Post on 21-Dec-2015
transcript
GPS/Dead Reckoning GPS/Dead Reckoning Navigation with Kalman Navigation with Kalman
Filter IntegrationFilter IntegrationPaul BakkerPaul Bakker
Kalman 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 using measurements linearly related to the state but corrupted by white noise. the state but corrupted by white noise. The resulting estimator is statistically The resulting estimator is statistically optimal with respect to any quadratic optimal with respect to any quadratic function of estimation error” [1]function of estimation error” [1]
Kalman Filter UsesKalman Filter Uses
EstimationEstimation• Estimating the State of Dynamic Estimating the State of Dynamic
SystemsSystems• Almost all systems have some dynamic Almost all systems have some dynamic
componentcomponent Performance 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 system
Basic Discrete Kalman Filter Basic Discrete Kalman Filter EquationsEquations
http://www.cs.unc.edu/~welch/media/pdf/kalman_intro.pdf
Automobile Voltimeter ExampleAutomobile Voltimeter Example
Time 50 SecondsTime 50 Seconds
Time 100 SecondsTime 100 Seconds
Global Positioning SystemGlobal Positioning System
GPSGPS
24 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 satellitessatellites
Radii of 26,560 and orbital period of Radii of 26,560 and orbital period of 11.976 hours11.976 hours
Four or more satellites required to Four or more satellites required to calculate user’s positioncalculate user’s position
GPS Satellite SignalsGPS Satellite Signals
GPS code sync AnimationGPS code sync Animation
http://www.colorado.edu/geography/gcrafthttp://www.colorado.edu/geography/gcraft/notes/gps/gif/bitsanim.gif/notes/gps/gif/bitsanim.gif
When 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 satellite
Receiver Block DiagramReceiver Block Diagram
Navigation PictorialNavigation Pictorial
Position Estimates with Noise and Position Estimates with Noise and Bias InfluencesBias Influences
Differential GPS ConceptDifferential GPS Concept
Reduce 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 signals
Then send this Then send this error information error information to receivers to receivers
GPS Error SourcesGPS Error Sources
GDOPGDOP
Example of Importance of Satellite Example of Importance of Satellite ChoiceChoice
The 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 orbitorbit
Satellites have Satellites have orbital periods of orbital periods of 43,08243,082
Right Ascension Angular Location
GDOP (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 equations
RMS X ErrorRMS X Error
Graphed 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 Filter
Optimal and Non-Optimal are similarOptimal and Non-Optimal are similar
RMS Y ErrorRMS Y Error
Covariance analysis for RMS north position Covariance analysis for RMS north position errorerror
RMS Z ErrorRMS Z Error
Covariance analysis for vertical position Covariance analysis for vertical position errorerror
Clock Bias ErrorClock Bias Error
Covariance analysis for Clock bias errorCovariance analysis for Clock bias error
Clock Drift ErrorClock Drift Error
Covariance analysis for Clock drift errorCovariance analysis for Clock drift error
Questions & 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, 2001