Tactical and Strategic Missile Guidance

1098 Pages·2012·26.488 MB·English

by  Zarchan Paul

Tactical and Strategic
Missile Guidance
Sixth Edition
Paul Zarchan
MIT Lincoln Laboratory
Lexington, Massachusetts
Volume 239
Progress in Astronautics and Aeronautics
Timothy C. Lieuwen, Interim Editor-in-Chief
Georgia Institute of Technology
Atlanta, Georgia
Publishedby
AmericanInstituteofAeronauticsandAstronautics,Inc.
1801AlexanderBellDrive,Reston,VA20191-4344
MATLABwisaregisteredtrademarkofTheMathWorks,Inc.
AmericanInstituteofAeronauticsandAstronautics,Inc.,Reston,Virginia
1 2 3 4 5
Cover art courtesy of Air University Press, Air Force Research Institute. From an illustration by
DanielArmstrongforColonelMikeCorbett,USAF,Retired,andPaulZarchan,“TheRoleofAirpower
inActiveMissileDefense,”AirandSpacePowerJournal24,no.2(Summer2010):57–71.
Copyright # 2012 by the American Institute of Aeronautics and Astronautics, Inc. All rights
reserved.ReproductionortranslationofanypartofthisworkbeyondthatpermittedbySections
107and108oftheU.S.CopyrightLawwithoutthepermissionofthecopyrightownerisunlawful.
Thecodefollowingthisstatementindicatesthecopyrightowner’sconsentthatcopiesofarticlesin
thisvolumemaybemadeforpersonalorinternaluse,onconditionthatthecopierpaytheper-
copyfee($2.50)plustheper-pagefee($0.50)throughtheCopyrightClearanceCenter,Inc.,222
RosewoodDrive,Danvers,Massachusetts01923.Thisconsentdoesnotextendtootherkindsof
copying, for which permission requests should be addressed to the publisher. Users should
employthefollowingcodewhenreportingcopyingfromthevolumetotheCopyrightClearance
Center:
978-1-60086-894-8/12$2.50þ.50
Data and information appearing in this book are for informational purposes only. AIAA is not
responsibleforanyinjuryordamageresultingfromuseorreliance,nordoesAIAAwarrantthat
useorreliancewillbefreefromprivatelyownedrights.
ISBN978-1-60086-894-8
PREFACE TO THE SIXTH EDITION
Thesixtheditioncontainssixnewchaptersonimportanttopicsrelatedtoimprov-
ingmissileguidancesystemperformanceandunderstandingkeydesignconcepts
and tradeoffs. In addition, at the request of many readers, all of the FORTRAN
source code that appeared in the first five editions of Tactical and Strategic
Missile Guidance has been converted to MATLAB. Interested readers can
obtain all MATLAB source code—and the equivalent FORTRAN versions—in
electronic form on the AIAA Web site as mentioned at the back of this book
onthe Supporting Materials page.
The first new chapter, Chapter 29, presents two new applications of the
method of adjoints for mixed continuous-discrete systems. The first application
involves multiple samplers with each of the samplers operating at a different
sampling rate. The second application involves taking the adjoint of a three-
statediscrete Kalman filter in thehomingloop.
Thesecondnewchapter,Chapter30,introducesanewguidancelawthatcan
beusedtoshapetheinterceptortrajectoryagainstastationarytarget.Theunique
advantageofthisnewguidanceapproachoverexistingguidancelawsisthattime-
to-goinformationisnotrequired.Thesecondpartofthischapterconsidersboth
theproblemoffindingtheminimumachievablemissdistanceofaradarhoming
missile and exploring practical techniques that can be used to achieve the
minimum possiblemiss.
Thethirdnewchapter,Chapter31,introducesthedifferentialgameguidance
lawwithboundedcontrolsanddemonstratesitsperformanceimprovementsover
conventionalguidancelawsunderchallengingconditionsinwhichthemissile-to-
target acceleration advantage is very low.
The fourth new chapter, Chapter 32, introduces techniques for graphically
presentingstrategicinformationonsuccessfulinterceptsofanimpulsiveballistic
targetbeingpursuedbyanimpulsiveinterceptor.Thethreegraphicaltechniques
discussed are the operational area method, the launch area denied method, and
the defended area method. Examples are presented illustrating the utility of
each method.
