD is zero.
The proof is identical to the M-T game.
97](https://image.slidesharecdn.com/d0df201a-e3c2-46c0-9eef-17e7d604c5fa-150722113059-lva1-app6892/85/Dissertation-A-Three-Players-Pursuit-and-Evasion-Conflict-113-320.jpg)
![22.4 M-T-D VG4 Game Strategy
Similarly to VG1, divide the game into two different phases: the evasion phase and the pursuit
phase. Define the fail-safe function
C tMD
go = +
1
2
(ρu + ρw) tMD
go
2
(22.84)
Analogically to VG1 (refer to Subsection 19.2), by making yMD reach C, the Missile guarantees
a M-D miss distance of , as depicted in Fig. 22.8.
||yMT|| ||yMD||
Evasion Pursuit
{ue
* ,w
* } {up
*
,w *
}
t*
tf
MD
tf
MT
Time, t
ℓ
||yMT
cr ||
||ZEM||
Figure 22.8: Functions A, C, yMT and yMD
Assuming ρu > ρw, the optimal ZEM trajectory, generated by the pair {u∗
e, w∗
}, is parallel to A
when t ∈ [0, t∗
], and in the worst case (u∗
e = −u∗
p) collides with C when t ∈ t∗
, tMT
f , allowing the
Missile to evade the Defender. However, this is true only if yMD
0 is not inside the area bounded
by A. Hence, since we demand
yMD tMD
go ≥ A tMD
go = −
1
2
(ρu − ρw) tMD
go
2
∀tMD
go ∈ 0, tMD
f
the initial condition must satisfy
yMD
0 ≥ −
1
2
(ρu − ρw) tMD
f
2
(22.85)
thus
ρu ≥ ρw +
2 − yMD
0
tMD
f
2 (22.86)
Note that (22.86) provides the same evasion condition as (19.36) in VG1 section; hence, the Missile-
Defender game is similar to VG1. However, the Missile-Target game conceptually differs from VG1.
98](https://image.slidesharecdn.com/d0df201a-e3c2-46c0-9eef-17e7d604c5fa-150722113059-lva1-app6892/85/Dissertation-A-Three-Players-Pursuit-and-Evasion-Conflict-114-320.jpg)
![||yMT|| ||yMD||
t*
tf
MD
tf
MT
Time, t
ℓ
||ZEM||
Figure 23.1: Functions yMT , yMD , and C
Therefore, at t = t∗
we have
yMT (t∗
) =
1
2
(ρu + ρv) tMT
f
2
−
1
2
(ρu + ρv) tMT
f − t∗ 2
(23.21)
=
1
2
(ρu − ρv) tMT
f − t∗ 2
Solving (23.21) for tMT
f yields,
tMT
f =
√
2t∗
√
2 − 1 + ρv
ρu
(23.22)
Algorithm. Approximate tMD
f from
tMD
f = arg min
tMD
f
rMD
0 + tMD
f V MD
0 +
1
2
(ρu − ρw) tMD
f
2
(23.23)
Approximate t∗
from
t∗
= tMD
f −
(ρu − ρw) tMD
f
2
− 2 + 2 rMD
0 + tMD
f V MD
0
2ρu
(23.24)
The estimated game time is
tMT
f =
√
2t∗
√
2 − 1 + ρv
ρu
(23.25)
23.4.2 Simulation Results
For Example 24.1 provided in the following section, the actual game time is tMT
f = 11.67 [sec],
while the estimated game time is tMT
f = 11.72 [sec].
103](https://image.slidesharecdn.com/d0df201a-e3c2-46c0-9eef-17e7d604c5fa-150722113059-lva1-app6892/85/Dissertation-A-Three-Players-Pursuit-and-Evasion-Conflict-119-320.jpg)
![24 VG4 Simulations
24.1 Basic VG4 Simulations
Example 24.1. Consider the following parameters,
ρu = 170
m
sec2
, ρv = 30
m
sec2
, ρw = 60
m
sec2
, m = 0.5 [m] , = 150 [m]
The simulation result is depicted in Fig. 24.1.
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
Figure 24.1: VG4 Planar Simulation 1
It is clear that
1. The required M-D miss distance is almost identical to the requested one.
2. The Missile intercepts the Target.
3. The interception time is about 12 seconds.
In order to see the advantage of VG4 over VG1, let us now compare both by running a planar
simulation with identical parameters and initial conditions. Such a comparison is depicted in Fig.
24.2.
105](https://image.slidesharecdn.com/d0df201a-e3c2-46c0-9eef-17e7d604c5fa-150722113059-lva1-app6892/85/Dissertation-A-Three-Players-Pursuit-and-Evasion-Conflict-121-320.jpg)
![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
(a) VG1
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
(b) VG4
Figure 24.2: VG1 Vs. VG4
In both cases the Missile evades the Defender and intercepts the Target. However, VG4 provides
a much smaller intercept time. We can also compare the relative distances rMT (t) , as depicted
in Fig. 24.3.
0 5 10 15 20 25
0
1000
2000
3000
4000
5000
6000
Time, t
M-TRange,||rMT(t)||
(a) VG1
0 2 4 6 8 10
0
1000
2000
3000
4000
5000
6000
Time, t
M-TRange,||rMT(t)||
(b) VG4
Figure 24.3: Relative M-T Distances, rMT (t)
By using VG4 the Missile captures the Target so that rMT (t) has no local minima.
Previously, we have seen that in M-D game, VG1 and VG4 introduce similar time-to-go under
certain conditions. Denote VG14 as VG1 for M-D game and VG4 for M-T game. Fig. 24.4
provides a comparison between VG4 and VG14.
106](https://image.slidesharecdn.com/d0df201a-e3c2-46c0-9eef-17e7d604c5fa-150722113059-lva1-app6892/85/Dissertation-A-Three-Players-Pursuit-and-Evasion-Conflict-122-320.jpg)
![Missile
Target
Defender
tf
MD
t*
0 1000 2000 3000 4000 5000 6000
0
1000
2000
3000
4000
x [m]
y[m]
Miss MD = 156 , tf
MD
= 6.73
Miss MT = 0.5 , tf
MT
= 12.9
(a) VG14
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
(b) VG4
Figure 24.4: VG14 Vs. VG4
VG4 provides slightly better results. Another reasonable question is what happens if the Missile
uses VG4 and the Target uses VG1. The comparison in Fig. 24.5 demonstrates such a situation.
Missile
Target
Defender
tf
MD
t*
0 1000 2000 3000 4000 5000 6000
0
1000
2000
3000
4000
x [m]
y[m]
Miss MD = 150 , tf
MD
= 6.96
Miss MT = 0.5 , tf
MT
= 12.1
(a) Target Using VG1
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
(b) Both Players Use VG4
Figure 24.5: Demonstration of the Target using VG1
The Target is intercepted in both cases.
We have seen in Subsections 22.2 and 22.3 that when a pair of players plays optimal, Missile’s
acceleration angle in fixed coordinates is constant. Therefore, when the Missile uses the provided
guidance law and the Target and the Defender play optimal, Missile’s acceleration angle (in fixed
coordinates), χ(t), is piecewise constant. This can be seen in Fig. 24.6.
107](https://image.slidesharecdn.com/d0df201a-e3c2-46c0-9eef-17e7d604c5fa-150722113059-lva1-app6892/85/Dissertation-A-Three-Players-Pursuit-and-Evasion-Conflict-123-320.jpg)
![0 2 4 6 8 10
-100
-50
0
50
100
Time, t
AccelerationAngle,χ(t)
(a) Acceleration Angle, χ(t)
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
(b) Planar Simulation
Figure 24.6: Acceleration Angle, χ(t) vs. Planar Simulation
A three dimensional simulation of VG4 is described in Fig. 24.7.
Missile Target Defender tf
MD t*
0
2000
4000
6000
x [m]
0
1000
2000
3000
y [m]
0
500
1000
1500
z [m]
Figure 24.7: VG4 3D Simulation 1
According to the simulations provided in the present section, VG4 provides substantially improved
results over VG1. Consequently, it is possible to decrease ρu and still get considerable results.
108](https://image.slidesharecdn.com/d0df201a-e3c2-46c0-9eef-17e7d604c5fa-150722113059-lva1-app6892/85/Dissertation-A-Three-Players-Pursuit-and-Evasion-Conflict-124-320.jpg)
![Example 24.2. Decreasing ρu to 120 yields the result depicted in Fig. 24.8.
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
Figure 24.8: VG4 Planar Simulation 2
The Missile is still capable of capturing the Target in reasonable time. A three dimensional version
of such a conflict is depicted in Fig. 24.9.
Missile Target Defender tf
MD t*
0
2000
4000
6000x [m]
0
2000
4000
6000
y [m]
0
500
1000
1500
2000
z [m]
Figure 24.9: VG4 3D Simulation 2
109](https://image.slidesharecdn.com/d0df201a-e3c2-46c0-9eef-17e7d604c5fa-150722113059-lva1-app6892/85/Dissertation-A-Three-Players-Pursuit-and-Evasion-Conflict-125-320.jpg)
![Example 24.3. Consider the same parameters as in Ex. 24.1. The simulation in Fig. 24.10 shows
the difference between VG4 and VG4∗
.
Missile
Target
Defender
tf
MD
t*
0 1000 2000 3000 4000 5000 6000
0
500
1000
1500
2000
2500
x [m]
y[m]
Miss MD = 151.8 , tf
MD
= 5.47
Miss MT = 0.5 , tf
MT
= 10.75
(a) VG4∗
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
(b) VG4
Figure 24.10: VG4 vs. VG4∗
By performing the maneuver described in (24.6), the Missile has managed to shorten its evasive
maneuver and intercept the Target in a smaller time. Although, in this example, there is no
dramatic difference between VG4 and VG4∗
, situation changes as tMD
f increases.
Example 24.4. Consider the values of Ex. 24.1, but with a a different geometry (Fig. 24.11).
Missile
Target
Defender
tf
MD
t*
-4000 -2000 0 2000 4000 6000
0
500
1000
1500
2000
2500
3000
x [m]
y[m]
Miss MD = 152.8 , tf
MD
= 7.39
Miss MT = 0.5 , tf
MT
= 12.88
(a) VG4∗
Missile
Target
Defender
tf
MD
t*
-4000 -2000 0 2000 4000 6000
0
1000
2000
3000
4000
5000
x [m]
y[m]
Miss MD = 159.9 , tf
MD
= 9.85
Miss MT = 0.5 , tf
MT
= 17.49
(b) VG4
Figure 24.11: VG4 vs. VG4∗
Obviously, the difference between VG4 and VG4∗
is substantial.
111](https://image.slidesharecdn.com/d0df201a-e3c2-46c0-9eef-17e7d604c5fa-150722113059-lva1-app6892/85/Dissertation-A-Three-Players-Pursuit-and-Evasion-Conflict-127-320.jpg)
![25.1.5 PVG4 Simulations
Example 25.1. Consider the following numerical values
ρu = 170
m
sec2
, ρv = 30
m
sec2
, ρw = 60
m
sec2
, m = 0.5 [m] , = 150 [m]
Fig. 25.1 shows a comparison between LMG and PVG4.
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
(a) PVG4
Missile
Target
Defender
tf
MD
t*
0 1000 2000 3000 4000 5000 6000
0
1000
2000
3000
4000
x [m]
y[m]
Miss MD = 325 , tf
MD
= 5.74
Miss MT = 0.1 , tf
MT
= 15.37
(b) LMG
Figure 25.1: PVG4 vs. LMG
By analyzing the simulations we conclude:
1. By using PVG4, the Target applies relatively small lateral acceleration and manages to es-
cape. In contrast, by adopting LMG it applies strong lateral evasive maneuver and eventually
gets intercepted. In fact, by analyzing a similar game10
in Part II (Example 13.7 on Pages
59–60), we have concluded that with this geometry, it is best for the Target to apply small
lateral acceleration (in Example 13.7 the Target applies v = 0). PVG4 supports the results
of Example 13.7; however, unlike LMG, PVG4 has no approximation factors11
.
2. PVG4 provides the Missile with the exactly required M-D miss distance, while in LMG
scenario it is substantially bigger.
3. A slight increase in the Missile’s maneuvering capability (from 170 [m/sec2
] to 180 [m/sec2
]),
enables the Missile intercepts the Target in PVG4 scenario (Example 13.7). In contrast, a
substantial increase in ρu is needed in the scenario of Example 13.7.
10
The initial geometry, and players’ capabilities are the same. Only the required M-D miss distance is different;
namely in Example 13.7 we had = 10 [m], while in the present example we have = 150 [m].
11
Recall that in Example 13.7, we had to add a factor of te(kv) in order to account for the addition to ∆t, resulted
from the Target’s evasive maneuver. In general, this factor can be obtained only from simulations, which makes
LMG hardly implementable.
116](https://image.slidesharecdn.com/d0df201a-e3c2-46c0-9eef-17e7d604c5fa-150722113059-lva1-app6892/85/Dissertation-A-Three-Players-Pursuit-and-Evasion-Conflict-132-320.jpg)
![Example 25.2. For the following parameters, Fig. 25.2 depicts the outcome of PVG4 game.
ρu = 180
m
sec2
, ρv = 30
m
sec2
, ρw = 60
m
sec2
, m = 0.5 [m] , = 150 [m]
Missile
Target
Defender
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
= 6.22
Miss MT = 0.5 , tf
MT
= 13.18
Figure 25.2: PVG4 Planar Simulation
Example 25.3. A three dimensional version of such a conflict is shown in Fig. 25.3.
Missile Target Defender tf
MD t*
0
2000
4000
6000
x [m]
0
500
1000
1500
y [m]
0
1000
2000
z [m]
Figure 25.3: PVG4 3D Simulation
117](https://image.slidesharecdn.com/d0df201a-e3c2-46c0-9eef-17e7d604c5fa-150722113059-lva1-app6892/85/Dissertation-A-Three-Players-Pursuit-and-Evasion-Conflict-133-320.jpg)
![25.2.4 TVG4 Simulations
Example 25.5. Consider the following numerical values
ρu = 170
m
sec2
, ρv = 30
m
sec2
, ρw = 60
m
sec2
m = 0.5 [m] , = 150 [m]
σ⊥1 = 1, σ⊥2 = 1
and σa = {0, 0.25, 0.5, 1}. Observe the planar simulation results in Fig. 25.5.
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
(a) σ = 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
(b) σ = 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
(c) σ = 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
(d) σ = 0
Figure 25.5: TVG4 Planar Simulations
121](https://image.slidesharecdn.com/d0df201a-e3c2-46c0-9eef-17e7d604c5fa-150722113059-lva1-app6892/85/Dissertation-A-Three-Players-Pursuit-and-Evasion-Conflict-137-320.jpg)
![Also, Fig. 25.6 shows the outcome of a similar three dimensional conflict.
Missile Target Defender tf
MD t*
0
2000
4000
6000
x [m]
0
1000
2000
3000
y [m]
0
1000
2000
z [m]
(a) σ = 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]
(b) σ = 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]
(c) σ = 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]
(d) σ = 0
Figure 25.6: TVG4 3D Simulations
122](https://image.slidesharecdn.com/d0df201a-e3c2-46c0-9eef-17e7d604c5fa-150722113059-lva1-app6892/85/Dissertation-A-Three-Players-Pursuit-and-Evasion-Conflict-138-320.jpg)
![26.1.7 White Noise Model
Since it is practically possible to measure the M-T range and 2 LOS angles, we assume that
r =
rx
ry
rz
=
R sin θ cos λ
R sin θ sin λ
R cos θ
(26.49)
where R ∈ R+
is the range, and θ ∈ [0, 2π] , λ ∈ [0, π] are the azimuthal and elevation LOS angles.
Since the range can be measured much more accurately then LOS angles, we assume that R is
perfectly known, while θ, λ are noisy. Since noisy signals are multiplied by perfectly known range,
we approximate the noise model as range dependent,
n(t) = r(t)knρnnw(t) (26.50)
where nw(t) is a unit intensity white noise, ρ2
n = E n2
w(t) is the noise variance, and kn is a scaling
factor. Thus, the measured relative position is
rm(t) = r(t) + n(t) = r(t) (1 + knρnnw(t)) (26.51)
Note that knρnnw(t) has a power spectral density of
ϕn = k2
nρ2
n (26.52)
An important property of this model is that rm approaches r when tgo approaches zero.
26.1.8 White Noise Bound
Generally, we have
eyi
(t) ≤ er/v(t) + er/n(t) + tgoeV/v(t) + tgoeV/n(t) (26.53)
where i = x, y, z. Assume worst case bound for the Target’s maneuver; hence,
er/v(t) = ρv
ˆ t
0
|gs11 (τ)| dτ (26.54)
eV/v(t) = ρv
ˆ t
0
|gs21 (τ)| dτ (26.55)
and derive a white noise bound for er/n and eV/n. Recall that
n(t) = r(t)knρnnw(t) (26.56)
By using convolution, one has
er/n(t) =
ˆ t
0
gs12 (τ)ri(t − τ)knρnnw(t − τ)dτ (26.57)
eV/n(t) =
ˆ t
0
gs22 (τ)ri(t − τ)knρnnw(t − τ)dτ (26.58)
129](https://image.slidesharecdn.com/d0df201a-e3c2-46c0-9eef-17e7d604c5fa-150722113059-lva1-app6892/85/Dissertation-A-Three-Players-Pursuit-and-Evasion-Conflict-145-320.jpg)
![26.1.11 Estimator Based Two Players VG4 Simulations
Let the players have, ρu = 50 m
sec2 , ρv = 20 m
sec2 , and let Target perform v∗
= ρvyMT / yMT
when tgo > ts, and v = ρvVT⊥/ VT⊥ when tgo ≤ ts, where VT⊥ is normal to VT , and ts = 0.5 [sec]
is the switch time. Using the Kalman Filter configuration of 26.1.3, we obtain the results in Fig.
26.3 for σn = {10−4
, 10−5
, 10−6
}.
σn Planar Simulation ZEM Trajectory
10−4
Missile
Target
0 1000 2000 3000 4000 5000 6000
0
200
400
600
x [m]
y[m]
Miss Distance = 0.415
||yMT||
||y
MT||
Bound
tf
MT
Time, t
||ZEM||
Guaranteed Miss Distance = 0.426 [m]
10−5
Missile
Target
0 1000 2000 3000 4000 5000 6000
0
200
400
600
x [m]
y[m]
Miss Distance = 0.133
||yMT||
||y
MT||
Bound
tf
MT
Time, t
||ZEM||
Guaranteed Miss Distance = 0.151 [m]
10−6
Missile
Target
0 1000 2000 3000 4000 5000 6000
0
200
400
600
x [m]
y[m]
Miss Distance = 0.078
||yMT||
||y
MT||
Bound
tf
MT
Time, t
||ZEM||
Guaranteed Miss Distance = 0.064 [m]
Figure 26.3: Estimator Based Two Players VG4 Simulations
Remark 26.4. ˆyMT is not inside its bound at the beginning because of initial conditions.
133](https://image.slidesharecdn.com/d0df201a-e3c2-46c0-9eef-17e7d604c5fa-150722113059-lva1-app6892/85/Dissertation-A-Three-Players-Pursuit-and-Evasion-Conflict-149-320.jpg)
![26.2 Missile – Defender Game
26.2.1 Optimal Maneuver Approximation
To simplify notations, denote rMD as r, VMD as V , and tMD
f as tf . Similarly, when the Missile
evades the Defender, the optimal relative displacement is,
r∗
(t) =
t2
− t2
f + T 2
r0 + t (T 2
− (tf − t) · tf ) V0
T 2 − t2
f
(26.79)
where T = 2q
ρu−ρw
[sec], and q is the nominal expected miss distance. Thus,
r∗
(t − τ) = r∗
(t) − V ∗
(t) · τ +
1
2
a∗
(t) · τ2
(26.80)
where
V ∗
(t) = ˙r∗
(t) =
2t · r0 + 2t · tf − t2
f + T 2
V0
T 2 − t2
f
and
a∗
(t) = ¨r∗
(t) =
2 (r0 + tf V0)
T 2 − t2
f
Similarly, e∗
r/n(t) satisfies,
e∗
r/n(t) ≤ knρn r∗
(t) · α1 − V ∗
(t) · β1 +
1
2
a∗
(t) · γ1 (26.81)
where
α1 =
ˆ t
0
g2
12(τ)dτ (26.82)
β1 =
ˆ t
0
(τ · g12(τ))2
dτ (26.83)
γ1 =
ˆ t
0
(τ2 · g12(τ))2
dτ (26.84)
and kn is the noise deviation scaling factor, so that e∗
r/n(t) bounds kn of the noise standard devia-
tion, ρn. Also,
e∗
V/n(t) ≤ knρn r∗
(t) · α2 − V ∗
(t) · β2 +
1
2
a∗
(t) · γ2 (26.85)
where α2 =
´ t
0
g2
22(τ)dτ, β2 =
´ t
0
(τ · g22(τ))2
dτ, and γ2 =
´ t
0
(τ2 · g22(τ))2
dτ. Hence, the
approximated deviation from the nominal ZEM trajectory is
e∗
yi
(t) = e∗
r/v(t) + e∗
r/n(t) + tgoe∗
V/v(t) + tgoe∗
V/n(t) (26.86)
134](https://image.slidesharecdn.com/d0df201a-e3c2-46c0-9eef-17e7d604c5fa-150722113059-lva1-app6892/85/Dissertation-A-Three-Players-Pursuit-and-Evasion-Conflict-150-320.jpg)
![26.2.2 Miss Distance Bound Approximation for VG4
Similarly to M-T game, define
J1(t) = q −
1
2
(ρu − ρw) t2
go + e∗
yi
(t) (26.87)
J2(t) = q −
1
2
(ρu − ρw) t2
go − e∗
yi
(t) (26.88)
and obtain the bounds,
Jui
(t) = J1(t) (26.89)
Jli
(t) = max {0, J2(t)} (26.90)
As a result,
Jli
(tf ) ≤ qi ≤ Jui
(tf ) (26.91)
Similarly to the M-T game, Ju = Jui
√
N, Jl = Jli
√
N, and q = qi
√
N
26.2.3 Estimator Based Two Players VG4 Simulations
Consider the following numerical values
ρu = 50
m
sec2
, ρw = 20
m
sec2
Both players play optimal. Set σn = 10−5
and obtain the results depicted in Fig. 26.4.
Missile
Defender
0 1000 2000 3000 4000 5000 6000
-1000
-500
0
500
x [m]
y[m]
Miss Distance = 1180
(a) Planar Simulation
||yMD||
||y
MD||
Bound
tf
MD
Time, t
||ZEM||
Guaranteed Miss Distance = 1178 [m]
(b) ZEM Trajectories
Figure 26.4: Estimator Based Two Players VG4 Simulations
135](https://image.slidesharecdn.com/d0df201a-e3c2-46c0-9eef-17e7d604c5fa-150722113059-lva1-app6892/85/Dissertation-A-Three-Players-Pursuit-and-Evasion-Conflict-151-320.jpg)
![lim
tMD
go →0
˙ψ tMD
go = 0 (27.91)
lim
tMD
go →∞
˙ψ tMD
go =
1
τM
(27.92)
Now examine the limits of ˙g tMD
go .
lim
tMD
go →0
˙g tMD
go =
rMD ˙rMD
rMD
< 0 (27.93)
lim
tMD
go →∞
˙g tMD
go = ∞ > 0 (27.94)
Therefore, since g tMD
go is continuous, it has at least one local minimum in the time interval
tMD
go ∈ [0, ∞).
Example 27.1. For some rMD and ˙rMD, and different values of q , the function g tMD
go is shown
in Fig. 27.8.
q = 100 [m] q = 1830 [m] q = 3550 [m]
2 4 6 8 10
tgo
MD
-2000
2000
4000
6000
8000
10000
g(tgo
MD)
Figure 27.8: Function g tMD
go For Different Values of q
Similarly, we end up with the same conclusions as for the ideal players:
1. If g tMD
go = 0 has no positive real solutions, then q is too small, and such a miss distance is
unachievable.
2. If g tMD
go = 0 has exactly one positive real solution, then q = qmin is the minimal achievable
M-D miss distance. Clearly, this occurs when rMD reaches its minimum.
3. If g tMD
go = 0 has more than one positive real solution, then q is too big, and by choosing the
biggest of the solutions for tMD
go , we are aiming for a higher trajectory than required (while
the smaller solutions are not physical, as rMD still decreases). This would require more
time for evasion than necessary and produce bigger M-D miss distance than desired.
149](https://image.slidesharecdn.com/d0df201a-e3c2-46c0-9eef-17e7d604c5fa-150722113059-lva1-app6892/85/Dissertation-A-Three-Players-Pursuit-and-Evasion-Conflict-165-320.jpg)
![Therefore, it is reasonable to define tMD
f = tMD
f (qmin); hence, tMD
go = tMD
go (qmin). Similarly, we
must find q for which g tMD
go = 0 has exactly one positive real root. Since q does not change the
shape of g tMD
go , rather it changes its vertical displacement, the function g tMD
go has one real root
when its smallest local minimum is tangent to the horizontal axis. Therefore, instead of looking
for q = qmin we can find tMD
go which minimizes g tMD
go , and obtain tMD
go (qmin). Thus,
tMD
go = arg min
tMD
go
g tMD
go (27.95)
= arg min
tMD
go
rMD + tMD
go ˙rMD − τ2
M ψ tMD
go u − q +
1
2
(ρu − ρw) tMD
go
2
− ρuτ2
M ψ tMD
go
= arg min
tMD
go
rMD + tMD
go ˙rMD − τ2
M ψ tMD
go u +
1
2
(ρu − ρw) tMD
go
2
− ρuτ2
M ψ tMD
go
27.4.3 M-T-D Game
Similarly to ideal players game, divide the game into two different phases: the evasion phase and
the pursuit phase. Recall the fail-safe function
C tMD
go = +
1
2
(ρu + ρw) tMD
go
2
− ρuτ2
M ψ tMD
go (27.96)
When yMD reaches C, the Missile guarantees a M-D miss distance of , as depicted in Fig. 27.9.
||yMT|| ||yMD|| (t) (t)
Evasion Pursuit
{ue
* ,w
* } {up
*
,w *
}
t*
tf
MD
tf
MT
Time, t
ℓ
||yMT
cr
||
||ZEM||
Figure 27.9: Functions A, C, and yMD
Assuming ρu > ρw, the optimal ZEM trajectory, generated by the pair {u∗
e, w∗
}, is parallel to A
when t ∈ [0, t∗
], and in the worst case (u∗
e = −u∗
p) collides with C when t ∈ t∗
, tMT
f , allowing the
150](https://image.slidesharecdn.com/d0df201a-e3c2-46c0-9eef-17e7d604c5fa-150722113059-lva1-app6892/85/Dissertation-A-Three-Players-Pursuit-and-Evasion-Conflict-166-320.jpg)
![27.5 Simulations
Example 27.2. Consider the following parameters,
ρu = 170
m
sec2
, ρv = 30
m
sec2
, ρw = 60
m
sec2
, m = 0.5 [m] , = 150 [m] , τM = 0.1 [sec]
Note that here m∗
= 0.0075 [m]; hence, we choose m > m∗
. The simulation result is depicted in
Fig. 27.10.
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
(a) VG4, First Order Lag
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
(b) VG4, Zero Order Lag
Figure 27.10: First Order Lag Vs. Zero Order Lag
Notice that
1. The non-ideal Missile cannot maneuver as fast as the ideal one; thus, it takes more time to
reach C, and more time to intercept the Target.
