planes in Figure 2.2 contains four or more satellites. The European Galileo and Chinese BeiDou use three orbital planes separated in longitude by 120°. Galileo has plans for 10 MEO satellites in each orbital plane. Glonass also used three orbital planes while maintaining a nominal 24 satellite constellation. GNSS augmentation or regional satellite‐based navigation systems are most often placed in GEO or IGSO over the region to be services; these satellites will have an orbital period of 24 hours.
Figure 2.1 Parameters defining satellite orbit geometry.
Figure 2.2 Six GPS orbit planes inclined 55° from the equatorial plane.
Regardless of the type of radionavigation (satellite‐based or terrestrial‐based) system, most often the solution method requires the determination of a unknown user antenna location with known transmitter locations and measured range (or pseudorange) observations.
2.3.2 Navigation Solution (Two‐Dimensional Example)
Antenna location in two dimensions can be calculated by using range measurements [3].
2.3.2.1 Symmetric Solution Using Two Transmitters on Land
In this case, the receiver and two transmitters are located in the same plane, as shown in Figure 2.3, with known positions of the two transmitters: x1, y1 and x2, y2. Ranges R1 and R2 from the two transmitters to the user position are calculated as
(2.1)
(2.2)
where
| c | = | speed of light (0.299 792 458 m/ns) |
| ΔT1 | = | time taken for the radiowave to travel from transmitter 1 to the user (ns) |
| ΔT2 | = | time taken for the radiowave to travel from transmitter 2 to the user (ns) |
| X, Y | = | unknown user position to be solved for (m) |
The range to each transmitter can be written as
Figure 2.3 Two transmitters with known 2D positions.
Expanding R1 and R2 in Taylor series expansion with small perturbation in X by Δx and Y by Δy yields
where u1 and u2 are higher‐order terms. The derivatives of Eqs. (2.3) and (2.4) with respect to X, Y are substituted into Eqs. (2.5) and (2.6), respectively.
Thus, for the symmetric case, we obtain
(2.7)
(2.8)
To obtain the least‐squares estimate of (X, Y), we need to minimize the quantity
(2.9)
which is
(2.10)
The solution for the minimum can be found by setting ∂J/∂Δx = 0 = ∂J/∂Δy, then solving for Δx and Δy:
(2.11)
(2.12)
with solution
(2.13)
The solution for Δy may be found in similar fashion as
(2.14)
2.3.2.2 Navigation Solution Procedure
Transmitter positions x1, y1, x2, y2 are known; the transmitter locations are typically provided by the GNSS satellites or known to be at a fixed surveyed location for a terrestrial source. Signal travel times ΔT1, ΔT2 are measured by the system. Initially, the user positions