Thefifthnewchapter,Chapter33,examinestwofilteringoptionsfortheboost
phaseinterceptofastrategictarget.Thetwofilterscomparedareatemplate-based
Kalman filter that has perfect a priori information and a linear three-state poly-
nomialKalmanfilterthatdoesnotrequiresuchinformation.Thefiltersarecom-
pared under both ideal and realistic conditions. The results of this chapter may
surprise some readers.
Finally,thesixthnewchapter,Chapter34,addressessomeoftheguidanceand
controlissuesinvolvedinenablinganair-launchedinterceptorcarryingahighly
maneuverable kinetic kill vehicle to perform an exoatmospheric intercept of a
boosting threat target that is capable of traveling many thousands of kilometers.
This chapter takes the reader through the first iteration of the multi-iteration
xix
xx PREFACETOTHESIXTHEDITION
design process in order to show how much divert and acceleration may be
requiredbythekinetickillvehiclesothatitcanhitthetarget.Simplifiedexamples
arepresentedtoindicatehowconventionalguidanceandfilteringtechniquescan
be used as a starting point in the iterative design process for this important
problem in missile defense.
ThesesixnewchaptersarebasedonmaterialfromtheAIAAcontinuingedu-
cation short course “Fundamentals of Tactical and Strategic Missile Guidance”
that I have been teaching since 1990. The new topics are treated from both an
analytical and a simulation point of view so that readers with different back-
grounds and learning styles can benefit from thenew material.
Readers of previous editions of Tactical and Strategic Missile Guidance will
notice that even though the sixth edition has six new chapters, the book is
approximately the same size as the fifth edition. This was achieved by deleting
fifth edition Chapters 10 and 27; half of Chapters 16, and 29, and Appendix A;
andallofAppendicesB–E.Inaddition,Chapter20hasbeenrewritteninresponse
to questions by readers. Although the material has been deleted from the sixth
edition, interested readers can find the fifth edition chapters and appendices in
PDFformat onAIAA’s Web site.
Onapersonalnote,itcontinuestobeverygratifyingformetolearnthatmany
peopleworkingwithorneedingtolearnaboutmissileguidancehavefoundTac-
ticalandStrategicMissileGuidanceuseful.Overtheyears,manyreadershavecon-
tacted me and have asked questions when the book’s material was not clear to
them. Material in the existing chapters has been clarified so that all will benefit
from my interaction with the readers. It is still my hope that this sixth edition
will be of value not only to new readers, but will also be worthwhile to those
whohave read previous editions.
Paul Zarchan
MITLincolnLaboratory
February2012
TABLE OF CONTENTS
Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvii
Preface to the Sixth Edition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xix
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxi
Chapter 1 Numerical Techniques . . . . . . . . . . . . . . . . . . . . . . . . . 1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
Laplace Transforms and Differential Equations . . . . . . . . . . . . . . . . . . . . . . 1
Numerical Integration of Differential Equations. . . . . . . . . . . . . . . . . . . . . . 4
ZTransforms and Difference Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
Chapter 2 Fundamentals ofTactical Missile Guidance . . . . . . . . . 13
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
WhatIs Proportional Navigation? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
Simulation of Proportional Navigation in TwoDimensions . . . . . . . . . . . . 14
Two-Dimensional Engagement Simulation . . . . . . . . . . . . . . . . . . . . . . . . 18
Linearization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
Linearized EngagementSimulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
Important Closed-Form Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
Proportional Navigation and Zero Effort Miss . . . . . . . . . . . . . . . . . . . . . . 32
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
Chapter 3 Method of Adjoints and the Homing Loop . . . . . . . . . 35
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
Homing Loop. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
Single Time Constant Guidance System . . . . . . . . . . . . . . . . . . . . . . . . . . 37
How toConstruct an Adjoint . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
Adjoint Mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
Adjoints for Deterministic Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
Deterministic Adjoint Example. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
Adjoint Closed-Form Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
Normalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
vii
viii TABLEOFCONTENTS
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
Chapter 4 Noise Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
Basic Definitions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
Gaussian Noise Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
Computational Issues. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
More Basic Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
Response of Linear System to White Noise . . . . . . . . . . . . . . . . . . . . . . . . 70
Low-Pass-Filter Example. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
Adjoints for Noise-DrivenSystems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
Shaping Filters and Random Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
Example of aStochastic Adjoint . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
Closed-Form Solution for Random Target Maneuver. . . . . . . . . . . . . . . . . 86
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
Chapter 5 Covariance Analysis and the HomingLoop . . . . . . . . . 89
Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
Low-Pass Filter Example. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
Numerical Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
Homing Loop Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
Acceleration Adjoint . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
Chapter 6 Proportional Navigationand Miss Distance . . . . . . . 107
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
System Order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
Design Relationships . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
Optimal Target Evasive Maneuvers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
Practical Evasive Maneuvers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
Saturation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
Parasitic Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
