Global Navigation Satellite Systems, Inertial Navigation, and Integration. Mohinder S. Grewal

target="_blank" rel="nofollow" href="#ulink_e4591029-f89b-5961-8336-a39863c32d97">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

(2.3)

Figure 2.3 Two transmitters with known 2D positions.

(2.4)

Expanding R₁ and R₂ in Taylor series expansion with small perturbation in X by Δx and Y by Δy yields

(2.5)

(2.6)

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 are assumed. Set position coordinates X, Y equal to their initial estimates:

      Compute the range errors:

(2.15)

      (2.16)

      Compute the θ angle (see Figure 2.3):

      (2.17)

      Compute update to user position:

      (2.18)

      (2.19)

      Compute a new estimate of position using the update:

(2.20)

      Continue to compute θ, ΔR₁, and ΔR₂ from these equations