Localization with GPS - Computer Sciencermartin/teaching/spring05/local/slides/gps.pdf ·...
Transcript of Localization with GPS - Computer Sciencermartin/teaching/spring05/local/slides/gps.pdf ·...
![Page 1: Localization with GPS - Computer Sciencermartin/teaching/spring05/local/slides/gps.pdf · Localization with GPS From GPS Theory and Practice Fifth Edition Presented by Martin Constantine](https://reader033.fdocuments.in/reader033/viewer/2022042708/5abbe59e7f8b9a297f8d75c6/html5/thumbnails/1.jpg)
Localization with GPSLocalization with GPS
From GPS Theory andPractice Fifth Edition
Presented by MartinConstantine
![Page 2: Localization with GPS - Computer Sciencermartin/teaching/spring05/local/slides/gps.pdf · Localization with GPS From GPS Theory and Practice Fifth Edition Presented by Martin Constantine](https://reader033.fdocuments.in/reader033/viewer/2022042708/5abbe59e7f8b9a297f8d75c6/html5/thumbnails/2.jpg)
Introduction
w GPS = Global Positioning System
w Three segments:1. Space (24 satellites)
2. Control (DOD)
3. User (civilian and military receivers)
![Page 3: Localization with GPS - Computer Sciencermartin/teaching/spring05/local/slides/gps.pdf · Localization with GPS From GPS Theory and Practice Fifth Edition Presented by Martin Constantine](https://reader033.fdocuments.in/reader033/viewer/2022042708/5abbe59e7f8b9a297f8d75c6/html5/thumbnails/3.jpg)
GPS Overview
w Satellites transmit L1 and L2 signalsw L1--two pseudorandom noise signals
– Protected (P-)code– Course acquisition (C/A) code (most civilian
receivers)
w L2--P-code onlyw Anti-spoofing adds noise to the P-code,
resulting in Y-code
![Page 4: Localization with GPS - Computer Sciencermartin/teaching/spring05/local/slides/gps.pdf · Localization with GPS From GPS Theory and Practice Fifth Edition Presented by Martin Constantine](https://reader033.fdocuments.in/reader033/viewer/2022042708/5abbe59e7f8b9a297f8d75c6/html5/thumbnails/4.jpg)
Observables
w Code pseudoranges
![Page 5: Localization with GPS - Computer Sciencermartin/teaching/spring05/local/slides/gps.pdf · Localization with GPS From GPS Theory and Practice Fifth Edition Presented by Martin Constantine](https://reader033.fdocuments.in/reader033/viewer/2022042708/5abbe59e7f8b9a297f8d75c6/html5/thumbnails/5.jpg)
Observables
w Phase pseudoranges– N = number of cycles between satellite and
receiver
![Page 6: Localization with GPS - Computer Sciencermartin/teaching/spring05/local/slides/gps.pdf · Localization with GPS From GPS Theory and Practice Fifth Edition Presented by Martin Constantine](https://reader033.fdocuments.in/reader033/viewer/2022042708/5abbe59e7f8b9a297f8d75c6/html5/thumbnails/6.jpg)
Observables
w Doppler Data– Dots indicate derivatives wrt time.
![Page 7: Localization with GPS - Computer Sciencermartin/teaching/spring05/local/slides/gps.pdf · Localization with GPS From GPS Theory and Practice Fifth Edition Presented by Martin Constantine](https://reader033.fdocuments.in/reader033/viewer/2022042708/5abbe59e7f8b9a297f8d75c6/html5/thumbnails/7.jpg)
Observables
w Biases and Noise
![Page 8: Localization with GPS - Computer Sciencermartin/teaching/spring05/local/slides/gps.pdf · Localization with GPS From GPS Theory and Practice Fifth Edition Presented by Martin Constantine](https://reader033.fdocuments.in/reader033/viewer/2022042708/5abbe59e7f8b9a297f8d75c6/html5/thumbnails/8.jpg)
Combining Observables
w Generally
w Linear combinations with integers
w Linear combinations with real numbers
w Smoothing
![Page 9: Localization with GPS - Computer Sciencermartin/teaching/spring05/local/slides/gps.pdf · Localization with GPS From GPS Theory and Practice Fifth Edition Presented by Martin Constantine](https://reader033.fdocuments.in/reader033/viewer/2022042708/5abbe59e7f8b9a297f8d75c6/html5/thumbnails/9.jpg)
Mathematical Models forPositioning
w Point positioningw Differential positioning
– With code ranges– With phase ranges
w Relative positioning– Single differences– Double differences– Triple differences
![Page 10: Localization with GPS - Computer Sciencermartin/teaching/spring05/local/slides/gps.pdf · Localization with GPS From GPS Theory and Practice Fifth Edition Presented by Martin Constantine](https://reader033.fdocuments.in/reader033/viewer/2022042708/5abbe59e7f8b9a297f8d75c6/html5/thumbnails/10.jpg)
Point PositioningWith Code Ranges
With Carrier Phases
With Doppler Data
![Page 11: Localization with GPS - Computer Sciencermartin/teaching/spring05/local/slides/gps.pdf · Localization with GPS From GPS Theory and Practice Fifth Edition Presented by Martin Constantine](https://reader033.fdocuments.in/reader033/viewer/2022042708/5abbe59e7f8b9a297f8d75c6/html5/thumbnails/11.jpg)
Differential Positioning
Two receivers used:•Fixed, A: Determines PRC and RRC•Rover, B: Performs point pos’ning with PRC and RRC
from AWith Code Ranges
![Page 12: Localization with GPS - Computer Sciencermartin/teaching/spring05/local/slides/gps.pdf · Localization with GPS From GPS Theory and Practice Fifth Edition Presented by Martin Constantine](https://reader033.fdocuments.in/reader033/viewer/2022042708/5abbe59e7f8b9a297f8d75c6/html5/thumbnails/12.jpg)
Differential Positioning
With Phase Ranges
![Page 13: Localization with GPS - Computer Sciencermartin/teaching/spring05/local/slides/gps.pdf · Localization with GPS From GPS Theory and Practice Fifth Edition Presented by Martin Constantine](https://reader033.fdocuments.in/reader033/viewer/2022042708/5abbe59e7f8b9a297f8d75c6/html5/thumbnails/13.jpg)
Relative PositioningAim is to determine the baseline vector A->B.
