Antti Lehtinen Doppler Positioning with GPS - Matematiikan laitos

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4.5 Standard Numerical Methods In this section, a new algorithm is developed for the GPS Doppler positioning. The algorithm is based on well known ideas from the numerical analysis. However, the method has not earlier been applied to the GPS Doppler positioningproblem. 4.5.1 Problem Formulation In the following, we will consider iterative methods that begin from an initial guess for the solution and continue to gradually enhance the estimate. Notation 1. The receiver position and drift estimate ⎡ ⎤ ˆxu ˆru ⎢ ˆyu ⎥ ˆx = ˆd = ⎢ ⎥ ⎣ ˆzu ⎦ ∈ R4 ˆd is the current estimate for the receiver position and drift vector x Notation 2. The delta range measurement vector ˙ρ =[˙ρ1 ˙ρ2 ... ˙ρn] T is a vector formed of the delta range measurements from the n satellites used in the calculations. The equation 4.13 gives a physical model for the delta range measurements. Assumingthat the noise components ɛ ˙ρi are zero-mean gives rise to the following definition: Definition 2. The expected delta range vector ⎡ ⎢ ˆ˙ρ = ⎢ ⎣ v1 • r1 − ˆru r1 − ˆru + ˆ d v2 • r2 − ˆru r2 − ˆru + ˆ d . vn • rn − ˆru rn − ˆru + ˆ d is the expected delta range measurement vector assuming the receiver to be located at ˆru and the clock drift to be ˆ d. The positioningsolution procedure should aim at findinga point ˆx such that the expected delta range measurement vector would be equal to the real delta range measurement vector, that is ˆ˙ρ = ˙ρ. In order to have the problem in a mathematically standard form we will define the followinghelpful function: 24 ⎤ ⎥ ⎦
Definition 3. Delta range residuals function ⎡ ⎢ p(ˆx) =ˆ˙ρ − ˙ρ = ⎢ ⎣ v1 • r1 − ˆru r1 − ˆru + ˆ d − ˙ρ1 v2 • r2 − ˆru r2 − ˆru + ˆ d − ˙ρ2 . vn • rn − ˆru rn − ˆru + ˆ d − ˙ρn is a vector valued function with the receiver position and drift estimate ˆx as an argument. The components of the vector function are the differences between the predicted delta ranges assuming x = ˆx and the measured delta ranges ˙ρi. p(ˆx) ∈ R n ,wheren is the number of satellites used in the calculations. From the system of equations 4.16 one immediately notices that if the estimate ˆx is equal to the real position and drift vector x then the value of the function p(ˆx) must be a zero vector. Thus our hope for findingthe real receiver position and drift vector x lies in findinga zero of the function p(ˆx). The problem can be formulated by the followingequation ⎡ ⎢ p(ˆx) = ⎢ ⎣ 4.5.2 The Newton Method v1 • r1 − ˆru r1 − ˆru + ˆ d − ˙ρ1 v2 • r2 − ˆru r2 − ˆru + ˆ d − ˙ρ2 . vn • rn − ˆru rn − ˆru + ˆ d − ˙ρn ⎤ ⎥ ⎦ ⎤ ⎥ ⎡ ⎤ ⎥ 0 ⎥ ⎢ ⎥ ⎢ 0. ⎥ ⎥ = ⎢ ⎥ ⎥ ⎣ ⎦ ⎥ ⎦ 0 (4.17) As should be clear from Subsection 4.4.2, there is no guarantee that the system of equations 4.16 has only one solution. Consequently, the equation 4.17 may well have solutions other than ˆx = x. In the followingwe will assume that we have an initial guess x0 that is sufficiently close to the real position and drift vector x. Then our solution algorithm should converge to the correct zero of the function p(ˆx). Here we will use the Newton method to find the solution. The solution procedure is based on the idea of linearly approximatingthe nonlinear function [Kaleva2000, p. 25]. This has to be done because there are no general methods for directly solvinga nonlinear equation. The linear approximation is accurate only in a small neighbourhood of the linearization point. However, as the linearization procedure is repeated, the solution estimate becomes increasingly accurate. Let 25
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 and 30: navigation message. Here, however,
Page 31 and 32: where wi is the satellite velocity,
Page 33 and 34: the receiver clock is runningfast a
Page 35: Let us now turn into our main goal,
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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