Antti Lehtinen Doppler Positioning with GPS - Matematiikan laitos

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the Transit, the receiver was lucky to be able to track even one satellite. Most of the time there were no satellites in the view. On the contrary, there are usually several GPS satellites in the view at the same time. Even if the receiver is indoors, most of the GPS signals can be received, although attenuated. The high number of satellites allows a GPS receiver to have simultaneous Doppler measurements from several satellites. This is a huge increase in the amount of information that can be used for the positioning. In addition, the GPS frequency is much higher than the Transit frequencies. This is a benefit, because relative frequency measurement errors become smaller. Furthermore, the higher frequency causes smaller atmospheric errors. As a conclusion, it cannot be said which is better: the Transit or the GPS Doppler positioning. Both have their advantages and drawbacks. A summary of these is shown in Table 4.1. After all, one can say that the GPS Doppler positioning may well be of at least the same accuracy as the Transit. Thus, there is a reason for further GPS Doppler positioningresearch. More thorough positioningerror analysis will be provided in Chapter 5. Table 4.1: Comparison between GPS Doppler positioningand Transit GPS Transit +Better satellite availability +Two different frequencies +Higher frequency +Higher satellite velocities +Lower satellite orbits 4.3 The Governing Equations Let us now begin to build a physical model for the GPS Doppler positioning. This will eventually lead to an algorithm, that can be used in position estimation. 4.3.1 The Doppler Shift Equation The frequency of the received GPS signal differs from the frequency transmitted by the satellite. This easily measurable frequency offset is mainly due to the Doppler effect. The Doppler effect is caused by the relative motion of the transmittingsatellite with respect to the receiver. The Doppler Shift can be modelled with the dot product equation wi − wu Di = − • c ri − ru L1 (4.1) ri − ru 18
where wi is the satellite velocity, wu is the receiver velocity, c is the speed of light, ri is the satellite location, ru is the receiver location and L1 is the frequency transmitted by the satellite. The Doppler shift equation 4.1 can be modified to achieve a range rate equation. For notational simplicity let us define the satellite’s relative velocity with the followingequation: (4.2) vi = wi − wu After algebraic manipulation one obtains vi • ru − ri ru − ri = Dic L1 (4.3) We notice that the left hand side of the equation 4.3 is the rate at which the satellite is approachingthe user. Thus, the equation can also be written as a range rate equation as − d dt ru − ri = Dic (4.4) Assume that the user knows the current time as well as his own velocity exactly, and that he has the valid ephemeris. Consequently he can calculate satellites’ position vectors ri and relative velocities vi. Also assume that the receiver knows the speed of light constant c, the real transmitted frequency L1 and the real Doppler shift Di. With precedingassumptions, the only unknown in the equation 4.3 is the user position vector ru. Interpretingthe dot product geometrically with the formula a • b = a·b·cos ∠(a, b) and noticingthat the vector ru − ri is of unit ru − ri length, the equation 4.3 can be written as L1 vi·cos α = Dic L1 (4.5) where α = ∠(vi, ru − ri) is the angle between the relative satellite velocity vi and the satellite-to-user vector ru − ri. Solvingfor the angle α in the equation 4.5 gives us Dic α = arccos (4.6) L1vi Thus the receiver must be located such that the satellite-to-user vector ru − ri forms an angle α with the satellite relative velocity vector vi. The possible receiver positions ru form a surface of a right circular cone whose vertex is at the satellite position ri. The cone is shown in Figure 4.1. The cone has a semiinfinite axis in the direction of the satellite velocity vi when the Doppler shift Di is positive and in the direction opposite to vi when Di is negative. The angle between the surface of the cone and the satellite relative velocity vector vi is 19
Page 1 and 2: TAMPERE UNIVERSITY OF TECHNOLOGY De
Page 3 and 4: Contents Abstract v Tiivistelmä vi
Page 5 and 6: 5.2.3 Applications of the Doppler D
Page 7 and 8: Tiivistelmä TAMPEREEN TEKNILLINEN
Page 9 and 10: Abbreviations and Acronyms C/A coar
Page 11 and 12: σ2 variance, covariance a vector i
Page 13 and 14: Chapter 1 Introduction Satellite po
Page 15 and 16: Chapter 2 The Doppler Effect In thi
Page 17 and 18: in Figure 2.2. The shape of the so
Page 19 and 20: Chapter 3 The Global Positioning Sy
Page 21 and 22: sufficiently many satellites, the r
Page 23 and 24: The satellite signal is very accura
Page 25 and 26: The estimated position and time vec
Page 27 and 28: in weak signal conditions. The key
Page 29: navigation message. Here, however,
Page 33 and 34: the receiver clock is runningfast a
Page 35 and 36: Let us now turn into our main goal,
Page 37 and 38: Definition 3. Delta range residuals
Page 39 and 40: method [Fletcher, p. 110], [Kaleva1
Page 41 and 42: product formula a × (b × c) =(a
Page 43 and 44: Chapter 5 Positioning Performance I
Page 45 and 46: eplaced by vi −△wu. Let us rewr
Page 47 and 48: Thus, the time bias effect to the p
Page 49 and 50: Let us first calculate the expected
Page 51 and 52: • Doppler drift dilution of preci
Page 53 and 54: Let us now make the even stronger a
Page 55 and 56: Chapter 6 Numerical Results In Chap
Page 57 and 58: 6.2.2 Numerical Testing of Normalit
Page 59 and 60: 6.2.4 Magnitude of Errors The actua
Page 61 and 62: Histogram of positioning errors wit
Page 63 and 64: used for testinghow the algorithm b
Page 65 and 66: Chapter 7 Conclusions This thesis d
Page 67 and 68: The positioningalgorithm was develo
Page 69 and 70: [Hill] Hill, Jonathan. The Principl

Antti Lehtinen Doppler Positioning with GPS - Matematiikan laitos

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