A is known,B is the reference point
Assumptions: A, B are simultaneously observed
Single Differences:•two points and one satellite•Phase equation of each point is differenced to yield
![Page 14: Localization with GPS - Computer Sciencermartin/teaching/spring05/local/slides/gps.pdf · Localization with GPS From GPS Theory and Practice Fifth Edition Presented by Martin Constantine](https://reader033.fdocuments.in/reader033/viewer/2022042708/5abbe59e7f8b9a297f8d75c6/html5/thumbnails/14.jpg)
Relative Positioning
w Double differences– Two points and two satellites
– Difference of two single-differences gives
![Page 15: Localization with GPS - Computer Sciencermartin/teaching/spring05/local/slides/gps.pdf · Localization with GPS From GPS Theory and Practice Fifth Edition Presented by Martin Constantine](https://reader033.fdocuments.in/reader033/viewer/2022042708/5abbe59e7f8b9a297f8d75c6/html5/thumbnails/15.jpg)
Relative Positioning
w Triple-Differences– Difference of double-differences across two
epochs
![Page 16: Localization with GPS - Computer Sciencermartin/teaching/spring05/local/slides/gps.pdf · Localization with GPS From GPS Theory and Practice Fifth Edition Presented by Martin Constantine](https://reader033.fdocuments.in/reader033/viewer/2022042708/5abbe59e7f8b9a297f8d75c6/html5/thumbnails/16.jpg)
Adjustment of MathematicalModels
w Models above need adjusting so that theyare in a linear form.w Idea is to linearize the distance metrics
which carry the form:
![Page 17: Localization with GPS - Computer Sciencermartin/teaching/spring05/local/slides/gps.pdf · Localization with GPS From GPS Theory and Practice Fifth Edition Presented by Martin Constantine](https://reader033.fdocuments.in/reader033/viewer/2022042708/5abbe59e7f8b9a297f8d75c6/html5/thumbnails/17.jpg)
Adjustment of MathematicalModels
w Each coordinate is decomposed as follows:
Allowing the Taylor series expansion of f
![Page 18: Localization with GPS - Computer Sciencermartin/teaching/spring05/local/slides/gps.pdf · Localization with GPS From GPS Theory and Practice Fifth Edition Presented by Martin Constantine](https://reader033.fdocuments.in/reader033/viewer/2022042708/5abbe59e7f8b9a297f8d75c6/html5/thumbnails/18.jpg)
Adjustment of MathematicalModels
w Computing the partial derivatives andsubstituting preliminary equations yieldsthe linear equation:
![Page 19: Localization with GPS - Computer Sciencermartin/teaching/spring05/local/slides/gps.pdf · Localization with GPS From GPS Theory and Practice Fifth Edition Presented by Martin Constantine](https://reader033.fdocuments.in/reader033/viewer/2022042708/5abbe59e7f8b9a297f8d75c6/html5/thumbnails/19.jpg)
Linear Models
w Point Positioning with Code Ranges– Recall:– Substitution of the linearized term (prev. slide) and
rearranging all unknowns to the left, gives:
![Page 20: Localization with GPS - Computer Sciencermartin/teaching/spring05/local/slides/gps.pdf · Localization with GPS From GPS Theory and Practice Fifth Edition Presented by Martin Constantine](https://reader033.fdocuments.in/reader033/viewer/2022042708/5abbe59e7f8b9a297f8d75c6/html5/thumbnails/20.jpg)
Linear Models
w Point Positioning with Code Rangesw Four unknowns implies the need for four
satellites. Let:
![Page 21: Localization with GPS - Computer Sciencermartin/teaching/spring05/local/slides/gps.pdf · Localization with GPS From GPS Theory and Practice Fifth Edition Presented by Martin Constantine](https://reader033.fdocuments.in/reader033/viewer/2022042708/5abbe59e7f8b9a297f8d75c6/html5/thumbnails/21.jpg)
Linear Models
w Point Positioning with Code Ranges
w Assuming satellites numbered from 1 to 4
Superscripts denote satellite numbers, not indices.