Thrust Vector Control. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
TABLEOFCONTENTS ix
Chapter 7 DigitalFading Memory Noise Filters in the
Homing Loop . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
FadingMemory Filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
FadingMemory Filter in HomingLoop . . . . . . . . . . . . . . . . . . . . . . . . . . 138
Mixed Continuous Discrete Adjoint Theory . . . . . . . . . . . . . . . . . . . . . . . 146
Replace n by N 2 n inthe Arguments of AllVariable Coefficients . . . . 146
Using Adjoints to Evaluate Filter Performance. . . . . . . . . . . . . . . . . . . . . 148
Some Properties of FadingMemory Filters . . . . . . . . . . . . . . . . . . . . . . . 155
Estimating Target Maneuver . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
Chapter 8 Advanced Guidance Laws . . . . . . . . . . . . . . . . . . . . . 163
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
Review of Proportional Navigation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
Augmented Proportional Navigation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
Derivation of Augmented ProportionalNavigation . . . . . . . . . . . . . . . . . 171
Influence of Time Constants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174
Optimal Guidance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184
Chapter 9 Kalman Filters and the Homing Loop . . . . . . . . . . . . 187
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187
Theoretical Equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187
Application toHoming Loop . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189
Kalman Gains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192
Numerical Examples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193
Experiments WithOptimal Guidance. . . . . . . . . . . . . . . . . . . . . . . . . . . . 204
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211
Chapter 10 Tactical Zones . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213
Velocity Computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213
Drag. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216
Acceleration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221
Gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223
x TABLEOFCONTENTS
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227
Chapter 11 Strategic Considerations . . . . . . . . . . . . . . . . . . . . 229
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229
Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229
Gravitational Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230
Polar Coordinate System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237
Closed-Form Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242
Hit Equation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249
Flight Time. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255
Chapter 12 Boosters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257
Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257
Staging. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259
Booster Numerical Example. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262
Gravity Turn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272
Chapter 13 Lambert Guidance . . . . . . . . . . . . . . . . . . . . . . . . . 273
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273
Statement of Lambert’s Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273
Solution to Lambert’s Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274
Numerical Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277
Speeding Up Lambert Routine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281
Booster Steering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284
General Energy Management (GEM) Steering . . . . . . . . . . . . . . . . . . . . . 290
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 298
Reference. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 298
Chapter 14 Strategic Intercepts . . . . . . . . . . . . . . . . . . . . . . . . 299
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299
Guidance Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299
Ballistic EngagementSimulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301
TABLEOFCONTENTS xi
Boosting Target Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323
Reference. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323
Chapter 15 Miscellaneous Topics . . . . . . . . . . . . . . . . . . . . . . . 325
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325
Gravity Compensation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325
Predictive Guidance. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 329
Pulsed Guidance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 338
Chapter 16 Ballistic Target Properties . . . . . . . . . . . . . . . . . . . 349
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 349
Ballistic Target Model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 349
Ballistic Target Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 351
Closed-Form Solutions forBallistic Targets . . . . . . . . . . . . . . . . . . . . . . . 356
MissileAerodynamics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 359
Intercepting aBallistic Target . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 362
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 370
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 371
Chapter 17 Extended Kalman Filtering and Ballistic
Coefficient Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373
Theoretical Equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373
Differential Equation for One-Dimensional Ballistic Target . . . . . . . . . . . 375
Extended KalmanFilter for One-Dimensional Ballistic Target . . . . . . . . . 376
Numerical Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 379
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 387
Reference. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 387
Chapter 18 Ballistic Target Challenges . . . . . . . . . . . . . . . . . . . 389
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 389
MissDistance Due toNoise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 389
Fifth-Order Binomial Guidance System Miss Distances . . . . . . . . . . . . . . 394
Minimum Guidance System Time Constant. . . . . . . . . . . . . . . . . . . . . . . 399
MissileTurningRate Time Constant. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 400
Checking Minimum Guidance System Time
Constant Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 401
MissDue to Noise for Aircraft and Ballistic Targets . . . . . . . . . . . . . . . . . 406