date post

23-Nov-2014Category

## Documents

view

115download

0

Embed Size (px)

Introduction to Modern Navigation Systems This page intentionally left blank This page intentionally left blankNE WJ E RSE Y L ONDON SI NGAP ORE BE I J I NG SHANGHAI HONGKONG TAI P E I CHE NNAI World ScientifcEsmat BekirIntroduction to ModernNavigation SystemsBritish Library Cataloguing-in-Publication DataA catalogue record for this book is available from the British Library.Forphotocopyingofmaterialinthisvolume,pleasepayacopyingfeethroughtheCopyrightClearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission tophotocopy is not required from the publisher.ISBN-13 978-981-270-765-9ISBN-10 981-270-765-4ISBN-13 978-981-270-766-6 (pbk)ISBN-10 981-270-766-2 (pbk)All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means,electronic or mechanical, including photocopying, recording or any information storage and retrievalsystem now known or to be invented, without written permission from the Publisher.Copyright 2007 by World Scientific Publishing Co. Pte. Ltd.Published byWorld Scientific Publishing Co. Pte. Ltd.5 Toh Tuck Link, Singapore 596224USA office:27 Warren Street, Suite 401-402, Hackensack, NJ 07601UK office:57 Shelton Street, Covent Garden, London WC2H 9HEPrinted in Singapore.INTRODUCTION TO MODERN NAVIGATION SYSTEMSKim - Intro to Modern.pmd 8/28/2007, 11:55 AM 1To my Lord with humility and love This page intentionally left blank This page intentionally left blankviiPreface Thegoalofthistextistoconciselypresentthemathematicalblocks neededforimplementingthemainbodyofastrappeddownInertial NavigationSystem(INS)inamannerthatprovidesamentalimageof thecontributionofeachblockandtheir interrelation.Thetextdescribes themakeupofeachblockandprovidesthederivationofitsequations. Towardsthisobjective,whentheneedforclarifyingorjustifyinga certainideaarises,itispresentedinanappendixsoasnottointerfere with the flow of the main ideas. This treatment should benefit both the novice as well as a practitioner in the field. For a journeyman in the area of navigation, this book can be usedtopinpointtheequationsthatarethebasisofsuchasystem,how theyaredevelopedandhowtheyareimplemented.Thosewithmore experience may use this book as a quick reference guide.What is navigation anyway? It is the ability to set the course of a ship to move between two desired locations. To do that the navigator must be able to know his location and set the velocity vector towards the desired destination.Thustheprimefunctionofanavigationsystemis determining the crafts position and velocity. Wewillbeprimarilyconcernedhereinwithaspecialtypeof navigation: inertial navigation. And why inertial navigation in particular?Inertialsystemsareself-contained:theyareindependentofweather conditions and are operable anywhere in seas, underwater, lands, tunnels, orinair.Shortofareliablesourceofpower,theycanworkalmost indefinitely. Introduction to Modern Navigation SystemsviiiIf the Earth were flat, inertial navigation algorithms would have been aloteasier.BecausenavigationusuallyisonorclosetoEarth,a sphericalbodythatrotatesaboutitself,wewillsoonfindourselves entangledindiscussingtwodifferentelementsatthesametime: developingthemathematicalalgorithmsanddescribingthepertinent physics of the Earth. Wehavedevotedthefirstthreechapterstointroducethe mathematicalfoundationfordevelopingthealgorithms.These algorithmsrelyheavilyonvectorandmatrixnotations,andforthat, vector and matrix properties are introduced in Chapter 1. On developing theequations,wewilldiscoverthatourvariablesofinterestare representedindifferentcoordinatesystems.Obviouslythiscreatesthe needofmovingfromonecoordinatesystemtoanotherandthusthis conceptisdiscussedinChapter2.Forfurtherclarity,Chapter3 introducesthemostcommonapproachesusedinperformingcoordinate transformations.Therewediscusstheseapproachesandtheir relationships amongst one another. Wediscuss thephysical propertiesoftheEarthinChapter4.At this point,armedwiththemathematicaltools and thegeometricalproperties ofEarth,wedeveloptheinertialnavigationequationsfromfirst principles inChapter5.Thisyieldsasetofcontinuoustimedifferential equationsthatshouldbesolvedtoyieldthenavigationsolution. Mechanizingandimplementingtheseequationsona digitalcomputer is introducedinChapter6.Wediscoveroneofthedrawbacksofinertial navigationsystems:unreliableverticalchannel.Thismeansthatthey cannot be relied on to provide altitude or vertical velocity. Typically, the navigation system for aircraft is usually complemented with some sort of altimeter. Integrating the altimeter measurements with theINSisdiscussedtherein.Ourrelianceonairdataforaidingthe navigationsystemdoesnotendatthispoint:theINScannotestimate windspeed.Chapter7isdevotedtothediscussionofairdata computationsandtheiruseforcomputingrateofclimb/decentand relative airspeed. Usingthelegendarylinesoflongitudeandlatitudetolocateacraft locationonthesurfaceoftheEarthintroducesapeculiarnavigational Prefaceixphenomenon:alllongitudelinesmeetatthetwopolarpointsofthe