![Page 22: Localization with GPS - Computer Sciencermartin/teaching/spring05/local/slides/gps.pdf · Localization with GPS From GPS Theory and Practice Fifth Edition Presented by Martin Constantine](https://reader033.fdocuments.in/reader033/viewer/2022042708/5abbe59e7f8b9a297f8d75c6/html5/thumbnails/22.jpg)
Linear Models
Point Positioning with Code Ranges
•We can now express the model in matrix form as l = Ax where
![Page 23: Localization with GPS - Computer Sciencermartin/teaching/spring05/local/slides/gps.pdf · Localization with GPS From GPS Theory and Practice Fifth Edition Presented by Martin Constantine](https://reader033.fdocuments.in/reader033/viewer/2022042708/5abbe59e7f8b9a297f8d75c6/html5/thumbnails/23.jpg)
Linear Models
Point Positioning with Carrier Phases
•Similarly computed.•Ambiguities in the model raise the number of unknowns from 4 to 8•Need three epochs to solve the system. It produces 12 equations with 10 unknowns.
![Page 24: Localization with GPS - Computer Sciencermartin/teaching/spring05/local/slides/gps.pdf · Localization with GPS From GPS Theory and Practice Fifth Edition Presented by Martin Constantine](https://reader033.fdocuments.in/reader033/viewer/2022042708/5abbe59e7f8b9a297f8d75c6/html5/thumbnails/24.jpg)
Linear Models
Point Positioning with Carrier PhasesLinear Model
![Page 25: Localization with GPS - Computer Sciencermartin/teaching/spring05/local/slides/gps.pdf · Localization with GPS From GPS Theory and Practice Fifth Edition Presented by Martin Constantine](https://reader033.fdocuments.in/reader033/viewer/2022042708/5abbe59e7f8b9a297f8d75c6/html5/thumbnails/25.jpg)
Linear Models
Point Positioning with Carrier Phasesl = Ax
![Page 26: Localization with GPS - Computer Sciencermartin/teaching/spring05/local/slides/gps.pdf · Localization with GPS From GPS Theory and Practice Fifth Edition Presented by Martin Constantine](https://reader033.fdocuments.in/reader033/viewer/2022042708/5abbe59e7f8b9a297f8d75c6/html5/thumbnails/26.jpg)
Linear Models
Relative Positioning
•Carrier phases considered•Double-differences treated•Recall: DD equation * _
•The second term on the lhs is expanded and linearized as in previous models to yield:
![Page 27: Localization with GPS - Computer Sciencermartin/teaching/spring05/local/slides/gps.pdf · Localization with GPS From GPS Theory and Practice Fifth Edition Presented by Martin Constantine](https://reader033.fdocuments.in/reader033/viewer/2022042708/5abbe59e7f8b9a297f8d75c6/html5/thumbnails/27.jpg)
Linear Models
Relative Positioning•The second term on the lhs is expanded and linearized as in previous models to yield ( [9.133]…see paper pg 262)• l’s:
![Page 28: Localization with GPS - Computer Sciencermartin/teaching/spring05/local/slides/gps.pdf · Localization with GPS From GPS Theory and Practice Fifth Edition Presented by Martin Constantine](https://reader033.fdocuments.in/reader033/viewer/2022042708/5abbe59e7f8b9a297f8d75c6/html5/thumbnails/28.jpg)
Linear Models
Relative Positioning• The right hand side is abbreviated as follows (a’s):
![Page 29: Localization with GPS - Computer Sciencermartin/teaching/spring05/local/slides/gps.pdf · Localization with GPS From GPS Theory and Practice Fifth Edition Presented by Martin Constantine](https://reader033.fdocuments.in/reader033/viewer/2022042708/5abbe59e7f8b9a297f8d75c6/html5/thumbnails/29.jpg)
Linear Models
Relative Positioning•Since the coordinates of A must be known, the number of unknowns is reduced by three. Now, 4 satellites (j,k,l,m) and two epochs are needed to solve the system.
![Page 30: Localization with GPS - Computer Sciencermartin/teaching/spring05/local/slides/gps.pdf · Localization with GPS From GPS Theory and Practice Fifth Edition Presented by Martin Constantine](https://reader033.fdocuments.in/reader033/viewer/2022042708/5abbe59e7f8b9a297f8d75c6/html5/thumbnails/30.jpg)
Extra References
w Introduction and overview:http://www.gpsy.org/gpsinfo/gps-faq.txt