PART41 » History » Version 34

COLIN, Tony, 03/23/2016 10:45 AM

1 3 COLIN, Tony
h1. PART 4 : Position Estimation.
2 2 COLIN, Tony
3 2 COLIN, Tony
{{toc}}
4 2 COLIN, Tony
5 13 COLIN, Tony
p(. Once the navigation bits from at least 4 satellites have been retrieved from the acquisition/tracking part, it is possible to estimate the desired position of the receiver.
6 2 COLIN, Tony
7 2 COLIN, Tony
---
8 2 COLIN, Tony
9 7 COLIN, Tony
h2. 1 - Ephemeris.
10 2 COLIN, Tony
11 16 COLIN, Tony
GPS uses a particular algorithm in order to characterise satellite position. In comparison with GLONASS, this method requires more parameters, but less complexity.
12 16 COLIN, Tony
13 32 COLIN, Tony
h3. a - Introduction of satellite orbit from [2].
14 1 COLIN, Tony
15 29 COLIN, Tony
p=. !OrbitalPlanePositioningMin.png!
16 29 COLIN, Tony
*Figure 4.1 :* Orbital plane positioning.
17 29 COLIN, Tony
18 1 COLIN, Tony
Orbital plane positioning parameters :
19 32 COLIN, Tony
20 30 COLIN, Tony
p=. !Parameters1.PNG!
21 30 COLIN, Tony
22 1 COLIN, Tony
p=. !OrbitPositioningInTheOrbitalPlaneMin.png!
23 31 COLIN, Tony
*Figure 4.2 :* Orbit positioning in the orbital plane.
24 1 COLIN, Tony
25 1 COLIN, Tony
Orbit positioning in the orbital plane :
26 32 COLIN, Tony
27 30 COLIN, Tony
p=. !Parameters2.PNG!
28 30 COLIN, Tony
29 29 COLIN, Tony
p=. !SatellitePositioningMin.png!
30 29 COLIN, Tony
*Figure 4.3 :* Orbital plane positioning.
31 1 COLIN, Tony
32 1 COLIN, Tony
Shape of the orbit :
33 32 COLIN, Tony
34 30 COLIN, Tony
p=. !Parameters3.PNG!
35 1 COLIN, Tony
36 1 COLIN, Tony
Positioning of the satellite on the orbit :
37 32 COLIN, Tony
38 30 COLIN, Tony
p=. !Parameters4.PNG!
39 1 COLIN, Tony
40 1 COLIN, Tony
Induced parameters :
41 32 COLIN, Tony
42 30 COLIN, Tony
p=. !Parameters5.PNG!
43 29 COLIN, Tony
44 29 COLIN, Tony
h3. b - GPS satellite ephemeris data.
45 29 COLIN, Tony
46 33 COLIN, Tony
GPS uses previous classical ephemeris data for orbit and satellite position determination, and decompose them into elementary parameters to be implemented in the navigation frame :
47 33 COLIN, Tony
48 1 COLIN, Tony
p=. !Eph12min.png!
49 29 COLIN, Tony
*Figure 4.4 :* List of ephemeris parameters included in GPS frames.
50 1 COLIN, Tony
51 29 COLIN, Tony
h3. c - GPS satellite position calculation algorithm.
52 1 COLIN, Tony
53 34 COLIN, Tony
Starting from the GPS ephemeris present in the navigation frame - subframe 1 - it is now possible to compute the satellite position via the following algorithm :
54 34 COLIN, Tony
55 28 COLIN, Tony
p=. !Alg12min.png!
56 29 COLIN, Tony
*Figure 4.5 :* Description of the algorithm step by step.
57 18 COLIN, Tony
58 29 COLIN, Tony
These tables are extracted from GPS Interface Control Document *[3]*
59 7 COLIN, Tony
60 1 COLIN, Tony
---
61 7 COLIN, Tony
62 7 COLIN, Tony
h2. 2 - Navigation computation.
63 1 COLIN, Tony
64 19 COLIN, Tony
h3. a - Reminder about the range impairments.