Earth.Solvingnavigationequationsnearthesetwopointscanproveto bemathematicallycumbersome.Despitetherarityoftheseevents, considerableattentionhas beengiventoaverttheconsequencesofsuch an occurrence. The wander azimuth angle is one such classical technique anditiscritiquedinChapter8.Aninnovativesimplealgorithmfor navigating in the polar circle is introduced shortly thereafter. Two problems remain to be solved. The first addresses the alignment problem.Simplystated,itisdeterminingtheinitialconditionsofthe differential equations that were developed in Chapter 5. The second deals with the real life factors: no matter how expensive the inertial sensors are theyhaveerrorsthatmustbeestimated.Solutionstotheseproblems depends on the inertial sensor level accuracy onboard the specific craft. Toelaborate,wemayhavenoticedthatimplementingtheINS equationsnotonlyprovidethecraftlocationandspeed,butalsoits attitudeandheading.Someapplicationsusenavigationgradesensorsto estimatealltheaboveparameters.Butothersutilizelow-grade inexpensive sensors focused on estimating only the attitude and heading. Insodoing,theseapplicationscalledAttitudeand HeadingReference Systems (AHRS) forgo estimating the location and velocity. Chapter 9 addresses the alignment problem for navigation systems. In Chapter10,wediscusstheAHRSsystems,introducetheirpertinent alignmentalgorithms,andestimatesensorerrors.Often,anAHRS complementsitsinertialsystemwithmagneticdetectorsandutilizes mathematical algorithms to estimate the relatively large errors introduced by the inertial systems. Asystemthatutilizesnavigationgradesensorscouldenhanceits performancebyusingaidssuchasaGlobalPositioningSystem(GPS). LiketheINS,theGPSprovidespositionandvelocity.Thesetwo,INS andGPS,whenmathematicallyfusedwithaKalmanfilter,canbeused toestimatetheinaccuraciesduetothesensorsoftheformersystem. These equations are developed in Chapter 11. Thisbookcouldnotbearealitywithoutthehelpandsupportfrommanyfriendsandcolleagues.IamverygratefultoA.BekirandH. Aleem for reviewing the script and suggesting many improvements. I Introduction to Modern Navigation SystemsxwouldliketothankDr.M.Hafez,Dr.M.A.M.Aliforsupportand encouragements.SpecialthankstoMs.KimTanforherguidance throughout the publishing process. Last but not least I am very grateful to myfamilyfortheirloveandsupportduringtheentiretimeforwriting the book. xiContents Prefacevii Introduction1 1.Vectors and Matrices7 1.1Introduction7 1.2Vector Inner Product9 1.3Vector Cross Products and Skew Symmetric Matrix Algebra10 2.Coordinate Transformation between Orthonormal Frames17 2.1Introduction17 2.2Direction Cosine Matrices18 2.3The Direction Cosine Matrix is a Unitary Matrix20 2.4The Direction Cosine Matrix is a Transformation Matrix21 2.5DCM Fixed Axis24 2.6The Rotation Matrix26 2.7Inner and Outer Transformation Matrices29 2.8The Quaternion32 3.Forms of the Transformation Matrix35 3.1 Introduction353.2 Simple Frame Rotations363.3 Euler Angles373.4 Rotation Vector383.5 Quaternion393.6 Simple Quaternions433.7 Conversion between Forms453.7.1Conversion between DCM and Euler453.7.2Conversion between DCM and Quaternion453.7.3Conversion between Euler Angles and Quaternion47Introduction to Modern Navigation Systemsxii3.8Dynamics of the Transformation Matrix47 3.8.1DCM Differential Equation483.8.2Quaternion Differential Equation503.8.3Rotation Vector Differential Equation523.8.4Euler Angles Differential Equation554.Earth and Navigation58 4.1 Introduction584.2 Earth, Geoid and Ellipsoid59 4.3 Radii of Curvature634.4 Earth, Inertial and Navigation Frames654.5 Earth Rate674.6 The Craft Rate nen 674.7 Solution of the DCMneC 704.8 Gravitational and Gravity Fields70 5.The Inertial Navigation System Equations75 5.1 Introduction755.2 Body Frame of Reference765.3 Inertial Sensors775.3.1The Accelerometer775.3.2The Rate Gyro785.4 The Attitude Equation785.5 The Navigation Equation805.6 Navigation Equations Computational Flow Diagram835.7 The Navigation Equation in Earth Frame84 6.Implementation86 6.1 Introduction866.2 The Rotation Vector Differential Equation876.3 The Attitude Equation926.4 The Craft Velocity Equation956.5 The Craft Position Equation996.6 The Vertical Channel101 7.Air Data Computer104 7.1Introduction104 7.2US Standard Atmosphere 1976105 7.3Pressure Altitude107 7.4Vertical Channel Parameter Estimation Using Inertial andAir Data111 7.5Density Altitude116 Contentsxiii7.6Altitude (Descend /Climb) Rate117 7.7Air Speed117 7.8Indicated Air Speed (IAS)119 8.Polar Navigation121 8.1 Introduction1218.2 The Wander Azimuth Navigation1238.3 Prospective of the Wander Azimuth Approach1268.4 Polar Circle Navigation Algorithm1288.5 Alternative Polar Circle Navigation Frame132 9.Alignment136 9.1 Introduction1369.2 IMU Alignment1379.3 Alternative Algorithm for bnC 1449.4 Estimation of the Accelerometer and Gyro Biases1499.5 Effects of Biases on Estimate of bnC 150

10.Attitude and Heading Reference System152 10.1 Introduction15210.2 Attitude