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UCGE ReportsNumber 20156Department of Geomatics EngineeringAccuracy Improvement of Low CostINS/GPS for Land Applications(URL: http:/www.geomatics.ucalgary.ca/links/GradTheses.html)byEun-Hwan ShinDecember, 20012THE UNIVERSITY OF CALGARYAccuracy Improvement of Low Cost INS/GPS for Land ApplicationsbyEun-Hwan ShinA THESISSUBMITTED TO THE FACULTY OF GRADUATE STUDIESIN PARTIAL FULFILLMENT OF THE REQUIREMENTSFOR THE DEGREE OF MASTER OF SCIENCEDEPARTMENT OF GEOMATICS ENGINEERINGCALGARY, ALBERTADECEMBER, 2001c Eun-Hwan Shin 2001AbstractSince the usage of high performance inertial navigation systems (INSs) is limited by their highprice and the regulation by the government, low cost INSs are used for general applicationareas. However, low cost INSs can experience large positioning errors in very short time dueto the low quality of the inertial measuring unit (IMU). To overcome the limitations of lowcost INSs, several techniques were developed and tested using the NovAtel Black DiamondSystem (BDSTM).A new eld calibration method was developed and tested successfully. It is not needed toalign the IMU to the local level frame with the method. Furthermore, the bias estimationof the calibration method is not aected by the reference gravity error. Almost half of thepositioning error could be removed with the accelerometer calibration information.The mechanization and navigation Kalman lter were implemented based on the navigationframe to test the velocity matching alignment and non-holonomic constraints. The veloc-ity matching alignment technique was tested for the IMUs to which stationary alignmenttechnique cannot be applied. All attitude components converged within three minutes withRMS 0.03. Non-holonomic constraints dramatically reduced the horizontal positioning er-ror, within 40 m for 20 minutes operation. Therefore, low cost INSs can be used as astand-alone positioning system during the GPS outages of over 10 minutes.iiiAcknowledgementsFirst of all, I would like to express my sincere gratitude to my supervisor, Dr. Naser El-Sheimy. He continuously encouraged me to go through all the researches during my graduatestudies. Special thanks go to Dr. Bruno Scherzinger for reviewing the new calibration methoddeveloped in this thesis and providing invaluable suggestions. I would like to acknowledgeall the examining committee members for taking time to read my thesis draft. I wouldlike to thank Waypoint Consulting Inc. for providing the GrafNav Software that was usedextensively in this thesis.Kai-Wei Chiang is thanked for his willingness to participate in all the eld tests. BruceWright, Sameh Nassar, Changlin Ma are also thanked for helping me in preparing the eldtests. I would like to acknowledge Mike Bobye for his willingness to provide detailed infor-mation about the data format of BDSTM system. I would like to acknowledge Dr. AlexanderBruton for many discussions and advices. Sandra Kennedy and Darren Cosandier are alsothanked for the discussion about GPS velocity errors. Michael Kern is thanked for helpingme get used to LATEX.Finally thanks go to my lovely wife, Su Nam Lee, for her dedication and endurance.ivContentsApproval PageAbstractAcknowledgementsContentsList of TablesList of FiguresList of Symbols, Abbreviations, Nomenclature1 Introductioniiiiiivvixxixiv11.11.2Background and Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . .Thesis Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .152 Terrestrial Inertial Navigation Mechanization72.1Reference Frames and Transformations . . . . . . . . . . . . . . . . . . . . .v82.22.3Inertial Navigation Equations . . . . . . . . . . . . . . . . . . . . . . . . . .INS Mechanization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .14.22.3.3Error Compensation . . . . . . . . . . . . . . . . . . . . . . . . . . .Attitude Integration . . . . . . . . . . . . . . . . . . . . . . . . . . .Velocity and Position Integration . . . . . . . . . . . . . . . . . . . .2122233 Development of INS/GPS Integration Kalman Filter263.1Perturbation Analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .23.5Position Error Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Velocity Error Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Attitude Error Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Implementation of the INS/GPS Kalman Filter . . . . . . . . . . . . . . . .282933354 Accuracy Improvement of Low Cost INS/GPS474.1Field Calibration Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . .4.2Consideration