THC Publishing

Contact

1. Introduction

This section is about how to use satellite data to compute receiver potion and time. Assume we know positions of 3 satellites and transmit time t_TX (or SV`s, space vehicles) and unknown receiver position and known receive time t_RX , so we have 3 unknowns x, y, and z. We know from elementary algebra that we need 3 equations to solve these 3 unknowns

(Eq.1)

where

(Eq.2)

with constant c is speed of light (300,000 km/s)

To be valid, (Eq.1) requires a straight line (line of sight) between SV and receiver. However, (Eq.2) gives a curve which is exaggerated in the figure above as signal transmitted in a curve due to, say, refraction through media with different index like light. That`s why r_i is called pseudo-range or pseudo-distance as range or distance implies straight line.

To count for incorrect time at user position, (Eq.1) is modified to be

(Eq.3)

where additional equation is for unkown b as user clock bias error expressed in distance

(Eq.4)

Therefore, to solve for user position and time, we need to solve the simultaneous equations below

(Eq.5)

These equations are transcendental, so cannot be solved algebraically. They must be solved numerically and the more equations involved, the quicker in getting to the solution, so we need at least 4 SV`s for solution. If they were solved algebraically, we need only 4 equations, ie 4 SV`s.

The SV data, like position and time, are carried in SV message called ephemeris (ephemeredes in plural). The following sections are about SV data and several methods to solve (Eq.5) from basic to advanced, like using Kalman filter.

2. Navigation Data

GPS satellite sends navigation data in frames to GPS receiver to in solving (Eq.5). These data are in spherical coordinates and required to transformed into Cartesian ones required in (Eq.5).

2.1. Signal Structure

GPS receivers get navigation data from satellite in frames with structures below.

Every subframe has 10 words, 30 bits each, so 300 bits long. Each frame has 1^st word for TLM (telemetry) with subframe boundary identifier (0x8b) and 2^nd for HOW (Hand-Over Word) with time of week (TOW) in unit of 6 seconds where 0 is at mid-night Saturday and Sunday. For 1 week, we have 7*24*3600 = 604,800 seconds; so maximum TOW is 604,800/6 = 100,800; 100,799 actually.

Frame 1

	Description	Location	Numberof Bits	ScaleFactor	Unit
Week Number	61~70	10	1	week

Frame 2

	Description	Location	Numberof Bits	ScaleFactor	Unit
Crs	Amplitude of the sine harmonic correction terms to the orbit radius	69~84	16	2-5	meter
Dn	Mean motion difference from computed value	91~106	16	2-43	Semicircle/sec
M0	Mean anomaly at reference time	107~114 121~144	16	2-31	radian
Cuc	Amplitude of the cosine harmonic correction terms to the argument of latitude	151~166	16	2-29
e	Eccentricity	167~174 181~204		2-33
Cus	Amplitude of the sine harmonic correction terms to the argument of latitude	211~226	16	2-29	radian
	Square root of the semi-major axis	227~234 241~264	32	2-19	Meter1/2
toe	Ephemeris reference time (seconds in the GPS week)	277~286	16	24	second

Frame 3

	Description	Location	Numberof Bits	ScaleFactor	Unit
Cic	Amplitude of the cosine harmonic correction terms to the angle of inclination	61~76	16	2-29	radians
W0	Right Ascension at reference time	77~84 91~114	32	2-31	Semicircle
Cis	Amplitude of the sine harmonic correction terms to the angle of inclination	121~136	16	2-29	radians
i0	Inclination angle at reference time	137~144 151~174	32	2-31	Semicircle
Crc	Amplitude of the cosine harmonic correction terms to the orbit radius	181~196	16	2-5	meter
w	Argument of Perigee	197~204 211~234	32	231	Semicircle
	Rate of change of Right Ascension angle	241~264	24	2-43	Semicircle/sec
IDOT	Rate of change of inclination angle	279~292	14	2-43	Semicircle/sec

2.2. Ephemeris Processing

GPS navigation Ephemeris data from satellite are converted to Cartesian coordinates required in (Eq.5).

WGS 84 value of the earth`s universal gravitational parameter

(Eq.6)

WGS 84 value of the earth`s universal gravitational parameter

(Eq.7)

Computed mean motion rad/sec

(Eq.8)

where a is semimajor axis.

Time from ephemeris reference epoch

(Eq.9)

where t is GPS trasmit time, i.e. range/speed_of_light; and t_k must account for week crossover.

Corrected mean motion

(Eq.10)

Mean anomaly

(Eq.11)

Kepler`s equation foe eccentric anomaly

(Eq.12)

True anomaly

(Eq.13)

Eccentricity anomaly

(Eq.14)

Argument of latitude

(Eq.15)

2^nd harmonic pertubations

(Eq.16)

Corrected argument of latitude

(Eq.17)

Corrected radius

(Eq.18)

Corrected inclination

(Eq.19)

Position in orbital plane

(Eq.20)

Corrected of longitude of ascending node

(Eq.21)

ECEF (Earth-Centered Earth-Fixed) coordinates

(Eq.22)

3. Basic Numerical Method

Linearizing Eq.5 gives

(Eq.23)

where

(Eq.24)

Eq.23 can be written in matrix form as

(Eq.25)

where

(Eq.26)

The solution of Eq.25 is

(Eq.27)

where [A^TA]^-1A^T is the pseudo-inverse of A as non-square matrix has no true inverse A^-1.

Based on Eq.27, we have the following algorithm to solve Eq.5

Choose initial values for x, y, z, and b; Compute xk, yk, and zk using Eq.22 and Ephemeris data from GPS satellite; Compute r using Eq.5; Compute matrix A using Eq.24; Compute dx using Eq.27; Stop if is less than an arbitrarily chosen threshold, otherwise continue the following Step (7); Update x using x = x + dx; and repeat back to Step (3)