65 7 COLIN, Tony
66 4 COLIN, Tony
The following figure gives the impairments affecting the range in case of the GPS system as well as the correction process :
67 4 COLIN, Tony
68 4 COLIN, Tony
p=. !003.PNG!
69 29 COLIN, Tony
*Figure 4.6 :* Pseudo-range measurement extracted from *[4]*
70 7 COLIN, Tony
71 7 COLIN, Tony
h3. b - Demonstration of the Pseudo-ranges with Least Square method.
72 7 COLIN, Tony
73 7 COLIN, Tony
Starting from the fact that can determine most of the elements within the pseudo-range measurement PR_sat(i) from the information provided by each satellite, we have the equation : 
74 7 COLIN, Tony
75 7 COLIN, Tony
p=. !Pos1.png!
76 7 COLIN, Tony
*Equation 1*
77 7 COLIN, Tony
78 7 COLIN, Tony
or put in another way,
79 7 COLIN, Tony
80 7 COLIN, Tony
p=. !Pos2.png!
81 7 COLIN, Tony
*Equation 2*
82 7 COLIN, Tony
83 7 COLIN, Tony
Indeed 4 measurements are needed, providing 4 equations with 4 unknows which are the receiver coordinates and the clock bias of the receiver. As the equation is highly non-linear, it is important to proceed to a linearization such as the Taylor expansion :
84 7 COLIN, Tony
85 7 COLIN, Tony
p=. !Pos3.png!
86 7 COLIN, Tony
*Equation 3*
87 7 COLIN, Tony
88 7 COLIN, Tony
Hence,
89 7 COLIN, Tony
90 7 COLIN, Tony
p=. !Pos4.png!
91 7 COLIN, Tony
*Equation 4*
92 7 COLIN, Tony
93 24 COLIN, Tony
In practise, for a receiver located e.g. in France PR (t_0) can be described by Paris location as initialization for the algorithm.
94 21 COLIN, Tony
In vectorial form the equation becomes :
95 7 COLIN, Tony
96 7 COLIN, Tony
p=. !Pos5.png!
97 7 COLIN, Tony
*Equation 5*
98 7 COLIN, Tony
99 7 COLIN, Tony
which can be expressed as :
100 7 COLIN, Tony
101 7 COLIN, Tony
p=. !Pos6.png!
102 7 COLIN, Tony
*Equation 6*
103 7 COLIN, Tony
104 7 COLIN, Tony
with the Least Square solution :
105 7 COLIN, Tony
106 7 COLIN, Tony
p=. !Pos7.png!
107 8 COLIN, Tony
*Equation 7*
108 8 COLIN, Tony
109 8 COLIN, Tony
Thus, it is possible to retrieve the receiver position.
110 5 COLIN, Tony
111 25 COLIN, Tony
_Note that all unknowns are depicted in red color._
112 25 COLIN, Tony
113 25 COLIN, Tony
114 25 COLIN, Tony
h3. c - Kalman filter.
115 1 COLIN, Tony
116 1 COLIN, Tony
Another position estimation method is Kalman filter i.e. an algorithm that uses a series of measurements observed over time, containing statistical noise and other inaccuracies, and produces estimates of unknown variables that tend to be more precise than those based on a single measurement alone.
117 1 COLIN, Tony
In this project, a single measurement will be used for "simplicity" purposes, therefore, the Least Square method is more appropriate for this issue.
118 25 COLIN, Tony
119 25 COLIN, Tony
---
120 5 COLIN, Tony
121 5 COLIN, Tony
*References :* 
122 5 COLIN, Tony
*[1]* K. Borre, D. M. Akos, N. Bertelsen, P. Rinder, S. H. Jensen, A software-defined GPS and GALILEO receiver
123 29 COLIN, Tony
*[2]* M. Bousquet, Orbits and Satellite Platforms lecture script, January 2016
124 29 COLIN, Tony
*[3]* GPS Interface Control Document under http://www.gps.gov/technical/icwg/IS-GPS-200H.pdf
125 29 COLIN, Tony
*[4]* Position Estimation Workshop, March 2016