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