2. The Missile eventually intercepts the Target.
3. The M-D miss distance is close to the required.
152](https://image.slidesharecdn.com/d0df201a-e3c2-46c0-9eef-17e7d604c5fa-150722113059-lva1-app6892/85/Dissertation-A-Three-Players-Pursuit-and-Evasion-Conflict-168-320.jpg)
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Dissertation - A Three Players Pursuit and Evasion Conflict
- 1. A Three Player Pursuit and Evasion Conflict Sergey Rubinsky
- 3. A Three Player Pursuit and Evasion Conflict Research Thesis Submitted In Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy Sergey Rubinsky Submitted to the Senate of the Technion – Israel Institute of Technology Nisan, 5775 Haifa April 2015 i
- 5. Supervision This research thesis was done under the supervision of Prof. Shaul Gutman in the department of Mechanical Engineering. Acknowledgments I am heartily thankful to my supervisor, Prof. Shaul Gutman, for his patient and devoted guidance throughout this research. It was my absolute privilege to be inspired by his unique passion towards true science. The Generous Financial Help of the Technion is Gratefully Acknowledged iii
- 7. Publication List Journals • S. Rubinsky, S. Gutman, “Three Player Pursuit and Evasion Conflict”. Journal of Guidance, Control, and Dynamics, Vol. 37, No. 1 (2014), pp. 98-110. DOI: 10.2514/1.61832. • S. Rubinsky, S. Gutman, “Vector Guidance Approach to a Three Player Conflict in Exo- Atmospheric Interception”. Journal of Guidance, Control, and Dynamics, In Press. DOI: 10.2514/1.G000942. • S. Gutman, S. Rubinsky, “Exoatmospheric Thrust Vector Interception Via Time-to-Go Anal- ysis”. Journal of Guidance, Control, and Dynamics, In Press. DOI: 10.2514/1.G001268. • S. Gutman, S. Rubinsky, “3D-Nonlinear Vector Guidance and Exo-Atmospheric Intercep- tion”. IEEE Trans. on aerospace and electronic systems, Accepted for publication. • S. Gutman, O. Goldan, S. Rubinsky, “Guaranteed Miss-Distance in Guidance Systems with Bounded Controls and Bounded Noise”. Journal of Guidance, Control, and Dynamics Vol. 35, No. 3 (2012), pp. 816-823. DOI: 10.2514/1.55723. Conferences • S.Rubinsky, S. Gutman, “Three Body Guaranteed Pursuit and Evasion”. AIAA GNC Con- ference, August 13-16, 2012, Minneapolis, Minnesota. • S. Gutman, S. Rubinsky, “Linear Optimal Guidance”. 52nd Annual Conference on Aerospace Sciences, March 1, 2012, Haifa, Israel. • S. Gutman, S. Rubinsky, “Exo-Atmospheric Mid-Course Guidance”, AIAA SciTech Confer- ence, 5-9 Jan. 2015, Orlando, FL. • S. Gutman, S. Rubinsky, “3D Nonlinear Vector Guidance and Exo-Atmospheric Interception”, 55-Israel Annual Conference on Aerospace Sciences, 25-26 Feb., 2015, Haifa, Israel. • S. Gutman, S. Rubinsky, “Exo-Atmospheric Thrust Vector Interception: Translation Only”, EuroGNC, 13-15 April, 2015, Toulouse, France. • S. Gutman, S. Rubinsky, T. Shima, M. Levi, “Single vs Two-Loop Integrated Guidance Systems”. CEAS EuroGNC Conference, April 10-12, 2013, Deft University, Netherlands. v
- 9. Contents 1 Introduction 7 1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.2 Noticeable Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.3 Main Results and Contribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 I Linear Model Guidance 11 2 Problem Overview 11 2.1 Basic Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.2 Linearization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 3 Dynamic Model and Zero Effort Miss 14 4 A Game of Three Ideal Players 18 5 Differential Game Definition 19 6 Game Formulation 22 7 Simple Differential Game Solution 23 7.1 Basic Concept . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 7.2 Fail-safe Function C tMD go . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 7.3 Various Evasion Strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 7.3.1 Basic Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 7.3.2 Evasive Maneuver Gain ku . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 7.3.3 The Impact of ku . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 7.4 Optimality Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 8 Optimality Analysis 31 9 Nonlinear Simulations 36 10 Discussion 40 11 Conclusions 40 II LMG Analysis 41 12 Parametric Analysis 41 12.1 Initial Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 12.2 Target’s and Defender’s Maneuver Capabilities . . . . . . . . . . . . . . . . . . . . . 44 12.3 Required M-D and M-T miss distances . . . . . . . . . . . . . . . . . . . . . . . . . 46 12.4 The final times tMD f and tMT f . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 vii
- 10. 13 Optimality Analysis 51 13.1 Linear Kinematics Scenario . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 13.1.1 Constant Gain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 13.1.2 Variable Gain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 13.2 Optimality in the nonlinear kinematics scenario . . . . . . . . . . . . . . . . . . . . 54 13.3 Intermediate conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 13.4 The Uncertainty Area Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 13.4.1 The M-T bound function revised . . . . . . . . . . . . . . . . . . . . . . . . 59 13.4.2 Function d(·) Revised . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 14 Conclusions 64 III Vector Guidance Approach 65 15 Preface 65 16 A game of players controlling their acceleration vectors 65 17 A Differential Game of Two Players 68 17.1 General Differential Game . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 17.2 Simple Differential Game . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 18 Vector Guidance Based On 1st Order Time-to-go (VG1) 70 19 Optimal Strategies for VG1 72 19.1 Basic Optimal Strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 19.2 Fail-safe Function: C tMD go . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 19.3 Various Evasion Strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 19.4 Algebraic Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 20 VG1 Simulations 78 21 Vector Guidance Based On 4th Order Time-to-go (VG4) 83 22 Optimal Strategies for VG4 84 22.1 Basic Optimal Strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 22.2 Missile – Target Game . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 22.2.1 M-T Game VG4 Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 22.2.2 M-T Game VG4 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 22.3 Missile – Defender Game . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 22.3.1 M-D Game VG4 Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 22.3.2 M-D Game VG4 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 22.4 M-T-D VG4 Game Strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 23 Time Optimal M-T-D Game 100 23.1 Evasion Phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 23.2 Pursuit Phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 23.3 M-T-D Time Optimal Guidance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 viii
- 11. 23.4 Time-Bound Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 23.4.1 Basic Derivations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 23.4.2 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 23.4.3 Time-Bounded Game . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 24 VG4 Simulations 105 24.1 Basic VG4 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 24.2 VG4 with Optimal Start-time (VG4∗ ) . . . . . . . . . . . . . . . . . . . . . . . . . . 110 25 Modified Vector Guidance 112 25.1 Projected Vector Guidance (PVG) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 25.1.1 Model Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 25.1.2 A Simple Projected Differential Game . . . . . . . . . . . . . . . . . . . . . 114 25.1.3 M-T-D Projected Game . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 25.1.4 M-T-D Projected Endo-Atmospheric Game . . . . . . . . . . . . . . . . . . . 115 25.1.5 PVG4 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 25.2 Generalization – Transformed Vector Guidance (TVG) . . . . . . . . . . . . . . . . 118 25.2.1 Elliptical Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 25.2.2 A Simple Transformed Differential Game . . . . . . . . . . . . . . . . . . . . 120 25.2.3 M-T-D Projected Game . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 25.2.4 TVG4 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 26 Estimator Based Vector Guidance 123 26.1 Missile – Target Game . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 26.1.1 Model Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 26.1.2 Luenberger Observer and Pole Placement . . . . . . . . . . . . . . . . . . . . 124 26.1.3 Kalman Filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 26.1.4 Schematic Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 26.1.5 Estimation Error . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 26.1.6 Worst Case Bound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 26.1.7 White Noise Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 26.1.8 White Noise Bound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 26.1.9 Optimal Maneuver Approximation . . . . . . . . . . . . . . . . . . . . . . . 130 26.1.10Miss Distance Bound Approximation for VG4 . . . . . . . . . . . . . . . . . 132 26.1.11Estimator Based Two Players VG4 Simulations . . . . . . . . . . . . . . . . 133 26.2 Missile – Defender Game . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 26.2.1 Optimal Maneuver Approximation . . . . . . . . . . . . . . . . . . . . . . . 134 26.2.2 Miss Distance Bound Approximation for VG4 . . . . . . . . . . . . . . . . . 135 26.2.3 Estimator Based Two Players VG4 Simulations . . . . . . . . . . . . . . . . 135 27 A Non-Ideal Players Game 136 27.1 Basic definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 27.2 A Differential Game of Two Players . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 27.2.1 General Differential Game . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 27.2.2 Isotropic Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 27.2.3 First Order Isotropic Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . 141 27.3 Optimal Strategies for Constant Final Times . . . . . . . . . . . . . . . . . . . . . . 143 ix
- 12. 27.3.1 Basic Optimal Strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 27.3.2 Fail-safe Function C tMD go . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144 27.3.3 Guaranteed Cost Game . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 27.4 Optimal Strategies for Varying Final Times (VG4) . . . . . . . . . . . . . . . . . . 146 27.4.1 M-T Game . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146 27.4.2 M-D Game . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 27.4.3 M-T-D Game . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150 27.5 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152 28 Conclusions 153 x
- 13. List of Figures 2.1 Planar Interception Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 3.1 Linearized Open Loop . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 4.1 Zero Order Lag Open Loop . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 5.1 Missile-Defender ZEM Optimal Trajectories . . . . . . . . . . . . . . . . . . . . . . 20 5.2 Missile-Target ZEM Optimal Trajectories . . . . . . . . . . . . . . . . . . . . . . . . 21 6.1 Missile-Defender and Missile-Target ZEM Bounds . . . . . . . . . . . . . . . . . . . 22 7.1 1st Case Linear Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 7.2 fail-safe Function C(tgo) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 7.3 2nd Case Linear Simulation (Aggressive Law) . . . . . . . . . . . . . . . . . . . . . . 26 7.4 2nd Case Linear Simulation (Minimal Maneuver) . . . . . . . . . . . . . . . . . . . . 26 7.5 Two Phases of Guidance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 7.6 Functions ycr MT t∗ go and B t∗ go . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 7.7 Cost Function d t∗ go . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 8.1 Intersection of the Cost Function d t∗ go . . . . . . . . . . . . . . . . . . . . . . . . . 31 8.2 Linear Simulation. ku = 100% of ρu . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 8.3 Linear Simulation. ku = 67% of ρu . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 8.4 d(t∗ go) > 0 ∀ 0 ≤ t∗ go ≤ t∗ gomax . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 8.5 d(t∗ go) < 0 ∀ 0 ≤ t∗ go ≤ t∗ gomax . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 8.6 Linear Simulation with ρumin . ku = 100% of ρu . . . . . . . . . . . . . . . . . . . . . 35 9.1 Nonlinear Simulation 1 (ku = ρu) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 9.2 Measured tMD go as a function of time . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 9.3 Nonlinear Simulation 2 (ku = ku,1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 9.4 Estimated tMD go as a function of time . . . . . . . . . . . . . . . . . . . . . . . . . . 37 9.5 Nonlinear Simulation 3 (ku = ρu) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 9.6 Nonlinear Simulation 4 (ku = ku,1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 12.1 Plot and Contour Plot of ρumin yMT 0 , yMD 0 . . . . . . . . . . . . . . . . . . . . . 42 12.2 Section Plots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 12.3 Linear Simulations for Different Initial Conditions . . . . . . . . . . . . . . . . . . . 43 12.4 Plot and Contour Plot of ρumin (ρv, ρw) . . . . . . . . . . . . . . . . . . . . . . . . . . 44 12.5 Section Plots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 12.6 Linear Simulations for Different Values of ρv and ρw . . . . . . . . . . . . . . . . . . 45 12.7 Plot and Contour Plot of ρumin (m, ) . . . . . . . . . . . . . . . . . . . . . . . . . . 46 12.8 Section Plots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 12.9 Linear Simulations for Different Values of m and . . . . . . . . . . . . . . . . . . . 47 12.10Plot and Contour Plot of ρumin (tf , ∆t) . . . . . . . . . . . . . . . . . . . . . . . . . . 48 12.11Section Plots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 12.12Linear Simulations for Different Values of tf . . . . . . . . . . . . . . . . . . . . . . 50 13.1 Function d (kv, kw) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 13.2 Riemann’s Series of ´ t∗ 0 kv(ξ)dξ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 13.3 Bounds and Different Possibilities of |yMD(t)| . . . . . . . . . . . . . . . . . . . . . 53 13.4 Nonlinear simulation for kv = −ρv, −0.5ρv, 0.5ρv and ρv . . . . . . . . . . . . . . . 54 13.5 Nonlinear simulations for kv = −ρv and kv = ρv . . . . . . . . . . . . . . . . . . . . 55 13.6 Linear simulations for kv = −ρv and kv = ρv . . . . . . . . . . . . . . . . . . . . . . 56 13.7 Results of Fig. 13.6, presented on the same plot . . . . . . . . . . . . . . . . . . . . 57 13.8 Nonlinear Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 xi
- 14. 13.9 Linear simulations for kv = −ρv and kv = ρv . . . . . . . . . . . . . . . . . . . . . . 59 13.10Function dv(t∗ go, kv) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 13.11Function dv (kv) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 13.12Linear simulation for kv = 0, te = 1 [sec] . . . . . . . . . . . . . . . . . . . . . . . . 62 13.13Nonlinear simulations for kv = −ρv and kv = ρv . . . . . . . . . . . . . . . . . . . . 63 13.14Nonlinear simulation for kv = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 16.1 Planar Interception Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 17.1 Optimal ZEM P-E Trajectories for amax P > amax E (left) and amax P < amax E (right) . . . 69 19.1 Optimal Missile-Defender ZEM Trajectories . . . . . . . . . . . . . . . . . . . . . . 72 19.2 Optimal Missile-Target ZEM Trajectories . . . . . . . . . . . . . . . . . . . . . . . . 73 19.3 Bound Functions A and B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 19.4 fail-safe Function C in addition toA and B . . . . . . . . . . . . . . . . . . . . . . . 75 20.1 VG1 Planar Simulation 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 20.2 VG1 Vs. LMG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 20.3 Planar Simulation and ZEM Trajectories . . . . . . . . . . . . . . . . . . . . . . . . 79 20.4 VG1 Planar Simulation 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 20.5 Relative Distance rMT (t) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 20.6 VG1 3D Simulation 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 20.7 VG1 Planar Simulation 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 20.8 Relative Distance rMT (t) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 20.9 VG1 3D Simulation 2 and ZEM Trajectories . . . . . . . . . . . . . . . . . . . . . . 82 22.1 Missile-Target Game Optimal ZEM Trajectory . . . . . . . . . . . . . . . . . . . . . 85 22.2 Missile-Defender Optimal ZEM Trajectories . . . . . . . . . . . . . . . . . . . . . . 90 22.3 Function g tMD go For Different Values of q . . . . . . . . . . . . . . . . . . . . . . . 92 22.4 Missile-Defender Relative Distance . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 22.5 Function ˙g tMD go . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 22.6 Function g tMD go . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 22.7 Evaluation of tMD go for VG1 and VG4 . . . . . . . . . . . . . . . . . . . . . . . . . . 96 22.8 Functions A, C, yMT and yMD . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 23.1 Functions yMT , yMD , and C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 24.1 VG4 Planar Simulation 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 24.2 VG1 Vs. VG4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 24.3 Relative M-T Distances, rMT (t) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 24.4 VG14 Vs. VG4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 24.5 Demonstration of the Target using VG1 . . . . . . . . . . . . . . . . . . . . . . . . 107 24.6 Acceleration Angle, χ(t) vs. Planar Simulation . . . . . . . . . . . . . . . . . . . . . 108 24.7 VG4 3D Simulation 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 24.8 VG4 Planar Simulation 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 24.9 VG4 3D Simulation 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 24.10VG4 vs. VG4∗ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 24.11VG4 vs. VG4∗ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 25.1 PVG4 vs. LMG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 25.2 PVG4 Planar Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 25.3 PVG4 3D Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 25.4 Elliptical Bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 25.5 TVG4 Planar Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 25.6 TVG4 3D Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 xii
- 15. 26.1 Estimator Based VG Block Diagram . . . . . . . . . . . . . . . . . . . . . . . . . . 126 26.2 Nominal ZEM and its Bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 26.3 Estimator Based Two Players VG4 Simulations . . . . . . . . . . . . . . . . . . . . 133 26.4 Estimator Based Two Players VG4 Simulations . . . . . . . . . . . . . . . . . . . . 135 27.1 Planar Interception Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 27.2 Open Loop State Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 27.3 ZEM Trajectories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 27.4 Functions A and B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144 27.5 Functions A(t), B(t), and C(t) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 27.6 Missile-Target Game Optimal ZEM Trajectory . . . . . . . . . . . . . . . . . . . . . 147 27.7 Missile-Defender Optimal ZEM Trajectories . . . . . . . . . . . . . . . . . . . . . . 148 27.8 Function g tMD go For Different Values of q . . . . . . . . . . . . . . . . . . . . . . . 149 27.9 Functions A, C, and yMD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150 27.10First Order Lag Vs. Zero Order Lag . . . . . . . . . . . . . . . . . . . . . . . . . . . 152 xiii
- 17. Abstract This research deals with a three player conflict. In addition to the standard pursuit – evasion game, in which the pursuer minimizes and the evader maximizes the miss-distance, the evader launches a short range missile (Defender) to defend itself. The Missile’s objective is to evade the Defender and intercept the Target. The Defender’s objective is to intercept the Missile and prevent it from capturing the Target. The Target’s objective is to escape the Missile. In this work, hard bounds are placed on players’ maneuvering capabilities, which leads to nonlinear strategies. This research suggests that the switch time, at which the Missile switches from evasion to pursuit, occurs before the Missile-Defender pass time; hence, the missile can start pursuing the Target before it passes by the Defender. This research is divided into three parts. The first, discusses a LOS linearized kinematics game, in which the equations of motion are set in a perpendicular to initial LOS direction, which leads to a one dimensional game. The problem is presented and discussed under linearization assumptions, and a guaranteed cost solution is obtained. In addition, the obtained solution is optimized with respect to a robustness measure, and an algebraic condition, under which the intercepting missile can evade the defending missile and capture the evading target, is derived. This enables the designer to perform a parameter analysis and compute the sufficient requirements at the early stages of the design. The second part introduces a deep analysis of the solution presented in Part I. In addition to the parametric analysis and optimality proof for the linearized model scenario, Part II presents the main problem of the linearized model and the contradiction resulted by this solution. This problem leads to a severe uncertainty of the linear model guidance in the real, nonlinear scenario, which leads to a need of looking for a different solution. Such a solution, called the Vector Guidance (VG) approach, is presented in Part III. In the Vector Guidance scenario, the players can apply bounded acceleration in any direction in 3D space. In addition, the VG kinematics is defined in the Cartesian coordinate system and does not suffer any linearization. In order to account for endo-atmospheric interception scenario, where the aerodynamic forces are dominant, a Transformed Vector Guidance approach is derived. Furthermore, in order to account for noisy measurements, an estimator based guidance algorithm is presented in Part III. Also, Part III introduces an analysis of a first order isotropic dynamics of the intercepting missile, and derives the optimal strategies for this scenario. 1
- 19. Nomenclature Interception Missile-Defender miss distance A tMD go Missile-Defender bound function B tMD go Missile-Target bound function Bv tMD go Missile-Target pseudo bound function C tMD go Missile’s fail-safe function d(·) Game robustness measure dv(·) Modified robustness measure m Interception Missile-Target miss distance u Part 1: Missile’s acceleration perpendicular to LOS. Part 3: Missile’s acceleration vector. ue Missile’s evasive strategy. up Missile’s pursuit strategy. v Part 1: Target’s acceleration perpendicular to LOS. Part 3: Target’s acceleration vector. w Part 1: Defender’s acceleration perpendicular to LOS. Part 3: Defender’s acceleration vector. ˙λij LOS rate beteen i and j, where i, j = M, T, D ˆy Estimated ZEM. λij LOS angle beteen i and j, where i, j = M, T, D |ycr MT | The maximal value of |yMT | Vij Zero-Effort-Miss norm between i and j, where i, j = M, T, D Jij Cost function of i and j, where i, j = M, T, D rij Part 1: Closing range beteen i and j, where i, j = M, T, D Part 3: Vector range beteen i and j, where i, j = M, T, D yij Zero-Effort-Miss (ZEM) between i and j, where i, j = M, T, D γi Vehicle’s heading angle, i = M, T, D ρumin Minimal maneuver capability that allows the Missile to evade the Defender and intercept the target. ρi Vehicle’s maneuver capability, i = M, T, D 3
- 20. ai Vehicle’s acceleration, i = M, T, D ki Vehicle’s suboptimal maneuver gain, i = u, v, w ku,1 Minimal maneuver gain that allows the Missile to evade the Defender and intercept the Target. kumin Minimal maneuver gain that allows the Missile to evade the Defender. Pi Vehicle’s projection matrix, i = M, T, D ri Vehicle’s position, i = M, T, D Ti Vehicle’s transformation matrix, i = M, T, D Vi Vehicle’s velocity, i = M, T, D Φ Missile-Target transition matrix Ψ Missile-Defender transition matrix ∆t The difference between tMT f and tMD f tMD f Missile-Defender final time tMT f Missile-Target final time t∗ go1 The minimal t∗ go required for evasion and interception. t∗ go The intersection time-to-go of |yMD| with the fail-safe function C tMD go Missile-Defender time-to-go tMT go Missile-Target time-to-go VC Closing speed D Defender E Abstract evader. M Missile P Abstract pursuer. T Target V G1 Vector Guidance based on first order time-to-go. V G14 Vector Guidance based on first order time-to-go for M-D game andfourth order time-to- go for M-T game. V G4 Vector Guidance based on fourth order time-to-go. V G4∗ Vector Guidance based on fourth order time-to-go, with optimal start time. 4
- 21. GM (s) Missile’s dynamics transfer matrix. XMD Missile’s controller dynamic function in M-D game. XMT Missile’s controller dynamic function in M-T game. YMD Target’s controller dynamic function in M-D game. YMT Target’s controller dynamic function in M-T game. ZMD Defender’s controller dynamic function in M-D game. ZMT Defender’s controller dynamic function in M-T game. 5
- 23. 1 Introduction 1.1 Motivation THE protection of an airborne vehicle against a homing missile has become a significant issue, since a modern interceptor carries a substantial threat to such a vehicle. As interceptor missiles become more sophisticated, the current passive countermeasure systems are not sufficient. There- fore, a more advanced countermeasure system is needed. Such possible countermeasure is a short range homing missile (Defender), aimed at the interception of the interceptor. In such a scenario, the protected aircraft (Target) can use both its own evasive maneuver and the defender, in order to evade the missile. In generating guidance strategies, a common practice is a linearization with respect to a collision course, which implies simplified linear kinematics. However, in a game of three players, linearization assumptions can be unrealistic. As a result, generated guidance strate- gies can be inaccurate. Thus, this research provides an alternative approach which is not based on linearization. This research is based on Differential Game (DG) theory [1, 2], as a natural way to describe conflicts. In formulating a DG, there are two main approaches. In the first, the Linear Quadratic Differential Games (LQDG) approach, the cost is formed of a terminal quadratic state to account for the miss distance, and a quadratic control integral to account for the control effort, [3, 4, 5]. As a result, the optimal strategies are linear. This approach suffers several drawbacks. First, it violates the saturation limit every actuator has. Second, it does not guarantee a miss-distance value. Third, in game theory, the players must “agree” on the cost. However, the linear strategies generated by LQDG imply that on a collision course (except at the terminal time), both strategies are identically zero. While for the pursuer this is acceptable, no rational evader can agree to use such a cost. Indeed, close to termination, the evader has in many cases the potential to increase the miss-distance. In the second approach [6–7], called Differential Game Guidance Law (DGL), hard bounds are imposed on the controls and the cost is purely terminal to account for the miss- distance. As a result, the optimal strategies are nonlinear. Moreover, the saddle-point property implies a guaranteed miss-distance to each player. In classical terms, the navigation gain increases with time, and at a certain time before termination the guidance law becomes pure bang-bang. 1.2 Noticeable Contributions In the field of active aircraft defense against an attacking missile, some noticeable contributions have been made. In [8], a closed form relation was derived for the initial missile-target range ratio as well as at interception for the missile-defender conflict, under the assumption of a constant collision course. Later, [9] finds the requirements on the defender firing angle and the distance it will run to intercept the attacker as a function of the game geometry and the point at which the target launches the defender. In that paper, the author derives the location of intercept point in the target-centered coordinates. This work assumes a constant collision course and therefore suffers many drawbacks, as in a real battle situation the vehicles do maneuver. In [10], a discretized and linearized solution to the three player differential game is presented, under the assumptions that the target is fixed or slowly moving (a battleship for example), the defender is launched from the target to intercept the missile, while the missile’s objective is to intercept the target. However, in this scenario, the missile has no knowledge about the defender and therefore will not revise its collision course with respect to the defender. This study suggests that the missile should use a random pursuit strategy; otherwise, its trajectory is predictable and can be easily intercepted 7
