Thesis - Leigh Moody.pdf - Bad Request - Cranfield University

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Chapter 7 / Missile Trajectory Optimisation _ _ 7.12 Discussion The guidance module in the missile simulator provides the building blocks for PN and CLOS guidance laws. If none are selected the program invokes on-line trajectory optimisation. The optimisation process is latched to the host’s processor clock to study the effect of limiting the available processing time. For example if the next control to be taken by the autopilot last for 0.5 s re-optimisation of the missile controls is allowed for precisely this amount of time. For this to be meaningful the algorithms must be written efficiently. Critical parameters must be stored at the beginning of each cycle in case the reoptimisation is prematurely terminated. Because the algorithms are to be executed in real-time, global optimisation algorithms currently in favour such as simulated annealing and shooting methods are inappropriate. Given the recent increase in computing capacity some of the simpler early optimisation techniques have been selected. Gradient techniques have proven to be robust for real-time optimisation given a reasonable set of initial controls, <strong>Moody</strong> [M.12] . Propagating the dynamic equations using ATPN with the missile initially pointing at the target providing the initial trajectory. The optimiser state space selected to define the missile dynamics was Cartesian position, speed and direction with respect to the earth referenced Alignment frame. The terms in the cost function and adjoined dynamic constraints can be weighted depending on target characteristics providing a more flexible system. Gradient projection in function-space along the direction of steepest-descent with an Armijo univariate search minimising the cost function is the primary algorithm. If cost reduction stalls a conjugate directions are explored using an exact Fibonnacci search. The line search step-length is doubled, or halved, depending on cost function values to accelerate the search for a local minimum. 7-20 SELECT INITIAL SET OF ADMISSIBLE CONTROLS AND IMPACT TIME STATE EQUATION FORWARD INTEGRATION FROM CURRENT TO IMPACT TIME COST EVALUATION IMPACT POINT ADJOINTS FROM TRANSVERSALITY CONDITIONS BACKWARD INTEGRATION OF EULER-LAGRANGE EQUATIONS HAMILTONIAN VARIATION WRT CONTROLS, STATES AND ADJOINTS ( GRADIENTS ) GRDIENT SCALING AND SEARCH DIRECTION UNIVARIATE SEARCH AND CONTROL UPDATE WITHIN ADMISSABLE REGION COST EVALUATION PRE AND POST LAUNCH EXIT CONDITIONS Figure 7-2 Optimisation Flowchart
Chapter 7 / Missile Trajectory Optimisation _ _ UPDATE GRADIENTS ( GRADIENTS ) INITIALISE CONTROLS BASED ON PN LAW ( I_OP_CONTROL ) INITIALISE COST FUNCTIONAL ( COST_FUNCTION ) COST FUNCTIONAL WITH ADJOINTED PENALTY FUNCTIONS ( COST_FUNCTION ) 2 1 1ST ORDER STATE VECTOR INTEGRATION STATE JACOBIAN WRT TO CONTROLS AND TIME ( OP_MS_DXDU ) CONTROLS, SCALING AND PERFORMANCE IINDEX UPDATE ( CG_SW = 0 ) STATE, CONTROL AND TIME CONSTRAINTS ( OP_LIMITS ) SET NUMBER OF REMAINING STEPS IN OPTIMISATION UPDATE INITIAL AND BOUNDARY CONDITIONS USING FF PARAMETERS STATE, CONTROL AND TIME BASED GRADIENTS ( GRADIENTS ) COST FUNCTIONAL WITH PENALTY FUNCTIONS ( COST_FUNCTION ) CURRENT CONTROL CPU USED OPTIMAL_GU_LAW 2 3 DIRECTION OF STEEPEST DESCENT WITH DESENSITISATION OF LIMITING CONTROLS ( STEEPEST_DD ) UNIVARIATE LINE SEARCH ( ARMIJO_LS ) PI REDUCION TOO SMALL AND CG_SW = 0 7-21 INITIALISE STEEPEST DESCENT GRADIENTS ( GRADIENTS ) 1 5 4 STATE VECTOR JACOBIAN WITH RESPECT TO TIME ( OP_MS_DXDT ) 1 2 STATE JACOBIAN WRT TO STATES ( OP_MS_DXDX ) LINE SEARCH STEP LENGTH CONTROL ( WOLFE_STEP ) NO YES RE-INSTATE PREVIOUS CONTROLS, GRADIENTS, SCALING AND PI ( CG_SW = 1 ) RESET ARMIJO STEP LENGTH CONTROLLER CURRENT CONTROL CPU USED YES YES RETURN INITIAL GRADIENT SCALING ( GRADIENT_SC ) COST FUNCTIONAL WITH PENALTY FUNCTIONS ( COST_FUNCTION ) UNIVARIATE LINE SEARCH ( FIBONA_LS ) CONJUGATE GRADIENT DESCENT DIRECTION ( CONJ_GRADIENTS ) Figure 7-3 : Trajectory Optimisation Simulator
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CRANFIELD UNIVERSITY LEIGH MOODY SE
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ABSTRACT When considering advances
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Contents _ _ LIST OF CONTENTS 1 INT
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Contents _ _ 8.2 The Scope of Moder
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Contents _ _ 17.4 Inertial Velocity
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Tables _ _ LIST OF TABLES Table 1-1
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Figures _ _ LIST OF FIGURES Figure
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Figures _ _ Figure 5-1 : Centralise
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Figures _ _ Figure 19-3 : Air Densi
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Acronyms _ _ Acronyms And Abbreviat
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Acronyms _ _ HDOP - NAVSTAR GPS Hor
