PART41 » History » Version 9
COLIN, Tony, 03/19/2016 06:59 PM
1 | 3 | COLIN, Tony | h1. PART 4 : Position Estimation. |
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2 | 2 | COLIN, Tony | |
3 | 2 | COLIN, Tony | {{toc}} |
4 | 2 | COLIN, Tony | |
5 | 2 | COLIN, Tony | p(. ....................... |
6 | 2 | COLIN, Tony | |
7 | 2 | COLIN, Tony | --- |
8 | 2 | COLIN, Tony | |
9 | 7 | COLIN, Tony | h2. 1 - Ephemeris. |
10 | 2 | COLIN, Tony | |
11 | 9 | COLIN, Tony | h3. Parameters. |
12 | 9 | COLIN, Tony | |
13 | 9 | COLIN, Tony | p=. !Eph12min.png! |
14 | 9 | COLIN, Tony | |
15 | 9 | COLIN, Tony | h3. Algorithm. |
16 | 9 | COLIN, Tony | |
17 | 9 | COLIN, Tony | p=. !Alg12min.png! |
18 | 1 | COLIN, Tony | |
19 | 7 | COLIN, Tony | --- |
20 | 1 | COLIN, Tony | |
21 | 7 | COLIN, Tony | h2. 2 - Navigation computation. |
22 | 7 | COLIN, Tony | |
23 | 7 | COLIN, Tony | h3. a - Reminder about the impairments. |
24 | 7 | COLIN, Tony | |
25 | 4 | COLIN, Tony | The following figure gives the impairments affecting the range in case of the GPS system as well as the correction process : |
26 | 4 | COLIN, Tony | |
27 | 4 | COLIN, Tony | p=. !003.PNG! |
28 | 1 | COLIN, Tony | *Figure 4.1 :* Pseudo-range measurement extracted from *[2]* |
29 | 7 | COLIN, Tony | |
30 | 7 | COLIN, Tony | h3. b - Demonstration of the Pseudo-ranges with Least Square method. |
31 | 7 | COLIN, Tony | |
32 | 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 : |
33 | 7 | COLIN, Tony | |
34 | 7 | COLIN, Tony | p=. !Pos1.png! |
35 | 7 | COLIN, Tony | *Equation 1* |
36 | 7 | COLIN, Tony | |
37 | 7 | COLIN, Tony | or put in another way, |
38 | 7 | COLIN, Tony | |
39 | 7 | COLIN, Tony | p=. !Pos2.png! |
40 | 7 | COLIN, Tony | *Equation 2* |
41 | 7 | COLIN, Tony | |
42 | 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 : |
43 | 7 | COLIN, Tony | |
44 | 7 | COLIN, Tony | p=. !Pos3.png! |
45 | 7 | COLIN, Tony | *Equation 3* |
46 | 7 | COLIN, Tony | |
47 | 7 | COLIN, Tony | Hence, |
48 | 7 | COLIN, Tony | |
49 | 7 | COLIN, Tony | p=. !Pos4.png! |
50 | 7 | COLIN, Tony | *Equation 4* |
51 | 7 | COLIN, Tony | |
52 | 7 | COLIN, Tony | or in matrix equation form, |
53 | 7 | COLIN, Tony | |
54 | 7 | COLIN, Tony | p=. !Pos5.png! |
55 | 7 | COLIN, Tony | *Equation 5* |
56 | 7 | COLIN, Tony | |
57 | 7 | COLIN, Tony | which can be expressed as : |
58 | 7 | COLIN, Tony | |
59 | 7 | COLIN, Tony | p=. !Pos6.png! |
60 | 7 | COLIN, Tony | *Equation 6* |
61 | 7 | COLIN, Tony | |
62 | 7 | COLIN, Tony | with the Least Square solution : |
63 | 7 | COLIN, Tony | |
64 | 7 | COLIN, Tony | p=. !Pos7.png! |
65 | 7 | COLIN, Tony | *Equation 7* |
66 | 7 | COLIN, Tony | |
67 | 8 | COLIN, Tony | Thus, it is possible to retrieve the receiver position. |
68 | 8 | COLIN, Tony | |
69 | 8 | COLIN, Tony | _Note that all unknowns are depicted in red color._ |
70 | 5 | COLIN, Tony | |
71 | 5 | COLIN, Tony | --- |
72 | 5 | COLIN, Tony | |
73 | 5 | COLIN, Tony | *References :* |
74 | 5 | COLIN, Tony | *[1]* K. Borre, D. M. Akos, N. Bertelsen, P. Rinder, S. H. Jensen, A software-defined GPS and GALILEO receiver |
75 | 5 | COLIN, Tony | *[2]* Position Estimation Workshop, March 2016 |