- 24. by the defender (assuming the defender has a greater maneuvering capability). Recently, [11] has presented a solution to the three player problem, using a linearized model. In his research, the author has defined a quadratic cost function that represents the player’s objectives and is formed of a terminal quadratic state to account for the miss-distance, and a quadratic integral to account for the control effort. That work presents a solution to the full-knowledge differential game, however the LQDG solution suffers the mentioned drawbacks. More recently, [12] has presented a cooperative target-defender guidance strategy against a pursuing missile. That article is based on a two team LQDG and provides an optimal analytic solution for the target-defender pair. Moreover, a parametric analysis has been done to study the conditions for existence of a saddle point. The authors have provided numerical simulations to prove their theoretical analysis. That article implies that all optimal strategies are linear, and therefore, suffers the drawbacks mentioned above. Using a different approach, [13–14] have presented a multiple model adaptive guidance strategy to defend the target from the missile. That work applies a multiple model adaptive estimator with measurement fusion, where each model represents a possible guidance law and guidance parameters of the incoming homing missile. Thus, under the assumption that the homing missile uses one of the known guidance strategies, the defender may anticipate the missile’s maneuver, as the target maneuver is known. That article provides a very interesting insight into the three player differential game strategy but cannot guarantee any result if the homing missile doesn’t use any of the known linear strategies. Moreover it cannot guarantee a miss-distance value. Articles [15–17] have also made some noticeable contributions on this problem. However, the obtained guidance laws in these articles are still linear, and suffer the same drawbacks mentioned above. Other noticeable contributions can be found at [20–41]. 1.3 Main Results and Contribution This research is divided into three main parts. In Part I, one finds the Linear Model Guidance (LMG) approach for the three players conflict, in which the kinematics is linear, the controls are bounded, and the cost is the miss distance. The LMG approach suggests that in certain regions of the state space, the missile can perform an evasive maneuver with respect to the defender, without losing its pursuit capabilities. Moreover, sufficient conditions under which a missile can hit a target while evading a defender launched by the target, are derived. Moreover, the guaranteed cost strategies are optimized with respect to a robustness measure. However, Part I is based on the linearized model; as a result, the obtained guidance strategies do not always accurately reflect the actual situation. A detailed analysis of the linearization problem is provided in Part II. There, one finds the contradiction of the optimal guidance strategies in the linear kinematics scenario, and the real, nonlinear world. The reason of such contradiction is described in Part II, as well as a partial solution. In addition, Part II provides a deep analysis of all parameters relevant to the problem, and an optimality proof for the target and the defender. Part III continues the study presented in Part I. While it relies on similar principles, Part III is based on a three dimensional Vector Guidance (VG) instead of the Linearized Model Guidance (LMG) provided in Part I. A detailed discussion about the VG in a two player scenario can be found in [18]. As a result, the obtained strategies are much better than in Part I, as they reflect the actual situation instead of the linearized one. Planar and three dimensional simulations are provided in order to confirm the results. In order to account for endo-atmospheric interception conflict, where the aerodynamic forces are dominant, a Transformed Vector Guidance approach is derived in Part III. This approach suggests that by using a transformation matrix, one can account for the difference between the lateral and axial acceleration capabilities of the players. In addition, in order to account for noisy 8
- 25. measurements, an estimator based guidance algorithm is presented in Part III. This algorithm introduces an analytically computable miss-distance bound approximation, which accounts for noisy measurements and physical disturbances, and can be used in the early design stages. Also, Part III introduces an analysis of a non-ideal players games, in which the intercepting missile has a first order isotropic dynamics. Game strategies are modified and re-derived to fit this scenario. 9
- 27. Part I Linear Model Guidance 2 Problem Overview 2.1 Basic Definitions Consider a three player problem as depicted in Fig. 2.1. aM VM M aT VT T aD VD D γM λMT λMD γT γT rMT rMD rTD Figure 2.1: Planar Interception Geometry Given three players (M – Missile, T – Target, D – Defender). Denote players’ velocity vectors as VM , VT and VD. All three players can apply a velocity-vector-perpendicular acceleration. The Missile’s objective is to evade the Defender and intercept the Target. The Defender’s objective is to intercept the Missile and prevent it from capturing the Target. The Target’s objective is to escape the Missile. Denote aM , aT and aD as the corresponding Missile’s, Target’s, and Defender’s lateral accelerations. Consider hard bounds on players’ accelerations, |aM | ≤ amax M (2.1) |aT | ≤ amax T (2.2) |aD| ≤ amax D (2.3) The line of sight (LOS) between the Missile and the Target is denoted as LOSMT , between the Missile and the Defender is denoted as LOSMD, and between the Target and the Defender is denoted as LOSTD. The Missile-Target (M-T), Missile-Defender (M-D), and Target-Defender (T- D) closing ranges are denoted as rMT , rMD and rTD respectively. The range rate geometric 11
- 28. relations are, ˙rMT = VM cos (γM − λMT ) + VT cos (γT + λMT ) (2.4) ˙rMD = VM cos (γM − λMD) + VD cos (γD + λMD) (2.5) ˙rTD = VD cos (γD − λTD) − VT cos (γT − λTD) (2.6) Also given the LOS rate relations, ˙λMT = VT sin (γT + λMT ) − VM sin (γM − λMT ) rMT (2.7) ˙λMD = VD sin (γD + λMD) − VM sin (γM − λMD) rMD (2.8) ˙λTD = VD sin (λTD − γD) − VM sin (λTD − γT ) rTD (2.9) For an aerodynamically maneuvering Missile, the heading angle rate is, ˙γM = aM VM (2.10) ˙γT = aT VT (2.11) ˙γD = aD VD (2.12) Missile’s acceleration perpendicular to LOSMD is denoted as uMD (t), and its acceleration perpen- dicular to LOSMT is denoted as uMT (t). Target’s acceleration perpendicular to LOSMT is v(t), and the Defender’s acceleration perpendicular to LOSMD is w(t). Missile’s LOS perpendicular accelerations are uMD = aM cos (γM − λMD) (2.13) uMT = aM cos (γM − λMT ) (2.14) Target’s and Defender’s LOS perpendicular accelerations are v = aT cos (γT + λMT ) (2.15) w = aD cos (γD + λMD) (2.16) Define perpendicular to initial LOS distances, • xMD − distance perpendicular to LOSMD0 • xMT − distance perpendicular to LOSMT0 and the relative accelerations ¨xMD(t) = w(t) − uMD(t) (2.17) ¨xMT (t) = v(t) − uMT (t) (2.18) 12
- 29. Rename the Missile’s acceleration as following uMD(t) = u(t) uMT (t) = aM cos (γM − λMT ) = cos (γM − λMT ) cos (γM − λMD) · u(t) Define hTD(t) = cos (γM − λMT ) cos (γM − λMD) (2.19) and obtain uMD(t) = u(t) (2.20) uMT (t) = hTD(t) · u(t) (2.21) Denote the bounds on u(t), v(t), and w(t) as, |u(t)| ≤ ρu |v(t)| ≤ ρv |w(t)| ≤ ρw 2.2 Linearization In order to obtain a linear and time invariant system, one makes the following assumptions 1. ˙λMD , ˙λMT 1. Thus, both LOS’s rotation speed is small, and all three players are close to the corresponding collision triangles (as depicted in Fig. 2.1). 2. hTD(t) ≈ hTD = const. Thus, the interception geometry doesn’t chance much. 3. ˙rMD, ˙rMT ≈ const. Thus, along LOS the closing speeds are approximately constant. Define the closing speeds, V MD C = − ˙rMD (2.22) V MT C = − ˙rMT (2.23) and obtain the game dynamics along LOS rMD(t) = V MD C tMD go (2.24) rMT (t) = V MT C tMT go (2.25) where the time-to-go variables are defined as tMD go = tMD f − t tMT go = tMT f − t and the final times tMD f and tMT f are constant. As a result, the dynamic equations become linear and time invariant (LTI), ¨xMD(t) = w(t) − u(t) (2.26) ¨xMT (t) = v(t) − hTD · u(t) (2.27) It is important to say that our linearization assumptions impose serious limitations on the game dynamics, and may cause inaccurate results. This problem is explored in Part II and resolved in Part III. 13
- 30. 3 Dynamic Model and Zero Effort Miss Consider the following Missile’s dynamics (In this discussion, the Target and Defender are ideal). GM (s) = u(s) uC(s) = AM bM cM dM (3.1) The state equations of GM (s) are ˙η(t) = AM η(t) + bM uC(t) (3.2) u(t) = cM η(t) + dM uC(t) (3.3) Using (2.26), (2.27), and (3.3) one has ¨xMD(t) = w(t) − u(t) = w(t) − cM η(t) − dM uC(t) (3.4) ¨xMT (t) = v(t) − hTDu(t) = v(t) − hTDcM η(t) − hTDdM uC(t) (3.5) The following state space model is obtained ˙xMT (t) ¨xMT (t) ˙xMD(t) ¨xMD(t) ˙η(t) = 0 1 0 0 0 0 0 0 0 −hTDcM 0 0 0 1 0 0 0 0 0 −cM 0 0 0 0 AM xMT (t) ˙xMT (t) xMD(t) ˙xMD(t) η(t) + 0 −hTDdM 0 −dM bM uC(t) (3.6) + 0 1 0 0 0 v(t) + 0 0 0 1 0 w(t) In Fig. 3.1, one finds a block diagram of the linearized open guidance loop. GM (s) hTD 1 s 1 s 1 s 1 s uC _u w _ v ˙xMD ˙xMT xMD xMT Figure 3.1: Linearized Open Loop Since the Defender comes out of the Target, the initial position is such that rMD , rMT rTD. Therefore, λMT ≈ λMD and hTD ≈ 1. If this isn’t true, similar results can be easily obtained for 14
- 31. any constant hTD = 1. The state space realization becomes, ˙xMT (t) ¨xMT (t) ˙xMD(t) ¨xMD(t) ˙η(t) = 0 1 0 0 0 0 0 0 0 −cM 0 0 0 1 0 0 0 0 0 −cM 0 0 0 0 AM A xMT (t) ˙xMT (t) xMD(t) ˙xMD(t) η(t) + 0 −dM 0 −dM bM b uC(t) (3.7) + 0 1 0 0 0 c v(t) + 0 0 0 1 0 d w(t) (3.8) where the state vector is x(t) = xMT (t) ˙xMT (t) xMD(t) ˙xMD(t) η(t) Assuming linearization, define two final times tMD f = rMD(0) V MD C (3.9) tMT f = rMT (0) V MT C (3.10) two cost functions JMT = 1 0 0 0 0 g x tMT f = gx tMT f (3.11) JMD = 0 0 1 0 0 h x tMD f = hx tMD f (3.12) and two Zero Effort Miss (ZEM) variables, yMT (t) = gΦ tMT f , t x(t) (3.13) yMD(t) = hΨ tMD f , t x(t) (3.14) where Φ tMT f , t and Ψ tMD f , t are the transition matrices of A regarding the final times tMT f and tMD f respectively, ˙Φ tMT f , t = −Φ tMT f , t A , Φ tMT f , tMT f = I (3.15) ˙Ψ tMD f , t = −Ψ tMD f , t A , Ψ tMD f , tMD f = I (3.16) 15
- 32. Differentiate the ZEM variables ˙yMT (t) = gΦ tMT f , t b u(t) + gΦ tMT f , t c v(t) + gΦ tMT f , t d w(t) = XMT tMT f , t u(t) + YMT tMT f , t v(t) + ZMT tMT f , t w(t) (3.17) ˙yMD(t) = hΨ tMD f , t b u(t) + hΨ tMD f , t c v(t) + hΨ tMD f , t d w(t) = XMD tMD f , t u(t) + YMD tMD f , t v(t) + ZMD tMD f , t w(t) (3.18) At this point, we find the explicit form of the ZEM variables. Consider the first transition matrix, Φ tMT f , t . Change the running time, t, to the time-to-go, tMT go , tMT go = tMT f − t (3.19) dtMT go = −dt (3.20) Equation (3.15) becomes, ˙Φ tMT go = Φ tMT go A , Φ(0) = I (3.21) Multiply (3.21) by the output vector g and obtain g ˙Φ tMT go = gΦ tMT go A , Φ(0) = I (3.22) thus ˙ϕ11 ˙ϕ12 ˙ϕ13 ˙ϕ14 ˙ϕ15 = ϕ11 ϕ12 ϕ13 ϕ14 ϕ15 0 1 0 0 0 0 0 0 0 −cM 0 0 0 1 0 0 0 0 0 −cM 0 0 0 0 AM (3.23) Equation (3.23) provides the following differential equations. ˙ϕ11 = 0 , ϕ11(0) = 1 (3.24) ˙ϕ12 = ϕ11 , ϕ12(0) = 0 (3.25) ˙ϕ13 = 0 , ϕ13(0) = 0 (3.26) ˙ϕ14 = ϕ13 , ϕ14(0) = 0 (3.27) ˙ϕ15 = −ϕ12cM − ϕ14cM + ϕ15AM , ϕ15(0) = 0 (3.28) Solving these equations yields ϕ11 = 1 (3.29) ϕ12 = tMT go (3.30) ϕ13 = 0 (3.31) ϕ14 = 0 (3.32) ϕ15 = −L−1 MT cM (sI − AM )−1 s2 (3.33) where L−1 MT operator stands for inverse Laplace transform from the Laplace variable, s, to the time domain variable tMT go . Using (3.13) and (3.29–3.33), one obtains the Missile-Target ZEM variable, 16
- 33. yMT (t) = xMT (t) + tMT go ˙xMT (t) − L−1 MT cM (sI − AM )−1 s2 η(t) (3.34) as well as XMT tMT go = −L−1 MT GM (s) s2 (3.35) YMT tMT go = tMT go (3.36) ZMT tMT go = 0 (3.37) Similarly, the M-D ZEM is, yMD(t) = xMD(t) + tMD go ˙xMD(t) − L−1 MD cM (sI − AM )−1 s2 η(t) (3.38) as well as, XMD tMD go = −L−1 MD GM (s) s2 (3.39) YMD tMD go = hΨ tMD go c = 0 (3.40) ZMD tMD go = hΨ tMD go d = tMD go (3.41) Define the ZEM norms, VMT (t) = yMT (t) (3.42) VMD(t) = yMD(t) (3.43) Differentiate VMT and VMD, ˙VMT = yMT yMT (XMT u + YMT v + ZMT w) (3.44) ˙VMD = yMD yMD (XMDu + YMDv + ZMDw) (3.45) Since both ZEM variables are scalars, (3.44) and (3.45) reduce to ˙VMT = sign(yMT ) (XMT u + YMT v + ZMT w) (3.46) ˙VMD = sign(yMD) (XMDu + YMDv + ZMDw) (3.47) 17
- 34. 4 A Game of Three Ideal Players When all three players are ideal, (3.7) reduces to ˙xMT (t) ¨xMT (t) ˙xMD(t) ¨xMD(t) = 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 A xMT (t) ˙xMT (t) xMD(t) ˙xMD(t) + 0 −1 0 −1 b u(t) + 0 1 0 0 c v(t) + 0 0 0 1 d w(t) (4.1) The open loop block diagram becomes as described in Fig. 4.1. 1 s 1 s 1 s 1 s _u _ w v ˙xMD ˙xMT xMD xMT Figure 4.1: Zero Order Lag Open Loop Recall the ZEM norm derivatives ˙VMT = sign(yMT ) (XMT u + YMT v + ZMT w) (4.2) ˙VMD = sign(yMD) (XMDu + YMDv + ZMDw) (4.3) For GM (s) = 1 , we have, L−1 MD {GM (s)/s2 } = tMD go and L−1 MT {GM (s)/s2 } = tMT go . Thus, for ideal players, (3.35–3.37) and (3.39–3.41) reduce to, XMT = −tMT go , YMT = tMT go , ZMT = 0 (4.4) XMD = −tMD go , YMD = 0 , ZMD = tMD go (4.5) the ZEM projected dynamics reduces to, ˙VMT (t) = tMT go sign(yMT ) (−u + v) (4.6) ˙VMD(t) = tMD go sign(yMD) (−u + w) (4.7) and, the explicit form of ZEM variables becomes, yMT = xMT + tMT go ˙xMT (4.8) yMD = xMD + tMD go ˙xMD (4.9) 18
- 35. 5 Differential Game Definition The Target maximizes ˙VMT (t) = d dt |yMT (t)| with its controller v(t). Therefore, from (4.6), its optimal strategy is1 v∗ = ρvsign(yMT ) (5.1) The Defender, minimizes ˙VMD(t) = d dt |yMD(t)| with its controller w(t). Analogically, from (4.7), its optimal guidance law is w∗ = −ρwsign(yMD) (5.2) The Missile has two objectives: Defender evasion and Target pursuit. To derive the game bounds, two separate game situations are analyzed. 1. Missile-Defender Game − The Missile evades the Defender by maximizing ˙VMD(t). In such case, by (4.7), its optimal guidance law is u∗ e = −ρusign(yMD) (5.3) Substituting u∗ e and w∗ into (4.7) gives, ˙V∗ MD(t) = tMD go (ρu − ρw) (5.4) Integration yields |y∗ MD(t)| = |y∗ MD(t = 0)| + ˆ t 0 tMD f (ρu − ρw) dξ − ˆ t 0 ξ (ρu − ρw) dξ = |y∗ MD(t = 0)| + tMD f (ρu − ρw) ξ|t 0 − 1 2 (ρu − ρw) ξ2 t 0 (5.5) = |y∗ MD(t = 0)| + tMD f t (ρu − ρw) − 1 2 (ρu − ρw) t2 Define y∗ MD t = tMD f = (5.6) where is the minimal desired M-D miss distance. Consequently, y∗ MD t = tMD f = = |y∗ MD(t = 0)| + 1 2 (ρu − ρw) tMD f 2 (5.7) |y∗ MD(t = 0)| = − 1 2 (ρu − ρw) tMD f 2 (5.8) thus |y∗ MD(t)| = − 1 2 (ρu − ρw) tMD f 2 + tMD f t (ρu − ρw) − 1 2 (ρu − ρw) t2 = − 1 2 (ρu − ρw) tMD f − t 2 (5.9) From here, we have the final form of the first bound. y∗ MD tMD go = − 1 2 (ρu − ρw) tMD go 2 (5.10) 1 For a complete derivation of DGL refer to [6] 19
- 36. Fig. 5.1, shows the Missile-Defender ZEM optimal trajectories. tgo MD yMD Figure 5.1: Missile-Defender ZEM Optimal Trajectories Define A tMD go y∗ MD tMD go = − 1 2 (ρu − ρw) tMD go 2 . When the Missile and the Defender play optimal, yMD tMD go is parallel to A tMD go ; therefore, if yMD tMD go < A tMD go , the Defender can guarantee a miss distance smaller than which the Missile cannot endure. Hence, A tMD go is the evasion bound. 2. Missile-Target Game − The Missile pursues the Target by minimizing ˙VMT (t). In such case, by (4.6), its optimal guidance strategy is u∗ p(t) = ρusign(yMT ) (5.11) Similarly to (5.4), ˙V∗ MT (t) = tMT go (−ρu + ρv) (5.12) Integrate and obtain, |y∗ MT (t)| = |y∗ MT (t = 0)| + ˆ t 0 tMT f (−ρu + ρv) dξ − ˆ t 0 ξ (−ρu + ρv) dξ = |y∗ MT (t = 0)| + tMT f (−ρu + ρv) ξ|t 0 − 1 2 (−ρu + ρv) ξ2 t 0 (5.13) = |y∗ MT (t = 0)| + tMT f t (−ρu + ρv) − 1 2 (−ρu + ρv) t2 Define y∗ MT t = tMT f = m 20
- 37. where m is the maximal desired M-T miss distance. Hence, y∗ MT t = tMT f = m = |y∗ MT (t = 0)| + 1 2 (−ρu + ρv) tMT f 2 (5.14) |y∗ MT (t = 0)| = m − 1 2 (−ρu + ρv) tMT f 2 (5.15) thus |y∗ MT (t)| = m − 1 2 (−ρu + ρv) tMT f 2 + tMT f t (−ρu + ρv) − 1 2 (−ρu + ρv) t2 = m + 1 2 (ρu − ρv) tMT f − t 2 (5.16) This leads to the final form of the second bound y∗ MT tMT go = m + 1 2 (ρu − ρv) tMT go 2 (5.17) Missile-Target ZEM optimal trajectories are described in Fig. 5.2. tgo MT yMT Figure 5.2: Missile-Target ZEM Optimal Trajectories Define B tMT go y∗ MT tMT go = m + 1 2 (ρu − ρv) tMT go 2 . Analogically, if the Missile and the Target play optimal, yMT tMT go is parallel to B tMT go , so if yMT tMT go > B tMT go , the Missile cannot guarantee a miss distance of m. Thus, B tMT go is the pursuit bound. In this three player differential game, there are two ZEM variables, yMT tMT go and yMD tMD go . In order to succeed, the Missile must ensure that yMD tMD go > A tMD go for tMD go ∈ 0, tMD f , and yMT tMT go < B tMT go for tMT go ∈ 0, tMT f . After tMD go = 0, the game becomes a “two player game” for which, the optimal strategies are u∗ p and v∗ . 21
- 38. 6 Game Formulation Given the functions A tMD go and B tMT go ; player maneuver capabilities ρu, ρv, and ρw; the fi- nal times tMD f and tMT f ; the desired miss distances and m; and the initial conditions yMD 0 = |yMD(t = 0)| and yMT 0 = |yMT (t = 0)| as depicted in Fig. 6.1, (t) ℬ(t) ℓ m tf MD tf MT Time, t ||ZEM|| Figure 6.1: Missile-Defender and Missile-Target ZEM Bounds Objectives: 1. Obtain a guidance law for the Missile controller u(t) which guarantees yMD t = tMD f ≥ yMT t = tMT f ≤ m and derive sufficient conditions for which this guidance law holds. 2. Optimize this guidance law for maximum robustness. 3. Obtain the optimal guidance strategies for Target-Defender team. 22
- 39. 7 Simple Differential Game Solution 7.1 Basic Concept Recall the Missile’s optimal evasion strategy, u∗ e(t) = −ρusign(yMD) (7.1) and its optimal pursuit strategy u∗ p(t) = ρusign(yMT ) (7.2) This leads us to discuss two possible cases: 1. Opposite ZEM signs. In this case, yMD and yMT have opposite signs, sign(yMD) = −sign(yMT ) (7.3) From (7.1) and (7.2) we have, u∗ e(t) = u∗ p(t) (7.4) Clearly, the optimal evasion law is the same as the pursuit law. Therefore, the Missile’s optimal controller is u(t) = u∗ e(t) = u∗ p(t), as it is optimal for both ZEM variables. Example 7.1. Case No 1 is depicted in Fig. 7.1. ||yMT|| ||yMD|| ℬ tf MD tf MT Time, t ℓ ||ZEM|| Figure 7.1: 1st Case Linear Simulation This is the simplest case because the obtained law satisfies every initial conditions inside the area defined by A(t) and B(t). However, this case is a product of initial conditions and the other players’ strategies; therefore, the Missile cannot enforce it. 23
- 40. 2. Same ZEM signs. Here, yMD and yMT have the same signs, sign(yMD) = sign(yMT ) (7.5) and the optimal guidance laws are opposite to each other u∗ e(t) = −u∗ p(t) (7.6) Hence, by using u∗ e(t) to evade the Defender, the Missile simultaneously makes the worst possible pursuit maneuver towards the Target. The opposite is also true, by using u∗ p(t) to pursue the Target, it makes the worst possible maneuver regarding the Defender evasion. From this point, only case No 2 will be discussed as the first case is trivial. 7.2 Fail-safe Function C tMD go Let the Missile pursue the Target with u∗ p = ρusign(yMT ), and the Defender pursue the Missile with w∗ = −ρwsign(yMD). Using (4.7) we have, ˙V∗∗ MD(t) = tMD go sign(yMD) −u∗ p + w∗ = tMD go sign(yMD) (−ρusign(yMT ) − ρwsign(yMD)) = −tMD go (ρusign(yMD)sign(yMT ) + ρw) (7.7) Equation (7.5) yields, sign(yMD)sign(yMT ) = 1 (7.8) Substitute (7.8) into (7.7) and obtain, ˙V∗∗ MD(t) = −tMD go (ρu + ρw) (7.9) Integration yields, |y∗∗ MD(t)| = |y∗∗ MD(t = 0)| − ˆ t 0 tMD f (ρu + ρw) dξ + ˆ t 0 ξ (ρu + ρw) dξ = |y∗∗ MD(t = 0)| − tMD f (ρu + ρw) ξ|t 0 + 1 2 (ρu + ρw) ξ2 t 0 (7.10) = |y∗∗ MD(t = 0)| − tMD f t (ρu + ρw) + 1 2 (ρu + ρw) t2 Require y∗∗ MD t = tMD f = (7.11) Substitute and obtain y∗∗ MD t = tMD f = = |y∗∗ MD(t = 0)| − 1 2 (ρu + ρw) tMD f 2 (7.12) |y∗∗ MD(t = 0)| = + 1 2 (ρu + ρw) tMD f 2 (7.13) 24
- 41. thus |y∗∗ MD(t)| = + 1 2 (ρu + ρw) tMD f − t 2 (7.14) and the final form of y∗∗ MD tMD go is y∗∗ MD tMD go = + 1 2 (ρu + ρw) tMD go 2 (7.15) This function implies that yMD tMD go which reduces due to Defender and Missile strategies, cannot reduce more rapidly than y∗∗ MD tMD go . Hence, we choose: y∗∗ MD t = tMD f = , so that even in the worst case yMD tMD go cannot fall below . This function is defined as the fail-safe: C tMD go y∗∗ MD tMD go = + 1 2 (ρu + ρw) tMD go 2 (7.16) The function C tMD go reduces to when tMD go = 0, so that if yMD tMD go ≥ C tMD go for any tMD go ≥ 0, Missile’s strategy can be safely switched to u∗ p(t), and a miss distance of is guaranteed. Graphically, C(tgo) is described in Fig. 7.2. (t) ℬ(t) (t) ℓ m tf MD tf MT Time, t ||ZEM|| Figure 7.2: fail-safe Function C(tgo) Thus, Missile’s strategy is to evade the Defender until |yMD| reaches C, and then switch to u∗ p to pursue the Target. u = ue , |yMD| < C u∗ p , |yMD| ≥ C (7.17) where ue stands for some evasion strategy. 25
- 42. 7.3 Various Evasion Strategies 7.3.1 Basic Examples In order to reach C tMD go , the Missile can use a variety of evasive maneuvers. Example 7.2. The aggressive law (Fig. 7.3) uses u∗ e until |yMD| reaches C, then switches to u∗ p. ||yMT|| ||yMD|| ℬ tgo * t* tf MD tf MT Time, t ℓ ||yMT cr || ℬ(t* ) ||ZEM|| Figure 7.3: 2nd Case Linear Simulation (Aggressive Law) Example 7.3. On the contrary, a minimal evasive maneuver, umin e , enables the Missile to reach C tMD go at the time point tMD go = 0 (Fig. 7.4). ||yMT|| ||yMD|| ℬ t* =tf MD tf MT Time, t ℓ ||yMT cr || ℬ(t* ) ||ZEM|| Figure 7.4: 2nd Case Linear Simulation (Minimal Maneuver) 26
- 43. In both examples the Missile reaches a M-D miss distance of ; however, the M-T miss distance dramatically differs. The entire spectrum of maneuver gains between umin e and u∗ e can guarantee a M-D miss distance of ; however, the M-T miss distance is obviously affected by the choice of ue. 7.3.2 Evasive Maneuver Gain ku Let the Missile evade the Defender using ue = −kusign(yMD) for some ku ≤ ρu. Also, let the Target evade the Missile using its optimal controller v∗ = ρvsign(yMT ), and the Defender pursue the Missile using w∗ = −ρwsign(yMD). Using (4.6) one has, ˙VMT (t) = tMT go sign(yMT ) (−ue + v∗ ) = tMT go sign(yMT ) (kusign(yMD) + ρvsign(yMT )) = tMT go (kusign(yMD)sign(yMT ) + ρv) (7.18) Recall that sign(yMD)sign(yMT ) = 1 and obtain, ˙VMT (t) = tMT go (ku + ρv) (7.19) Integration gives, |yMT (t)| = yMT 0 + ˆ t 0 tMT f (ku + ρv) dξ − ˆ t 0 ξ (ku + ρv) dξ = yMT 0 + tMT f (ku + ρv) ξ|t 0 − 1 2 (ku + ρv) ξ2 t 0 = yMT 0 + tMT f t (ku + ρv) − 1 2 (ku + ρv) t2 (7.20) = yMT 0 + 1 2 (ku + ρv) tMT f 2 − 1 2 (ku + ρv) tMT f 2 + tMT f t (ku + ρv) − 1 2 (ku + ρv) t2 = yMT 0 + 1 2 (ku + ρv) tMT f 2 − 1 2 (ku + ρv) tMT f − t 2 The final form of yMT tMT go is yMT tMT go = yMT 0 + 1 2 (ku + ρv) tMT f 2 − 1 2 (ku + ρv) tMT go 2 (7.21) Rename some of our variables in order to work with a single time-to-go variable. Define tMD go = tgo (7.22) tMD f = tf (7.23) tMT go = tgo + ∆t (7.24) tMT f = tf + ∆t (7.25) Equation (7.21) becomes |yMT (tgo)| = yMT 0 + 1 2 (ku + ρv) (tf + ∆t)2 − 1 2 (ku + ρv) (tgo + ∆t)2 (7.26) 27
- 44. Similarly, for the second ZEM variable ˙VMD(t) = tgosign(yMD) (−ue + w∗ ) = tgosign(yMD) (kusign(yMD) − ρwsign(yMD)) = tgo (ku − ρw) (7.27) Similarly to (7.20), integration yields |yMD(tgo)| = yMD 0 + 1 2 (ku − ρw) t2 f − 1 2 (ku − ρw) t2 go (7.28) Recall that C(tgo) = + 1 2 (ρu + ρw) t2 go (7.29) 7.3.3 The Impact of ku Equate (7.28) and (7.29) to find the intersection of |yMD(tgo)| and C(tgo). We have, t∗ go (ku) = t2 f (ku − ρw) − 2 + 2 |yMD 0 | ku + ρu (7.30) or alternatively, ku t∗ go = 2 + t∗ go 2 ρu − 2 yMD 0 + t2 f ρw t2 f − t∗ go 2 (7.31) where t∗ go is the intersection time-to-go of |yMD| with C, and ku t∗ go is the appropriate maneuver gain. Since t∗ go ∈ R, we obtain an essential condition for evasion: ku ≥ ρw + 2 − yMD 0 t2 f (7.32) Therefore, ku must satisfy ρw + 2 − yMD 0 t2 f kumin ≤ ku ≤ ρu (7.33) Otherwise the Defender can guarantee a miss distance smaller than . By substituting (7.33) into (7.30), one obtains 0 ≤ t∗ go ≤ t2 f (ρu − ρw) − 2 + 2 |yMD 0 | 2ρu t∗ gomax (7.34) Note that kumin produces the evasive maneuver umin e , which makes |yMD| reach C at t∗ go = 0, and is illustrated in Example 7.3. While ku = ρu produces u∗ e, for which |yMD| reaches C at t∗ go = t∗ gomax . It is illustrated in Example 7.2. Substituting (7.31) into (7.26) yields ycr MT t∗ go = 1 2 tf + 2∆t + t∗ go t∗ go 2 (ρu − ρv) + t2 f (ρw + ρv) + 2 − 2 yMD 0 tf + t∗ go + yMT 0 (7.35) 28
- 45. From (7.35) one can see the maximal value of |yMT | as a function of the intersection time t∗ go. This is indeed the maximum as at this point the Missile’s guidance law becomes u∗ p(t), and the variable |yMT | starts decreasing. In Fig. 7.5, one finds a qualitative plot of the two phases of guidance (Evasion and Pursuit). Since maxt {|yMT (t)|} = |ycr MT |, the Missile guarantees a miss distance of from the Defender and a miss distance of m from the Target if ycr MT t∗ go ≤ B t∗ go . |yMT| |yMD| ℬ Evasion Pursuit tgo * t* tf MD tf MT Time, t ℓ |yMT cr | ℬ(t* ) |ZEM| Figure 7.5: Two Phases of Guidance Example 7.4. Fig. 7.6 shows the functions ycr MT t∗ go and B t∗ go . |yMT cr (tgo * )| ℬ(tgo * ) tgomax * tgo * |ZEM| d(tgo * ) Figure 7.6: Functions ycr MT t∗ go and B t∗ go 29
- 46. 7.4 Optimality Definition Define d t∗ go B t∗ go − ycr MT t∗ go = m + 1 2 (ρu − ρv) t∗ go + ∆t 2 (7.36) − 1 2 tf + 2∆t + t∗ go t∗ go 2 (ρu − ρv) + t2 f (ρw + ρv) + 2 − 2 yMD 0 tf + t∗ go − yMT 0 In order to maximize the robustness, the Missile must keep |yMT | as far from the bound, B, as possible. Thus, the optimal maneuver gain kOpt u is the one that maximizes d t∗ go in the appropriate interval 0 ≤ t∗ go ≤ t∗ gomax . Therefore, the optimal cost is dOpt = max t∗ go d t∗ go (7.37) Example 7.5. For the same set of parameters as in Example 7.4, the function d t∗ go is presented in Fig. 7.7. tgomax * tgo * d(tgo * ) Figure 7.7: Cost Function d t∗ go Clearly, in this example the maximal value of d t∗ go is at t∗ gomax which corresponds to ku = ρu; therefore, the guidance law that maximizes d t∗ go is u∗ = u∗ e , |yMD| < C u∗ p , |yMD| ≥ C (7.38) 30
- 47. 8 Optimality Analysis In this section, the optimal maneuver gain, kOpt u , and sufficient conditions for the three players game are derived. Theorem 8.1. The function d(t∗ go) is monotonically increasing. Proof. Differentiate (7.36) with respect to t∗ go, simplify, and obtain d dt∗ go d t∗ go = ∆t (ρu + ρw) t2 f + 2 − 2 yMD 0 tf + t∗ go 2 (8.1) The denominator of (8.1) is always positive. The numerator is also positive if, (ρu + ρw) t2 f + 2 − 2 yMD 0 ≥ 0 (8.2) thus ρu ≥ −ρw + 2 − yMD 0 t2 f (8.3) From (7.32) we understand that the Missile can guarantee evasion only if ρu ≥ ρw + 2 − yMD 0 t2 f (8.4) Assuming (8.4) holds2 , (8.3) also must hold. Hence, d t∗ go is monotonically increasing. Denote the intersection time-to-go of d t∗ go with the horizontal axis as t∗ go1 . In Fig. 8.1, the function d t∗ go and its intersection point t∗ go1 with the time axis are depicted. tgo1 * tgomax * tgo * d(tgo * ) Figure 8.1: Intersection of the Cost Function d t∗ go 2 if not, the Missile is unable to evade the Defender and this entire discussion is pointless 31
- 48. Since d(t∗ go) is monotonically increasing, the proposed guidance strategy (7.17) provides the entire spectrum of controls for the 1st phase of evasion. t∗ go1 ≤ t∗ go ≤ t∗ gomax (8.5) Substituting (8.5) into (7.31) yields the desired set of controls ku,1 ≤ ku ≤ ρu (8.6) where ku,1 matches the intersection time t∗ go1 , and ρu matches t∗ gomax . By equating d t∗ go to zero, analytical solution for t∗ go1 is obtained. t∗ go1 = − (ρv + ρw) t3 f − 2∆t (ρv + ρw) t2 f + (ρu − ρv) ∆t2 + 2 m − + yMD 0 − yMT 0 tf + 4∆t yMD 0 − (ρv + ρw) t2 f − 2∆t (ρu − ρv) tf − (ρu − ρv) ∆t2 − 2 (m − + |yMD 0 | − |yMT 0 |) (8.7) Theorem 8.2. Let t∗ go1 ≤ t∗ gomax . Any value of ku which satisfies (8.6) can be used by the Missile in order to evade the Defender and intercept the Target. Proof. Since d t∗ go is monotonically increasing, and d t∗ go1 = 0, we have d t∗ go ≥ 0 ∀t∗ go ≥ t∗ go1 (8.8) Therefore, ycr MD t∗ go ≤ B t∗ go ∀ku ≥ ku,1 (8.9) Hence, the Missile can guarantee a M-T miss distance of m and M-D miss distance of . Example 8.1. Here, ku = ρu can be used to obtain a solution, as presented in Fig. 8.2. ||yMT|| ||yMD|| ℬ tgo * t* tf MD tf MT Time, t ℓ ||yMT cr || ℬ(t* ) ||ZEM|| Figure 8.2: Linear Simulation. ku = 100% of ρu 32
- 49. Example 8.2. Alternatively, instead of using its full capability, the Missile can apply the minimal allowed evasive maneuver, ku = ku,1 (= 0.67ρu in this example) as shown in Fig. 8.3. Moreover, any value of ku in the range 0.67ρu ≤ ku ≤ ρu can be used to guarantee a M-D miss distance of and a M-T miss distance of m. ||yMT|| ||yMD|| ℬ tgo * t* tf MD tf MT Time, t ℓ ||yMT cr || ||ZEM|| Figure 8.3: Linear Simulation. ku = 67% of ρu The advantage in ku,1 is that it allows the Missile to complete its task with minimal evasion. However, one must keep in mind that in such a case, |ycr MT | = B(t∗ go); thus, the robustness of this strategy is zero. Theorem 8.3. The optimal value of the evasive maneuver gain ku (which maximizes d(t∗ go), and provides maximum robustness) is always kOpt u = ρu. Proof. Since d t∗ go is monotonically increasing in the interval t∗ go1 ≤ t∗ go ≤ t∗ gomax , it has its maxi- mum at t∗ go = t∗ gomax . Hence, the corresponding maneuver gain is k∗ u = ρu. Proposition 8.1. If t∗ go1 < 0, then d t∗ go is greater than zero in the range 0 ≤ t∗ go ≤ t∗ gomax and any value of ku, such that kumin ≤ ku ≤ ρu can be used. As can be seen from Fig. 8.4, even at t∗ go = 0, the robustness criterion d(t∗ go) is positive. 33
- 50. tgomax * tgo * d(tgo * ) Figure 8.4: d(t∗ go) > 0 ∀ 0 ≤ t∗ go ≤ t∗ gomax Proposition 8.2. If t∗ go1 > t∗ gomax (Fig. 8.5), then d t∗ go < 0 in the range 0 ≤ t∗ go ≤ t∗ gomax , and the Missile cannot evade the Defender and intercept the Target. tgomax * tgo * d(tgo * ) Figure 8.5: d(t∗ go) < 0 ∀ 0 ≤ t∗ go ≤ t∗ gomax Remark 8.1. According to Theorem 8.3, the optimal Missile’s guidance law, which maximizes d t∗ go , is u∗ = u∗ e , |yMD| < C u∗ p , |yMD| ≥ C (8.10) 34
- 51. where u∗ e = −ρusign(yMD) and u∗ p = ρusign(yMT ). Also, the optimal guidance laws for the Target- Defender team is v∗ = ρvsign(yMT ) (8.11) w∗ = −ρwsign(yMD) (8.12) Condition 1. Rewrite Theorem 8.2 explicitly to impose a sufficient condition for the three players problem. In order to have a solution; namely, enable the Missile to evade the Defender with a miss distance greater or equal to , and intercept the Target with a miss distance smaller or equal to m, the inequality (8.13) must hold. − (ρv + ρw) t3 f − 2∆t (ρv + ρw) t2 f + (ρu − ρv) ∆t2 + 2 m − + yMD 0 − yMT 0 tf + 4∆t yMD 0 − (ρv + ρw) t2 f − 2∆t (ρu − ρv) tf − (ρu − ρv) ∆t2 − 2 (m − + |yMD 0 | − |yMT 0 |) (8.13) ≤ t2 f (ρu − ρw) − 2 + 2 |yMD 0 | 2ρu Remark 8.2. Analytic solution for the minimal ρu which guarantees success ,ρumin , is possible, though the expression is very complicated. Example 8.3. Substituting this value of ρumin into the linear simulation yields the solution de- scribed in Fig. 8.6. ||yMT|| ||yMD|| ℬ tgo * t* tf MD tf MT Time, t ℓ ||yMT cr || ||ZEM|| Figure 8.6: Linear Simulation with ρumin . ku = 100% of ρu It is readily seen that with its full capability, the ZEM |yMT (tgo)| hits the bound B(t). 35
- 52. 9 Nonlinear Simulations Example 9.1. Simulation results for ku = ρu and the following parameters is shown on Fig. 9.1. ρu = 120 m sec2 , ρv = 60 m sec2 , ρw = 70 m sec2 , m = 0.5 [m] , = 150 [m] Missile Target Defender tf MD t* 0 1000 2000 3000 4000 5000 6000 7000 0 1000 2000 3000 4000 x [m] y[m] Miss MD = 206 , tf MD = 5.66 Miss MT = 0.17 , tf MT = 15.35 Figure 9.1: Nonlinear Simulation 1 (ku = ρu) The actual M-D miss distance is greater than the required. This happens because the actual time- to-go isn’t linear since the Missile evades the Defender and “breaks” the collision triangle which is the base for our linearization assumptions. Fig. 9.2 shows the measured tgo as a function of the simulation time t. 0 1 2 3 4 5 0 1 2 3 4 5 Time, t Estimatedtgo MD t* Figure 9.2: Measured tMD go as a function of time 36
- 53. Clearly, the time-to-go is nonlinear until the switch point. Example 9.2. One can use the proposed guidance law for ku,1 in order to reduce Missile’s maneuver so that the collision triangle would suffer less distortion. To obtain ku,1 it is necessary to know the final times tMD f , tMT f . Since these values are unknown, it is possible compute them online by substituting tMD go , tMT go instead of tMD f , tMT f and updating it every time step. In such a case, (Fig. 9.3) a much closer result is obtained. Missile Target Defender tf MD t* 0 1000 2000 3000 4000 5000 6000 7000 0 1000 2000 3000 4000 x [m] y[m] Miss MD = 156.5 , tf MD = 5.6 Miss MT = 0.05 , tf MT = 15.3 Figure 9.3: Nonlinear Simulation 2 (ku = ku,1) and the time-to-go is closer to linear as shown in Fig. 9.4. 0 1 2 3 4 5 0 1 2 3 4 5 Time, t Estimatedtgo MD Figure 9.4: Estimated tMD go as a function of time 37