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Acronyms _ _ RCS - Radar Cross Sect
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Glossary _ _ GLOSSARY A challenge w
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Glossary _ _ 0.1 Predicate Calculus
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Glossary _ _ ]A,B] Real number rang
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Glossary _ _ 0.9 Matrix Operators [
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Glossary _ _ 0.14.2 Special Vectors
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Glossary _ _ P C a , b ≡ XC YC ZC
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Glossary _ _ V ≡ V : = V • V Th
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Glossary _ _ 0.16.4 Matrix Partitio
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Glossary _ _ Only the subset of qua
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Glossary _ _ 0.19 Systeme Internati
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1 Chapter 1 / Introduction _ _ Chap
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Chapter 1 / Introduction _ _ Wherea
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Chapter 1 / Introduction _ _ • Cr
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Chapter 1 / Introduction _ _ genera
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Chapter 1 / Introduction _ _ radar
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Chapter 1 / Introduction _ _ simple
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Chapter 1 / Introduction _ _ optimi
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Chapter 1 / Introduction _ _ 1.3.9
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2 Chapter 2 / Target Modelling _ _
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Chapter 2 / Target Modelling _ _ 2.
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Chapter 2 / Target Modelling _ _
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Chapter 2 / Target Modelling _ _ t
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Chapter 2 / Target Modelling _ _ P&
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Chapter 2 / Target Modelling _ _ GO
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Chapter 2 / Target Modelling _ _ th
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Chapter 2 / Target Modelling _ _
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Chapter 2 / Target Modelling _ _ VX
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Chapter 2 / Target Modelling _ _ ma
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3 Chapter 3 / Sensors _ _ Chapter 3
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Chapter 3 / Sensors _ _ 3.1 Simulat
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Chapter 3 / Sensors _ _ Table 3-4 :
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Chapter 3 / Sensors _ _ Reference D
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Chapter 3 / Sensors _ _ 3.2-9 s s i
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Chapter 3 / Sensors _ _ 3.2-11 t o
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Chapter 3 / Sensors _ _ Figure 3-5
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Chapter 3 / Sensors _ _ Table 3-7 :
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Chapter 3 / Sensors _ _ IF ϕ LIM
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Chapter 3 / Sensors _ _ The phase e
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Chapter 3 / Sensors _ _ Taking expe
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Chapter 3 / Sensors _ _ Expressing
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Chapter 3 / Sensors _ _ INPUT; AA,
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Chapter 3 / Sensors _ _ The functio
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Chapter 3 / Inertial Navigation _ _
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Chapter 3 / Sensors / Gyroscopes _
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Chapter 3 / Sensors / Accelerometer
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Chapter 3 / Sensors / Barometric Al
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Chapter 3 / Sensors / Barometric Al
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Chapter 3 / Sensors / Radar Altimet
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Chapter 3 / Sensors / Radar _ _ 3.8
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Chapter 3 / Sensors / Radar _ _ LF
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Chapter 3 / Sensors / Radar _ _ ( )
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Chapter 3 / Sensors / Radar _ _ S :
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Chapter 3 / Sensors / Radar _ _ SN
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Chapter 3 / Sensors / Radar _ _ 2
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Chapter 3 / Sensors / Radar _ _ ⎛
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Chapter 3 / Sensors / Seeker _ _ 3.
