GPS/Dead Reckoning Navigation with Kalman Filter IntegrationKalman FilterKalman Filter UsesBasic Discrete Kalman Filter EquationsAutomobile Voltimeter ExampleTime 50 SecondsTime 100 SecondsGlobal Positioning SystemGPSGPS Satellite SignalsGPS code sync AnimationReceiver Block DiagramNavigation PictorialPosition Estimates with Noise and Bias InfluencesDifferential GPS ConceptGPS Error SourcesGDOPExample of Importance of Satellite ChoiceGDOP (1,2,3,4) vs. (1,2,3,5)RMS X ErrorRMS Y ErrorRMS Z ErrorClock Bias ErrorClock Drift ErrorQuestions & ReferencesGPS/Dead Reckoning GPS/Dead Reckoning Navigation with Kalman Navigation with Kalman Filter IntegrationFilter IntegrationPaul BakkerPaul BakkerKalman FilterKalman Filter““The Kalman Filter is an estimator for what The Kalman Filter is an estimator for what is called the linear-quadratic problem, is called the linear-quadratic problem, which is the problem of estimating the which is the problem of estimating the instantaneous ‘state’ of a linear dynamic instantaneous ‘state’ of a linear dynamic system perturbed by white noise – by system perturbed by white noise – by using measurements linearly related to the using measurements linearly related to the state but corrupted by white noise. The state but corrupted by white noise. The resulting estimator is statistically optimal resulting estimator is statistically optimal with respect to any quadratic function of with respect to any quadratic function of estimation error” [1]estimation error” [1]Kalman Filter UsesKalman Filter UsesEstimationEstimation•Estimating the State of Dynamic Estimating the State of Dynamic SystemsSystems•Almost all systems have some dynamic Almost all systems have some dynamic componentcomponentPerformance AnalysisPerformance Analysis•Determine how to best use a given set Determine how to best use a given set of sensors for modeling a systemof sensors for modeling a systemBasic Discrete Kalman Filter Basic Discrete Kalman Filter EquationsEquationshttp://www.cs.unc.edu/~welch/media/pdf/kalman_intro.pdfAutomobile Voltimeter ExampleAutomobile Voltimeter ExampleTime 50 SecondsTime 50 SecondsTime 100 SecondsTime 100 SecondsGlobal Positioning SystemGlobal Positioning SystemGPSGPS24 or more satellites (28 operational 24 or more satellites (28 operational in 2000)in 2000)6 circular orbits containing 4 or more 6 circular orbits containing 4 or more satellitessatellitesRadii of 26,560 and orbital period of Radii of 26,560 and orbital period of 11.976 hours11.976 hoursFour or more satellites required to Four or more satellites required to calculate user’s positioncalculate user’s positionGPS Satellite SignalsGPS Satellite SignalsGPS code sync AnimationGPS code sync Animationhttp://www.colorado.edu/geography/gcraft/nhttp://www.colorado.edu/geography/gcraft/notes/gps/gif/bitsanim.gifotes/gps/gif/bitsanim.gifWhen the Pseudo Random codes match up When the Pseudo Random codes match up the receiver is in sync and can determine the receiver is in sync and can determine its distance from the satelliteits distance from the satelliteReceiver Block DiagramReceiver Block DiagramNavigation PictorialNavigation PictorialPosition Estimates with Noise and Position Estimates with Noise and Bias InfluencesBias InfluencesDifferential GPS ConceptDifferential GPS ConceptReduce error by Reduce error by using a known using a known ground reference ground reference and determining and determining the error of the the error of the GPS signalsGPS signalsThen send this Then send this error information to error information to receivers receiversGPS Error SourcesGPS Error SourcesGDOPGDOPExample of Importance of Satellite Example of Importance of Satellite ChoiceChoiceThe satellites are The satellites are assumed to be at a assumed to be at a 55 degree 55 degree inclination angle inclination angle and in a circular and in a circular orbitorbitSatellites have Satellites have orbital periods of orbital periods of 43,08243,082Right AscensionAngular LocationGDOP (1,2,3,4) vs. (1,2,3,5)GDOP (1,2,3,4) vs. (1,2,3,5)Optimum GDOP for the satellitesOptimum GDOP for the satellites•The smaller the GDOP the betterThe smaller the GDOP the better“GDOP Chimney” (Bad) – 2 of the 4 satellites are too close to one another – don’t provide linearly independent equationsRMS X ErrorRMS X ErrorGraphed above is the covariance analysis for RMS Graphed above is the covariance analysis for RMS east position erroreast position error•Uses Riccati equations of a Kalman FilterUses Riccati equations of a Kalman FilterOptimal and Non-Optimal are similarOptimal and Non-Optimal are similarRMS Y ErrorRMS Y ErrorCovariance analysis for RMS north position Covariance analysis for RMS north position errorerrorRMS Z ErrorRMS Z ErrorCovariance analysis for vertical position Covariance analysis for vertical position errorerrorClock Bias ErrorClock Bias ErrorCovariance analysis for Clock bias errorCovariance analysis for Clock bias errorClock Drift ErrorClock Drift ErrorCovariance analysis for Clock drift errorCovariance analysis for Clock drift errorQuestions & ReferencesQuestions & References[1] M. S. Grewal, A. P. Andrews, [1] M. S. Grewal, A. P. Andrews, Kalman Filtering, Theory and Kalman Filtering, Theory and Practice Using MATLABPractice Using MATLAB, New York: , New York: Wiley, 2001Wiley,
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