- 54. One must understand that the greater ∆ρuw = ρu − ρw is, the more distortion suffers the M-D collision triangle; therefore, linearization assumptions become less valid. Example 9.3. Consider the parameters ρu = 170 m sec2 , ρv = 30 m sec2 , ρw = 60 m sec2 , m = 0.5 [m] , = 150 [m] The result, shown in Fig. 9.5, is the outcome of the nonlinear simulation, using the stated above parameters. Missile Target Defender tf MD t* 0 1000 2000 3000 4000 5000 6000 0 1000 2000 3000 4000 x [m] y[m] Miss MD = 325 , tf MD = 5.74 Miss MT = 0.1 , tf MT = 13.37 Figure 9.5: Nonlinear Simulation 3 (ku = ρu) Denote t∗ as the switch time (refer to Fig. 9.5). It is readily seen that t∗ < tMD f ; therefore, the Missile switches to pursuit strategy before it passes by the Defender. In fact, this is a big advantage of the proposed guidance strategy, as it allows the Missile to pursue the Target while it still plays against the Defender. One can also see a big difference between the requested M-D miss distance and the actual one. Example 9.4. It is possible to use ku = ku,1 in order to reduce the Missile’s evasive maneuver and cause less distortion to the collision triangle. The outcome of such simulation is shown in Fig. 9.6. 38
- 55. Missile Target Defender tf MD t* 0 1000 2000 3000 4000 5000 6000 0 1000 2000 3000 4000 x [m] y[m] Miss MD = 154.5 , tf MD = 5.57 Miss MT = 0.18 , tf MT = 14.57 Figure 9.6: Nonlinear Simulation 4 (ku = ku,1) As expected, the Missile – Defender miss distance is much closer to the linear kinematics simulation. 39
- 56. 10 Discussion In this part, a guaranteed-cost guidance strategy has been derived for the linearized model. Such a strategy enables the Missile to evade the Defender and intercept the Target, provided the derived algebraic condition holds. Also, optimal strategies for the Defender and the Target are presented, and the Missile’s strategy is optimized for maximum robustness. There are considerable differences between the linear and the nonlinear simulation results, as tMT go does not behave as a linear function of the real simulation time. In addition tMT f , which is fixed in linear simulations, changes during nonlinear simulations, since linearization assumptions do not hold. Therefore, this part outlines the differences between the linear kinematics, used to obtain the optimal strategies, and the nonlinear kinematics, typical to a real conflict. When using the maximal evasive gain, ku = ρu, one has no need to know the final times tMD f , tMT f ; thus, the real-time computations do not suffer any causality problems. However, when using the minimal gain, ku,1, one needs to know the values of tMD f , tMT f . These values are not constant in the nonlinear scenari; therefore, are not known apriori. It is possible to compute ku,1 in real-time and update it in every time step. In order to do this, one must use the values of tMD go , tMT go instead of tMD f , tMT f . In addition, one must use the values of |yMD| , |yMT | instead of yMD 0 , yMT 0 at every time step. However, there are difficulties in measuring the time-to-go variables correctly due to their nonlinear behavior. Due to this difficulty, it is impossible to reach the exact value of the Missile–Defender miss distance. Another problem of using ku,1 arises because the initial values of tMT go − tMD go are far from the final values of tMT f − tMD f . Therefore, one needs to add an approximated factor to the value of tMT go − tMD go . The cause to this problem is the Missile’s high gain evasive maneuver that distorts the collision triangle, provided that the Missile’s maneuver capability is much higher than that the Defender. As a result, the measured ZEM variables can be inacurate and introduce disturbances in the Missile’s control loop. Therefore, one must understand that the optimal solution; namely ku = ρu, also introduces the most significant disturbances. 11 Conclusions Unlike other approaches discussed in the Introduction, the current approach singles out the miss distance as the outcome of the conflict. Moreover, all three players have bounded controls, while in previous studies they are free. In particular, it suggests that the Missile wins the game if the Missile–Target miss distance is smaller than a prescribed value, while the Missile–Defender miss distance is bigger than a prescribed value. In an ideal Missile–Target conflict, a sufficient condition for capture is the Missile advantage in acceleration perpendicular to the LOS. In a three player conflict, while this becomes much more complicated, it is still an algebraic condition. It enables the designer to determine algebraically the necessary parameters at an early stage of the design. The present study suggests that the switch time, at which the Missile ceases to evade the Defender and starts pursuing the Target, occurs before the pass time, at which the Missile passes by the Defender. The switch time depends on the initial conditions and on various system parameters. Similar to the sufficient capture condition presented here for the Missile, it is possible to generate a sufficient evasion condition for the Target. Similar to the two player conflict, this study can be extended to the non-ideal scenario. In such a scenario, every player has its own dynamics which plays an important role in the outcome of the conflict. This research has been performed in the end of Part III for Vector Guidance approach (refer to Part III) and 1st order dynamics, while high order dynamics is left for future research. 40
- 57. Part II LMG Analysis This part has two main purposes: • It provides deep parametric analysis of the results obtained in Part I. Also, it proves opti- mality for the Target’s and Defender’s maneuvers. • Analyzes the problem caused by linearization. This analysis emphasizes the need for a different approach discussed in Part III. 12 Parametric Analysis Recall the inequality derived in Section 8 of Part I. − (ρv + ρw) t3 f − 2∆t (ρv + ρw) t2 f + (ρu − ρv) ∆t2 + 2 m − + yMD 0 − yMT 0 tf + 4∆t yMD 0 − (ρv + ρw) t2 f − 2∆t (ρu − ρv) tf − (ρu − ρv) ∆t2 − 2 (m − + |yMD 0 | − |yMT 0 |) ≤ t2 f (ρu − ρw) − 2 + 2 |yMD 0 | 2ρu (12.1) Provided (12.1) holds, a M-D miss distance of , and a M-T miss distance of m can be guaranteed. From (12.1), one obtains the solution for ρumin , the minimal maneuver capability required by the Missile to complete its task. However, analytic solution for ρumin is too long to be written here; therefore, qualitative and quantitative properties of ρumin yMT 0 , yMD 0 , ρv, ρw, m, , tf , ∆t and its dependence on the various parameters is explored. Remark 12.1. For yMD 0 = yMT 0 = m = 0 we have a simpler solution, ρumin = ∆t (∆t3 ρv − tf (tf (3∆t + 2tf ) (ρv + ρw) + 4 )) + 3∆t2 − √ 8∆t (∆t + tf ) tf ρv (∆t + tf ) + t2 f ρw + 2 2 − ρv∆t2 t2 f ρw + 2 ∆t2 (∆t2 − 4tf (∆t + tf )) 12.1 Initial Conditions The first topic to explore is the influence of the initial conditions, yMT 0 and yMD 0 , on ρumin . Example 12.1. For the following numerical values: ρv = 30 m Sec2 , ρw = 50 m Sec2 , tf = 3 [Sec] , ∆t = 4 [Sec] , m = 0.5 [m] , = 150 [m] the plot of ρumin yMT 0 , yMD 0 is shown in Fig. 12.1. 41
- 58. 0 50 100 150 200 |y0 MT | 0 50 100 150 200 |y0 MD | 110 120 130 140 ρumin (a) Plot of ρumin yMT 0 , yMD 0 0 50 100 150 200 0 50 100 150 200 |y0 MT | |y0 MD | ρumin (|y0 MT |, |y0 MD |) 105 115 125 135 145 (b) Contour Plot of ρumin yMT 0 , yMD 0 Figure 12.1: Plot and Contour Plot of ρumin yMT 0 , yMD 0 and the section plots of ρumin yMT 0 and ρumin yMD 0 are depicted in Fig. 12.2. We conclude that ρumin yMT 0 , yMD 0 behaves almost as a linear function of yMT 0 and yMD 0 . |y0 MT | = 0 |y0 MT | = 100 |y0 MT | = 200 50 100 150 200 |y0 MD| 110 120 130 140 ρumin (a) ρumin yMT 0 |y0 MD | = 0 |y0 MD | = 100 |y0 MD | = 200 50 100 150 200 |y0 MT | 110 120 130 140 ρumin (b) ρumin yMD 0 Figure 12.2: Section Plots Obviously, bigger yMT 0 complicates the Missile’s task, while bigger yMD 0 simplifies it. This makes sense because the bigger yMT 0 is, the closer is |yMT | to the bound B at the beginning. On the other hand, starting from yMD 0 > 0 lets |yMD| start closer to the fail-safe function C; hence, the bigger yMD 0 is, the easier it is for the Missile to evade the Defender. 42
- 59. Example 12.2. Consider the following values, ρu = 170 m sec2 , ρv = 30 m sec2 , ρw = 50 m sec2 , tf = 3 [sec] , ∆t = 4 [sec] , m = 0.5 [m] , = 150 [m] The simulations in Fig. 12.3 demonstrate the above analysis. ||yMT|| ||yMD|| ℬ tgo * t* tf MD tf MT Time, t ℓ ||yMT cr || ℬ(t* ) ||ZEM|| (a) yMD 0 = 0, yMT 0 = 0 ||yMT|| ||yMD|| ℬ tgo * t* tf MD tf MT Time, t ℓ ||yMT cr || ℬ(t* ) ||ZEM|| (b) yMD 0 = 200, yMT 0 = 0 ||yMT|| ||yMD|| ℬ tgo * t* tf MD tf MT Time, t ℓ ||yMT cr || ||ZEM|| (c) yMD 0 = 0, yMT 0 = 200 ||yMT|| ||yMD|| ℬ tgo * t* tf MD tf MT Time, t ℓ ||yMT cr || ℬ(t* ) ||ZEM|| (d) yMD 0 = 200, yMT 0 = 200 Figure 12.3: Linear Simulations for Different Initial Conditions Notice that the influence of yMD 0 is greater than of yMT 0 . From Fig. 12.3, we conclude that yMT 0 = yMD 0 = 200 is better for the Missile than yMT 0 = yMD 0 = 0. 43
- 60. 12.2 Target’s and Defender’s Maneuver Capabilities This subsection explores the influence of ρv and ρw on ρumin . Example 12.3. Consider the following numerical values, yMT 0 = yMD 0 = 0, tf = 3 [Sec] , ∆t = 4 [Sec] , m = 0.5 [m] , = 150 [m] The plot of ρumin (ρv, ρw) is shown in Fig. 12.4. 0 20 40 ρv 0 20 40 ρw 50 100 150 ρumin (a) Plot of ρumin (ρv, ρw) 0 10 20 30 40 50 0 10 20 30 40 50 ρv ρw ρumin (ρv, ρw) 60 100 140 180 (b) Contour Plot of ρumin (ρv, ρw) Figure 12.4: Plot and Contour Plot of ρumin (ρv, ρw) and the section plots of ρumin (ρv) and ρumin (ρw) are depicted in Fig. 12.5. ρw = 0 ρw = 25 ρw = 50 10 20 30 40 50 ρv 50 100 150 ρumin (a) ρumin (ρv) ρv = 0 ρv = 25 ρv = 50 10 20 30 40 50 ρw 50 100 150 ρumin (b) ρumin (ρw) Figure 12.5: Section Plots 44
- 61. Note that, ρumin (ρv, ρw) behaves almost as a linear function of ρv and ρw. As expected, the grater ρv and ρw are, the harder it is for the Missile to complete its task. Example 12.4. Consider the following numerical values, ρu = 170 m Sec2 , yMT 0 = yMD 0 = 0, tf = 3 [Sec] , ∆t = 4 [Sec] , m = 0.5 [m] , = 150 [m] Linear simulation results for different values of ρv and ρw are shown in Fig. 12.6. ||yMT|| ||yMD|| ℬ tgo * t* tf MD tf MT Time, t ℓ ||yMT cr || ||ZEM|| (a) ρv = 30, ρw = 50 ||yMT|| ||yMD|| ℬ tgo * t* tf MD tf MT Time, t ℓ ||yMT cr || ||ZEM|| (b) ρv = 40, ρw = 50 ||yMT|| ||yMD|| ℬ tgo * t* tf MD tf MT Time, t ℓ ||yMT cr || ||ZEM|| (c) ρv = 30, ρw = 60 ||yMT|| ||yMD|| ℬ tgo * t* tf MD tf MT Time, t ℓ ||yMT cr || ||ZEM|| (d) ρv = 40, ρw = 60 Figure 12.6: Linear Simulations for Different Values of ρv and ρw Indeed, the increase of ρv and ρw makes it harder for the Missile to achieve its goal. 45
- 62. 12.3 Required M-D and M-T miss distances While we impose our requirements on the miss distances m and , it is important to understand their impact on the Missile’s required capability, ρumin . Example 12.5. Consider the numerical values, ρv = 30 m Sec2 , ρw = 50 m Sec2 , tf = 3 [Sec] , ∆t = 4 [Sec] , yMT 0 = yMD 0 = 0 Fig. 12.7 depicts the plot of ρumin (m, ). 0 2 4 50 100 150 200ℓ 120 140 160 ρumin (a) Plot of ρumin (m, ) 0 1 2 3 4 5 50 100 150 200 ℓ ρumin (, ℓ) 120 130 140 150 160 (b) Contour Plot of ρumin (m, ) Figure 12.7: Plot and Contour Plot of ρumin (m, ) Also, the section plots of ρumin (m) and ρumin ( ) are shown in Fig. 12.8. ℓ = 10 ℓ = 50 ℓ = 100 2 4 6 8 10 115 120 125 130 135 ρumin (a) ρumin (m) = 0 = 50 = 100 50 100 150 200 ℓ 110 120 130 140 150 160 ρumin (b) ρumin ( ) Figure 12.8: Section Plots 46
- 63. Again, the dependence of ρumin on m and is close to linear. However, the required M-T miss distance, m, has small influence on ρumin . Example 12.6. Consider the numerical values ρu = 170 m sec2 , ρv = 30 m sec2 , ρw = 50 m sec2 , tf = 3 [sec] , yMT 0 = yMD 0 = 0, ∆t = 4 [sec] Linear simulations depicted in Fig. 12.9 illustrate the above analysis. ||yMT|| ||yMD|| ℬ tgo * t* tf MD tf MT Time, t ℓ ||yMT cr || ℬ(t* ) ||ZEM|| (a) m = 0, = 150 ||yMT|| ||yMD|| ℬ tgo * t* tf MD tf MT Time, t ℓ ||yMT cr || ℬ(t* ) ||ZEM|| (b) m = 10, = 150 ||yMT|| ||yMD|| ℬ tgo * t* tf MD tf MT Time, t ℓ ||yMT cr || ℬ(t* ) ||ZEM|| (c) m = 0, = 300 ||yMT|| ||yMD|| ℬ tgo * t* tf MD tf MT Time, t ℓ ||yMT cr || ℬ(t* ) ||ZEM|| (d) m = 10, = 300 Figure 12.9: Linear Simulations for Different Values of m and Hence, for any practical use, m = 0 can be chosen, as it simplifies the expressions and has small effect on the required capability. 47
- 64. 12.4 The final times tMD f and tMT f When we talk about the final times, we refer to tMD f and tMT f . However, in Subsection 7.3 of Part I, the following parameters were defined. tMD f = tf (12.2) tMT f = tf + ∆t (12.3) Therefore, we explore the influence of the final times in terms of tf and ∆t. Example 12.7. Consider the numerical values ρv = 30 m Sec2 , ρw = 50 m Sec2 , tf = 3 [Sec] , ∆t = 4 [Sec] , m = 0.5 [m] , = 150 [m] The plot of ρumin (tf , ∆t) is shown in Fig. 12.10, 1 2 3 4 5 Δt 2 3 4 5 tf 200 300 400 500 600 ρumin (a) Plot of ρumin (tf , ∆t) 1 2 3 4 5 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 Δt tf ρumin (Δt, tf ) 150 250 350 450 550 (b) Contour Plot of ρumin (tf , ∆t) Figure 12.10: Plot and Contour Plot of ρumin (tf , ∆t) and the section plots are depicted in Fig. 12.11. 48
- 65. Δt = 1 Δt = 2 Δt = 4 1 2 3 4 5 tf100 200 300 400 500 600 ρumin (a) ρumin (tf ) tf = 2 tf = 6 tf = 10 0 1 2 3 4 5 6 Δt100 200 300 400 500 600 ρumin (b) ρumin (∆t) Figure 12.11: Section Plots Unlike the dependence of ρumin on other parameters, the behavior of ρumin (tf , ∆t) is far from being linear. This function tends to have infinite values when tf or ∆t approach zero. This makes sense because the Missile needs infinite maneuver capability to complete its task in zero time. Another point is that for every value of ∆t there is an optimal value of tf which satisfies, tOpt f = arg min tf ρumin (12.4) Two main conclusions can be derived from the above: 1. The minimal maneuver capability,ρumin , is a decaying function of ∆t. It makes sense because ∆t gives the Missile more time to intercept the Target from the moment it passes by the Defender (Fig. 12.11 (b)). 2. If the Missile starts the game too early; namely, causes a large tf , it would have to evade the Defender for a long time; hence, get far away from the Target. This would increase ρumin (Fig. 12.11 (a)). On the other hand, if tf is very small, the Missile has a little time to evade the Defender, resulting again in high values of ρumin . The optimal value of tf is somewhere in the middle. There is no simple algebraic solution for tOpt f ; nevertheless, the Missile can obtain it numerically, and choose the best time to start the game, unless the Target releases the Defender close to engagement, resulting tf < tOpt f . 49
- 66. Example 12.8. Linear simulations in Fig. 12.12 Illustrate this analysis. Consider the numerical values: ρu = 170 m Sec2 , ρv = 30 m Sec2 , ρw = 50 m Sec2 , ∆t = 4 [Sec] , m = 0.5 [m] , = 150 [m] From Fig. 12.11 we have that for ∆t = 4 [sec], the optimal value of tf is tOpt f ≈ 2.5 [sec]. ||yMT|| ||yMD|| ℬ t* =tf MD tf MT Time, t ℓ ||yMT cr || ℬ(t* ) ||ZEM|| (a) tf = 1.5 ||yMT|| ||yMD|| ℬ tgo * t* tf MD tf MT Time, t ℓ ||yMT cr || ℬ(t* ) ||ZEM|| (b) tf = 2.5 ||yMT|| ||yMD|| ℬ tgo * t* tf MD tf MT Time, t ℓ ||yMT cr || ℬ(t* ) ||ZEM|| (c) tf = 3.5 ||yMT|| ||yMD|| ℬ tgo * t* tf MD tf MT Time, t ||yMT cr || ||ZEM|| (d) tf = 4.5 Figure 12.12: Linear Simulations for Different Values of tf At t∗ go, the ZEM |yMT | is most far from its bound B, at tf = 2.5 [sec]. 50
- 67. 13 Optimality Analysis 13.1 Linear Kinematics Scenario 13.1.1 Constant Gain In Subsection 7.4 of Part I, the following function was defined d(·) m + 1 2 (ρu − ρv) t∗ go + ∆t 2 (13.1) − 1 2 tf + 2∆t + t∗ go t∗ go 2 (ρu − ρv) + t2 f (ρw + ρv) + 2 − 2 yMD 0 tf + t∗ go − yMT 0 Recall that (13.1) is actually a “measure of success”, as the Missile can guarantee its success if d(·) > 0. Therefore, the Missile maximizes d (·) with its controller u, and the Target-Defender team minimizes it with v and w. The optimal value for ku (Section 8 of Part I) is kOpt u = ρu. Now, rewrite d(·) for some maneuvers v(t) = kvsign(yMT ) and w(t) = −kwsign(yMD), where |kv| ≤ ρv and |kv| ≤ ρw. Eq. (13.1) becomes, d(·) m + 1 2 (ρu − ρv) t∗ go + ∆t 2 (13.2) − 1 2 tf + 2∆t + t∗ go t∗ go 2 (ρu − kv) + t2 f (kw + kv) + 2 − 2 yMD 0 tf + t∗ go − yMT 0 Differentiate (13.2) with respect to kv and obtain, ∂d(·) ∂kv = − 1 2 (tf − t∗ go)(2∆t + tf + t∗ go) (13.3) It can be seen that d(·) is a monotonically decreasing function of kv; thus, to minimize d(·), the Target must choose kOpt v = ρv. Similar derivation is true for kw, ∂d(·) ∂kw = − 1 2 t2 f tf + 2∆t + t∗ go tf + t∗ go (13.4) The function d(·) is a monotonically decreasing function of kw; hence, kOpt w = ρw (Fig. 13.1). -ρv ρv kv -ρw ρw kw Figure 13.1: Function d (kv, kw) As a result, the optimal maneuvers are v∗ (t) = ρvsign(yMT ) and w∗ (t) = −ρwsign(yMD). 51
- 68. 13.1.2 Variable Gain If the maneuvers are not constant; namely, v(t) = kv(t)sign(yMT ) and, w(t) = −kw(t)sign(yMD), same results can be obtained by analyzing the ZEM variables. For general maneuver gains, the ZEM norm derivatives ˙VMT (t) = d dt |yMT (t)|, and ˙VMD(t) = d dt |yMD(t)| , at the evasion stage, become ˙VMT (t) = (tf − t + ∆t) (ρu + kv(t)) (13.5) ˙VMD(t) = (tf − t) (ρu − kw(t)) (13.6) for some |kv(t)| ≤ ρv, and |kw(t)| ≤ ρw. Integration in parts yields, |yMT (t)| = ¨ t 0 kv(ξ)dξdξ + (tf − t + ∆t) ˆ t 0 kv(ξ)dξ + f(t) (13.7) Therefore, for t∗ (the intersection time of |yMD| with C) yMT t∗ = |ycr MT | = ¨ t∗ 0 kv(ξ)dξdξ + (tf − t∗ + ∆t) ˆ t∗ 0 kv(ξ)dξ + f t∗ (13.8) Recall that by definition, d(·) B t∗ − |ycr MT | = 1 2 (ρu − ρv) t∗ go + ∆t 2 − ¨ t∗ 0 kv(ξ)dξdξ − t∗ go + ∆t ˆ t∗ 0 kv(ξ)dξ − f t∗ (13.9) where t∗ go = tf − t∗ . Thus, in order to minimize d(·), the Target must maximize ˜ t∗ 0 kv(ξ)dξ and ´ t∗ 0 kv(ξ)dξ. According to Riemann’s definition (Fig. 13.2), ˆ t∗ 0 kv(ξ)dξ = lim N→∞ N i=1 kv(ti)dt (13.10) t1 t2 t3 ... tN Time, t kv(t) Figure 13.2: Riemann’s Series of ´ t∗ 0 kv(ξ)dξ 52
- 69. where dt = ti − ti−1 ∀i = 1, 2, . . . , N. Therefore, maximizing (13.10) means max kv(t) ˆ t∗ 0 kv(ξ)dξ = lim N→∞ N i=1 max kv(ti) {kv(ti)} dt (13.11) where −ρv ≤ kv(t) ≤ ρv. Hence; maximizing the Riemann’s integral means maximizing the function kv(t) at each time point, ti. The maximizing value for kv(t) at each time point ti is kOpt v (ti) = ρv ∀i = 1, 2, . . . , N. The same conclusion can be made for ˜ t∗ 0 kv(ξ)dξ. Consequently, the optimal value of the Target’s maneuver gain is kv(t)=ρv. As for optimality of kw(t), recall (13.9), d(·) 1 2 (ρu − ρv) t∗ go + ∆t 2 − ¨ t∗ 0 kv(ξ)dξdξ − t∗ go + ∆t ˆ t∗ 0 kv(ξ)dξ − f (t∗ ) (13.12) One can see that d(·) doesn’t depend on kw(t) directly, rather it depends on t∗ which is the intersection time of |yMD(t)| = − ¨ t 0 kw(ξ)dξdξ − tgo ˆ t 0 kw(ξ)dξ + g(t) , tgo = tf − t (13.13) with the fail-safe function C(tgo) = + 1 2 (ρu + ρw) t2 go. Thus, d(·) is not affected by the shape of the function kw(t), rather it is only affected by t∗ . However, since −ρw ≤ kw(t) ≤ ρw , the function |yMD(t)| is bounded, yMIN MD (kw = ρw) ≤ yMD kw(t) ≤ yMAX MD (kw = −ρw) (13.14) Denote t∗ MIN as the intersection of yMAX MD with C, and t∗ MAX as the intersection of yMIN MD with C. Assuming continuity, the entire range of t∗ ∈ [t∗ MIN , t∗ MAX] is reachable by a constant maneuver gain kw ∈ [−ρw, ρw]. Hence, there always exists a constant maneuver kw that yields the same intersection time t∗ ; thus, the same function d(·) (Fig. 13.3). However, from (13.4) we know that if kw(t) = kw = const. then the optimal solution is: kw = ρw. Consequently, the optimal maneuver gain of the Defender is kw = ρw. |yMD MAX | |yMD MIN | |yMD| |yMD Equivalent (kw=Const)| tMIN * t* tMAX * Time,t |yMD(t* )| |ZEM| Figure 13.3: Bounds and Different Possibilities of |yMD(t)| 53
- 70. 13.2 Optimality in the nonlinear kinematics scenario For linear kinematics, the optimal maneuvers regarding the “measure of success” d(·), are u∗ = −ρusign(yMD) |yMD| < C ρusign(yMT ) |yMD| ≥ C (13.15) v∗ = ρvsign(yMT ) (13.16) w∗ = −ρwsign(yMD) (13.17) Example 13.1. Let the Target use v(t) = kvsign(yMT ). Consider the parameters, ρu = 170 m Sec2 , ρv = 30 m Sec2 , ρw = 60 m Sec2 , m = 0 [m] , = 120 [m], yMT 0 = yMD 0 = 0 Missile Target Defender tf MD t* 0 1000 2000 3000 4000 5000 6000 0 500 1000 1500 2000 x [m] y[m] Miss MD = 215 , tf MD = 5.81 Miss MT = 0.2 , tf MT = 8.3 (a) kv = −ρv Missile Target Defender tf MD t* 0 1000 2000 3000 4000 5000 6000 0 500 1000 1500 2000 2500 x [m] y[m] Miss MD = 215 , tf MD = 5.81 Miss MT = 0.2 , tf MT = 9.24 (b) kv = −0.5ρv Missile Target Defender tf MD t* 0 1000 2000 3000 4000 5000 6000 0 1000 2000 3000 4000 x [m] y[m] Miss MD = 215 , tf MD = 5.81 Miss MT = 0.2 , tf MT = 12 (c) kv = 0.5ρv Missile Target Defender tf MD t* 0 1000 2000 3000 4000 5000 6000 0 1000 2000 3000 4000 x [m] y[m] Miss MD = 215 , tf MD = 5.81 Miss MT = 0.2 , tf MT = 13.5 (d) kv = ρv Figure 13.4: Nonlinear simulation for kv = −ρv, −0.5ρv, 0.5ρv and ρv 54
- 71. From simulation (Fig. 13.4), it is readily seen that regardless of the Target’s strategy, it gets intercepted by the Missile. Therefore, one might think that optimal strategies for linear kinematics are indeed optimal in the real (nonlinear) scenario. Generally, since the M-D game takes place at the first phase of guidance, the collision triangle between them suffers relatively small distortion (assuming players are close to collision triangle at the beginning, and evasion doesn’t take too much time), the time-to-go is close to linear, and u∗ and w∗ are arguably justified (although the actual M-D miss distance considerably bigger than required). However, by evading the Defender, the Missile also “evades” the Target (recall that u∗ e = −u∗ p), while the Target evades the Missile (applies v∗ ). Consequently, the M-T collision triangle breaks and linearization assumptions fail to hold. Example 13.2. Now, consider the same parameters, except: = 150 [m], and a slightly different geometry. Nonlinear simulations for kv = −ρv and kv = ρv are depicted in Fig. 13.5. Missile Target Defender tf MD t* 0 1000 2000 3000 4000 5000 6000 -400 -200 0 200 400 600 800 x [m] y[m] Miss MD = 325 , tf MD = 5.74 Miss MT = 380 , tf MT = 7.37 (a) kv = −ρv Missile Target Defender tf MD t* 0 1000 2000 3000 4000 5000 6000 0 1000 2000 3000 4000 x [m] y[m] Miss MD = 325 , tf MD = 5.74 Miss MT = 0.1 , tf MT = 13.37 (b) kv = ρv Figure 13.5: Nonlinear simulations for kv = −ρv and kv = ρv Clearly, the Target gets intercepted when maneuvering optimally, but manages to escape by apply- ing the opposite guidance strategy, which by our analysis is the worst for it to choose. This refutes our optimality analysis for the Target. What went wrong? In order to explain this, observe again the nonlinear simulations in Fig. 13.5. Notice, that while the final time tMT f for kv = −ρv is about 7.4 [sec], it is about 13.4 [sec] for kv = ρv. Indeed, by applying kv = −ρv the Target “pursues” the Missile. Therefore, the M-T collision triangle suffers relatively small distortion and the final time tMT f suffers small change during the game. However, by applying kv = ρv the Missile and the Target maneuver at opposite directions, resulting the collision triangle to break. As a result, the value of tMT f is dramatically different from tMT go at t = 0. Recall that tMT f = tf + ∆t; thus, loosely speaking, Target’s evasive maneuver has “increased” ∆t. This is the main idea of this analysis: the harder the Target evades the Missile, the more it “increases” ∆t. 55
- 72. Example 13.3. Approximate the nonlinear simulations of Example 13.2 with linear simulations. Consider ρu = 170 m Sec2 , ρv = 30 m Sec2 , ρw = 60 m Sec2 , m = 0 [m] , = 325 [m] , tf = 5.74 [Sec] , yMD 0 = yMT 0 = 0 For kv = −ρv we set ∆t = 1.7 [sec], while for kv = ρv we set ∆t = 7.7 [sec]. Fig. 13.6 shows the results. ||yMT|| ||yMD|| ℬ tgo * t* tf MD tf MT Time, t ℓ ||yMT cr || ℬ(t* ) ||ZEM|| (a) kv = −ρv, ∆t = 1.7 [sec] ||yMT|| ||yMD|| ℬ tgo * t* tf MD tf MT Time, t ||yMT cr || ℬ(t* ) ||ZEM|| (b) kv = ρv, ∆t = 7.7 [sec] Figure 13.6: Linear simulations for kv = −ρv and kv = ρv Linear Simulations in Fig. 13.6 justify the results of the nonlinear simulations in Example 13.2. By using kv = −ρv, the Target minimizes |yMT | instead of maximizing it (which appears to be optimal); however, ∆t remains almost unchanged. As a result, small ∆t enables it to evade the Missile despite the opposite maneuver. On the other hand, by applying kv = ρv, the Target maximizes |yMT |; however, it also adds about 6 [sec] to ∆t. As a result the Target “increases” the bound B t∗ go by ∆B t∗ go = te (ρu − ρv) te 2 + t∗ go (13.18) where te is the addition to ∆t (in this example te = 6 [sec]). Consequently, the Target has let the Missile to intercept it, despite the maximization of |yMT | which appears to be optimal. To clarify even more, Fig. 13.7 presents the results of Example 13.3 on the same plot. 56
- 73. |yMT(kv=-ρv)| |yMT(kv=ρv)| ℬ(kv=-ρv) ℬ(kv=ρv) Δℬ(t* )Δ|yMT cr | t* tf MT tf MT Time, t |yMT cr | ℬ(t* ) |yMT cr | ℬ(t* ) |ZEM| Figure 13.7: Results of Fig. 13.6, presented on the same plot Clearly, ∆ |ycr MT | is smaller than ∆B(t∗ go). Hence, by performing an evasive maneuver, the Target has lost in general more than it gained from maximizing its ZEM. Remark 13.1. In order to intercept the Target in such a scenario (kv = −ρv), the Missile must have more maneuvering capability, or alternatively, the required M-D miss distance, has to be reduced. Fig. 13.8 demonstrates the idea. Missile Target Defender tf MD t* 0 1000 2000 3000 4000 5000 6000 -600 -400 -200 0 200 400 600 800 x [m] y[m] Miss MD = 241.8 , tf MD = 5.84 Miss MT = 0.4 , tf MT = 7.47 (a) ρu = 220 Missile Target Defender tf MD t* 0 1000 2000 3000 4000 5000 6000 -200 0 200 400 600 800 x [m] y[m] Miss MD = 28.5 , tf MD = 5.63 Miss MT = 0.3 , tf MT = 7.13 (b) ρu = 170 Figure 13.8: Nonlinear Simulations 57
- 74. 13.3 Intermediate conclusions Rewrite the inequality derived in Section 8 of Part I for some Target’s maneuver v = kvsign(yMT ), where −ρv ≤ kv ≤ ρv. − (kv + ρw) t3 f − 2∆t (kv + ρw) t2 f + (ρu − kv) ∆t2 + 2 m − + yMD 0 − yMT 0 tf + 4∆t yMD 0 − (kv + ρw) t2 f − 2∆t (ρu − kv) tf − (ρu − kv) ∆t2 − 2 (m − + |yMD 0 | − |yMT 0 |) ≤ t2 f (ρu − ρw) − 2 + 2 |yMD 0 | 2ρu (13.19) 1. If (13.19) holds for3 kv = ρv, the Missile can guarantee its success for any Target’s maneuver. Namely, even if ∆t doesn’t suffer an increase due to the non-linearity caused by Target’s evasive maneuver, the Missile is still able to intercept it. Moreover, if ∆t grows, or the Target applies a suboptimal maneuver gain, kv < ρv, it is even easier for the Missile to intercept it. Such a case is described in Example 13.1, where the Target is being intercepted regardless of its maneuver. 2. If (13.19) does not hold for4 kv = −ρv, the Target can evade the Missile using any kv ∈ [−ρv, ρv] if ∆t remains constant. However, we know that ∆t remains approximately constant only if kv = −ρv (again, assuming evasion doesn’t take too much time). Therefore, the Target can guarantee its safety by performing an opposite maneuver towards the Missile; namely, by applying kv = −ρv. It is important to understand that even in this case, kv = −ρv is not the optimal5 maneuver. However, this maneuver guarantees Target’s evasion, while other strategies have the chance to increase ∆t and enable interception. We can observe this case in Example 13.2. 3. If (13.19) does not hold for kv = ρv but holds for kv = −ρv, a further analysis (provided in Subsection 13.4) is required. In this case, the Target cannot apply neither kv = −ρv nor kv = ρv, because kv = −ρv leads to capture (as (13.19) holds), and kv = ρv makes ∆t grow and again, (usually) leads to capture. This case is called: The Uncertainty Area. 3 This statement implies that (13.19) also holds for any other kv ∈ [−ρv, ρv] 4 This statement implies that (13.19) doesn’t hold for any kv ∈ [−ρv, ρv] 5 A maneuver which maximizes the M-T miss distance. 58
- 75. 13.4 The Uncertainty Area Analysis 13.4.1 The M-T bound function revised As we know, the function B(tgo) = m + 1 2 (ρu − ρv) (tgo + ∆t)2 (13.20) describes the bound of the M-T singular area. However, we also know, that it is not always wise for the Target to use its maximal evasive maneuver; thus, let us modify (13.20). Consider a Target maneuvering with v = kvsign(yMT ), where −ρv ≤ kv ≤ ρv. In such a case (13.20) becomes, B(tgo) = m + 1 2 (ρu − kv) (tgo + ∆t)2 (13.21) In order to account for the non-linearity of time-to-go, define the M-T pseudo-singular area, Bv(tgo) = m + 1 2 (ρu − kv) (tgo + ∆t + te(kv))2 (13.22) where te(kv) is an approximated addition factor to ∆t resulted by the Target’s evasive maneuver. Although te(kv) cannot be determined analytically, as it would require knowing all players’ strate- gies during the entire game period, one can approximate it from simulations. Note that te has to be a monotonically increasing function of kv, since the bigger kv is, the bigger is the addition to ∆t. The function Bv(tgo) defines the M-T pseudo-singular area; namely, an area in which the Missile’s strategy is arbitrary, and the M-T miss distance is smaller than m, if the Target uses v = kvsign(yMT ). Example 13.4. Consider the following numerical values ρu = 170 m Sec2 , ρv = 30 m Sec2 , ρw = 60 m Sec2 , m = 0 [m] , = 30 [m] , tf = 5.5 [Sec] , yMD 0 = yMT 0 = 0 Linear simulations for kv = ρv and kv = −ρv are depicted in Fig. 13.9. In these simulations ∆t = 1.5 [sec]; however, for kv = ρv we set te = 4 [sec], and for kv = −ρv we set te = 0 [sec] (These parameters approximate nonlinear simulations which are discussed later). ||yMT|| ||yMD|| ℬv tgo * t* tf MD tf MT Time, t ||yMT cr || ||ZEM|| (a) kv = −ρv, te = 0 [sec] ||yMT|| ||yMD|| ℬv tgo * t* tf MD tf MT Time, t ||yMT cr || ||ZEM|| (b) kv = ρv, te = 4 [sec] Figure 13.9: Linear simulations for kv = −ρv and kv = ρv 59
- 76. For kv = ρv we have a normal singular area (with an addition of te) Bv(tgo) = 1 2 (ρu − ρv) (tgo + ∆t + te(ρv))2 (13.23) However, for kv = −ρv we have Bv(tgo) = 1 2 (ρu + ρv) (tgo + ∆t + te(−ρv))2 (13.24) Although Bv does not define the actual singular area (unless kv = ρv), |yMT | behaves inside it just like inside a singular area. Thus, if at the critical time tgo = t∗ go the ZEM yMT (t∗ go) > Bv(t∗ go), then the Target can guarantee a miss distance greater than m. It is wise for the Target to look for such a maneuver gain kv, that ensures yMT (t∗ go) > Bv(t∗ go). In this example, for both kv = ρv and kv = −ρv the Target gets intercepted by the Missile. This is exactly the uncertainty case when (13.19) does not hold for kv = ρv but holds for kv = −ρv. Let us try to find such a maneuver gain kv that keeps the Target safe. 13.4.2 Function d(·) Revised Let the Target apply some maneuver gain kv ∈ [−ρv, ρv]. Hence, (7.35) becomes |ycr MT | = 1 2 tf + 2∆t + t∗ go t∗ go 2 (ρu − kv) + t2 f (ρw + kv) + 2 − 2 yMD 0 tf + t∗ go + yMT 0 (13.25) In order to account for the addition to ∆t, rewrite (13.25) as follows ycr MTv = 1 2 tf + 2 (∆t + te(kv)) + t∗ go t∗ go 2 (ρu − kv) + t2 f (ρw + kv) + 2 − 2 yMD 0 tf + t∗ go + yMT 0 (13.26) Consider a revised “measure of success” function. Define dv(·) Bv(t∗ go) − ycr MTv = m + 1 2 (ρu − kv) t∗ go + ∆t + te(kv) 2 (13.27) − 1 2 tf + 2 (∆t + te(kv)) + t∗ go t∗ go 2 (ρu − kv) + t2 f (ρw + kv) + 2 − 2 yMD 0 tf + t∗ go − yMT 0 This function has the same meaning as d(·), only now it accounts for the addition to ∆t. Hence, if te(kv) can be approximated, then dv(·) can be minimized by kv and the optimal solution for the Target can be found. Example 13.5. Consider the same numerical values as in Example 13.4. In that example we had, yMD 0 = yMT 0 = m = 0; thus, (13.27) reduces to dv(·) = 1 2 (ρu − kv) t∗ go + ∆t + te(kv) 2 (13.28) − 1 2 tf + 2 (∆t + te(kv)) + t∗ go t∗ go 2 (ρu − kv) + t2 f (ρw + kv) + 2 tf + t∗ go 60