of Nonorthogonality . . . . . . . . . . . . . . . . . . .A New Calibration Method . . . . . . . . . . . . . . . . . . . . . . .vi495.44.1.5Adjustment Computation . . . . . . . . . . . . . . . . . . . . . . . .Calibration Methodology . . . . . . . . . . . . . . . . . . . . . . . . .Sensitivity of the Method . . . . . . . . . . . . . . . . . . . . . . . .5561634.2Alignment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .6.24.2.3Analytic Coarse Alignment . . . . . . . . . . . . . . . . . . . . . . . .Fine Alignment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Velocity Matching Alignment . . . . . . . . . . . . . . . . . . . . . .6872754.34.4Using Non-Holonomic Constraints . . . . . . . . . . . . . . . . . . . . . . . .Limiting Attitude Error Growth . . . . . . . . . . . . . . . . . . . . . . . . .78805 Tests and Results85.45.5System Conguration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Test Dataset . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Data Processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Field Calibration Method . . . . . . . . . . . . . . . . . . . . . . . . . . . .INS Mechanization Performance . . . . . . . . . . . . . . . . . . . . . . . . .vii83878890985.65.7Velocity Matching Alignment . . . . . . . . . . . . . . . . . . . . . . . . . . 100Test of Non-Holonomic Constraints . . . . . . . . . . . . . . . . . . . . . . . 1056 Conclusions and Recommendations10Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110Recommendations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113Referencesviii115List of Tables1.1INS performance (Schwarz and El-Sheimy, 1999; Greenspan, 1995; Gebre-Egziabher et al., 2001) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5.4Calibration with horizontal measurements . . . . . . . . . . . . . . . . . . .Sensitivity of the new calibration method to the reference gravity errors . . .Scale factor sensitivity of the simplest calibration method . . . . . . . . . . .The specication of HG1700 IMU (Honeywell, Inc.) . . . . . . . . . . . . . .Statistics of the rst calibration observations . . . . . . . . . . . . . . . . . .Statistics of the second calibration observations . . . . . . . . . . . . . . . .Calibration results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .646567869192945.5Attitude corrections after 5 minutes . . . . . . . . . . . . . . . . . . . . . . . 102ix5.6Velocity matching alignment residuals after 5 minutes . . . . . . . . . . . . . 104xList of Figures4.1The inertial frame . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .The Earth frame and the navigation frame . . . . . . . . . . . . . . . . . . .The body frame . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Sampling rate of HG1700 IMU . . . . . . . . . . . . . . . . . . . . . . . . . .The navigation frame INS mechanization . . . . . . . . . . . . . . . . . . . .GPS and IMU measurement time . . . . . . . . . . . . . . . . . . . . . . . .Leverarm eect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .The feedforward method . . . . . . . . . . . . . . . . . . . . . . . . . . . . .The feedback method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Misalignment to the local level frame . . . . . . . . . . . . . . . . . . . . . .xi8101022254244454554.84.9Nonorthogonality between x and y . . . . . . . . . . . . . . . . . . . . . . .Nonorthogonality of z-axis to xy plane . . . . . . . . . . . . . . . . . . . . .IMU measurement attitudes . . . . . . . . . . . . . . . . . . . . . . . . . . .An example of IMU rotation scheme . . . . . . . . . . . . . . . . . . . . . .Calibration with horizontal measurements . . . . . . . . . . . . . . . . . . .Scale Factor Error Sensitivity . . . . . . . . . . . . . . . . . . . . . . . . . .Coarse Alignment Error . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Fine alignment with the specic force measurements . . . . . . . . . . . . . .51526162636672734.10 Fine alignment with the velocity measurements . . . . . . . . . . . . . . . .4.11 Initial Heading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4.12 Velocity Matching Alignment . . . . . . . . . . . . . . . . . . . . . . . . . .4.13 Implementation of the non-holonomic constraints . . . . . . . . . . . . . . .73767680The BDSTM system, Courtesy of NovAtel Inc., Canada . . . . . . . . . . . .Instrument setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Test dataset . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .xii8585875.45.5Data processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .The reference GPS trajectories . . . . . . . . . . . . . . . . . . . . . . . . . .88905.6Eect of calibration on positioning (Test #1). . . . . . . . . . . . . . . . .955.7Eect of calibration on positioning (Test #2, 15 minutes) . . . . . . . . . . .965.8Eect of calibration on positioning (Test #2). . . . . . . . . . . . . . . . .975.9Mechanization performance (Test #1). . . . . . . . . . . . . . . . . . . . .995.10 Mechanization performance (Test #2). . . . . . . . . . . . . . . . . . . . . 1005.11 GPS velocity measurement noise . . . . . . . . . . . . . . . . . . . . . . . . . 1015.12 PDOP during the velocity matching alignment . . . . . . . . . . . . . . . . . 1015.13 Attitude corrections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1035.14 Velocity matching alignment residuals . . . . . . . . . . . . . . . . . . . . . . 1045.15 Misalignment of IMU to the vehicle forward direction . . . . . . . . . . . . . 1055.16 Test of nonholonomic constraints for the rst dataset . . . . . . . . . . . . . 1065.17 Test of nonholonomic constraints for the second dataset . . . . . . . . . . . . 107xiiiList of Symbols, Abbreviations,NomenclatureThe notations used in this thesis follow those widely used in Geomatics and navigation elds.1. Conventions(a) Matrices are denoted as upper case italic letters.(b) Vectors are denoted as underlined lower case italic letters.(c) The coordinate frames that are involved in the vector transformation are denotedas subscript and superscript. For instance, Cbn is the direction cosine matrix fromthe body frame to the navigation frame. For the angular rate vector subscriptdenotes the reference and target frame, and superscript denotes the projected orrealized frame. For example, nib represents the angular rate vector of the bodyframe with respect to the inertial frame projected to the navigation frame.xiv(d) Operators are dened as:( )1Ttime derivativeestimated or computed valuesmeasured valuesKalman predictionerror of, correction toDirac delta functionincrement ofmatrix inversematrix transpose inner product( )cross productcross product or skew symmetric form of a vectordiag() diagonal matrixE f ( )expectationis a function ofL1() inverse Laplace transform2. Symbols0333 3 zero matrixxvattitude error vectoraAbBCeeEfFnormal gravitygeodetic longitudeangular rate vectorcross product or skew symmetric matrix form of geodetic latituderolltransition matrixpitch, non-orthogonality of IMU axesheadingsemi-major axis of the reference ellipsoiddesign matrixbiasdesign matrixdirection cosine matrix, covariance matrixlinear eccentricity of the reference ellipsoidmeasurement noise vectorcross product or skew symmetric matrix form ofspecic force vectordynamics matrixxviggGhHII33KmMNPqQkQ(t)rRgravitational acceleration vectorgravity vectordesign matrix of system noise vectorellipsoidal heightdesign matrix for measurementsidentity matrix3 3 identity matrixKalman gain matrixmeasurementsorder of interpolation or extrapolationradius of curvature in meridianradius of curvature in prime vertical,coecient matrix of the normal equationcovariance matrix of state vectorquaternion vectorcovariance matrix of system noise sequence vectorspectral density matrixposition vector, residual vectorrotation matrix of coordinate systems or vectors,covariance matrix of measurement error vectorxviisuvvfwxyz3. AcronymsBDSTMDCMENUFOGGPSIMUINSMEMSNEDscale factorcontinuous time system noise vectorvelocity vectorvelocity increment vector which is not correctedfor the Coriolis and gravity forcesystem noise sequence vector, misclosure vectorstate vector, x-axisy-axismeasurement vector, z-axisBlack Diamond SystemDirection Cosine MatrixEast-North-UpFibre Optic GyroscopeGlobal Positioning SystemInertial Measuring UnitInertial Navigation SystemMicro Electrical Mechanical SystemsNorth-East-DownxviiiChapter 1Introduction1.1Background and ObjectiveThe integration of a navigation-grade inertial navigation system (INS) with the global posi-tioning system (GPS) has been done for the application areas in which attitude informationis indispensable and rapid collection of geographic information is required. In practice, in-tegration is necessary for navigation in urban areas where the signal from the satellites issusceptible to blocking by many obstacles (such as skyscrapers, trees, etc.). However, thereare two restrictions in using high performance INSs. One is their price, over US$100,000,and the other is a regulation by the government. Hence, a high performance INS is usuallyused in military applications and commercial airliners,