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 with new values of
2.3.3 User Solution and Dilution of Precision (DOP)
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.
The DOP can be interpreted as the dilution of the precision from the measurement (i.e. range) domain to the solution (i.e. PVT) domain.
The observation measurement equations in three dimensions for each satellite with known coordinates (xi, yi, zi) and unknown user coordinates (X, Y, Z) are given by
where
These are nonlinear equations that can be linearized using the Taylor series (see, e.g., chapter 5 of Ref. [4]). The satellite positions may be converted to east–north–up (ENU) from Earth‐centered, Earth‐fixed (ECEF) coordinates (see Appendix B).
Let the vector of ranges be Zρ = h(x), a nonlinear function h(x) of the four‐dimensional vector x representing user solution for position and receiver clock bias and expand the left‐hand side of this equation in a Taylor series about some nominal solution xnom for the unknown vector
(2.22)
of variables,
| X | = | east component of the user's antenna location (m) |
| Y | = | north component of the user's antenna location (m) |
| Z | = | upward vertical component of the user's antenna location (m) |
| C b | = | receiver clock bias (m) |
for which
(2.23)
where HOT stands for “higher‐order terms.”
These equations become
(2.24)
where H[1] is the first‐order term in the Taylor series expansion:
for vρ is the noise in receiver measurements. This vector equation can be written in scalar form where i is the satellite number as
for i = 1, 2, 3, 4 (i.e. four satellites).
We can combine Eqs. (2.25) and (2.26a) into the matrix equation with measurements as
(2.26b)
which we can write in symbolic form as
(2.27)
(see table 5.3 in Ref. [4]).
To calculate H[1], one needs satellite positions and the nominal value of the user's position in ENU coordinate frames.
To