- 77. Now, let us find the approximation factor. One thing about te(kv) is known for certain: te(kv = −ρv) ≈ 0 (13.29) This is true because when the Target maneuvers towards the Missile, the M-T collision trian- gle suffers relatively small distortion and the time-to-go acts close to linear. By running many simulations with similar numerical values, we obtain te(kv = 0) ≈ 1 (13.30) te(kv = −ρv) ≈ tf (13.31) This approximation is very rough in general, but it is pretty accurate in a certain range of values. Hence, we can try a parabolic approximation: te(kv) = tf − 2 2ρ2 v k2 v + tf 2ρv kv + 1 (13.32) Substitute (13.32) into (13.28) and obtain dv(·) = 1 2 (ρu − kv) t∗ go + ∆t + tf − 2 2ρ2 v k2 v + tf 2ρv kv + 1 2 (13.33) − 1 2 tf + 2 ∆t + tf −2 2ρ2 v k2 v + tf 2ρv kv + 1 + t∗ go t∗ go 2 (ρu − kv) + t2 f (ρw + kv) + 2 tf + t∗ go A plot of dv(t∗ go, kv) is depicted in Fig. 13.10. 0 - ρv ρv kv 0 tgomax tgo * 0 dv Figure 13.10: Function dv(t∗ go, kv) Assuming the Missile maximizes dv(·) with kOpt u = ρu, we have t∗ go = tgomax . In such a case, the function dv(kv) is shown in Fig. 13.11. 61
- 78. -ρv ρv kv dv Figure 13.11: Function dv (kv) It is readily seen that in order to minimize dv(·) it is best for the Target to apply a small maneuver, or approximately not to maneuver at all; namely, apply kv = 0. Also, for both kv = ρv and kv = −ρv we have dv(·) > 0 resulting Bv(t∗ go) > yMT (t∗ go) , meaning the Target gets intercepted. Indeed, our linear simulations in Example 13.4 approve these results. Example 13.6. Consider the same numerical values as in Example 13.4. Now, examine the linear simulation for kv = 0, resulting te = 1 [sec] (Fig. 13.12). ||yMT|| ||yMD|| ℬv tgo * t* tf MD tf MT Time, t ||yMT cr || ||ZEM|| Figure 13.12: Linear simulation for kv = 0, te = 1 [sec] Clearly, yMT t = tMT f > m = 0; hence, the Target evades the Missile. Example 13.7. Consider the following numerical values ρu = 170 m Sec2 , ρv = 30 m Sec2 , ρw = 60 m Sec2 , m = 0 [m] , = 10 [m] In addition, all players are on appropriate collision courses; hence, yMD 0 = yMT 0 = 0. Fig. 13.13 depicts nonlinear simulations for kv = ρv and kv = −ρv, while Fig. 13.14 shows a nonlinear simulation for kv = 0. In these simulations ∆t = 5.5 [sec] for kv ≈ ρv, ∆t ≈ 1.5 [sec] for kv = −ρv, 62
- 79. and ∆t ≈ 2.5 [sec] for kv = 0. Linear simulations in Example 13.4, approximate the nonlinear simulations presented in Fig. 13.13, while Example 13.6 approximates the nonlinear simulation in Fig. 13.14. Missile Target Defender tf MD t* 0 1000 2000 3000 4000 5000 6000 -200 0 200 400 600 800 x [m] y[m] Miss MD = 28.5 , tf MD = 5.63 Miss MT = 0.3 , tf MT = 7.13 (a) kv = −ρv Missile Target Defender tf MD t* 0 1000 2000 3000 4000 5000 6000 0 500 1000 1500 2000 2500 3000 x [m] y[m] Miss MD = 28.5 , tf MD = 5.63 Miss MT = 0.1 , tf MT = 11.54 (b) kv = ρv Figure 13.13: Nonlinear simulations for kv = −ρv and kv = ρv Missile Target Defender tf MD t* 0 1000 2000 3000 4000 5000 6000 0 500 1000 1500 x [m] y[m] Miss MD = 28.5 , tf MD = 5.63 Miss MT = 461 , tf MT = 8.07 Figure 13.14: Nonlinear simulation for kv = 0 These results justify our analysis, as for both kv = −ρv and kv = ρv the Target gets intercepted, while for kv = 0 it escapes. 63
- 80. 14 Conclusions In the current part, a parametric analysis was performed. From that analysis we conclude that the influence of most parameters on the Missile’s minimal maneuvering capability, ρumin , is very rational and not surprising. However, the influence of tf suggests a presence of an optimal final time, tOpt f , which minimizes ρumin . This parameter can be numerically evaluated, and serve as a tactical consideration for both, the Missile, and the Target. The present part also provides an optimality analysis with respect to d(·), which is a robust- ness measure (or “success measure”). Although equations (13.15), (13.16), and (13.17) perfectly describe the players’ optimal strategies in the linear kinematics scenario, the situation is much more complicated when the kinematics become nonlinear. Since the M-D collision triangle suf- fers relatively small distortion (assuming the players start close to the M-D collision triangle, and evasion doesn’t take much time), the Missile’s and the Defender’s optimal strategies, described by (13.15) and (13.17), are arguably justified for the nonlinear kinematics. However, by applying an evasive maneuver, the Target also “increases ∆t”, which is assumed to be constant by linearization. As a result, the Missile obtains more time to intercept it, and the advantage from the extra inter- cept time can overcome the disadvantage from the Target’s evasive maneuver. Hence, the optimal Target’s maneuver, described by (13.16), is not necessarily optimal in a real, nonlinear, conflict. By performing the analysis described earlier in this part, the following conclusions can be made. 1. If (13.19) holds for kv = ρv (also holds for any other kv ∈ [−ρv, ρv]), the Missile can guarantee its success for any Target’s maneuver. Namely, even if ∆t doesn’t suffer an increase due to the non-linearity caused by Target’s evasive maneuver, the Missile is still able to intercept it. Moreover, if ∆t grows, or the Target applies a suboptimal maneuver gain, kv < ρv, it is even easier for the Missile to intercept it. Such a case is described in Example 13.1, where the Target is being intercepted regardless of its maneuver. 2. If (13.19) does not hold for kv = −ρv (doesn’t hold for any kv ∈ [−ρv, ρv]), the Target can evade the Missile using any kv ∈ [−ρv, ρv] if ∆t remains constant. However, we know that ∆t remains approximately constant only if kv = −ρv (again, assuming evasion doesn’t take too much time). Therefore, the Target can guarantee its safety by performing an opposite maneuver towards the Missile; namely, by applying kv = −ρv. It is important to understand that even in this case, kv = −ρv is not the optimal maneuver. However, this maneuver guarantees Target’s evasion, while other strategies have the chance to increase ∆t and enable interception. We can observe this case in Example 13.2. 3. If (13.19) does not hold for kv = ρv but holds for kv = −ρv, the approximated addition factor, te(kv), must be obtained. Once te(kv) is found, the new robustness measure, dv(·), serves as the optimization criterion instead of d(·). Of course te(kv) is only a rough approximation and can be obtained only from simulations; therefore, the uncertainty in this case is very significant. This case is shown in Example 13.7. A very important conclusion we can make from the above is that LMG cannot provide accurate results in a nonlinear conflict. This is because LMG relies on linearization, which fails to hold in a conflict of three players. In addition, the improvements designed to account for the nonlinear nature of the game rely on rough approximations and are very hard for implementation. This leads us to the need of developing a new method, one that does not rely on linearization and provides accurate results in a real, nonlinear conflict. Such approach is discussed in Part III. 64
- 81. Part III Vector Guidance Approach 15 Preface As was mentioned in Part II, the LOS Linearized Model Guidance (LMG) provides reliable results only in the linear kinematics scenario. Otherwise, the optimal strategies lack reliability, and in some cases, can be considered as unacceptable. Although approximations, described in Subsection 13.4 of Part II can improve the situation, we should keep in mind that these approximations are not 100% reliable, and they do not resolve the root of the problem: LOS linearization is not acceptable in a conflict of three players. Therefore, a new approach is to be used, one that enables the players to account for the actual game kinematics when considering a guidance strategy. Such an approach is the Vector Guidance (VG), discussed in this part. 16 A game of players controlling their acceleration vectors Consider the following initial interception geometry as depicted in Fig. 16.1 u VM rM v VT rT w VD rD rMT rMD rTD Figure 16.1: Planar Interception Geometry In this exo-atmospheric scenario, the Missile (M), the Target (T) and the Defender (D) can apply a bounded acceleration of u, v and w respectively in any direction of their choice. Define rM , rT , rD and VM , VT , VD as the positions and the velocities of the Missile, the Target and the Defender 65
- 82. respectively. Also define the relative position and velocity vectors. rMT = rT − rM (16.1) rMD = rD − rM (16.2) VMT = VT − VM (16.3) VMD = VD − VM (16.4) Assuming ideal players, we have the following kinematic equations ˙rMT (t) = VMT (t) (16.5) ˙rMD(t) = VMD(t) (16.6) ˙VMT (t) = v(t) − u(t) (16.7) ˙VMD(t) = w(t) − u(t) (16.8) Therefore, the state space realization in terms of ˙x(t) = Ax(t) + bu(t) + cv(t) + dw(t) (16.9) becomes, ˙rMT (t) ˙VMT (t) ˙rMD(t) ˙VMD(t) = 0 In 0 0 0 0 0 0 0 0 0 In 0 0 0 0 rMT (t) VMT (t) rMD(t) VMD(t) + 0 −In 0 −In u(t)+ 0 In 0 0 v(t)+ 0 0 0 In w(t) (16.10) where the state vector is x(t) = rMT (t) VMT (t) rMD(t) VMD(t) and n = 2 or 3 is the number of the dimensions defined for the problem (generally, n = 3). Assume limited magnitude controllers, u ≤ ρu (16.11) v ≤ ρv (16.12) w ≤ ρw (16.13) Also define two terminal cost functions, with fixed final times JMT = rMT tMT f = In 0 0 0 x tMT f = gx tMT f (16.14) JMD = rMD tMD f = 0 0 In 0 x tMD f = hx tMD f (16.15) and two ZEM variables, yMT (t) = gΦ tMT f , t x(t) (16.16) yMD(t) = hΨ tMD f , t x(t) (16.17) 66
- 83. where the transition matrices satisfy, ˙Φ tMT f , t = −Φ tMT f , t A, Φ tMT f , tMT f = I (16.18) ˙Ψ tMD f , t = −Ψ tMD f , t A, Ψ tMD f , tMD f = I (16.19) Similarly to (3.17) and (3.18), differentiating the ZEM variables yields, ˙yMT (t) = XMT tMT f , t u(t) + YMT tMT f , t v(t) + ZMT tMT f , t w(t) (16.20) ˙yMD(t) = XMD tMD f , t u(t) + YMD tMD f , t v(t) + ZMD tMD f , t w(t) (16.21) where, XMT tMT go = gΦ tMT go b (16.22) YMT tMT go = gΦ tMT go c (16.23) ZMT tMT go = gΦ tMT go d = 0 (16.24) XMD tMD go = hΨ tMD go b (16.25) YMD tMD go = hΨ tMD go c = 0 (16.26) ZMD tMD go = hΨ tMD go d (16.27) and the time-to-go variables defined as tMD go = tMD f − t (16.28) tMT go = tMT f − t (16.29) It is important to note that in later sections the final times will no longer be fixed. If all players are ideal, then XMD tMD go = −tMD go In YMD tMD go = 0 ZMD tMD go = tMD go In XMT tMT go = −tMT go In YMT tMT go = tMT go In ZMT tMT go = 0 (16.30) and the explicit form of the ZEM variables is yMT (t) = rMT (t) + tMT go VMT (t) (16.31) yMD(t) = rMD(t) + tMD go VMD(t) (16.32) Also, the M-T and M-D ZEM projected state space realizations become ˙yMT (t) = tMT go (−u(t) + v(t)) (16.33) JMT = yMT tMT f (16.34) and ˙yMD(t) = tMD go (−u(t) + w(t)) (16.35) JMD = yMD tMD f (16.36) 67
- 84. 17 A Differential Game of Two Players 17.1 General Differential Game At this point, it makes sense to analyze a differential game of two hypothetical players. Given 2 players: the pursuer (P) and the evader (E). The pursuer can apply a bounded acceleration of aP ≤ amax P , and the evader can apply a bounded acceleration of aE ≤ amax E . The ZEM projected state space model is, ˙yPE = XPE tPE f , t aP + YPE tPE f , t aE (17.1) JPE = yPE tPE f (17.2) Define the ZEM norm, VPE = yPE (17.3) Differentiating (17.3) with respect to t yields, ˙VPE = yPE yPE XPE tPE f , t aP + YPE tPE f , t aE (17.4) Assuming ˙VPE(t) is Riemann integrable6 , the optimal controllers are, a∗ P = −amax P XPEyPE XPEyPE (17.5) a∗ E = amax E YPEyPE YPEyPE (17.6) Substitute into (17.4), and obtain ˙V∗ PE = yPE yPE (XPEa∗ P + YPEa∗ E) = yPE yPE −amax P XPE XPEyPE XPEyPE + amax E YPE YPEyPE YPEyPE = −amax P XPEyPE XPEyPE XPEyPE · yPE + amax E YPEyPE YPEyPE YPEyPE · yPE = −amax P XPEyPE 2 XPEyPE · yPE + amax E YPEyPE 2 YPEyPE · yPE = −amax P XPEyPE yPE + amax E YPEyPE yPE = −amax P XPE ˆyPE + amax E YPE ˆyPE (17.7) where ˆyPE = yP E yP E is a unit vector. 6 The integrability of ˙VP E(t) is in general complicated issue and needs further research 68
- 85. 17.2 Simple Differential Game For ideal players we have, XPE = −tPE go In (17.8) YPE = tPE go In (17.9) The optimal guidance laws, (17.5) and (17.6) reduce to, a∗ P = amax P yPE yPE (17.10) a∗ E = amax E yPE yPE (17.11) where the ZEM variable is yPE = rPE + tPE go VPE, so that, rPE and VPE are the relative distance and velocity respectively, and tPE go is the appropriate time-to-go. Also, (17.7) reduces to, ˙V∗ PE = −amax P −tPE go InyPE yPE + amax E tPE go InyPE yPE = tPE go −amax p + amax E (17.12) Integrating (17.12) yields the optimal trajectories in the ZEM plane, yPE tPE go = JPE + 1 2 (amax P − amax E ) tPE go 2 (17.13) where, yPE tPE go = rPE + tPE go VPE (17.14) The controllers {aP , aE} are optimal with respect to the saddle point inequality JPE a∗ P , aE ≤ JPE a∗ P , a∗ E ≤ JPE aP , a∗ E (17.15) Fig. 17.1 describes the optimal P-E ZEM Trajectories tgo PE ||yPE|| tgo PE ||yPE|| Figure 17.1: Optimal ZEM P-E Trajectories for amax P > amax E (left) and amax P < amax E (right) Clearly, (17.13) is analogous to LMG discussed in Part I. The difference, is that the ZEM variable is now a state of the real kinematics instead of the LOS linearized one. 69
- 86. 18 Vector Guidance Based On 1st Order Time-to-go (VG1) It is important to note that while Part I discusses a three player conflict, it is based on linearized kinematics in which, along the LOS, the range is r = Vc (tf − t), with a constant closing speed Vc. In such a linearization, the dynamics and the controls take place perpendicular to LOS. This is a one dimensional motion. In the present part, the dynamics and controls are allowed to be in any direction of 3D space; thus, the results of Part I are repeated in the following section for 3D space. A reader, who is well familiar with Part I may skip this analysis, review the VG1 simulations in Section 20, and go directly to VG4 analysis in Section 21. In Section 17 we have derived the ZEM variable for a two player differential game, as well as the optimal trajectories in the ZEM plane. Although many properties of the vector based guidance seem to be analogous to the linear guidance, we must keep in mind that vector guidance model is not based on linearization; therefore, the nature of the derived variables can differ from LMG. This happens because the time-to-go variable, which is well defined as tgo = r Vc in the linearized model, can be defined in many ways in the vector based guidance model. Different definitions of the time-to-go imply different nature of the game variables and properties; consequently, it is critical to explore this topic. This section explores the properties of 1st order time-to-go, defined by tPE go = rPE VPE (18.1) 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 unless they chatter. 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 (18.1) may cause the guidance law to enter sliding mode (chattering) at some point. This will be lifted with VG4 (Section 21) Since these items match the assumptions of the linearized model, the nature of the game variables and their properties are analogous to those of LMG. It is important to understand that VG1 differs from the analysis provided in Part I only by the game kinematics and ZEM definition; namely, unlike Part I which discusses a 1D kinematics perpendicular to LOS, the present paper accounts for the entire 3D kinematics. However, at this point, it is assumed that linearization assumptions still hold; thus, time-to-go is approximately r Vc , and tf is approximately constant. Obviously, these assumptions have small chance to hold over the entire game period; therefore, VG1 • Provides a connection between the linearized kinematics of Part I, and the full kinematics, VG4, provided later in the present work. • Warns the designer from using VG1, which is a common practice in Missile guidance, as VG4 provides better results, due to the presence of acceleration (thrust). Remark 18.1. When yPE = 0, the players are on a collision course, so that rPE and VPE are collinear. 70
- 87. Proof. By definition, when yPE = 0, we have yPE = rPE + tPE go VPE = rPE + rPE VPE VPE = 0 Thus VPE = − VPE rPE rPE and rPE rPE = − VPE VPE . (18.2) Remark 18.2. When both players play optimal, the ZEM trajectory is parallel to the optimal one. Remark 18.3. When amax p > amax E , there is a singular area in the ZEM plane, in which the optimal strategies are arbitrary, and the cost (miss distance) is zero. The advantage of the 1st order time-to-go is that it has the properties of the linearized model, and the further analysis is very similar to Part I. The disadvantage is that the actual Missile-Target game is far from being close to a collision course; therefore, this definition of the time-to-go is not realistic. In the following sections we will explore a different type of time-to-go, which resolves the problems of VG1. 71
- 88. 19 Optimal Strategies for VG1 19.1 Basic Optimal Strategies In our game there are 3 players; hence, two ZEM variables: yMD and yMT . Similarly to Part I, the Target maximizes yMT ; therefore, its optimal guidance law is v∗ = ρv YMT yMT YMT yMT = ρv yMT yMT = ρv rMT + tMT go VMT rMT + tMT go VMT (19.1) The Defender minimizes yMD , thus w∗ = −ρw ZMDyMD ZMDyMD = −ρw yMD yMD = −ρw rMD + tMD go VMD rMD + tMD go VMD (19.2) The Missile has two objectives: The optimal evasion law that maximizes yMD is, u∗ e = ρu XMDyMD XMDyMD = −ρu yMD yMD = −ρu rMD + tMD go VMD rMD + tMD go VMD (19.3) and the optimal pursuit law that minimizes yMT is, u∗ p = −ρu XMT yMT XMT yMT = ρu yMT yMT = ρu rMT + tMT go VMT rMT + tMT go VMT (19.4) Following Section 5 in Part I, we obtain the trajectories (depicted in Fig. 19.1) for the Missile- Defender game. y∗ MD tMD go = JMD − 1 2 (ρu − ρw) tMD go 2 (19.5) tgo MD ||yMD|| Figure 19.1: Optimal Missile-Defender ZEM Trajectories 72
- 89. while the trajectories in Fig. 19.2 depict the Missile-Target game. y∗ MT tMT go = JMT + 1 2 (ρu − ρv) tMT go 2 (19.6) tgo MT ||yMT|| Figure 19.2: Optimal Missile-Target ZEM Trajectories As in Part I, define as the desired M-D miss distance and m as the desired M-T miss distance and obtain the bound functions (Fig. 19.3) A tMD go = − 1 2 (ρu − ρw) tMD go 2 (19.7) B tMT go = m + 1 2 (ρu − ρv) tMT go 2 (19.8) (t) ℬ(t) ℓ m tf MD tf MT Time, t ||ZEM|| Figure 19.3: Bound Functions A and B 73
- 90. The functions described by (19.7) and (19.8) are the bounds, in which yMT and yMD are to be if the Missile wants to guarantee the desired M-T and M-D miss distances. Unlike the linearized model discussed in Part I, where only 2 cases were possible: sign(yMD) = sign(yMT ) and sign(yMD) = −sign(yMT ), here, in the vector model, there are infinitely many possibilities. However, the worst case is still when yMD yMD = yMT yMT (19.9) In such a case we have, u∗ e = −u∗ p (19.10) Hence, by pursuing the Target, the Missile makes the worst possible maneuver regarding the Defender, and by evading the Defender, it performs the worst possible maneuver regarding the Target. Note, that it is only the worst case and cannot predict the actual terminal cost; however, a guaranteed cost guidance law can be created. 19.2 Fail-safe Function: C tMD go Similarly to Part I, derive the fail-safe function for the M-D game. Let the Missile pursue the Target using its optimal strategy, u∗ p. In such a case one obtains ˙V∗∗ MD(t) = yMD yMD −ρuXMD XMT yMT XMT yMT − ρwZMD ZMDyMD ZMDyMD = yMD yMD ρutMD go −tMT go InyMT −tMT go InyMT − ρwtMD go tMD go InyMD tMD go InyMD (19.11) = yMD yMD −ρutMD go yMT yMT − ρwtMD go yMD yMD Recall that in worst case yMD yMD = yMT yMT . Therefore, (19.11) becomes, ˙V∗∗ MD(t) = yMD yMD −ρutMD go yMD yMD − ρwtMD go yMD yMD = −tMD go (ρu + ρw) = −tMD f (ρu + ρw) + t (ρu + ρw) (19.12) Integration yields, y∗∗ MD(t) = y∗∗ MD(t = 0) − ˆ t 0 tMD f (ρu + ρw) dξ + ˆ t 0 ξ (ρu + ρw) dξ = y∗∗ MD(t = 0) − tMD f (ρu + ρw) ξ|t 0 + 1 2 (ρu + ρw) ξ2 t 0 (19.13) = y∗∗ MD(t = 0) − tMD f t (ρu + ρw) + 1 2 (ρu + ρw) t2 Define y∗∗ MD t = tMD f = (19.14) Substitute (19.14) into (19.13), and obtain y∗∗ MD t = tMD f = = y∗∗ MD(t = 0) − 1 2 (ρu + ρw) tMD f 2 (19.15) 74
- 91. thus y∗∗ MD(t = 0) = + 1 2 (ρu + ρw) tMD f 2 (19.16) and y∗∗ MD(t) = + 1 2 (ρu + ρw) tMD f 2 − tMD f t (ρu + ρw) + 1 2 (ρu + ρw) t2 = + 1 2 (ρu + ρw) tMD f − t 2 (19.17) This yields the final form, y∗∗ MD tMD go = + 1 2 (ρu + ρw) tMD go 2 (19.18) This function implies that yMD , which decreases due to Defender and Missile strategies, cannot decrease more rapidly than y∗∗ MD(t) . Hence, we choose: y∗∗ MD t = tMD f = , so that even in the worst case, yMD tMD f will not fall below . This function is defined as the fail-safe: C tMD go = + 1 2 (ρu + ρw) tMD go 2 (19.19) Unlike Part I, where yMD tMD go is parallel to C tMD go if sign(yMD) = sign(yMT ) when the Missile and the Defender apply u∗ p and w∗ respectively; here, yMD tMD go is parallel to C tMD go only if yMD yMD = yMT yMT . Otherwise yMD tMD go decreases less rapidly and eventually the M-D miss dis- tance is greater than . The function C(t), along with A(t) and B(t), is depicted in Fig. 19.4. (t) ℬ(t) (t) ℓ m tf MD tf MT Time, t ||ZEM|| Figure 19.4: fail-safe Function C in addition toA and B 75
- 92. 19.3 Various Evasion Strategies The Missile can use various evasion strategies to reach the fail-safe function C tMD go . Consider the Missile using an evasive strategy, u = −ku yMD yMD , 0 ≤ ku ≤ ρu (19.20) while the Target and the Defender apply v∗ and w∗ respectively. The M-T ZEM norm derivative becomes, ˙VMT (t) = yMT yMT kutMT go yMD yMD + ρvtMT go yMT yMT (19.21) Recall that in worst case ˆyMD = ˆyMT . Thus, (19.21) becomes ˙VMT (t) = yMT yMT kutMT go yMT yMT + ρvtMT go yMT yMT = tMT go (ku + ρv) (19.22) Integration gives, yMT (t) = yMT (t = 0) + ˆ t 0 tMT f (ku + ρv) dξ − ˆ t 0 ξ (ku + ρv) dξ = yMT (t = 0) + tMT f (ku + ρv) ξ|t 0 − 1 2 (ku + ρv) ξ2 t 0 = yMT 0 + tMT f t (ku + ρv) − 1 2 (ku + ρv) t2 (19.23) = yMT 0 + 1 2 (ku + ρv) tMT f 2 − 1 2 (ku + ρv) tMT f 2 + tMT f t (ku + ρv) − 1 2 (u + ρv) t2 = yMT 0 + 1 2 (ku + ρv) tMT f 2 − 1 2 (ku + ρv) tMT f − t 2 Substituting tMT go = tMT f − t yields the final form, yMT tMT go = yMT 0 + 1 2 (ku + ρv) tMT f 2 − 1 2 (ku + ρv) tMT go 2 (19.24) Similarly, the M-D ZEM derivative is, ˙VMD(t) = yMD yMD kutMD go yMD yMD − ρwtMD go yMD yMD = tMD go (ku − ρw) (19.25) Consequently, integration yields yMD tMD go = yMD 0 + 1 2 (ku − ρw) tMD f 2 − 1 2 (ku − ρw) tMD go 2 (19.26) Define new variables: tMD go = tgo (19.27) tMD f = tf (19.28) tMT go = tgo + ∆t (19.29) tMT f = tf + ∆t (19.30) 76
- 93. 19.4 Algebraic Conditions Equations (19.24), (19.26), and (19.19) become, yMD(tgo) = yMD 0 + 1 2 (ku − ρw) t2 f − 1 2 (ku − ρw) t2 go (19.31) yMT (tgo) = yMT 0 + 1 2 (ku + ρv) (tf + ∆t)2 − 1 2 (ku + ρv) (tgo + ∆t)2 (19.32) C(tgo) = + 1 2 (ρu + ρw) t2 go (19.33) These equations are of the same form as (7.28), (7.26), and (7.29) in Part I. Therefore, by equating (19.31) and (19.33) we obtain the same intersection time as (7.30) in Part I, t∗ go(ku) = t2 f (ku − ρw) − 2 + 2 yMD 0 ku + ρu (19.34) and the same maneuver gain that causes this intersection time, ku t∗ go = 2 + t2 goρu − 2 yMD 0 + t2 f ρw t2 f − t2 go (19.35) Also, we have the same essential evasion condition as (7.32), ku ≥ ρw + 2 − yMD 0 t2 f (19.36) and the critical M-T ZEM ycr MT t∗ go also has the same form as (7.35). ycr MT t∗ go = 1 2 tf + 2∆t + t∗ go t∗ go 2 (ρu − ρv) + t2 f (ρw + ρv) + 2 − 2 yMD 0 tf + t∗ go + yMT 0 (19.37) Consequently, similarly to the linear model guidance (LMG) discussed in Part I, the Missile can evade the Defender and capture the Target if − (ρv + ρw) t3 f − 2∆t (ρv + ρw) t2 f + (ρu − ρv) ∆t2 + 2 m − + yMD 0 − yMT 0 tf + 4∆t yMD 0 − (ρv + ρw) t2 f − 2∆t (ρu − ρv) tf − (ρu − ρv) ∆t2 − 2 (m − + yMD 0 − yMT 0 ) (19.38) ≤ t2 f (ρu − ρw) − 2 + 2 yMD 0 2ρu 77
- 94. 20 VG1 Simulations One must understand that tMT f , and therefore ∆t, can change due to the player’s strategies. As a result, (19.38) provides only a sufficient condition regarding the initial value of ∆t. Thus, if (19.38) holds for the initial ∆t, then the Missile is able to intercept the Target in the worst case when ∆t remains unchanged. However, if (19.38) does not hold, the Missile might still be able to intercept the Target with a different tMT f . This analysis is similar to the one made for the linear model; however, the big difference is that all players can now account for the true nature of game kinematics, and therefore, adopt the optimal strategy regarding the actual ZEM. Example 20.1. Consider the following parameters, ρu = 170 m sec2 , ρv = 30 m sec2 , ρw = 60 m sec2 m = 0.5 [m] , = 150 [m] The M-T-D trajectories are depicted in Fig. 20.1. Missile Target Defender tf MD t* 0 1000 2000 3000 4000 5000 6000 0 500 1000 1500 2000 2500 x [m] y[m] Miss MD = 154.5 , tf MD = 5.57 Miss MT = 0.18 , tf MT = 14.57 Figure 20.1: VG1 Planar Simulation 1 Clearly, the Missile misses the Target. Actually, the simulation stops after d rMT dt changes sign. If not, the Missile is fully capable of intercepting the Target. This result shows that 1. VG1 is more reliable than LMG. As discussed in Part II, a considerable disadvantage of LMG is its dependance on linearization. Hence, by performing the optimal evasive maneuver regarding the linear model, the Target increases tMT f , which is assumed to be constant in LMG, and increases the bound B. As a result, the advantage gained from maximizing yMT is smaller then the disadvantage gained from the increase in tMT f . However, VG accounts for full kinematics; therefore, the Target’s evasive maneuver is more reliable. Note that the direction of the Target’s trajectory is similar to the one in Fig. 13.14 of Part II. However, now this result is obtained without any approximations. 78
- 95. 2. The actual M-D miss distance is almost identical to the required one, unlike LMG, where the actual M-D miss distance is much greater than required due to the nonlinear effects, which cannot be foreseen in the linearized model. In Fig. 20.2, one finds the nonlinear simulation of LMG (Fig. 9.5), and the vector guidance (VG1). Missile Target Defender tf MD t* 0 1000 2000 3000 4000 5000 6000 0 500 1000 1500 2000 2500 x [m] y[m] Miss MD = 154.5 , tf MD = 5.57 Miss MT = 0.18 , tf MT = 14.57 (a) VG1 Missile Target Defender tf MD t* 0 1000 2000 3000 4000 5000 6000 0 1000 2000 3000 4000 x [m] y[m] Miss MD = 325 , tf MD = 5.74 Miss MT = 0.1 , tf MT = 15.37 (b) LMG Figure 20.2: VG1 Vs. LMG Indeed, VG1 shows better results than LMG. In Fig. 20.3, one finds the trajectories plot vs. the ZEM plot. Missile Target Defender tf MD t* 0 1000 2000 3000 4000 5000 6000 0 500 1000 1500 2000 2500 x [m] y[m] Miss MD = 154.5 , tf MD = 5.57 Miss MT = 0.18 , tf MT = 14.57 (a) Planar Simulation ||yMT|| ||yMD|| ℬ tgo * t* tf MD tf MT Time, t ||yMT cr || ℬ(t* ) ||ZEM|| (b) ZEM Trajectories Figure 20.3: Planar Simulation and ZEM Trajectories 79
- 96. Note that tMD f and tMT f are relatively close to each other. However, this picture is not yet complete, as by using VG1, the Missile has the capability to pursue the Target even if rMT changes its sign. Example 20.2. Consider the same parameters as in Example 20.1; however, the simulation does not stop when the sign of d dt rMT changes (Fig. 20.4). 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 Figure 20.4: VG1 Planar Simulation 2 The Missile eventually intercepts the Target. In Fig. 20.5, one finds a local minimum in rMT (t) . 0 5 10 15 20 25 0 1000 2000 3000 4000 5000 6000 Time, t M-TRange,||rMT(t)|| Figure 20.5: Relative Distance rMT (t) 80
- 97. Example 20.3. A three dimensional version of such a conflict is shown in Fig. 20.6. Missile Target Defender tf MD t* 0 5000 x [m] 0 2000 4000 6000 8000y [m] 0 1000 2000 3000 z [m] Figure 20.6: VG1 3D Simulation 1 Since ρu = 170 is not enough to capture the Target, without having d dt rMT change its sign, in tMT f ≈ 8.5 [sec] as in Fig. 20.1, the Missile needs more maneuvering capability. Example 20.4. A simulation for the following parameters is depicted in Fig. 20.7. ρu = 270 m sec2 , ρv = 30 m sec2 , ρw = 60 m sec2 , m = 0.5 [m] , = 150 [m] 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 Figure 20.7: VG1 Planar Simulation 3 81
- 98. Also, rMT (t) becomes (Fig. 20.8). 0 2 4 6 8 10 0 1000 2000 3000 4000 5000 6000 Time, t M-TRange,||rMT(t)|| Figure 20.8: Relative Distance rMT (t) Clearly, rMT (t) has no local minima, and the Missile intercepts the Target without turning around and chasing it. Example 20.5. A 3D version of such a conflict is described in Fig. 20.9 (3D trajectories (left), ZEM trajectories (right)). Missile Target Defender tf MD t* 0 2000 4000 6000x [m] 0 1000 2000 3000y [m] 0 500 1000 z [m] ||yMT|| ||yMD|| ℬ tgo * t* tf MD tf MT Time, t ||yMT cr || ℬ(t* ) ||ZEM|| Figure 20.9: VG1 3D Simulation 2 and ZEM Trajectories Although yMD and yMT are distorted due to nonlinearities in tMD go and tMT go , yMT stays inside the bound B, and the Target is intercepted. 82
- 99. 21 Vector Guidance Based On 4th Order Time-to-go (VG4) Although VG1 introduces certain improvements over LMG, its nature does not fully describe the actual game situation, because: 1. The players do accelerate. 2. The players can be far collision triangle. 3. The final times don’t have to be constant. This requires a definition of a different type of time-to-go that accounts for the items above. Near Optimal Course Time-to-go Previously (17.13), we had that if both hypothetical players (P and E) play optimal, then the ZEM trajectory is yPE tPE go = rPE + tPE go VPE = JPE + 1 2 (amax P − amax E ) tPE go 2 (21.1) 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 (21.1) for a given set of rPE, VPE, JPE, amax P , and amax E . If players do not play optimal, then at every time step the time-to-go updates so that it fits the optimal ZEM trajectory. 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 . 83
- 100. 22 Optimal Strategies for VG4 22.1 Basic Optimal Strategies Clearly, players’ roles, regarding the relevant ZEM variable, do not change. Hence the Target is still the maximizer of yMT , v∗ = ρv yMT yMT = ρv rMT + tMT go VMT rMT + tMT go VMT (22.1) The Defender is the minimizer of yMD , w∗ = −ρw yMD yMD = −ρw rMD + tMD go VMD rMD + tMD go VMD (22.2) and the Missile has 2 objectives. To maximize yMD , u∗ e = −ρu yMD yMD = −ρu rMD + tMD go VMD rMD + tMD go VMD (22.3) and to minimize of yMT , u∗ p = ρu yMT yMT = ρu rMT + tMT go VMT rMT + tMT go VMT (22.4) 22.2 Missile – Target Game 22.2.1 M-T Game VG4 Basics Define m as the desired M-T miss distance. 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 (22.5) Square both sides of the equation, simplify and obtain that tMT go is the positive real root7 of the 4th order polynomial equation 1 4 (ρu − ρv)2 tMT go 4 + m (ρu − ρv) − VMT 2 tMT go 2 −2rMT VMT ·tMT go +m2 − rMT 2 = 0 (22.6) Note that solving numerically (22.5) is easier and more accurate than solving (22.6). Remark 22.1. When the players move on a collision course, such that rMT and VMT are collinear, the time to go is given by tMT go = − VMT + VMT 2 + 2 (ρu − ρv) ( rMT − m) ρu − ρv (22.7) 7 Squaring the equation can introduce irrelevant solutions; thus, one should use extra care when choosing the relevant solution. If not unique, it is usually the smallest positive real solution. 84
