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

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: x 1, y 1and x 2, y 2. Ranges R 1and R 2from the two transmitters to the user position are calculated as

(2.1)

(2.2)

where

The range to each transmitter can be written as

(2.3)

Figure 2.3Two transmitters with known 2D positions.

(2.4)

Expanding R 1and R 2in Taylor series expansion with small perturbation in X by Δ x and Y by Δ y yields

(2.5)

(2.6)

where u 1and u 2are 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 /∂Δ x = 0 = ∂J /∂Δ y , then solving for Δ x and Δ y :

(2.11)

(2.12)

with solution

(2.13)

The solution for Δ y may be found in similar fashion as

(2.14)

Transmitter positions x 1, y 1, x 2, y 2are 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 Δ T 1, Δ T 2are 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 , Δ R 1, and Δ R 2from these equations with new values of . Iterate Eqs. (2.15)– (2.20)and stop when Δ x and Δ y become less than the desired accuracy:

Just as in a land‐based system, better accuracy is obtained by using reference points (i.e. ranging sources) well separated in space. For example, the range measurements made to four reference points clustered together will yield nearly equal values. Position calculations involve range differences, and where the ranges are nearly equal, small relative errors are greatly magnified in the difference. This effect, brought about as a result of satellite geometry, is known as dilution of precision (DOP). This means that range errors that occur from other independent effects such as multipath, atmospheric delays, and/or satellite clock errors are also magnified by the geometric effect.