Dissertation - A Three Players Pursuit and Evasion Conflict

A Three Player Pursuit
and Evasion Conﬂict
Sergey Rubinsky

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

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

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

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

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

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

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
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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
(ρv, ρw) . . . . . . . . . . . . . . . . . . . . . . . . . . 44
12.5 Section Plots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
12.6 Linear Simulations for Different Values of ρv and ρw . . . . . . . . . . . . . . . . . . 45
(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

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
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
20.5 Relative Distance rMT (t) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
20.6 VG1 3D Simulation 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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
24.2 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
24.9 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

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
27.2 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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

where u∗
e = −ρusign(yMD) and u∗
p = ρusign(yMT ). Also, the optimal guidance laws for the Target-
Defender team is
v∗
w∗
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
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

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

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

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

Missile
Target
Defender
tf
MD
t*
0 1000 2000 3000 4000 5000 6000
0
1000
2000
3000
4000
x [m]
y[m]
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

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

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
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

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

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

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)
(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)
44

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

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, )
(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
( )
46

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

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)
(tf , ∆t)
and the section plots are depicted in Fig. 12.11.
48

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)
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

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

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

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

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

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∗
w∗
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

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

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

|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]
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

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

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

Dissertation - A Three Players Pursuit and Evasion Conflict

Dissertation - A Three Players Pursuit and Evasion Conflict

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