- 101. Remark 22.2. A positive real solution to (22.5) always exists. Proof. Define f tMT go = rMT + tMT go VMT − m − 1 2 (ρu − ρv) tMT go 2 (22.8) Examine the limits of f tMT go lim tMT go →0 f tMT go = rMT − m ≥ 0 (22.9) lim tMT go →∞ f tMT go = −∞ < 0 (22.10) Therefore, ∀ {rMT , VMT } ∈ Rn and m ≥ 0, there exists tMT go ∈ [0 , ∞) that satisfies f tMT go = 0. Thus, a positive real solution to (22.5) always exists. This implies that given ρu > ρv, the Missile can intercept the Target from any initial condition. In order to simplify matters, assume m = 0, and obtain that tMT go is the solution of rMT + tMT go VMT = 1 2 (ρu − ρv) tMT go 2 (22.11) Or equivalently, tMT go is the solution of the 4th order polynomial equation 1 4 (ρu − ρv)2 tMT go 4 − VMT 2 tMT go 2 − 2rMT VMT · tMT go − rMT 2 = 0 (22.12) Therefore, the ZEM norm, yMT is always placed on the function B tMT go = 1 2 (ρu − ρv) tMT go 2 (22.13) as depicted in Fig. 22.1. tgo MT ||yMT|| Figure 22.1: Missile-Target Game Optimal ZEM Trajectory 85
- 102. This definition is remarkable because: 1. There is no singular area, in which the optimal strategies are arbitrary, and the function B is not a bound; rather, it is the only possible optimal ZEM trajectory. 2. This guidance laws never chatters as the denominator never vanishes. 3. If ρu > ρv, a zero M-T miss distance can be achieved from any initial condition. 4. The ZEM variable has absolutely different meaning now. Unlike LMG, where the ZEM is the miss distance if both players do nothing until the end of the game, 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. Moreover, it is a function of the players strategies and 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. Define tMT f u∗ p, v∗ as the final time when both players play optimal. Since u∗ p is the minimizer, and v∗ is the maximizer of yMT , it will be proven later that if the Target does not play optimal, then yMT reaches zero before tMT f u∗ p, v∗ , and if the Missile doesn’t apply u∗ p, then yMT reaches zero after tMT f u∗ p, v∗ . A saddle point inequality can be formulated. tMT f u∗ p, v ≤ tMT f u∗ p, v∗ ≤ tMT f u, v∗ (22.14) This has practical meaning as the Missile’s engine has a limited burning time; therefore, the Missile has interest to minimize tMT f , and Target has interest to maximize it. 22.2.2 M-T Game VG4 Properties Theorem 22.1. The optimal acceleration direction for both Missile and Target in fixed coordinates is constant. Proof. Using (16.5) and (16.7), one obtains the following relative kinematic equation, ¨rMT (t) = v(t) − u(t) (22.15) Let both players use optimal maneuvers u(t) = u∗ p(t) = ρu rMT (t) + tMT go ˙rMT (t) rMT (t) + tMT go ˙rMT (t) (22.16) v(t) = v∗ (t) = ρv rMT (t) + tMT go ˙rMT (t) rMT (t) + tMT go ˙rMT (t) (22.17) Substitute into (22.15) and obtain, ¨rMT (t) = − (ρu − ρv) rMT (t) + tMT go ˙rMT (t) rMT (t) + tMT go ˙rMT (t) (22.18) 86
- 103. Also let both players use VG4. Hence, rMT (t) + tMT go ˙rMT (t) = 1 2 (ρu − ρv) tMT go 2 (22.19) Assuming that both players play optimal, we have tMT f = const. Substitute (22.19) into (22.18), and obtain ¨rMT (t) = − (ρu − ρv) rMT (t) + tMT go ˙rMT (t) 1 2 (ρu − ρv) tMT go 2 = − 2 rMT (t) + tMT go ˙rMT (t) tMT go 2 (22.20) Simplify and have, tMT go 2 ¨rMT (t) + 2tMT go ˙rMT (t) + 2rMT (t) = 0 rMT (0) = rMT 0 ˙rMT (0) = V MT 0 (22.21) In order to solve this ODE, we need to transform rMT (t) into rMT tMT go with all its derivatives. Define: tMT go = tMT f − t (22.22) dtMT go = −dt (22.23) Now, we have rMT tMT go = drMT tMT go dtMT go = − drMT (t) dt = − ˙rMT (t) (22.24) rMT tMT go = d2 rMT tMT go d tMT go 2 = d2 rMT (t) dt2 = ¨rMT (t) (22.25) By substituting (22.24) and (22.25) into (22.21), the following ODE is obtained tMT go 2 rMT tMT go − 2tMT go rMT tMT go + 2rMT tMT go = 0 rMT tMT f = rMT 0 rMT tMT f = −V MT 0 (22.26) Solving (22.26), and substituting back tMT go = tMT f − t yields rMT (t) = tMT f − t tMT f + t · rMT 0 + tMT f t · V MT 0 tMT f 2 = − yMT 0 tMT f 2 · t2 + V MT 0 · t + rMT 0 (22.27) where yMT 0 = rMT 0 + tMT f V MT 0 . Differentiate (22.27) with respect to t ˙rMT (t) = −2t · rMT 0 + tMT f tMT f − 2t V MT 0 tMT f 2 = − 2yMT 0 tMT f 2 · t + V MT 0 (22.28) Therefore, the ZEM is (after substituting tMT go = tMT f − t) yMT (t) = rMT (t) + tMT go ˙rMT (t) = tMT go 2 rMT 0 + tMT f V MT 0 tMT f 2 = tMT go tMT f 2 · yMT 0 (22.29) 87
- 104. Hence, the ZEM direction remains constant, and its magnitude decays as a function of tMT go /tMT f 2 . Now, substitute (22.29) and (22.19) into the optimal guidance laws, (22.16) and (22.17), and obtain u∗ p(t) = ρu rMT (t) + tMT go ˙rMT (t) rMT (t) + tMT go ˙rMT (t) = ρu tMT go tMT f 2 · yMT 0 1 2 (ρu − ρv) tMT go 2 = ρu yMT 0 yMT 0 (22.30) v∗ (t) = ρv rMT (t) + tMT go ˙rMT (t) rMT (t) + tMT go ˙rMT (t) = ρv tMT go tMT f 2 · yMT 0 1 2 (ρu − ρv) tMT go 2 = ρv yMT 0 yMT 0 (22.31) Hence, both optimal laws are constant and pointed in the direction of the initial Zero Effort Miss: yMT 0 = rMT 0 + tMT f V MT 0 . Theorem 22.2. When both players play optimal and start on a collision course, such that rMT 0 and V MT 0 are collinear, the LOS direction is constant. Proof. Previously, in (22.27), we had rMT (t) = − tMT f − t tMT f + t · rMT 0 + tMT f t · V MT 0 tMT f 2 (22.32) Assuming collision course, we have that rMT 0 and V MT 0 are collinear, V MT 0 = α · rMT 0 , α ∈ R (22.33) Substitute into (22.32) and obtain, rMT (t) = − tMT f − t tMT f + αtMT f t + t tMT f 2 · rMT 0 = β(t) · rMT 0 , β : R → R (22.34) As a result, LOS direction is always the direction of LOS0. Theorem 22.3. When both players play optimal and start on collision course, such that rMT 0 and V MT 0 are collinear, Missile’s relative to LOS acceleration angle δMT is zero. Proof. Rewriting (22.30) yields, u∗ p(t) = ρu yMT 0 · rMT 0 + tMT f V MT 0 (22.35) Assuming collision course, and substituting V MT 0 = α · rMT 0 into (22.35), yields u∗ p(t) = ρu 1 + tMT f α yMT 0 · rMT 0 = γ · rMT 0 , γ ∈ R (22.36) Define rMT 0 = rMT 0 ∠λ0 MT , rMT 0 , λ0 MT ∈ R (22.37) 88
- 105. And obtain, ∠u∗ p(t) = ∠rMT (t) = λ0 MT (22.38) Therefore, δMT (t) = ∠u∗ p(t) − ∠rMT (t) = 0 (22.39) Hence, Missile’s acceleration angle remains pointed at LOS0 direction. The same is easy to prove for the Target. As a result, if the players play optimal and start on a collision course, then rMT , VMT , u∗ p and v∗ are collinear. Theorem 22.4. If both players use their optimal guidance strategies, rMT and VMT become collinear at t = tMT f . Proof. Recall that when both players play optimal we have rMT (t) = tMT f − t tMT f + t · rMT 0 + tMT f t · V MT 0 tMT f 2 (22.40) VMT (t) = −2t · rMT 0 + tMT f tMT f − 2t V MT 0 tMT f 2 (22.41) Now, examine the limits lim t→tMT f rMT (t) tMT f − t = 2rMT 0 tMT f + V MT 0 (22.42) lim t→tMT f VMT (t) = − 2rMT 0 tMT f + V MT 0 (22.43) Therefore, when t → tMT f rMT = − tMT f − t VMT (22.44) thus rMT rMT = VMT VMT (22.45) which implies that rMT and VMT are collinear. Remark 22.3. In case m = 0, the optimal Missile’s pursuit strategy u∗ p(t) and the optimal Target’s evasion strategy v∗ (t) can be rewritten as u∗ p = 2 1 − ρv ρu 1 tMT go 2 rMT + tMT go VMT = N∗ up tMT go 2 · yMT (22.46) v∗ = 2 ρu ρv − 1 1 tMT go 2 rMT + tMT go VMT = N∗ v tMT go 2 · yMT (22.47) where N∗ up = 2 1− ρv ρu and N∗ v = 2 ρu ρv −1 . Although (22.46) and (22.47) look like a linear guidance laws, they are not because tMT go is nonlinear in rMT and VMT . 89
- 106. 22.3 Missile – Defender Game 22.3.1 M-D Game VG4 Basics 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 (22.48) 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 (22.49) where Z tMD go is parallel to A tMD go = − 1 2 (ρu − ρw) tMD go 2 (22.50) Graphically, Z tMD go and A tMD go are depicted in Fig. 22.2. (tgo MD ) (tgo MD ) tgo MD ℓ q ||yMD|| Figure 22.2: Missile-Defender Optimal ZEM Trajectories The shape of Z is well defined by ρu and ρw, and its vertical displacement depends on the parameter q. Using (22.48), define: g tMD go = rMD + tMD go VMD − q + 1 2 (ρu − ρw) tMD go 2 (22.51) Solving (22.48) is equivalent to solving g tMD go = 0. 90
- 107. Theorem 22.5. The function g tMD go has exactly one local minimum Proof. Differentiate with respect to tMD go and obtain ˙g tMD go = (ρu − ρw) tMD go + rMDVMD + tMD go VMD 2 rMD + tMD go VMD (22.52) Note that rMD > 0 since the distances in all 3 directions are positive. In addition, assuming that both players are approaching, we have: VMD < 0. As a result, rMDVMD < 0 (22.53) Now examine the limits of ˙g tMD go . lim tMD go →0 ˙g tMD go = rMDVMD rMD < 0 (22.54) lim tMD go →∞ ˙g tMD go = ∞ > 0 (22.55) Therefore, since g tMD go is continuous, it has at least one local minimum point in the interval tMD go ∈ [0, ∞). Now, examine the second derivative of g tMD go . ¨g tMD go = (ρu − ρw) + rMD 2 VMD 2 − rMDVMD 2 rMD + tgoVMD 3 (22.56) From Cauchy–Schwarz inequality, one has rMD 2 VMD 2 ≥ rMDVMD 2 (22.57) Therefore ¨g tMD go > 0, resulting ˙g tMD go is monotonically increasing. Now, having 1. limtMD go →0 ˙g tMD go = rMDVMD rMD < 0 2. limtMD go →∞ ˙g tMD go = ∞ > 0 3. ¨g tMD go > 0 ∀tMD go ⇒ ˙g tMD go is monotonically increasing we conclude that ˙g tMD go has only one intersection point with the tMD go axis; therefore, g tMD go has exactly one local minimum. Since g tMD go is affine in q, its shape is not affected by q, only its vertical displacement. Using Theorem 22.5 we understand that only three possibilities exist: 1. The function g tMD go has no positive real roots. 2. The function g tMD go has one positive real root. 3. The function g tMD go has two positive real roots. 91
- 108. Example 22.1. For some rMD and VMD, and different values of q , the function g tMD go is shown in Fig. 22.3. q = 100 [m] q = 1050 [m] q = 2100 [m] 2 4 6 8 10 tgo MD 2000 4000 6000 8000 g(tgo MD) Figure 22.3: Function g tMD go For Different Values of q As a result, the following conclusions are to be made (keep in mind that in this analysis both players are assumed to play optimal): 1. If g tMD go = 0 has no positive real solutions, then q is too small, and such a miss distance is unachievable. Refer to the Example 22.2. Here the required q is smaller than qmin; thus, there is no value of the time, t, for which q < qmin. 2. If g tMD go = 0 has exactly one positive real solution, then q = qmin is the minimal achievable M-D miss distance. Clearly, this occurs when rMD reaches its minimum (again, refer to Example 22.2). 3. If g tMD go = 0 has more than one positive real solution (in fact, exactly 2 solutions), then q is too big, and by choosing the bigger of the two solutions for tMD go , we are aiming for a higher trajectory than required (while the smaller solution is not physical, as rMD still decreases). This would require more time for evasion than necessary and produce bigger M-D miss distance than desired. In Example 22.2 we see that the first solution of g tMD go = 0 is not physical as it occurs when the relative distance is still decreasing, while the second solution introduces a bigger evasion time than the required minimum, and a bigger distance than the actual miss-distance. 92
- 109. Example 22.2. The relative distance, rMD , in an optimal M-D game is depicted in Fig. 22.4. Not Physical tf MD (qmin) tf MD (q > qmin) Time, t qmin q > qmin ||rMD|| Figure 22.4: Missile-Defender Relative Distance Therefore, it is reasonable to define tMD f = tMD f (qmin); hence, tMD go = tMD go (qmin). As a result, we have to find the parameter q for which g tMD go = 0 has exactly one positive real root. Since q does not change the shape of g tMD go , rather it changes its vertical displacement, clearly g tMD go has one real root when its local minimum is tangent to the horizontal axis. Therefore, instead of looking for q = qmin we can find tMD go which minimizes g tMD go , and obtain tMD go (qmin). Thus, tMD go = arg min tMD go g tMD go (22.58) = arg min tMD go rMD + tMD go VMD − q + 1 2 (ρu − ρw) tMD go 2 (22.59) Since q doesn’t affect the solution, we can choose any value of it. In particular, q = 0. Thus, finally we have tMD go = arg min tMD go rMD + tMD go VMD + 1 2 (ρu − ρw) tMD go 2 (22.60) Remark 22.4. Although (22.60) can be minimized analytically, this method is hardly imple- mentable; therefore, it is preferable to use the numerical solution. Generally, a numerical routine that can minimize (22.60) in real-time needs further research. 93
- 110. 22.3.2 M-D Game VG4 Properties Theorem 22.6. If the Missile and the Defender are on a collision course and approaching each other, so that rMD and VMD are collinear and have opposite signs, and (22.61) holds, rMD ≤ VMD 2 ρu − ρw (22.61) Then VG1 and VG4 time-to-go is equal; namely, tMD go = rMD VMD . Proof. Recall that, g tMD go = rMD + tMD go VMD − q + 1 2 (ρu − ρw) tMD go 2 (22.62) ˙g tMD go = (ρu − ρw) tMD go + rMDVMD + tMD go VMD 2 rMD + tMD go VMD (22.63) When the Missile and the Defender are on a collision course and approaching each other, we have rMD rMD = − VMD VMD thus VMD = − VMD rMD · rMD (22.64) Substitute (22.64) into the expression for ˙g tMD go and obtain ˙g tMD go = (ρu − ρw) tMD go + −rMD VMD rMD · rMD + tMD go VMD 2 rMD − tMD go VMD rMD · rMD = (ρu − ρw) tMD go + − VMD rMD · rMD 2 + tMD go VMD 2 rMD 1 − tMD go VMD rMD = (ρu − ρw) tMD go + − rMD · VMD + tMD go VMD 2 rMD · 1 − tMD go VMD rMD = (ρu − ρw) tMD go − VMD · rMD − tMD go VMD rMD − tMD go VMD (22.65) Finally, we have ˙g tMD go = (ρu − ρw) tMD go − VMD · sign rMD − tMD go VMD (22.66) Now, check the limits of ˙g tMD go in the neighborhood of rMD VMD , and obtain lim tMD go → rMD VMD − ˙g tMD go = (ρu − ρw) rMD VMD − VMD (22.67) lim tMD go → rMD VMD + ˙g tMD go = (ρu − ρw) rMD VMD + VMD > 0 (22.68) 94
- 111. Using the condition (22.61) yields lim tMD go → rMD VMD − ˙g tMD go ≤ 0 (22.69) Hence, ˙g tMD go intersects the horizontal axis at tMD go = rMD VMD . Example 22.3. The function ˙g tMD go is depicted in Fig. 22.5. 2 4 6 8 10 tgo MD -1000 -500 500 1000 1500 g (tgo MD ) Figure 22.5: Function ˙g tMD go Since this is the only intersection of ˙g tMD go with the horizontal axis (Theorem 22.5), the minimum of g tMD go is exactly at tMD go = rMD VMD , as depicted in Fig. 22.6. 2 4 6 8 10 tgo MD 1000 2000 3000 4000 5000 6000 7000 g(tgo MD ) Figure 22.6: Function g tMD go Consequently, the 4th order time-to-go is tMD go = rMD VMD . 95
- 112. Example 22.4. For some numerical values, Fig. 22.7 shows a comparison between 1st and 4th order time-to-go. VG4 VG1 0 1 2 3 4 5 6 0 1 2 3 4 5 6 Time, t Estimatedtgo MD t* Figure 22.7: Evaluation of tMD go for VG1 and VG4 It is readily seen that when tMD go ∈ [0, ts], both VG1 and VG4 produce similar results. Theorem 22.7. The optimal acceleration direction for both Missile and Defender in fixed coordi- nates is constant. Proof. The proof is similar to that of M-T game. The kinematic equation is, ¨rMD(t) = w(t) − u(t) (22.70) Let both players use optimal maneuvers u∗ e(t) = −ρu rMD(t) + tMD go ˙rMD(t) rMD(t) + tMD go ˙rMD(t) (22.71) w∗ (t) = −ρw rMD(t) + tMD go ˙rMD(t) rMD(t) + tMD go ˙rMD(t) (22.72) Therefore, ¨rMD(t) = (ρu − ρw) rMD(t) + tMD go ˙rMD(t) rMD(t) + tMD go ˙rMD(t) (22.73) Also let both players use VG4. Hence, rMD(t) + tMD go ˙rMD(t) = q − 1 2 (ρu − ρw) tMD go 2 (22.74) Assuming that both players play optimal, we have tMD f = const. Substitute, and obtain ¨rMD(t) = (ρu − ρw) rMD(t) + tMD go ˙rMD(t) q − 1 2 (ρu − ρw) tMD go 2 = 2 rMD(t) + tMD go ˙rMD(t) 2q ρu−ρw − tMT go 2 (22.75) 96
- 113. Simplify and obtain tMD f − t 2 − T 2 ¨rMD(t) + 2tMD go ˙rMD(t) + 2rMD(t) = 0 rMD(0) = rMD 0 ˙rMD(0) = V MD 0 (22.76) where T = 2q ρu−ρw [sec]. The solution of (22.76) is rMD(t) = t2 − tMD f 2 + T 2 rMD 0 + t T 2 − tMD f − t · tMD f V MD 0 T 2 − tMD f 2 (22.77) Differentiate with respect to t, ˙rMD(t) = 2t · rMD 0 + 2t · tMD f − tMD f 2 + T 2 V MD 0 T 2 − tMD f 2 (22.78) Therefore, the ZEM is yMD(t) = rMD(t) + tMD go ˙rMD(t) = T 2 − tMD go 2 T 2 − tMD f 2 · yMD 0 (22.79) Now, substitute into the optimal guidance laws, u∗ e(t) = −ρu rMD(t) + tMD go ˙rMD(t) rMD(t) + tMD go ˙rMD(t) = −ρu T 2−(tMD go ) 2 T 2−(tMD f ) 2 · yMD 0 q − 1 2 (ρu − ρw) tMD go 2 = −ρu yMD 0 yMD 0 (22.80) w∗ (t) = −ρw rMD(t) + tMD go ˙rMD(t) rMD(t) + tMD go ˙rMD(t) = −ρw T 2−(tMD go ) 2 T 2−(tMD f ) 2 · yMD 0 q − 1 2 (ρu − ρw) tMD go 2 = −ρw yMD 0 yMD 0 (22.81) Hence, both optimal laws are constant and pointed in the direction of yMD 0 = rMD 0 +tMD f V MD 0 . Theorem 22.8. When both players play optimal and start on a collision course, so that rMD 0 and V MD 0 are collinear, the LOS direction is constant. Proof. Assuming collision course, we have that rMD 0 and V MD 0 are collinear. V MD 0 = α · rMD 0 , α ∈ R (22.82) Substitute into (22.77) and obtain, rMD(t) = (1 + αt) T 2 − tMD go t + (1 + αt) tMD f T 2 − tMD f 2 · rMD 0 = β(t) · rMD 0 , β : R → R (22.83) Hence the LOS direction is always the direction of LOS0. Theorem 22.9. When both players play optimal and start on collision course, such that rMD 0 and V MD 0 are collinear, Missile’s relative to LOS acceleration angle δMD is zero. The proof is identical to the M-T game. 97
- 114. 22.4 M-T-D VG4 Game Strategy Similarly to VG1, divide the game into two different phases: the evasion phase and the pursuit phase. Define the fail-safe function C tMD go = + 1 2 (ρu + ρw) tMD go 2 (22.84) Analogically to VG1 (refer to Subsection 19.2), by making yMD reach C, the Missile guarantees a M-D miss distance of , as depicted in Fig. 22.8. ||yMT|| ||yMD|| Evasion Pursuit {ue * ,w * } {up * ,w * } t* tf MD tf MT Time, t ℓ ||yMT cr || ||ZEM|| Figure 22.8: Functions A, C, yMT and yMD Assuming ρu > ρw, the optimal ZEM trajectory, generated by the pair {u∗ e, w∗ }, is parallel to A when t ∈ [0, t∗ ], and in the worst case (u∗ e = −u∗ p) collides with C when t ∈ t∗ , tMT f , allowing the Missile to evade the Defender. However, this is true only if yMD 0 is not inside the area bounded by A. Hence, since we demand yMD tMD go ≥ A tMD go = − 1 2 (ρu − ρw) tMD go 2 ∀tMD go ∈ 0, tMD f the initial condition must satisfy yMD 0 ≥ − 1 2 (ρu − ρw) tMD f 2 (22.85) thus ρu ≥ ρw + 2 − yMD 0 tMD f 2 (22.86) Note that (22.86) provides the same evasion condition as (19.36) in VG1 section; hence, the Missile- Defender game is similar to VG1. However, the Missile-Target game conceptually differs from VG1. 98
- 115. Since ρu > ρv, the Missile can always enforce any M-T final time, tMT f , such that eventually the M-T miss distance is zero (Fig. 22.8). Therefore, there is no upper bound that yMT has to stay in, and the Missile can always intercept the Target if it successfully evades the Defender. Hence, instead of (19.38), we have now a much simpler condition. Define the Missile’s optimal strategy as a combination of its optimal pursuit and evasion strategies, u∗ = u∗ e = −ρu yMD yMD , yMD < C tMD go u∗ p = ρu yMT yMT , yMD ≥ C tMD go (22.87) Also recall the other players optimal strategies, v∗ = ρv yMT yMT (22.88) w∗ = −ρw yMD yMD (22.89) Theorem 22.10. Let the Missile use its optimal guidance strategy, u∗ , and let the other players use any bounded maneuver, such that v ≤ ρv and w ≤ ρw. The Missile can evade the Defender and capture the Target if 1. ρu ≥ ρw + 2( − yMD 0 ) t2 f 2. ρu > ρv Theorem 22.11. Let all the players apply their optimal strategies, {u∗ , v∗ , w∗ }. The Missile can evade the Defender and capture the Target if and only if 1. ρu ≥ ρw + 2( − yMD 0 ) t2 f 2. ρu > ρv Since conditions 1 and 2 are assumed to hold, it seems that the Missile is always capable of achieving its goal. However, we must remember that the Missile has limited fuel; therefore, will not be capable of pursuing the Target as long as it needs. As a result, game time should be considered. 99
- 116. 23 Time Optimal M-T-D Game We already know that by making yMD reach the fail-safe function C, the Missile can guarantee a M-D miss distance of . However, in VG4 scenario the Missile has no limitation on how, where, and when to reach C. Moreover, it can even reach any point yp, such that yp ≥ C, and still be able to intercept the Target. Since ρu > ρv, the Missile can ignore the increase in yMT while performing the evasive maneuver, because it can guarantee any M-T miss distance from any point of the ZEM plane. Yet, since its fuel is limited, the Missile must complete the game in tMT f ≤ tb, where tb is the Missile’s engine burning time. Thus, it is best for the Missile to complete the game in minimum time; however, the solution of the global minimum time problem is not trivial and not implementable in real-time guidance systems. Nevertheless, since the Missile’s game is constructed out of two phases: evasion and pursuit, we can locally minimize each phase, and obtain a suboptimal solution. 23.1 Evasion Phase Minimizing the evasion phase, means minimizing t∗ , where t∗ t : yMD t∗ = C t∗ . Recall the M-D ZEM norm derivative ˙VMD(t) = tMD go yMD(t) yMD(t) (−u(t) + w(t)) One can see that {ue, w} satisfy the saddle point inequality for ˙VMD, ˙VMD ue, w∗ ≤ ˙VMD u∗ e, w∗ ≤ ˙VMD u∗ e, w ∀t ∈ 0, tMD f (23.1) where u∗ e = −ρu yMD yMD and w∗ = −ρw yMD yMD . Recall that yMD(t) = yMD 0 + ˆ t 0 ˙VMD(ξ)dξ (23.2) Define the integral as Riemann’s series ˆ t 0 ˙VMD(ξ)dξ = lim dξ→0 N i=0 ˙VMD(ξi)dξ where N = t/dξ. Having that (23.1) holds ∀ξi, we conclude that yMD ue, w∗ ≤ yMD u∗ e, w∗ ≤ yMD u∗ e, w ∀t ∈ 0, tMD f (23.3) In particular, (23.3) is true for t = t∗∗ , where t∗∗ t : yMD t∗∗ = C t∗∗ (23.4) given {u∗ e, w∗ }. Consequently, for some {ue, w∗ } where ue = u∗ e, we have yMD t∗∗ ≤ C t∗∗ (23.5) Namely, the Missile has not reached the fail-safe yet, and the evasion phase is not yet complete. As a result, u∗ e and w∗ (the optimality for w∗ is easy to prove the same way) are optimal regarding the saddle point t∗ ue, w∗ ≤ t∗ u∗ e, w∗ ≤ t∗ u∗ e, w 100
- 117. 23.2 Pursuit Phase Similarly, ˙VMT (t) = tMT go yMT (t) yMT (t) (−u(t) + v(t)) (23.6) The controllers {up, v} satisfy, ˙VMT u∗ p, v ≤ ˙VMT u∗ p, v∗ ≤ ˙VMT up, v∗ ∀t ∈ 0, tMT f (23.7) where u∗ p = ρu yMT yMT and v∗ = ρv yMT yMT . Similarly to Subsection 23.1, integration yields, yMT u∗ p, v ≤ yMT u∗ p, v∗ ≤ yMT up, v∗ ∀t ∈ 0, tMT f (23.8) In particular, (23.8) is true for t = tMT∗ f , where tMT∗ f t : yMT tMT∗ f = m (23.9) given u∗ p, v∗ . Hence, for some {up, v∗ } where up = u∗ p we have yMT tMT∗ f ≥ m (23.10) Namely, the Missile has not yet intercepted the Target. The same is easy to prove for the opti- mality of v∗ . Thus, u∗ p and v∗ are optimal regarding the saddle point inequality, tMT f u∗ p, v ≤ tMT f u∗ p, v∗ ≤ tMT f up, v∗ (23.11) 23.3 M-T-D Time Optimal Guidance For conclusion, the optimal guidance law which minimizes t∗ and tMT f is u∗ = u∗ e = −ρu yMD yMD , yMD < C tMD go u∗ p = ρu yMT yMT , yMD ≥ C tMD go (23.12) and, the optimal Target’s and Defender’s guidance laws are v∗ = ρv yMT yMT (23.13) w∗ = −ρw yMD yMD (23.14) 101
- 118. 23.4 Time-Bound Approximation 23.4.1 Basic Derivations Since in VG4 scenario both final times are not assumed to be constant, it is hard to compute them a priori. However, if all players maneuver optimally, the final times remain constant, and the game properties of LMG and VG1 are preserved. Assumption 1. All players perform optimal maneuvers; namely, u = u∗ e , yMD < C tMD go u∗ p , yMD ≥ C tMD go (23.15) v = v∗ (23.16) w = w∗ (23.17) Therefore, tMD f and tMT f are constant. Assumption 2. Worst case ZEM direction; namely, ˆyMT = −ˆyMD. In this case, u∗ p = −u∗ e. Assumption 3. The vectors rMT 0 and V MT 0 are collinear. In fact, this assumption can be lifted, but then yMT 0 = 0, and analytical solution is impossible. Assumption 4. Assume m = 0 (this can be also lifted). Having these assumptions, we obtain (derived in Part I and reproduced in Part III for VG1) yMT (t) = 1 2 (ρu + ρv) tMT f 2 − 1 2 (ρu + ρv) tMT f − t 2 , t ≤ t∗ 1 2 (ρu − ρv) tMT f − t 2 , t ≥ t∗ (23.18) where the switch time is t∗ = tMD f − (ρu − ρw) tMD f 2 − 2 + 2 rMD 0 + tMD f V MD 0 2ρu (23.19) and the M-D final time can be obtained from the VG4 algorithm for rMD 0 and V MD 0 ; namely, tMD f = arg min tMD f rMD 0 + tMD f V MD 0 + 1 2 (ρu − ρw) tMD f 2 (23.20) If all assumptions hold, then we get the well-known picture in the ZEM plane (Fig. 23.1). 102
- 119. ||yMT|| ||yMD|| t* tf MD tf MT Time, t ℓ ||ZEM|| Figure 23.1: Functions yMT , yMD , and C Therefore, at t = t∗ we have yMT (t∗ ) = 1 2 (ρu + ρv) tMT f 2 − 1 2 (ρu + ρv) tMT f − t∗ 2 (23.21) = 1 2 (ρu − ρv) tMT f − t∗ 2 Solving (23.21) for tMT f yields, tMT f = √ 2t∗ √ 2 − 1 + ρv ρu (23.22) Algorithm. Approximate tMD f from tMD f = arg min tMD f rMD 0 + tMD f V MD 0 + 1 2 (ρu − ρw) tMD f 2 (23.23) Approximate t∗ from t∗ = tMD f − (ρu − ρw) tMD f 2 − 2 + 2 rMD 0 + tMD f V MD 0 2ρu (23.24) The estimated game time is tMT f = √ 2t∗ √ 2 − 1 + ρv ρu (23.25) 23.4.2 Simulation Results For Example 24.1 provided in the following section, the actual game time is tMT f = 11.67 [sec], while the estimated game time is tMT f = 11.72 [sec]. 103
- 120. 23.4.3 Time-Bounded Game The computed tMT f is only an approximation for the scenario in which all players play optimal; however, if the Target-Defender team does not play optimal, the game time is smaller. Hence, the obtained is an approximated bound for the game time. Theorem 23.1. Let the Missile use its optimal guidance strategy, u∗ , and let the other players use any bounded maneuver, such that v ≤ ρv and w ≤ ρw. The Missile can evade the Defender and intercept the Target if 1. ρu ≥ ρw + 2( − yMD 0 ) t2 f 2. ρu > ρv 3. tb ≥ √ 2t∗ √ 2− √ 1+ ρv ρu where tb is the Missile’s engine burning time. 104
- 121. 24 VG4 Simulations 24.1 Basic VG4 Simulations Example 24.1. Consider the following parameters, ρu = 170 m sec2 , ρv = 30 m sec2 , ρw = 60 m sec2 , m = 0.5 [m] , = 150 [m] The simulation result is depicted in Fig. 24.1. 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 Figure 24.1: VG4 Planar Simulation 1 It is clear that 1. The required M-D miss distance is almost identical to the requested one. 2. The Missile intercepts the Target. 3. The interception time is about 12 seconds. In order to see the advantage of VG4 over VG1, let us now compare both by running a planar simulation with identical parameters and initial conditions. Such a comparison is depicted in Fig. 24.2. 105
- 122. 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 (a) VG1 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 (b) VG4 Figure 24.2: VG1 Vs. VG4 In both cases the Missile evades the Defender and intercepts the Target. However, VG4 provides a much smaller intercept time. We can also compare the relative distances rMT (t) , as depicted in Fig. 24.3. 0 5 10 15 20 25 0 1000 2000 3000 4000 5000 6000 Time, t M-TRange,||rMT(t)|| (a) VG1 0 2 4 6 8 10 0 1000 2000 3000 4000 5000 6000 Time, t M-TRange,||rMT(t)|| (b) VG4 Figure 24.3: Relative M-T Distances, rMT (t) By using VG4 the Missile captures the Target so that rMT (t) has no local minima. Previously, we have seen that in M-D game, VG1 and VG4 introduce similar time-to-go under certain conditions. Denote VG14 as VG1 for M-D game and VG4 for M-T game. Fig. 24.4 provides a comparison between VG4 and VG14. 106
- 123. Missile Target Defender tf MD t* 0 1000 2000 3000 4000 5000 6000 0 1000 2000 3000 4000 x [m] y[m] Miss MD = 156 , tf MD = 6.73 Miss MT = 0.5 , tf MT = 12.9 (a) VG14 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 (b) VG4 Figure 24.4: VG14 Vs. VG4 VG4 provides slightly better results. Another reasonable question is what happens if the Missile uses VG4 and the Target uses VG1. The comparison in Fig. 24.5 demonstrates such a situation. Missile Target Defender tf MD t* 0 1000 2000 3000 4000 5000 6000 0 1000 2000 3000 4000 x [m] y[m] Miss MD = 150 , tf MD = 6.96 Miss MT = 0.5 , tf MT = 12.1 (a) Target Using VG1 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 (b) Both Players Use VG4 Figure 24.5: Demonstration of the Target using VG1 The Target is intercepted in both cases. We have seen in Subsections 22.2 and 22.3 that when a pair of players plays optimal, Missile’s acceleration angle in fixed coordinates is constant. Therefore, when the Missile uses the provided guidance law and the Target and the Defender play optimal, Missile’s acceleration angle (in fixed coordinates), χ(t), is piecewise constant. This can be seen in Fig. 24.6. 107
- 124. 0 2 4 6 8 10 -100 -50 0 50 100 Time, t AccelerationAngle,χ(t) (a) Acceleration Angle, χ(t) 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 (b) Planar Simulation Figure 24.6: Acceleration Angle, χ(t) vs. Planar Simulation A three dimensional simulation of VG4 is described in Fig. 24.7. Missile Target Defender tf MD t* 0 2000 4000 6000 x [m] 0 1000 2000 3000 y [m] 0 500 1000 1500 z [m] Figure 24.7: VG4 3D Simulation 1 According to the simulations provided in the present section, VG4 provides substantially improved results over VG1. Consequently, it is possible to decrease ρu and still get considerable results. 108
- 125. Example 24.2. Decreasing ρu to 120 yields the result depicted in Fig. 24.8. 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 Figure 24.8: VG4 Planar Simulation 2 The Missile is still capable of capturing the Target in reasonable time. A three dimensional version of such a conflict is depicted in Fig. 24.9. Missile Target Defender tf MD t* 0 2000 4000 6000x [m] 0 2000 4000 6000 y [m] 0 500 1000 1500 2000 z [m] Figure 24.9: VG4 3D Simulation 2 109
- 126. 24.2 VG4 with Optimal Start-time (VG4∗ ) In Part II, we have derived the following expression for ρumin , assuming yMT 0 = yMD 0 = m = 0, ρumin = ∆t (∆t3 ρv − tf (tf (3∆t + 2tf ) (ρv + ρw) + 4 )) + 3∆t2 − √ 8∆t (∆t + tf ) tf ρv (∆t + tf ) + t2 f ρw + 2 2 − ρv∆t2 t2 f ρw + 2 ∆t2 (∆t2 − 4tf (∆t + tf )) (24.1) which is the minimal maneuvering capability required from the Missile to complete its task in LMG scenario. One of the results, obtained from the parametric analysis provided in Part II (ref. Page 45), is that for every set of parameters, there exists an optimal value of tf = tMD f , which satisfies tOpt f = arg min tf ρumin (24.2) Namely, tf = tOpt f brings the Missile’s required capability to minimum. Although VG4 does not assume constant final-times and provides a much simpler sufficient condition8 than LMG, it makes sense for the Missile to start the game at tMD go = tOpt f , assuming the final-times, tMD f and tMT f , are approximately constant and close to the ones predicted in Subsection 23.4. Namely, tMD f ≈ arg min tMD f rMD 0 + tMD f V MD 0 + 1 2 (ρu − ρw) tMD f 2 (24.3) tMT f ≈ √ 2t∗ √ 2 − 1 + ρv ρu (24.4) where t∗ ≈ tMD f − (ρu − ρw) tMD f 2 − 2 + 2 rMD 0 + tMD f V MD 0 2ρu (24.5) Although in general the approximation for tOpt f is pretty rough in VG4 scenario9 and needs a final numerical tuning, it improves the results of the standard VG4 guidance law. Denote VG4∗ as the following set of guidance strategies u = u∗ p tOpt f ≤ tMD go ≤ tMD f u∗ e t∗ go ≤ tMD go ≤ tOpt f u∗ p tMD go ≤ t∗ go (24.6) v = v∗ (24.7) w = w∗ (24.8) 8 In fact, for any reasonable set of initial conditions and the parameter , the following sufficient conditions will do 1. ρu > ρv 2. ρu > ρw 9 These approximations are true under the assumption that the Missile starts evading the Defender at tMD go = tMD f , and not tMD go = tOpt f . As a result, the true values of tMD f and tMT f are different from the computed approximation. Thus, a numerical routine which converges to the proper values of tMD f and tMT f should improve this approximation. This is left for future research. 110