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Chapter 3 / Sensors / Seeker _ _ Th
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Chapter 3 / Sensors / Seeker _ _ Th
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Chapter 3 / Sensors / Fins _ _ 3.10
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Chapter 3 / Sensors / Fins _ _ thro
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Chapter 3 / Sensors / NAVSTAR GPS _
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Chapter 3 / Sensors / Helmet Mounte
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Chapter 3 / Sensors / Air Data Syst
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Chapter 4 / Target Tracking _ _ 3.1
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Chapter 4 / Target Tracking _ _ Ine
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Chapter 4 / Target Tracking _ _ ran
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Chapter 5 / Missile State Observer
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5 Chapter 5 / Missile State Observe
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6 Chapter 6 / Missile Guidance _ _
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Chapter 6 / Missile Guidance _ _ 6.
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Chapter 6 / Missile Guidance _ _ As
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Chapter 6 / Missile Guidance _ _
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Chapter 6 / Missile Guidance _ _ CD
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Chapter 6 / Missile Guidance _ _ ge
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Chapter 9 / Performance _ _ AR_BOM
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Chapter 9 / Performance Chapter 9 /
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Initially the position error falls
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values of CN are not always a relia
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crude initialisation introduces err
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9.5.3 Singer Filter Tuning The Sing
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9.6.2 CLOS Study The CLOS study sho
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Chapter 10 / Conclusions _ _ 10-2
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Chapter 10 / Conclusions _ _ 10.3 T
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Chapter 10 / Conclusions _ _ 10.7 M
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Chapter 11 / Future Research _ _ 11
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Chapter 11 / Future Research _ _
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Chapter 11 / Future Research _ _ To
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References _ _ B.4 BAZARAA M.S., SH
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References _ _ F.1 FOX J.E., “A C
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References _ _ H.5 #HULL D.G., RADK
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References _ _ L.2 LOOZE D.P., HSU
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References _ _ N.4 NABAA N., BISHOP
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References _ _ R.3 RUSNACK I., “O
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References _ _ S.15 SINGER R.A.,
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References _ _ Y.2 YUAN P., CHERN J
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Bibliography _ _ B.13 BABA Y., YAMA
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Bibliography _ _ B.40 BECKER J.C.,
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Bibliography _ _ C.33 CORTINA E., O
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Bibliography _ _ E.8 ERSHOV A.A.,
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Bibliography _ _ G.34 GUTMAN P., VE
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Bibliography _ _ H.35 HUSSAIN A.M.,
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Bibliography _ _ K.14 KATZIR S., CL
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Bibliography _ _ L.19 LEFAS C.C,
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Bibliography _ _ M.16 MAYNE D.Q., P
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Bibliography _ _ P.1 PAINTER J.H.,
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Bibliography _ _ R.32 RONG LI X., Z
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Bibliography _ _ S.47 SPEYER J.L.,
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Bibliography _ _ V.5 VIAN J.L., MOO
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Bibliography _ _ W.29 WATSON G.A.,
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Appendix A / Geometric Points _ _ 1
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Appendix A / Geometric Points _ _ 1
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Appendix B / Frames of Reference _
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Appendix C / Axis Transforms _ _ 16
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Appendix C / Axis Transforms _ _ YP
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Appendix C / Axis Transforms _ _ B
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Appendix C / Axis Transforms _ _ YB
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Appendix C / Axis Transforms _ _ Us
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Appendix C / Axis Transforms _ _ No
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Appendix C / Axis Transforms _ _ B
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Appendix C / Axis Transforms _ _
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Appendix D / Point Mass Dynamics _
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Appendix E / Earth Geometry _ _ 18-
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Appendix E / Earth Geometry _ _ ELL
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Appendix E / Earth Geometry _ _ (dx
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Appendix E / Earth Geometry _ _ Int
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Appendix E / Earth Geometry _ _ 18.
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Appendix E / Earth Geometry _ _ 19-
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Appendix E / Earth Geometry _ _ P S
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Appendix E / Earth Geometry _ _ 19.
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Appendix E / Earth Geometry _ _ ∂
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Appendix G / Gravity _ _ 20-2
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Appendix G / Gravity _ _ g : = g +
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Appendix G / Gravity _ _ d −6 2
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Appendix G / Gravity _ _ 20-8
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Appendix H / Steady State Tracking
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Appendix I / Utilities _ _ 22.1-2
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Appendix I / Utilities _ _ Cartesia
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Appendix I / Utilities _ _ Matrix P
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Appendix I / Utilities _ _ 22.1-8
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Appendix I / Utilities / General Ut
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Appendix I / Utilities / WGS 84 Tar
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Appendix I / Utilities / Point Mass
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Appendix I / Utilities / Earth, Atm
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Appendix I / Utilities / Digital Ma
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Appendix I / Utilities / Matrices _
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Appendix I / Utilities / Quaternion
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