- 127. Example 24.3. Consider the same parameters as in Ex. 24.1. The simulation in Fig. 24.10 shows the difference between VG4 and VG4∗ . Missile Target Defender tf MD t* 0 1000 2000 3000 4000 5000 6000 0 500 1000 1500 2000 2500 x [m] y[m] Miss MD = 151.8 , tf MD = 5.47 Miss MT = 0.5 , tf MT = 10.75 (a) VG4∗ 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 (b) VG4 Figure 24.10: VG4 vs. VG4∗ By performing the maneuver described in (24.6), the Missile has managed to shorten its evasive maneuver and intercept the Target in a smaller time. Although, in this example, there is no dramatic difference between VG4 and VG4∗ , situation changes as tMD f increases. Example 24.4. Consider the values of Ex. 24.1, but with a a different geometry (Fig. 24.11). Missile Target Defender tf MD t* -4000 -2000 0 2000 4000 6000 0 500 1000 1500 2000 2500 3000 x [m] y[m] Miss MD = 152.8 , tf MD = 7.39 Miss MT = 0.5 , tf MT = 12.88 (a) VG4∗ Missile Target Defender tf MD t* -4000 -2000 0 2000 4000 6000 0 1000 2000 3000 4000 5000 x [m] y[m] Miss MD = 159.9 , tf MD = 9.85 Miss MT = 0.5 , tf MT = 17.49 (b) VG4 Figure 24.11: VG4 vs. VG4∗ Obviously, the difference between VG4 and VG4∗ is substantial. 111
- 128. 25 Modified Vector Guidance 25.1 Projected Vector Guidance (PVG) 25.1.1 Model Definition In all previous sections of this part, ideal VG was discussed. Namely, the players could point their acceleration vector into any direction of R3 space. However, this situation is not always true; as all R3 might not be reachable for some vehicles. For example, endo-atmospheric vehicles can apply acceleration only perpendicular to their velocity vector. Therefore, it is important to determine the optimal guidance strategies under these restrictions and explore the properties of the solution. Consider the VG state space model (derived in Section 16). ˙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 uα + 0 In 0 0 vα + 0 0 0 In wα (25.1) where uα ∈ U, vα ∈ V, wα ∈ W, where U, V, W ⊂ R3 . Also consider the orthonormal projection matrices, Pu = Mu MuMu −1 Mu (25.2) Pv = Mv MvMv −1 Mv (25.3) Pw = Mw MwMw −1 Mw (25.4) where the columns of Mu, Mv, and Mw span U, V and W respectively. As a result, Puu ∈ U (25.5) Pvv ∈ V (25.6) Pww ∈ W (25.7) for any u, v, w ∈ R3 . Hence, (25.1) can be rewritten as ˙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 −Pu 0 −Pu u + 0 Pv 0 0 v + 0 0 0 Pw w (25.8) where u, v, w ∈ R3 . Indeed, since Pu, Pv and Pw are orthonormal projection matrices, they elim- inate the components orthogonal to U, V and W, which are not reachable by uα, vα and wα. Similarly to Section 16, define JMT = rMT tMT f = In 0 0 0 x tMT f = gx tMT f (25.9) JMD = rMD tMD f = 0 0 In 0 x tMD f = hx tMD f (25.10) 112
- 129. and the ZEM variables, yMT (t) = gΦ tMT f , t x(t) (25.11) yMD(t) = hΨ tMD f , t x(t) (25.12) where ˙Φ tMT f , t = −Φ tMT f , t A , Φ tMT f , tMT f = I ˙Ψ tMD f , t = −Ψ tMD f , t A , Ψ tMD f , tMD f = I Differentiating the ZEM variables yields, ˙yMT (t) = XMT tMT f , t u(t) + YMT tMT f , t v(t) + ZMT tMT f , t w(t) (25.13) ˙yMD(t) = XMD tMD f , t u(t) + YMD tMD f , t v(t) + ZMD tMD f , t w(t) (25.14) However, now we have XMT tMT go = gΦ tMT go b = −tMT go Pu (25.15) YMT tMT go = gΦ tMT go c = tMT go Pv (25.16) ZMT tMT go = gΦ tMT go d = 0 (25.17) XMD tMD go = hΨ tMD go b = −tMD go Pu (25.18) YMD tMD go = hΨ tMD go c = 0 (25.19) ZMD tMD go = hΨ tMD go d = tMD go Pw (25.20) while the ZEM variables remain unchanged. yMT (t) = rMT (t) + tMT go VMT (t) (25.21) yMD(t) = rMD(t) + tMD go VMD(t) (25.22) Also, the M-T and M-D ZEM projected state space realizations become ˙yMT (t) = tMT go (−Puu(t) + Pvv(t)) (25.23) JMT = yMT tMT f (25.24) and ˙yMD(t) = tMD go (−Puu(t) + Pww(t)) (25.25) JMD = yMD tMD f (25.26) It is important to note that since the projection matrices are functions of the state; namely, Pi = Pi(x) where i = M, T, D, the derivatives of the ZEM variables are also functions of the state. As a result, integration is possible only for a specific set of initial conditions. 113
- 130. 25.1.2 A Simple Projected Differential Game Similarly to Section 17, we analyze a differential game of two hypothetical players: the pursuer (P) and the evader (E). The pursuer can apply a bounded acceleration of aP ≤ amax P , projected by the matrix PP , and the evader can apply a bounded acceleration of aE ≤ amax E , projected by PE, so that PP aP ∈ AP ⊂ R3 and PEaE ∈ AE ⊂ R3 for any aP , aE ∈ R3 . For these players we have XPE tPE f , t = −tPE go PP (25.27) YPE tPE f , t = tPE go PE (25.28) The ZEM projected state space model is, ˙yPE = (−PP aP + PEaE) tPE go (25.29) JPE = yPE tPE f (25.30) Define the ZEM norm, VPE = yPE (25.31) Differentiating (25.31) with respect to t yields, ˙VPE = yPE yPE (−PP aP + PEaE) tPE go (25.32) Therefore, the optimal controllers are, a∗ P = amax P PP yPE PP yPE (25.33) a∗ E = amax E PEyPE PEyPE (25.34) Substitute into (25.32), and obtain ˙V∗ PE = tPE go yPE yPE (−PP a∗ P + PEa∗ E) = tPE go yPE yPE −amax P PP PP yPE PP yPE + amax E YPE PEyPE PEyPE = −amax P PP yPE PP yPE PP yPE · yPE + amax E PEyPE PEyPE PEyPE · yPE = −amax P PP yPE 2 PP yPE · yPE + amax E PEyPE 2 PEyPE · yPE = −amax P PP yPE yPE + amax E PEyPE yPE = −amax P PP ˆyPE + amax E PE ˆyPE (25.35) Since 0 = σmin PP ≤ PP ˆyPE ≤ σmax PP = 1 and 0 = σmin PE ≤ PE ˆyPE ≤ σmax PE = 1, where σi PP and σi PE are the singular values of PP and PE respectively, neither the pursuer nor the evader can guarantee the sign of ˙V∗ PE, and therefore, cannot guarantee the miss distance. 114
- 131. 25.1.3 M-T-D Projected Game Using the same logic as before, the optimal guidance strategies are u∗ = −ρu PuyMD PuyMD , yMD < C tMD go ρu PuyMT PuyMT , yMD ≥ C tMD go (25.36) v∗ = ρv PvyMT PvyMT (25.37) w∗ = −ρw PwyMD PwyMD (25.38) Since PVG does not guarantee terminal cost, there are no sufficient conditions which guarantee the outcome of the game; however, the fail-safe function C tMD go , defines the worst case situation for the Missile. Therefore, once yMD reaches C tMD go , M-D miss distance of is guaranteed. 25.1.4 M-T-D Projected Endo-Atmospheric Game Consider a game where all three players can apply bounded acceleration perpendicular to their velocity vector (Fig. 2.1). Denote Vi = vi1 vi2 vi3 where i = M, T, D as the players’ velocity vectors, and Vi⊥ as the subspaces of R3 orthogonal to Vi. In this case we have, Vi⊥ ∈ span −vi2 vi1 0 , −vi3 0 vi1 (25.39) Define Mi = −vi2 −vi3 vi1 0 0 vi1 (25.40) and obtain the projection matrices Pi = Mi Mi Mi −1 Mi = 1 v2 i1 + v2 i2 + v2 i3 v2 i2 + v2 i3 −vi1vi2 −vi1vi3 −vi1vi2 v2 i1 + v2 i3 −vi2vi3 −vi1vi3 −vi2vi3 v2 i1 + v2 i2 (25.41) Note that Pi are symetric; therefore, Pi = Pi. As a result, the guidance strategies become u∗ = −ρu Mu MuMu −1 MuyMD Mu(MuMu) −1 MuyMD , yMD < C tMD go ρu Mu MuMu −1 MuyMT Mu(MuMu) −1 MuyMT , yMD ≥ C tMD go (25.42) v∗ = ρv Mv MvMv −1 MvyMT Mv (MvMv)−1 MvyMT (25.43) w∗ = − ρw Mw MwMw −1 MwyMD Mw (MwMw)−1 MwyMD (25.44) 115
- 132. 25.1.5 PVG4 Simulations Example 25.1. Consider the following numerical values ρu = 170 m sec2 , ρv = 30 m sec2 , ρw = 60 m sec2 , m = 0.5 [m] , = 150 [m] Fig. 25.1 shows a comparison between LMG and PVG4. 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 (a) PVG4 Missile Target Defender tf MD t* 0 1000 2000 3000 4000 5000 6000 0 1000 2000 3000 4000 x [m] y[m] Miss MD = 325 , tf MD = 5.74 Miss MT = 0.1 , tf MT = 15.37 (b) LMG Figure 25.1: PVG4 vs. LMG By analyzing the simulations we conclude: 1. By using PVG4, the Target applies relatively small lateral acceleration and manages to es- cape. In contrast, by adopting LMG it applies strong lateral evasive maneuver and eventually gets intercepted. In fact, by analyzing a similar game10 in Part II (Example 13.7 on Pages 59–60), we have concluded that with this geometry, it is best for the Target to apply small lateral acceleration (in Example 13.7 the Target applies v = 0). PVG4 supports the results of Example 13.7; however, unlike LMG, PVG4 has no approximation factors11 . 2. PVG4 provides the Missile with the exactly required M-D miss distance, while in LMG scenario it is substantially bigger. 3. A slight increase in the Missile’s maneuvering capability (from 170 [m/sec2 ] to 180 [m/sec2 ]), enables the Missile intercepts the Target in PVG4 scenario (Example 13.7). In contrast, a substantial increase in ρu is needed in the scenario of Example 13.7. 10 The initial geometry, and players’ capabilities are the same. Only the required M-D miss distance is different; namely in Example 13.7 we had = 10 [m], while in the present example we have = 150 [m]. 11 Recall that in Example 13.7, we had to add a factor of te(kv) in order to account for the addition to ∆t, resulted from the Target’s evasive maneuver. In general, this factor can be obtained only from simulations, which makes LMG hardly implementable. 116
- 133. Example 25.2. For the following parameters, Fig. 25.2 depicts the outcome of PVG4 game. ρu = 180 m sec2 , ρv = 30 m sec2 , ρw = 60 m sec2 , m = 0.5 [m] , = 150 [m] Missile Target Defender 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 = 6.22 Miss MT = 0.5 , tf MT = 13.18 Figure 25.2: PVG4 Planar Simulation Example 25.3. A three dimensional version of such a conflict is shown in Fig. 25.3. Missile Target Defender tf MD t* 0 2000 4000 6000 x [m] 0 500 1000 1500 y [m] 0 1000 2000 z [m] Figure 25.3: PVG4 3D Simulation 117
- 134. 25.2 Generalization – Transformed Vector Guidance (TVG) 25.2.1 Elliptical Transformation Since players’ maneuver capability along and perpendicular to velocity vector is not equal, another way to describe a more realistic bound is an ellipsoid. Such a bound reflects the difference between axial and lateral acceleration capabilities. Consider a unitary matrix describing each player’s velocity vector framed orthonormal coordinate system. 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 (25.45) 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. Since the columns of Ui span an orthonormal basis of R3 , the matrices Ui are unitary and satisfy Ui = U−1 i . Consider a scaling matrix Σ = σa 0 0 0 σ⊥1 0 0 0 σ⊥2 (25.46) where σa scales each player’s axial acceleration capability and σ⊥1, σ⊥2 scale their lateral acceler- ation capability. Now, consider each player’s transformation matrix Ti = UiΣUi (25.47) The expression UiΣUi is a singular value decomposition (SVD) of Ti (here, the input and output directions of Ti are identical). As a result, the · 2 ball, which defines each player’s maneuvering capability in the standard VG scenario, transforms into a velocity vector framed ellipsoid. Note that when σa = 1 we have a standard · 2 bound, and when σa = 0, the players have only lateral maneuvering capability, equally described by PVG. Example 25.4. For σ⊥1 = 1, σ⊥2 = 1 and σa = {0, 0.25, 0.5, 1}, the acceleration bounds are depicted in Fig. 25.4. As a result of this transformation, we have the following state space model ˙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 −Tu 0 −Tu u + 0 Tv 0 0 v + 0 0 0 Tw w (25.48) where Tu = UuΣUu (25.49) Tv = UvΣUv (25.50) Tw = UwΣUw (25.51) 118
- 135. (a) σ = 1 (b) σ = 0.5 (c) σ = 0.25 (d) σ = 0 Figure 25.4: Elliptical Bounds By applying the same analysis as in Subsection 25.1, we obtain XMT tMT go = gΦ tMT go b = −tMT go Tu (25.52) YMT tMT go = gΦ tMT go c = tMT go Tv (25.53) ZMT tMT go = gΦ tMT go d = 0 (25.54) XMD tMD go = hΨ tMD go b = −tMD go Tu (25.55) YMD tMD go = hΨ tMD go c = 0 (25.56) ZMD tMD go = hΨ tMD go d = tMD go Tw (25.57) and the ZEM variables are the same, yMT = rMT + tMT go VMT (25.58) yMD = rMD + tMD go VMD (25.59) 119
- 136. 25.2.2 A Simple Transformed Differential Game Identically to Subsection 25.1, we analyze a differential game of two hypothetical players: the pursuer (P) and the evader (E). The pursuer can apply a bounded acceleration of aP ≤ amax P , transformed by the matrix TP , and the evader can apply a bounded acceleration of aE ≤ amax E , transformed by TE, for any aP , aE ∈ R3 . For these players we have ˙VPE = yPE yPE (−TP aP + TEaE) tPE go (25.60) Therefore, the optimal controllers are, a∗ P = amax P TP yPE TP yPE (25.61) a∗ E = amax E TEyPE TEyPE (25.62) Substitute into (25.60), and obtain ˙V∗ PE = −amax P TP ˆyPE + amax E TE ˆyPE (25.63) Since σmin ≤ TP ˆyPE ≤ σmax and σmin ≤ TE ˆyPE ≤ σmax, the pursuer can guarantee the sign of ˙V∗ PE if and only if σminamax P > σmaxamax E (25.64) where σmin = min {σa, σ⊥1, σ⊥2} (25.65) σmax = max {σa, σ⊥1, σ⊥2} (25.66) Hence, the pursuer can guarantee a miss distance value if and only if its minimal acceleration capability is greater then the maximal capability of the evader. Note that if σmin ≈ 0, then the pursuer’s minimal capability is small; hence, a miss distance value cannot be guaranteed. 25.2.3 M-T-D Projected Game Using the same logic as before, we have u∗ = −ρu TuyMD TuyMD , yMD < C tMD go ρu TuyMT TuyMT , yMD ≥ C tMD go (25.67) v∗ = ρv TvyMT TvyMT (25.68) w∗ = −ρw TwyMD TwyMD (25.69) 120
- 137. 25.2.4 TVG4 Simulations Example 25.5. Consider the following numerical values ρu = 170 m sec2 , ρv = 30 m sec2 , ρw = 60 m sec2 m = 0.5 [m] , = 150 [m] σ⊥1 = 1, σ⊥2 = 1 and σa = {0, 0.25, 0.5, 1}. Observe the planar simulation results in Fig. 25.5. 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 (a) σ = 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 (b) σ = 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 (c) σ = 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 (d) σ = 0 Figure 25.5: TVG4 Planar Simulations 121
- 138. Also, Fig. 25.6 shows the outcome of a similar three dimensional conflict. Missile Target Defender tf MD t* 0 2000 4000 6000 x [m] 0 1000 2000 3000 y [m] 0 1000 2000 z [m] (a) σ = 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] (b) σ = 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] (c) σ = 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] (d) σ = 0 Figure 25.6: TVG4 3D Simulations 122
- 139. 26 Estimator Based Vector Guidance Thus far, we assumed a full knowledge game; namely, the players had perfect information about rMT , rMD, VMT and VMD. However, such definition is not practical since the players don’t have all information. It is more reasonable to assume that the players have noisy measurements of rMT and rMD, while VMT and VMD are to be estimated. This section discusses an estimator based VG model and derives approximate bounds on the miss distance. 26.1 Missile – Target Game Since the actual three players’ game, discussed in this research, can actually be decomposed into two separate games, it makes sense to handle a classic game of two players. To simplify notifications, denote rMT and VMT as r and V respectively. 26.1.1 Model Definition Given the relative Missile-Target kinematics, ˙r(t) = V (t) (26.1) ˙V (t) = v(t) − u(t) (26.2) where r ∈ R3+ and V ∈ R3 are the relative position and velocity vectors, and u ∈ R3 and v ∈ R3 are the Missile’s and Target’s accelerations. Let the Missile have a noisy measurement of r(t), rm(t) = r(t) + I3n(t) (26.3) where n ∈ R3 is the additive noise. Note that we assume independent noise in all three directions. The state space realization is ˙r(t) ˙V (t) = 03 I3 03 03 r(t) V (t) + 03 −I3 u(t) + 03 I3 v(t) (26.4) rm(t) = I3 03 r(t) V (t) + I3n(t) (26.5) In this scenario, the Missile is able to measure only the noisy relative position vector rm(t) (probably by measuring the range and two angles). Therefore, we must • Estimate the relative velocity vector V (t) in order to compute the ZEM, ˆy(t) = ˆr(t) + tgo ˆV (t) (26.6) where ˆr and ˆV are the estimation of r and V , and ˆy is the estimated ZEM. • Account for noise, as it might have destructive effect on the guidance system. Generally, a wide range of noise models can be analyzed in order to simulate the actual noise present in the guidance system. This work overviews the worst-case noise model, and discusses the white noise model. 123
- 140. 26.1.2 Luenberger Observer and Pole Placement The basic Luenberger Observer equation is ˙ˆr(t) ˙ˆV (t) = 03 I3 03 03 ˆr(t) ˆV (t) + 03 −I3 u(t) + L rm(t) − I3 03 ˆr(t) ˆV (t) (26.7) where, L = Lr ∈ R3×3 LV ∈ R3×3 (26.8) is our design parameter set. Although Ackerman’s formula is incompatible with MIMO systems, in this particular case, we place two poles over three independent dimensions. Thus, Ackerman’s formula can be used as follows, = r V = ˆχcl (As) M−1 O 0 1 (26.9) where the SISO realization matrices are As = 0 1 0 0 (26.10) Cs = 1 0 (26.11) the gain satisfies, = r ∈ R V ∈ R (26.12) the observability matrix is, MO = Cs CsAs = 1 0 0 1 = I2 (26.13) and ˆχcl(s) = s2 + a1s + a0 is the desired characteristic polynomial of AL = As − Cs. As a result, (26.9) becomes r V = ˆχcl (As) 0 1 = a1 a0 (26.14) Thus, the characteristic polynomial becomes, ˆχcl(s) = s2 + rs + V , and the observer gain reduces to L = Lr LV = rI3 V I3 (26.15) where r and V are the design parameters12 . Although Luenberger observer provides a simple way to deal with the observer’s performance, pole placement does not provide any indication about noise compensation or disturbance attenuation. Thus, an alternative model can be examined. 12 Ackerman’s formula is just one possible way to place poles. The alternatives are equivalent. 124
- 141. 26.1.3 Kalman Filter Another possibility is to use the Kalman Filter form. Consider the following state space model ˙r(t) ˙V (t) = 03 I3 03 03 r(t) V (t) + 03 −I3 u(t) + 03 03 03 I3 vw(t) (26.16) rm(t) = I3 03 r(t) V (t) + I3 √ σnnw(t) (26.17) where vw = vw1 ∈ R3 vw2 ∈ R3 and nw ∈ R3 are unit intensity white noise signals, and the weight- ing matrices Bv = 03 03 03 I3 and Sn = I3 √ σn are the covariance of the disturbances, vw, and measurement noise, nw. In such a case, the observer equation is still, ˙ˆr(t) ˙ˆV (t) = 03 I3 03 03 ˆr(t) ˆV (t) + 03 −I3 u(t) + L rm(t) − I3 03 ˆr(t) ˆV (t) (26.18) and the optimal observer gain is L = 1 σn QC (26.19) where Q is the solution of CARE QA + AQ − 1 σ QC CQ = 0 (26.20) This way, we are left with only one design parameter, σn. By increasing it we require faster disturbance (Target maneuver) attenuation at the cost of higher noise sensitivity. Note that here we assume that vw(t) and nw(t) are white; however, in real guidance systems the actual disturbance (Target maneuver), v(t), and measurement noise, n(t), are not necessarily white. Therefore, if any additional information is known about v(t) or n(t), shaping filters can be used in order to improve the observer’s performance. For example, let the actual measurement noise, n(t), have a power spectral density Φn(ω) ≥ 0, then there exists a rational Wn(s) having no poles and zeros in Re s > 0 satisfying Φn(ω) = W∗ n (jω)Wn(jω) (26.21) In such a case, nw = W−1 n (s)n is white with unit intensity. Also, let Wn(s) have a (minimal) state space realization ˙xn(t) = Anxn(t) + Bnnw(t) (26.22) n(t) = Cnxn(t) + nw(t) (26.23) where xn ∈ Rl . Then, the augmented state space realization of the model and noise becomes ˙r(t) ˙V (t) ˙xn(t) = 03 I3 03×l 03 03 03×l 0l×3 0l×3 An r(t) V (t) xn(t) + 03 −I3 0l×3 u(t) + 03 03 03 03 I3 03 0l×3 0l×3 Bn vw1(t) vw2(t) nw(t) rm(t) = I3 03 √ σnCn r(t) V (t) xn(t) + I3 √ σnnw(t) (26.24) 125
- 142. 26.1.4 Schematic Model The estimated ZEM is ˆy(t) = ˆr(t) + tgo ˆV (t) (26.25) Thus, the optimal Missile’s guidance strategy becomes u∗ = ρu ˆy(t) ˆy(t) (26.26) In Fig. 26.1 one has a schematic block diagram of an estimator based two players vector guidance. 1 sI3 1 sI3 C1 sI6−B Guidance Law A L × Kinematics u v v − u V r n rm ˙ˆr ˙ˆV ˆr ˆV ˆr ˆV tgo ˆr rm − ˆr ˆy − u Figure 26.1: Estimator Based VG Block Diagram Here, a noisy measurement of r(t), and the Missile’s acceleration u(t), are passed through an observer. The estimated relative position and velocity vectors, ˆr(t) and ˆV (t), are used to compute the estimated ZEM, ˆy(t). In some configurations, one might prefer to use the noisy rm(t) instead of ˆr(t), and obtain ˆy(t) = rm(t) + tgo ˆV (t). However, simulations show that ˆy(t) = ˆr(t) + tgo ˆV (t) yields better results, although, this topic can use further research. 126
- 143. 26.1.5 Estimation Error Observer estimation error, e(t) = x(t) − ˆx(t) ,is ˙e(t) = (A − LC) e(t) − Bv(t) − Ln(t) (26.27) where13 L = Lr LV = rI3 V I3 (26.28) and r, V ∈ R. The explicit form of (26.27) is ˙er(t) ˙eV (t) = −Lr I3 −LV 03 er(t) eV (t) − 03 −I3 v(t) − Lr LV n(t) (26.29) Alternatively, rewrite (26.29) as ˙er(t) ˙eV (t) = −Lr I3 −LV 03 AL er(t) eV (t) − 03 Lr −I3 LV BL v(t) n(t) (26.30) As a result, the transfer matrix G : v n → er eV is G(s) = AL BL I6 06 = 1 s2+ rs+ V I3 − rs+ V s2+ rs+ V I3 s+ r s2+ rs+ V I3 vs s2+ rs+ V I3 (26.31) The matrix G(s) is symmetric in all three directions; therefore, this problem can be analyzed as a one dimensional problem14 ; namely, in each direction of the 3D space we have the transfer matrix Gs : vi ni → eri eVi , Gs(s) = 1 s2+ rs+ V − rs+ V s2+ rs+ V s+ r s2+ rs+ V vs s2+ rs+ V where i = x, y, z. Recall the estimated ZEM is ˆy(t) = ˆr(t) + tgo ˆV (t) (26.32) while the real ZEM is y(t) = r(t) + tgoV (t) (26.33) Therefore, the estimation error, ey = ˆy − y, is ey(t) = (ˆr(t) − r(t)) + tgo ˆV (t) − V (t) (26.34) = er(t) + tgoeV (t) (26.35) 13 In both Luenberger Observer and Kalman Filter configurations we obtain the form described in (26.28) due to the independence of each direction. 14 In order to separate the dimensions, we also have to bound the disturbance (Target’s maneuver) and noise with · ∞, which is more conservative than · 2 127
- 144. 26.1.6 Worst Case Bound By using convolution, we have ey(t) = ˆ t 0 g11(t − τ)v(τ)dτ + ˆ t 0 g12(t − τ)n(τ)dτ (26.36) + tgo ˆ t 0 g21(t − τ)v(τ)dτ + ˆ t 0 g22(t − τ)n(τ)dτ where g(t) = L−1 {G(s)}. Alternatively, in each direction (26.36) can be written as eyi (t) = ˆ t 0 gs11 (τ)vi(t − τ)dτ + ˆ t 0 gs12 (τ)ni(t − τ)dτ (26.37) + tgo ˆ t 0 gs21 (τ)vi(t − τ)dτ + ˆ t 0 gs22 (τ)ni(t − τ)dτ where gs(t) = L−1 {Gs(s)}, and i = x, y, z. Assuming bounded maneuver, v 2 ≤ v ∞ ≤ ρv, and bounded noise, n 2 ≤ n ∞ ≤ ρn, in each direction we have ˆ t 0 gs11 (τ)vi(t − τ)dτ ≤ ρv ˆ t 0 |gs11 (τ)| dτ (26.38) ˆ t 0 gs12 (τ)ni(t − τ)dτ ≤ ρn ˆ t 0 |gs12 (τ)| dτ (26.39) ˆ t 0 gs21 (τ)vi(t − τ)dτ ≤ ρv ˆ t 0 |gs21 (τ)| dτ (26.40) ˆ t 0 gs22 (τ)ni(t − τ)dτ ≤ ρn ˆ t 0 |gs22 (τ)| dτ (26.41) where the worst case maneuvers and noises are v∗ 11 = ρvsign(gs11 ), n∗ 12 = ρnsign(gs12 ) (26.42) v∗ 21 = ρvsign(gs21 ), n∗ 22 = ρnsign(gs22 ) (26.43) As a result, eyi (t) ≤ er/v(t) + er/n(t) + tgoeV/v(t) + tgoeV/n(t) (26.44) where er/v(t) = ρv ˆ t 0 |gs11 (τ)| dτ (26.45) er/n(t) = ρn ˆ t 0 |gs12 (τ)| dτ (26.46) eV/v(t) = ρv ˆ t 0 |gs21 (τ)| dτ (26.47) eV/n(t) = ρn ˆ t 0 |gs22 (τ)| dτ (26.48) Although er/v and eV/v produce admissible results, er/n and eV/n are too conservative. Therefore, in this work we derive a white noise bound. 128
- 145. 26.1.7 White Noise Model Since it is practically possible to measure the M-T range and 2 LOS angles, we assume that r = rx ry rz = R sin θ cos λ R sin θ sin λ R cos θ (26.49) where R ∈ R+ is the range, and θ ∈ [0, 2π] , λ ∈ [0, π] are the azimuthal and elevation LOS angles. Since the range can be measured much more accurately then LOS angles, we assume that R is perfectly known, while θ, λ are noisy. Since noisy signals are multiplied by perfectly known range, we approximate the noise model as range dependent, n(t) = r(t)knρnnw(t) (26.50) where nw(t) is a unit intensity white noise, ρ2 n = E n2 w(t) is the noise variance, and kn is a scaling factor. Thus, the measured relative position is rm(t) = r(t) + n(t) = r(t) (1 + knρnnw(t)) (26.51) Note that knρnnw(t) has a power spectral density of ϕn = k2 nρ2 n (26.52) An important property of this model is that rm approaches r when tgo approaches zero. 26.1.8 White Noise Bound Generally, we have eyi (t) ≤ er/v(t) + er/n(t) + tgoeV/v(t) + tgoeV/n(t) (26.53) where i = x, y, z. Assume worst case bound for the Target’s maneuver; hence, er/v(t) = ρv ˆ t 0 |gs11 (τ)| dτ (26.54) eV/v(t) = ρv ˆ t 0 |gs21 (τ)| dτ (26.55) and derive a white noise bound for er/n and eV/n. Recall that n(t) = r(t)knρnnw(t) (26.56) By using convolution, one has er/n(t) = ˆ t 0 gs12 (τ)ri(t − τ)knρnnw(t − τ)dτ (26.57) eV/n(t) = ˆ t 0 gs22 (τ)ri(t − τ)knρnnw(t − τ)dτ (26.58) 129
- 146. 26.1.9 Optimal Maneuver Approximation Since er/n and eV/n depend on r(t), which depends on players’ strategies, the situation is unclear. However, if the players apply optimal maneuvers, r(t) becomes a solution of (22.26). r∗ (t) = − y0 t2 f t2 + tV0 + r0 (26.59) where y0 = r0 + tf V0. Let us assume that r∗ (t) ≈ r(t). Simulations justify this assumption even when the Target does not play optimal. Thus, r∗ (t − τ) = − y0 t2 f (t − τ)2 + (t − τ) V0 + r0 = − y0 t2 f t2 + tV0 + r0 − − 2y0 t2 f t + V0 τ − y0 t2 f τ2 (26.60) = r∗ (t) − V ∗ (t) · τ + 1 2 a∗ (t) · τ2 where V ∗ (t) = ˙r∗ (t) and a∗ (t) = ¨r∗ (t). Thus, (26.57) becomes e∗ r/n(t) = ˆ t 0 gs12 (τ)r∗ i (t − τ)knρnnw(t − τ)dτ = ˆ t 0 gs12 (τ) r∗ i (t) − V ∗ i (t) · τ + 1 2 a∗ i (t) · τ2 knρnnw(t − τ)dτ = r∗ i (t) ˆ t 0 gs12 (τ)knρnnw(t − τ)dτ (26.61) − V ∗ i (t) ˆ t 0 τ · gs12 (τ)knρnnw(t − τ)dτ + 1 2 a∗ i (t) ˆ t 0 τ2 gs12 (τ)knρnnw(t − τ)dτ Define: α1 = ˆ t 0 g2 s12 (τ)dτ (26.62) β1 = ˆ t 0 (τ · gs12 (τ))2 dτ (26.63) γ1 = ˆ t 0 (τ2 · gs12 (τ))2 dτ (26.64) and obtain the RMS bound 130
- 147. e∗ r/n(t) ≤ r∗ i (t) ˆ t 0 g2 s12 (τ)k2 nρ2 ndτ − V ∗ i (t) ˆ t 0 (τ · gs12 (τ))2 k2 nρ2 ndτ (26.65) + 1 2 a∗ i (t) ˆ t 0 (τ2 · gs12 (τ))2 k2 nρ2 ndτ = knρn r∗ i (t) · α1 − V ∗ i (t) · β1 + 1 2 a∗ i (t) · γ1 (26.66) Note that kn is the noise deviation scaling factor, so that e∗ r/n(t) bounds kn of the noise standard deviation. For example, by choosing kn = 3, we bound 3ρn of the noise, which is approximately 95%. Similarly, the second bound is e∗ V/n(t) ≤ knρn r∗ i (t) · α2 − V ∗ i (t) · β2 + 1 2 a∗ i (t) · γ2 (26.67) where α2 = ˆ t 0 g2 s22 (τ)dτ (26.68) β2 = ˆ t 0 (τ · gs22 (τ))2 dτ (26.69) γ2 = ˆ t 0 (τ2 · gs22 (τ))2 dτ (26.70) Therefore, the approximated deviation from the nominal ZEM trajectory is e∗ yi (t) ≈ e∗ r/v(t) + e∗ r/n(t) + tgoe∗ V/v(t) + tgoe∗ V/n(t) (26.71) Remark 26.1. The bounds e∗ r/n(t) and e∗ V/n(t) are precise only if both players play optimal. How- ever, simulations show that even when the Target does not play optimal, the relative displacement function r(t) is very close to the optimal function r∗ (t); therefore, er/n(t) and eV/n(t) are close to e∗ r/n(t) and e∗ V/n(t). Remark 26.2. It is impossible to bound white noise. However, since white noise is normally distributed we can bound a certain percentage of it. For example, by choosing kn = 3, we put our bound on approximately 95% of the white noise. Remark 26.3. The white noise bound is much more tight than the worst case bound; thus, provides more accurate results if the noise is actually white. However, some noise structures (such as slow wave noise) are much more dangerous for guidance systems than white noise. Therefore other noise structures should be also considered by the designer (out of the scope of this research). 131
- 148. 26.1.10 Miss Distance Bound Approximation for VG4 Under the assumption of optimal maneuver game, the final time, tf , is constant. Therefore, it is reasonable to wrap the nominal ZEM norm with a threshold that bounds ˆy(t) . Define J1(t) = 1 2 (ρu − ρv) t2 go + e∗ yi (t) (26.72) J2(t) = 1 2 (ρu − ρv) t2 go − e∗ yi (t) (26.73) J3(t) = 1 2 (ρu − ρv) t2 go − e∗ yi (t) + h(t) (26.74) where h(t) = inf 1 2 (ρu − ρv) t2 go − e∗ yi (t) . Thus, the upper and the lower bounds are Jui (t) = max {J1(t), J3(t)} (26.75) Jli (t) = max {0, J2(t)} (26.76) and the estimated ZEM satisfies Jli (t) ≤ ˆyi(t) ≤ Jui (t) (26.77) As a result, the miss distance bound in each direction is mi = Jui (tf ) (26.78) where i = x, y, z. Thus, the predicted M-T miss distance is m = mi √ N, where N = 2 or 3 for planar or 3D scenario respectively. Similarly, Ju = Jui √ N and Jl = Jli √ N. A qualitative plot of the nominal ZEM norm and its bounds is presented in Fig. 26.2. u ℓ 0.5(ρu -ρv )tgo 2 tf MT Time, t0 m ||ZEM|| Figure 26.2: Nominal ZEM and its Bounds 132
- 149. 26.1.11 Estimator Based Two Players VG4 Simulations Let the players have, ρu = 50 m sec2 , ρv = 20 m sec2 , and let Target perform v∗ = ρvyMT / yMT when tgo > ts, and v = ρvVT⊥/ VT⊥ when tgo ≤ ts, where VT⊥ is normal to VT , and ts = 0.5 [sec] is the switch time. Using the Kalman Filter configuration of 26.1.3, we obtain the results in Fig. 26.3 for σn = {10−4 , 10−5 , 10−6 }. σn Planar Simulation ZEM Trajectory 10−4 Missile Target 0 1000 2000 3000 4000 5000 6000 0 200 400 600 x [m] y[m] Miss Distance = 0.415 ||yMT|| ||y MT|| Bound tf MT Time, t ||ZEM|| Guaranteed Miss Distance = 0.426 [m] 10−5 Missile Target 0 1000 2000 3000 4000 5000 6000 0 200 400 600 x [m] y[m] Miss Distance = 0.133 ||yMT|| ||y MT|| Bound tf MT Time, t ||ZEM|| Guaranteed Miss Distance = 0.151 [m] 10−6 Missile Target 0 1000 2000 3000 4000 5000 6000 0 200 400 600 x [m] y[m] Miss Distance = 0.078 ||yMT|| ||y MT|| Bound tf MT Time, t ||ZEM|| Guaranteed Miss Distance = 0.064 [m] Figure 26.3: Estimator Based Two Players VG4 Simulations Remark 26.4. ˆyMT is not inside its bound at the beginning because of initial conditions. 133
- 150. 26.2 Missile – Defender Game 26.2.1 Optimal Maneuver Approximation To simplify notations, denote rMD as r, VMD as V , and tMD f as tf . Similarly, when the Missile evades the Defender, the optimal relative displacement is, r∗ (t) = t2 − t2 f + T 2 r0 + t (T 2 − (tf − t) · tf ) V0 T 2 − t2 f (26.79) where T = 2q ρu−ρw [sec], and q is the nominal expected miss distance. Thus, r∗ (t − τ) = r∗ (t) − V ∗ (t) · τ + 1 2 a∗ (t) · τ2 (26.80) where V ∗ (t) = ˙r∗ (t) = 2t · r0 + 2t · tf − t2 f + T 2 V0 T 2 − t2 f and a∗ (t) = ¨r∗ (t) = 2 (r0 + tf V0) T 2 − t2 f Similarly, e∗ r/n(t) satisfies, e∗ r/n(t) ≤ knρn r∗ (t) · α1 − V ∗ (t) · β1 + 1 2 a∗ (t) · γ1 (26.81) where α1 = ˆ t 0 g2 12(τ)dτ (26.82) β1 = ˆ t 0 (τ · g12(τ))2 dτ (26.83) γ1 = ˆ t 0 (τ2 · g12(τ))2 dτ (26.84) and kn is the noise deviation scaling factor, so that e∗ r/n(t) bounds kn of the noise standard devia- tion, ρn. Also, e∗ V/n(t) ≤ knρn r∗ (t) · α2 − V ∗ (t) · β2 + 1 2 a∗ (t) · γ2 (26.85) where α2 = ´ t 0 g2 22(τ)dτ, β2 = ´ t 0 (τ · g22(τ))2 dτ, and γ2 = ´ t 0 (τ2 · g22(τ))2 dτ. Hence, the approximated deviation from the nominal ZEM trajectory is e∗ yi (t) = e∗ r/v(t) + e∗ r/n(t) + tgoe∗ V/v(t) + tgoe∗ V/n(t) (26.86) 134
- 151. 26.2.2 Miss Distance Bound Approximation for VG4 Similarly to M-T game, define J1(t) = q − 1 2 (ρu − ρw) t2 go + e∗ yi (t) (26.87) J2(t) = q − 1 2 (ρu − ρw) t2 go − e∗ yi (t) (26.88) and obtain the bounds, Jui (t) = J1(t) (26.89) Jli (t) = max {0, J2(t)} (26.90) As a result, Jli (tf ) ≤ qi ≤ Jui (tf ) (26.91) Similarly to the M-T game, Ju = Jui √ N, Jl = Jli √ N, and q = qi √ N 26.2.3 Estimator Based Two Players VG4 Simulations Consider the following numerical values ρu = 50 m sec2 , ρw = 20 m sec2 Both players play optimal. Set σn = 10−5 and obtain the results depicted in Fig. 26.4. Missile Defender 0 1000 2000 3000 4000 5000 6000 -1000 -500 0 500 x [m] y[m] Miss Distance = 1180 (a) Planar Simulation ||yMD|| ||y MD|| Bound tf MD Time, t ||ZEM|| Guaranteed Miss Distance = 1178 [m] (b) ZEM Trajectories Figure 26.4: Estimator Based Two Players VG4 Simulations 135
- 152. 27 A Non-Ideal Players Game 27.1 Basic definitions Consider again the three players VG interception scenario as depicted in Fig. 27.1 u VM rM v VT rT w VD rD rMT rMD rTD Figure 27.1: Planar Interception Geometry Unlike previous sections, here the Missile’s dynamics is not ideal, and described by the transfer matrix, GM : uC → u, GM (s) = AM BM CM DM (27.1) where uC, u ∈ R3 , and GM (s) ∈ RH∞ . The state equations of GM (s) are ˙η(t) = AM η(t) + BM uC(t) u(t) = CM η(t) + DM uC(t) (27.2) The dynamic equations become ¨rMD(t) = w(t) − u(t) = w(t) − CM η(t) − DM uC(t) (27.3) ¨rMT (t) = v(t) − u(t) = v(t) − CM η(t) − DM uC(t) (27.4) and the following state space model is obtained ˙rMT (t) ¨rMT (t) ˙rMD(t) ¨rMD(t) ˙η(t) = 0 I3 0 0 0 0 0 0 0 −CM 0 0 0 I3 0 0 0 0 0 −CM 0 0 0 0 AM rMT (t) ˙rMT (t) rMD(t) ˙rMD(t) η(t) + 0 −DM 0 −DM BM uC(t)+ 0 I3 0 0 0 v(t)+ 0 0 0 I3 0 w(t) Note that in this discussion the Target and the Defender have ideal dynamics; namely, GT (s) = GD(s) = I3. 136
- 153. In Fig. 27.2, one finds a block diagram of the open guidance loop. GM (s) 1 s · I3 1 s · I3 1 s · I3 1 s · I3 uC _ _u w v ˙rMD ˙rMT rMD rMT Figure 27.2: Open Loop State Space Define the final times tMD f and tMT f , two cost functions JMT = I3 0 0 0 0 g x tMT f = gx tMT f (27.5) JMD = 0 0 I3 0 0 h x tMD f = hx tMD f (27.6) and two Zero Effort Miss variables. yMT (t) = gΦ tMT f , t x(t) (27.7) ˙Φ tMT f , t = −Φ tMT f , t A , Φ tMT f , tMT f = I (27.8) yMD(t) = hΨ tMD f , t x(t) (27.9) ˙Ψ tMD f , t = −Ψ tMD f , t A , Ψ tMD f , tMD f = I (27.10) Differentiate the ZEM variables ˙yMT (t) = XMT tMT f , t u(t) + YMT tMT f , t v(t) + ZMT tMT f , t w(t) (27.11) ˙yMD(t) = XMD tMD f , t u(t) + YMD tMD f , t v(t) + ZMD tMD f , t w(t) (27.12) Similarly to Part I, we obtain the Missile-Target ZEM variable, yMT (t) = rMT (t) + tMT go ˙rMT (t) − L−1 MT CM (sI − AM )−1 s2 η(t) (27.13) as well as, XMT tMT go = −L−1 MT GM (s) s2 (27.14) YMT tMT go = tMT go I3 (27.15) ZMT tMT go = 03 (27.16) 137
- 154. Similarly, the M-D ZEM is, yMD(t) = rMD(t) + tMD go ˙rMD(t) − L−1 MD CM (sI − AM )−1 s2 η(t) (27.17) as well as, XMD tMD go = −L−1 MD GM (s) s2 (27.18) YMD tMD go = 03 (27.19) ZMD tMD go = tMD go I3 (27.20) As a result, we have two ZEM projected systems. The M-T system: ˙yMT (t) = XMT tMT f , t u(t) + YMT tMT f , t v(t) (27.21) JMT = yMT tMT f (27.22) and the M-D system: ˙yMD(t) = XMD tMD f , t u(t) + ZMD tMD f , t w(t) (27.23) JMD = yMD tMD f (27.24) Define the ZEM norms. VMT (t) = yMT (t) (27.25) VMD(t) = yMD(t) (27.26) Differentiate them and obtain, ˙VMT = yMT yMT (XMT u + YMT v) (27.27) ˙VMD = yMD yMD (XMDu + ZMDw) (27.28) 138
- 155. 27.2 A Differential Game of Two Players 27.2.1 General Differential Game Similarly to Part I, observe two game situations Missile-Target Game In this game, the Missile pursues the Target, while the Target evades the Missile. From (27.27) we have the optimal guidance strategies: u∗ Cp = −ρu XMT yMT XMT yMT (27.29) v∗ = ρv YMT yMT YMT yMT (27.30) Substitute into (27.27), and obtain ˙V∗ MT = yMT yMT XMT u∗ p + YMT v∗ = = yMT yMT −ρuXMT XMT yMT XMT yMT + ρvYMT YMT yMT YMT yMT = −ρu (XMT yMT ) (XMT yMT ) XMT yMT · yMT + ρv (YMT yMT ) (YMT yMT ) YMT yMT · yMT = −ρu XMT yMT 2 XMT yMT · yMT + ρv YMT yMT 2 YMT yMT · yMT = −ρu XMT yMT yMT + ρv YMT yMT yMT = −ρu −L−1 MT {GM (s)/s2 } yMT yMT + ρvtMT go (27.31) Missile-Defender Game Here, the Missile evades the Defender, while the Defender pursues the Missile. From (27.28),the optimal strategies are: u∗ Ce = ρu XMDyMD XMDyMD (27.32) w∗ = −ρw ZMDyMD ZMDyMD (27.33) Similarly, we obtain ˙V∗ MD = yMD yMD (XMDu∗ e + ZMDw∗ ) = ρu XMDyMD yMD − ρw ZMDyMD yMD = ρu −L−1 MD {GM (s)/s2 } yMD yMD − ρwtMD go (27.34) 139
- 156. 27.2.2 Isotropic Dynamics In order to simplify matters, assume isotropic dynamics; namely, GM (s) = Gs(s) · I3. Thus, ux(s) uy(s) uz(s) = Gs(s) 0 0 0 Gs(s) 0 0 0 Gs(s) GM (s) uCx (s) uCy (s) uCz (s) (27.35) where Gs(s) = As bs cs ds ∈ RH∞ is a SISO transfer function. In such a case, we have XMT tMT go = −L−1 MT Gs(s)/s2 I3 (27.36) XMD tMD go = −L−1 MD Gs(s)/s2 I3 (27.37) YMT tMT go = tMT go I3 (27.38) ZMD tMD go = tMD go I3 (27.39) and the ZEM variables are, yMT (t) = rMT (t) + tMT go ˙rMT (t) − ηsx (t) ηsy (t) ηsz (t) L−1 MT cs (sI − As)−1 /s2 (27.40) yMD(t) = rMD(t) + tMD go ˙rMD(t) − ηsx (t) ηsy (t) ηsz (t) L−1 MD cs (sI − As)−1 /s2 (27.41) where ηsx , ηsy , and ηsz are the dynamic state vectors in x, y, and z directions. Thus, ˙V∗ MT = −ρu −L−1 MT {GM (s)/s2 } yMT yMT + ρvtMT go = −ρu −L−1 MT {Gs(s)/s2 } I3yMT yMT + ρvtMT go (27.42) = −ρu L−1 MT Gs(s)/s2 + ρvtMT go and, ˙V∗ MD = ρu −L−1 MD {GM (s)/s2 } yMD yMD − ρwtMD go = ρu −L−1 MD {Gs(s)/s2 } I3yMD yMD − ρwtMD go (27.43) = ρu L−1 MD Gs(s)/s2 − ρwtMD go 140
- 157. 27.2.3 First Order Isotropic Dynamics To simplify even more, assume first-order dynamics: Gs(s) = 1 τM s + 1 = − 1 τM 1 τM 1 0 (27.44) The dynamic equations become ¨rMD(t) = w(t) − u(t) (27.45) ¨rMT (t) = v(t) − u(t) (27.46) ˙u(t) = − I3 τM u(t) + I3 τM uC(t) (27.47) and 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) The ZEM projected variables become, XMT tMT go = −τM tMT go τM − 1 + e−tMT go /τM I3 (27.48) XMD tMD go = −τM tMD go τM − 1 + e−tMD go /τM I3 (27.49) YMT tMT go = tMT go I3 (27.50) ZMD tMD go = tMD go I3 (27.51) as well as, yMT (t) = rMT (t) + tMT go ˙rMT (t) − τ2 M tMT f − t τM − 1 + e−(tMT f −t)/τM u(t) (27.52) yMD(t) = rMD(t) + tMD go ˙rMD(t) − τ2 M tMD f − t τM − 1 + e−(tMD f −t)/τM u(t) (27.53) and the optimal ZEM derivatives become ˙V∗ MT = −ρu L−1 MT Gs(s)/s2 + ρvtMT go = (−ρu + ρv) tMT go + ρuτM 1 − e−tP E go /τM (27.54) ˙V∗ MD = ρu L−1 MD Gs(s)/s2 − ρwtMD go = (ρu − ρw) tMD go − ρuτM 1 − e−tP E go /τM (27.55) 141
- 158. Integration yields y∗ MT tMT go = m + 1 2 (ρu − ρv) tMT go 2 − ρuτ2 M tMT go τM − 1 + e−tMT go /τM (27.56) y∗ MD tMD go = − 1 2 (ρu − ρw) tMD go 2 + ρuτ2 M tMD go τM − 1 + e−tMD go /τM (27.57) In Fig. 27.3 one finds the optimal trajectories in the ZEM plane. tgos tgo MT m* ||yMT || (a) y∗ MT tMT go tgoc tgo MD ||yMD|| (b) y∗ MD tMD go Figure 27.3: ZEM Trajectories where the marked areas are the singular areas in which the optimal strategies are arbitrary, and m∗ is the minimal M-T miss distance, defined as m∗ = inf 1 2 (ρu − ρv) tMT go 2 − ρuτ2 M tMT go τM − 1 + e−tMT go /τM (27.58) 142
- 159. 27.3 Optimal Strategies for Constant Final Times 27.3.1 Basic Optimal Strategies Define the function ψ(tgo) = tgo τM − 1 + e−tgo/τM (27.59) The players’ roles regarding the pursuit-evasion strategies remain unchanged. Namely, the Target is still the maximizer of yMT , v∗ = ρv YMT yMT YMT yMT = ρv rMT + tMT go ˙rMT − τ2 M ψ tMT go u rMT + tMT go ˙rMT − τ2 M ψ tMT go u (27.60) The Defender is the minimizer of yMD , w∗ = −ρw ZMDyMD ZMDyMD = −ρw rMD + tMD go ˙rMD − τ2 M ψ tMD go u rMD + tMD go ˙rMD − τ2 M ψ tMD go u (27.61) and the Missile has 2 objectives. To maximize yMD , u∗ Ce = ρu XMDyMD XMDyMD = ρu −τM ψ tMD go I3yMD −τM ψ tMD go I3yMD (27.62) = −ρusign ψ tMD go rMD + tMD go ˙rMD − τ2 M ψ tMD go u rMD + tMD go ˙rMD − τ2 M ψ tMD go u and to minimize of yMT , u∗ Cp = −ρu XMT yMT XMT yMT = −ρu −τM ψ tMT go I3yMT −τM ψ tMT go I3yMT (27.63) = ρusign ψ tMT go rMT + tMT go ˙rMT − τ2 M ψ tMT go u rMT + tMT go ˙rMT − τ2 M ψ tMT go u It is easy to show that ψ(tgo) is a monotonically increasing function15 of tgo, and ψ(0) = 0. Therefore, sign (ψ(tgo)) > 0 ∀tgo > 0 (27.64) As a result, u∗ Cp and u∗ Ce reduce to u∗ Ce = −ρu rMD + tMD go ˙rMD − τ2 M ψ tMD go u rMD + tMD go ˙rMD − τ2 M ψ tMD go u (27.65) u∗ Cp = ρu rMT + tMT go ˙rMT − τ2 M ψ tMT go u rMT + tMT go ˙rMT − τ2 M ψ tMT go u (27.66) As before, define m as the desired M-T miss distance (note that now m ≥ m∗ ), and as the desired M-D miss distance, and from (27.56) and (27.57) obtain the bound functions, A and B (Fig. 27.4). A tMD go = − 1 2 (ρu − ρw) tMD go 2 + ρuτ2 M ψ tMD go (27.67) B tMT go = m + 1 2 (ρu − ρv) tMT go 2 − ρuτ2 M ψ tMT go (27.68) 15 In fact, since ψ(tgo) is a scaled ramp response of Gs(s), this statement is true ∀Gs(s) ∈ RH∞ , whose zeros are in OLHP. 143
- 160. (t) ℬ(t) ℓ m tf MD tf MT Time, t ||ZEM|| Figure 27.4: Functions A and B The functions, A and B are the bounds, in which yMT and yMD are to be if the Missile wants to guarantee the desired M-T and M-D miss distances. 27.3.2 Fail-safe Function C tMD go Similarly to Subsection 19.2, we derive a fail-safe function for the M-D game. Let the Missile pursue the Target with u∗ Cp, and let the Defender pursue the Missile with w∗ . Thus, one obtains ˙V∗∗ MD(t) = yMD yMD −ρuXMD XMT yMT XMT yMT − ρwZMD ZMDyMD ZMDyMD = yMD yMD −ρu −τM ψ tMD go −τM ψ tMT go I3yMT −τM ψ tMT go I3yMT − ρwtMD go tMD go I3yMD tMD go I3yMD (27.69) = yMD yMD −ρuτM ψ tMD go yMT yMT − ρwtMD go yMD yMD Recall that in worst case yMD yMD = yMT yMT , which yields, u∗ Cp = −u∗ Ce. Therefore, (27.69) becomes, ˙V∗∗ MD(t) = yMD yMD −ρuτM tMD go τM − 1 + e−tMD go /τM yMD yMD − ρwtMD go yMD yMD = −tMD go (ρu + ρw) + ρuτM 1 − e−tMD go /τM (27.70) Integration yields, y∗∗ MD tMD go = + 1 2 (ρu + ρw) tMD go 2 − ρuτ2 M ψ tMD go (27.71) This function implies that yMD , which decreases due to Defender’s and Missile’s strategies, cannot decrease more rapidly than y∗∗ MD(t) . Hence, we choose: y∗∗ MD t = tMD f = , so that 144
- 161. even in the worst case, yMD tMD f does not fall below . This function is defined as the fail-safe: C tMD go = + 1 2 (ρu + ρw) tMD go 2 − ρuτ2 M ψ tMD go (27.72) The functions: A(t), B(t), and C(t), are depicted in Fig. 27.5. (t) ℬ(t) (t) ℓ m tf MD tf MT Time, t ||ZEM|| Figure 27.5: Functions A(t), B(t), and C(t) 27.3.3 Guaranteed Cost Game As in the ideal players game, by making yMD reach C, the missile guarantees a M-D miss distance . Define t∗ as that intersection time. Simultaneously, if B t∗ − yMT t∗ > 0, then yMT stays inside the singular area defined by B, and a M-T miss distance m is guaranteed. Unfortunately, neither t∗ nor d(t∗ ) B t∗ − yMT t∗ have analytical solutions in this game of non-ideal players; thus, there is no sufficient condition for the Missile to evade the Defender and intercept the Target. A possible Missile’s maneuver, which guarantees a M-T miss distance m and M-D miss distance , provided d(t∗ ) > 0, is uC = u∗ Ce = −ρu rMD+tMD go ˙rMD−τ2 M ψ tMD go u rMD+tMD go ˙rMD−τ2 M ψ tMD go u , yMD < C tMD go u∗ Cp = ρu rMT +tMT go ˙rMT −τ2 M ψ tMT go u rMT +tMT go ˙rMT −τ2 M ψ tMT go u , yMD ≥ C tMD go (27.73) and, the optimal 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 (27.74) w∗ = −ρw rMD + tMD go ˙rMD − τ2 M ψ tMD go u rMD + tMD go ˙rMD − τ2 M ψ tMD go u (27.75) 145
- 162. 27.4 Optimal Strategies for Varying Final Times (VG4) 27.4.1 M-T Game Define m as the desired M-T miss distance (here m ≥ m∗ ). Recall that VG4 assumes optimal maneuver; therefore, the time-to-go is the solution of rMT + tMT go ˙rMT − τ2 M ψ tMT go u = m + 1 2 (ρu − ρv) tMT go 2 − ρuτ2 M ψ tMT go (27.76) Analytical solution is impossible; therefore, use numerical solution. Proposition 27.1. Newton’s method: Define the function f tMT go = rMT + tMT go ˙rMT − τ2 M ψ tMT go u − m − 1 2 (ρu − ρv) tMT go 2 + ρuτ2 M ψ tMT go (27.77) Note that lim tMT go →0 f tMT go = rMT − m > 0 (27.78) lim tMT go →∞ f tMT go = −∞ < 0 (27.79) Therefore f tMT go has at least one positive real root. Differentiate f tMT go and obtain ˙f tMT go = rMT + tMT go ˙rMT − τ2 M ψ tMT go u ˙rMT − τ2 M ˙ψ tMT go u rMT + tMT go ˙rMT − τ2 M ψ tMT go u (27.80) − (ρu − ρv) tMT go + ρuτ2 M ˙ψ tMT go where ˙ψ tMT go = 1 τM 1 − e−tMT go /τM . According to Newton’s method, t(n+1) go = t(n) go − f t (n) go ˙f t (n) go (27.81) where n is the iteration number, and tgo = tMT go . For fast convergence, use the computed tMT go at the time step k as the initial guess for Newton’s algorithm at time step k + 1. This way, the ZEM norm, yMT , is always placed on the function B tMT go = m + 1 2 (ρu − ρv) tMT go 2 − ρuτ2 M ψ tMT go (27.82) as depicted in Fig. 27.6. 146
- 163. tgos tgo MT m* ||yMT || Figure 27.6: Missile-Target Game Optimal ZEM Trajectory Some properties: 1. There is no singular area, in which the optimal strategies are arbitrary, and the function B is not a bound; rather, it is the optimal ZEM trajectory. 2. This guidance laws never chatters as the denominator never vanishes. 3. If ρu > ρv, a M-T miss distance m can be achieved from any initial condition. 4. When both players play optimal, m∗ is the minimal achievable miss distance. 27.4.2 M-D Game Analogically to the M-T game, we enforce tMD go , to be a solution of, rMD + tMD go ˙rMD − τ2 M ψ tMD go u = q − 1 2 (ρu − ρw) tMD go 2 + ρuτ2 M ψ tMD go (27.83) where q is the M-D miss distance (provided both players apply their optimal strategies in the 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 + ρuτ2 M ψ tMD go (27.84) where Z tMD go is parallel to A tMD go = − 1 2 (ρu − ρw) tMD go 2 + ρuτ2 M ψ tMD go (27.85) Graphically, Z tMD go and A tMD go are shown in Fig. 27.7. 147
- 164. (tgo MD ) (tgo MD ) tgo MD ℓ q ||yMD|| Figure 27.7: Missile-Defender Optimal ZEM Trajectories The shape of Z is well defined by ρu, ρw, and τM , although its vertical displacement depends on the parameter q. Define: g tMD go = rMD + tMD go ˙rMD − τ2 M ψ tMD go u − q + 1 2 (ρu − ρw) tMD go 2 − ρuτ2 M ψ tMD go (27.86) Theorem 27.1. The function g tMD go has at least one local minimum Proof. Differentiate with respect to tMD go and obtain ˙g tMD go = (ρu − ρw) tMD go − ρuτ2 M ˙ψ tMD go (27.87) + rMD + tMD go ˙rMD − τ2 M ψ tMD go u ˙rMD − τ2 M ˙ψ tMD go u rMD + tMD go ˙rMD − τ2 M ψ tMD go u Note that rMD > 0 since the distances in all three directions are positive. In addition, assuming that both players are approaching, we have: ˙rMD < 0. As a result, rMD ˙rMD < 0 (27.88) Recall that ψ tMD go = tMD go τM − 1 + e−tMD go /τM . Hence, lim tMD go →0 ψ tMD go = 0 (27.89) lim tMD go →∞ ψ tMD go = ∞ (27.90) Also, recall that ˙ψ tMD go = 1 τM 1 − e−tMD go /τM . Thus, 148
- 165. lim tMD go →0 ˙ψ tMD go = 0 (27.91) lim tMD go →∞ ˙ψ tMD go = 1 τM (27.92) Now examine the limits of ˙g tMD go . lim tMD go →0 ˙g tMD go = rMD ˙rMD rMD < 0 (27.93) lim tMD go →∞ ˙g tMD go = ∞ > 0 (27.94) Therefore, since g tMD go is continuous, it has at least one local minimum in the time interval tMD go ∈ [0, ∞). Example 27.1. For some rMD and ˙rMD, and different values of q , the function g tMD go is shown in Fig. 27.8. q = 100 [m] q = 1830 [m] q = 3550 [m] 2 4 6 8 10 tgo MD -2000 2000 4000 6000 8000 10000 g(tgo MD) Figure 27.8: Function g tMD go For Different Values of q Similarly, we end up with the same conclusions as for the ideal players: 1. If g tMD go = 0 has no positive real solutions, then q is too small, and such a miss distance is unachievable. 2. If g tMD go = 0 has exactly one positive real solution, then q = qmin is the minimal achievable M-D miss distance. Clearly, this occurs when rMD reaches its minimum. 3. If g tMD go = 0 has more than one positive real solution, then q is too big, and by choosing the biggest of the solutions for tMD go , we are aiming for a higher trajectory than required (while the smaller solutions are not physical, as rMD still decreases). This would require more time for evasion than necessary and produce bigger M-D miss distance than desired. 149
- 166. Therefore, it is reasonable to define tMD f = tMD f (qmin); hence, tMD go = tMD go (qmin). Similarly, we must find q for which g tMD go = 0 has exactly one positive real root. Since q does not change the shape of g tMD go , rather it changes its vertical displacement, the function g tMD go has one real root when its smallest local minimum is tangent to the horizontal axis. Therefore, instead of looking for q = qmin we can find tMD go which minimizes g tMD go , and obtain tMD go (qmin). Thus, tMD go = arg min tMD go g tMD go (27.95) = arg min tMD go rMD + tMD go ˙rMD − τ2 M ψ tMD go u − q + 1 2 (ρu − ρw) tMD go 2 − ρuτ2 M ψ tMD go = arg min tMD go rMD + tMD go ˙rMD − τ2 M ψ tMD go u + 1 2 (ρu − ρw) tMD go 2 − ρuτ2 M ψ tMD go 27.4.3 M-T-D Game Similarly to ideal players game, divide the game into two different phases: the evasion phase and the pursuit phase. Recall the fail-safe function C tMD go = + 1 2 (ρu + ρw) tMD go 2 − ρuτ2 M ψ tMD go (27.96) When yMD reaches C, the Missile guarantees a M-D miss distance of , as depicted in Fig. 27.9. ||yMT|| ||yMD|| (t) (t) Evasion Pursuit {ue * ,w * } {up * ,w * } t* tf MD tf MT Time, t ℓ ||yMT cr || ||ZEM|| Figure 27.9: Functions A, C, and yMD Assuming ρu > ρw, the optimal ZEM trajectory, generated by the pair {u∗ e, w∗ }, is parallel to A when t ∈ [0, t∗ ], and in the worst case (u∗ e = −u∗ p) collides with C when t ∈ t∗ , tMT f , allowing the 150
- 167. Missile to evade the Defender. However, this is true only if yMD 0 is not inside the area bounded by A. Hence, since we demand yMD tMD go ≥ A tMD go = − 1 2 (ρu − ρw) tMD go 2 + ρuτ2 M ψ tMD go ∀tMD go ∈ 0, tMD f The initial condition must satisfy yMD 0 ≥ − 1 2 (ρu − ρw) tMD f 2 + ρuτ2 M ψ tMD f (27.97) thus ρu ≥ 2 − 2 yMD 0 + ρw tMD f 2 tMD f 2 − τ2 M ψ tMD f = 2 − 2 yMD 0 + ρw tMD f 2 tMD f 2 − τ2 M tMD f τM − 1 + e−tMD f /τM (27.98) Since ρu > ρv, the Missile can always enforce any M-T final time, tMT f , such that eventually the M-T miss distance is m∗ (Fig. 27.9). Therefore, there is no upper bound that yMT has to stay in, and the Missile can always intercept the Target if it successfully evades the Defender. Define the Missile’s optimal strategy as a combination of its optimal pursuit and evasion strategies, u∗ C = u∗ Ce = −ρu rMD+tMD go ˙rMD−τ2 M ψ tMD go u rMD+tMD go ˙rMD−τ2 M ψ tMD go u , yMD < C tMD go u∗ Cp = ρu rMT +tMT go ˙rMT −τ2 M ψ tMT go u rMT +tMT go ˙rMT −τ2 M ψ tMT go u , yMD ≥ C tMD go (27.99) Also recall the other players optimal strategies, v∗ = ρv rMT + tMT go ˙rMT − τ2 M ψ tMT go u rMT + tMT go ˙rMT − τ2 M ψ tMT go u (27.100) w∗ = −ρw rMD + tMD go ˙rMD − τ2 M ψ tMD go u rMD + tMD go ˙rMD − τ2 M ψ tMD go u (27.101) Theorem 27.2. Let the Missile use its optimal guidance strategy, u∗ C, and let the other players use any bounded maneuver, such that v ≤ ρv and w ≤ ρw. The Missile can evade the Defender and capture the Target if 1. ρu ≥ 2 −2 yMD 0 +ρw(tMD f ) 2 (tMD f ) 2 −τ2 M tMD f τM −1+e −tMD f /τM 2. ρu > ρv Theorem 27.3. Let all the players apply their optimal strategies, {u∗ , v∗ , w∗ }. The Missile can evade the Defender and capture the Target if and only if 1. ρu ≥ 2 −2 yMD 0 +ρw(tMD f ) 2 (tMD f ) 2 −τ2 M tMD f τM −1+e −tMD f /τM 2. ρu > ρv 151
- 168. 27.5 Simulations Example 27.2. Consider the following parameters, ρu = 170 m sec2 , ρv = 30 m sec2 , ρw = 60 m sec2 , m = 0.5 [m] , = 150 [m] , τM = 0.1 [sec] Note that here m∗ = 0.0075 [m]; hence, we choose m > m∗ . The simulation result is depicted in Fig. 27.10. 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 (a) VG4, First Order Lag 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 (b) VG4, Zero Order Lag Figure 27.10: First Order Lag Vs. Zero Order Lag Notice that 1. The non-ideal Missile cannot maneuver as fast as the ideal one; thus, it takes more time to reach C, and more time to intercept the Target. 2. The Missile eventually intercepts the Target. 3. The M-D miss distance is close to the required. 152
- 169. 28 Conclusions Part III continues the work provided in Part I. However, unlike Part I, where the kinematic model was linearized in perpendicular to LOS direction, the model described in this part is based on the actual game kinematics. As a result, players’ strategies are precise in any geometrical condition regardless of their closeness to appropriate collision triangles. Furthermore, by adopting the methods described in Part I, one finds out that in some scenarios (Analyzed and described in Part II) it is best for the Target to apply a non optimal maneuver in order to evade the Missile. Moreover, in some scenarios the opposite to optimal maneuver allows the Target to evade the Missile, while the optimal one fails to do so. This is a result of linearization, as by evading the Missile, the Target also extends the M-T game final time (which is assumed to be constant in the linearized model), resulting an expansion of the singular area in ZEM plane where the Missile guarantees any desired miss distance. Consequently, Target’s advantage from maximizing the ZEM is smaller than the disadvantage from the extension of tMT f . In contrast, by adopting the techniques presented in Part III, one finds that all strategies depend on the true kinematic model, so that optimal maneuvers are indeed optimal. Part III discusses two approaches: VG1 and VG4. In each approach the time-to-go variables get different definition; therefore, game definitions and properties change as well. The VG1 approach assumes that players’ closing speed is approximately constant, so most of the definitions and properties are analogous to LMG described in Part I. This approach’s main advantages are that tgo is easily computed, and most of the algebraic conditions derived in Part I are still relevant. The disadvantages of VG1 are that it does not account for players’ acceleration, and like LMG, it also has the chattering problem when the M-T ZEM norm reaches zero. The second approach is VG4. This approach assumes optimal maneuvering for all players. Its main advantage over VG1 is that it accounts for players’ accelerations, does not result chattering, does not assume constant final times, and generally provides better results. However, the time-to-go variables are numerically harder to compute, and often require iterative conversion. As was shown in Fig. 24.4, VG14, which is defined as VG1 for M-D game and VG4 for M-T game, can provide a reasonable compromise between the two approaches. Part III also provides some interesting properties of the vector guidance. Unlike VG1, where the Missile attempts to reduce yMT to zero before tMT f ; and therefore, drive rMT and VMT to collinearity, VG4 drives yMT to zero only at tMT f . Furthermore, VG4 drives rMT and VMT to collinearity at tMT f . In addition, VG4 implies that both players should point their acceleration vectors in the initial ZEM direction; thus, when the players start from initial conditions where rMT and VMT collinear, the acceleration vectors are to be pointed in LOS direction. As a result, VG1 can be used in M-T game for the midcourse guidance in order to start the terminal guidance from advantageous initial conditions. VG1 assumes constant final times; therefore, optimization problems such as minimum time make no sense. The only criterion that makes sense is the robust measure d t∗ go , which is derived in Part I, and adapted for VG in Part III. In contrast, provided ρu > ρv and ρu > ρw (and the players start from any reasonable initial conditions that ensure yMD doesn’t start inside the area bounded by A), the Missile is well capable of evading the Defender and capturing the Target from any point in the ZEM plane. As a result, optimization problems make sense. However, because of the non-linearity in tMT go , global minimum time problem is impossible to solve analytically; therefore, a suboptimal solution has been provided. This solution optimizes the evasion phase and the pursuit phase separately, provided the evasion phase starts at t = 0. In addition, an estimation algorithm for tMT f is derived. 153
- 170. Another interesting property of VG4 is that when both players play optimal, the thrust direction remains constant. Therefore, since the game of three players can be separated into two pair games (Missile–Target game and Missile–Defender game), the obtained guidance law when the Target and the Defender play optimal is bang-bang in the thrust direction. A disadvantage of the Vector Guidance approach is the basic assumption that all players can direct their acceleration vectors at any direction of the three dimensional space. While this assump- tion is legitimate for exo-atmospheric interception scenario where the thrust is the only steering force driving the players, it is invalid for the endo-atmospheric interception scenario in which aero- dynamic forces are dominant. Therefore, Transformed Vector Guidance (TVG) algorithms are derived in Part III. These algorithms transform the standard · 2 bound into a velocity-vector- framed ellipsoid or even a disk, which reflects the difference between the lateral and the axial maneuvering capabilities. Another topic discussed in Part III is the estimation based guidance. Although the relative position vectors can be measured in practice (by using electro-optical and inertial systems), the relative velocity vectors must be estimated. Thus, estimation configurations have been proposed, and miss distance bounds are derived. These bounds are derived for the M-D and M-T games separately, while other relevant issues are left for future research. The final section of Part III discusses a game of non-ideal players. To be more specific, it discusses a game of a non-ideal missile, which has a first order isotropic dynamics. There are still many issues to be explored in this topic; however, this is left for future research. 154
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- 174. .מלאה קינמטיקה עם לינארי לא בתרחיש אופטימליות בהכרח אינן המתקבלות האסטרטגיות ,מכך וכתוצאה הלא בתרחיש מאוד גרועה תהיה הלינאריזציה במערכת האופטימלית שהאסטרטגיה להיות יכול ,כן על יתר בחלק למצוא ניתן ,ראשית .הראשון בחלק שהוצג הפתרון של מעמיק ניתוח מציג השני החלק .לינארי הוגדר אשר הרובסטיות לקריטריון ביחס הלינאריזציה מודל עבור האופטימליות והוכחת פרמטרי ניתוח השני אין ,העבודה בגוף המוצגת ההנחיה אסטרטגיית את מיישם המיירט הטיל כאשר כי לציין ראוי .הראשון בחלק המתחמק של האופטימליות האסטרטגיות העבודה בגוף שהוכח וכפי ,פעולה לשתף סיבה שום והמגן למתחמק לינאריזציה המודל של העיקרית הבעיה את מציג השני החלק ,לכך בנוסף .פעולה שיתוף כוללות אינן והמגן ,הלינאריזציה במערכת כקבוע מוגדר אשר ,המשחק שזמן הוא הבעיה עיקר .זה פתרון מייצר אשר והסתירה אל נמוג אופטימלי תמרון ביצוע ע"י המטרה שצברה היתרון ,מכך כתוצאה .לינארי לא בתרחיש מאוד משתנה בתרחיש ליניארי המודל של קשה וודאות לחוסר מובילה זו בעיה .המשחק זמן בהגדלת הכרוך החסרון מול ההנחיה הינו כזה פתרון .בלינאריזציה תלוי יהיה לא אשר ,אחר בפתרון לצורך שמוביל מה ,לינארי לא ,אמיתי חסומה תאוצה להפעיל יכולים השחקנים ,הווקטורית ההנחיה בגישת .שלישי בחלק מוצג אשר ,הווקטורית צירים מערכת על מסתמכת הווקטורית בהנחיה הקינמטיקה ,בנוסף .התלת־מימדי במרחב כיוון בכל בגודלה אשר מימדי תלת ווקטור הינו זו בגישה המתקבל הבקרה אות .מלינאריזציה סובלת ואינה תלת־מימדית קרטזית בגישת המשחק זמן כי ,לציין חשוב .האופטימיזציה תהליך של תוצאה וכיוונו המקסימלי הדחף כגודל גודלו מרחק למיירט מבטיחה הווקטורית ההנחיה .לינארית לא משוואה מתוך רקורסיבית מחושב הווקטורית ההנחיה מחיר לכן ,הרקטי המנוע של הבערה מזמן קטן המשחק שזמן בתנאי ,במרחב התחלה תנאי מכל אפסי החטאה ההנחיה אסטרטגיית כי ,הוכח העבודה בגוף .(אפס להיות מובטח )אשר ההחטאה מרחק ולא הזמן הינו המשחק לתרחיש הווקטורית ההנחיה את להתאים מנת על .זמן במינימום המטרה אל להגיע למיירט נותנת הווקטורית הינה השחקנים של התמרון יכולת ורוב הדומיננטיים הם אווירודינמיים הכוחות שבו ,אנדו־אטמוספרי יירוט בטרנספורמציה משתמשת זו גישה .המורחבת הווקטורית ההנחיה גישת פותחה ,המהירות לווקטור בניצב במישור ""כדור ידי על לא השחקנים תאוצת את לחסום לאפשר מנת על משקל ובמטריצת הגוף צירי למערכת והמאונכת הצירית התאוצה יכולת בין בהבדל להתחשב נוכל ,כך ידי על .אליפסויד ידי על אלא ,האוקלידי היטב מתארות הלינאריזציה גישת לבין המורחבת הווקטורית ההנחיה בין המשוות סימולציות .השחקנים של הנחיה אלגוריתם פותח ,רועשות במדידות להתחשב מנת על ,לכך בנוסף .הווקטורית השיטה של היתרון את אפשרות למתכנן נותנת זו שיטה .אלגברי חישוב ידי על השערוך שגיאת את לחסום ניתן כאשר ,משערך מבוסס השלישי החלק ,כן כמו .היחסי המיקום ווקטור את מודד שהוא בתנאי היחסית המהירות ווקטור את לשערך ראשון מסדר (הכיוונים בכל זהה )דינמיקה איזוטרופית דינמיקה המיירט לטיל כאשר משחק של ניתוח מציג בין למשחק ההרחבה .זה לתרחיש אופטימליות אסטרטגיות ומציג ,(אידאליים גופים הינם והמגן )המתחמק התאוצה פקודת שבין בדינמיקה ולהתחשב יותר ומדויק נכון משחק לתכנן מאפשרת אידאליים שאינם גופים לאורך המוצגות הגישות כל את מדגימות אשר מימדיות ותלת דו סימולציות ישנן זו בעבודה .עצמה לתאוצה את להבליט מנת על השונות הגישות בין השוואות ישנן ,כן כמו .התאורטיות התוצאות את ומאששות העבודה .גישה כל של והחסרונות היתרונות
- 175. תקציר מודרני שמיירט מכיוון ,האחרונות בשנים משמעותי לנושא הפכה ביות טילי מפני מוטסים כלים על ההגנה הפסיביות ההגנה ומערכות ,יותר למתוחכמים היירוט טילי הפכו ,הזמן עם .שכזה לכלי משמעותי איום נושא על משמעותי איום תהווה אשר יותר מתקדמות הגנה במערכות הצורך קם ,לכן .מספיקות אינן כבר הנוכחיות שבו ,הסטנדרטי רדיפה־התחמקות למשחק בנוסף .שחקנים שלושה בין בעימות עוסק זה מחקר .המיירט הטיל על להגן מנת על ,()מגן טווח קצר טיל משגר המתחמק ,ההחטאה מרחק את ממקסם והמתחמק ממזער הרודף את ליירט היא המגן של המטרה .המתחמק את וליירט מהמגן להתחמק היא המיירט הטיל של המטרה .עצמו בעבודה .המיירט מהטיל לברוח היא המתחמק של המטרה .המתחמק את ליירט ממנו למנוע ו/או המיירט הטיל זה מחקר .לינאריות לא לאסטרטגיות שמוביל מה ,שחקנים של התמרון יכולות על קשיחים אילוצים ישנם ,זו הטיל ההתחמקות בשלב .ורדיפה התחמקות :שלבים לשני שלו המשחק את לחלק יכול המיירט הטיל כי מראה זמן .המתחמק אחר לרדיפה עובר המיירט מכן ולאחר ,מעבר תנאי מתקיים אשר עד מהמגן מתחמק המיירט שאם היא ומשמעותה ,Zero Effort Miss (ZEM) ה־ במרחב המוגדרת ,אל־כשל פונקציית ע"י מוכתב המעבר עבר אל המיירט יתמרן בו ביותר הגרוע במקרה גם ,זו פונקציה אל הגיע והמגן המיירט הטיל של ZEM ה־ כתוצאה .המתכנן ידי על מהמוכתב קטן יהיה לא ההחטאה מרחק ,המיירט אל אופטימלית יתמרן והמגן המגן אחר רדיפה למצב מהמגן התחמקות של ממצב לעבור המיירט הטיל יכול ,מתקיים המעבר שתנאי ברגע ,מכך החליפה מזמן משמעותית קצר הינו לרדיפה מהתחמקות המעבר זמן .המגן ידי על שייתפס חשש ללא המתחמק על שחלף לפני עוד המתחמק אחר ברדיפה להתחיל יכול המיירט הטיל כלומר ,המגן פני על המיירט של שלו הרדיפה יכולת על שומר ובכך מיותרת התחמקות על זמן מבזבז אינו המיירט ,מכך וכתוצאה ,המגן פני עם במשחק דן ,הראשון .חלקים לשלושה מחולק זה מחקר .ZEM ה־ במרחב סינגולרי אזור ע"י המוגדרת הגופים ,מכך כתוצאה .""הרבה מסתובב אינו הראיה קו כי מניחים אנו שבו ,הלינאריזציה במערכת קינמטיקה מתארות התנועה ומשוואות ,ההתחלתי הראיה קו בכיוון (הסגירה מהירות נקראת )אשר קבועה במהירות נעים מסוימים באזורים כי מראה המחקר .מימדי חד למשחק שמוביל מה ,ההתחלתי הראיה לקו בניצב הדינמיקה את יכולת את לאבד מבלי התחמקותי תמרון לבצע יכול המיירט הטיל ,(סינגולריים )איזורים המצב מרחב של על (יותר גבוהה תמרון )יכולת יתרון יש המיירט לטיל שכאשר מכיוון מתרחשת זו תופעה .שלו הרדיפה שרירותיות הינן האופטימליות האסטרטגיות בהם ,ZEM ה־ במרחב סינגולריים איזורים קיימים ,המתחמק פני בתוך נמצא והמתחמק המיירט הטיל של ZEM ה־ משתנה עוד כל ,לכן .קבוע הינו (ההחטאה )מרחק והמחיר הבעיה .מהמגן התחמקות אסטרטגיית - ובפרט אסטרטגיה כל להפעיל יכול המיירט הטיל ,הסינגולרי האיזור תנאי מתקיים כאשר נקוב החטאה מרחק מבטיח המתקבל והפתרון ,הלינאריזציה הנחות תחת ומנותחת מוצגת ה־ משתנה ,לרדיפה מהתחמקות המעבר שבעת ההבנה מתוך מתקבל זה תנאי .המחקר בגוף המוצג אלגברי להבטיח המיירט הטיל יכול לא אחרת ,הסינגולרי באיזור להישאר חייב והמתחמק המיירט הטיל של ZEM ישירות נובע זה קריטריון ,רובסטיות לקריטריון ביחס אופטימלי הינו המוצג הפתרון ,בנוסף .החטאה מרחק מהתחמקות המעבר בעת הסינגולרי האיזור מגבול והמתחמק המיירט של ZEM ה־ משתנה של מהמרחק מותנה ,המתחמק את וליירט מהמגן להתחמק יכול המיירט שהטיל מבטיח אשר האלגברי והתנאי ,לרדיפה מהמערכת הדרישות את ולחשב מעמיק פרמטרי ניתוח לבצע למתכנן מאפשר זה .חיובי הינו זה שמרחק בכך ,הלינאריזציה מודל על מבוסס הראשון החלק כי לזכור יש ,זאת למרות .התכנון של המוקדמים בשלבים
- 176. הנחיה .מכונות להנדסת בפקולטה גוטמן שאול 'פרופ בהנחיית נעשה המחקר תודה הבעת במהלך והמסורה הסבלנית הנחייתו על ,גוטמן שאול 'פרופ ,שלי למנחה לב מקרב מודה אני .אמיתי מדע כלפי הייחודית מתשוקתו השראה לשאוב לזכות לי היה .המחקר .בהשתלמותי הנדיבה הכספית התמיכה על לטכניון מודה אני
- 177. שחקנים שלושה עם והתחמקות רדיפה עימות מחקר על חיבור התואר לקבלת הדרישות של חלקי מילוי לשם לפילוסופיה דוקטור רובינסקי סרגיי לישראל טכנולוגי מכון - הטכניון לסנט הוגש 2015 אפריל חיפה תשע"ה ניסן
- 178. שחקנים שלושה עם והתחמקות רדיפה עימות רובינסקי סרגיי