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Data Fusion via Kalman Filter 115

Due to the differencing of measurements, this estimate is noisy; since I a changes
very slowly, it can be low-pass filtered to remove the noise. For measure-
ments from single frequency receivers, it is possible to compensate part of the
ionospheric delay errors by an ionospheric delay model [36]. Alternatively,
differential operation using at least two receivers can effectively remove all
common mode errors (i.e., c t sv , I a , E cm ).
The methods discussed in the subsequent sections can be used for the pseu-
dorange or integer-resolved carrier phase measurements. We will not discuss
Doppler measurements. To avoid redundant text for the code and integer-
resolved carrier measurements, we will adopt the following general model for
the range measurement to the ith satellite

2 2 2
˜ ρ i = (X i − x) + (Y i − y) + (Z i − z) + b u + ε i (3.44)
ρ
where ˜ could represent the code pseudorange measurements or integer-
resolved carrier phase measurements. The variable b u represents the receiver
clock bias. The symbol ε represents the error terms appropriate for the different
measurements. When a GPS receiver has collected range measurements from
four or more satellites, it can calculate a navigation solution.

3.3.2 Single-Point GPS Navigation Solution

Thissection presentsthestandard GPS position solution method using nonlinear
least squares. In the process, we will introduce notation needed for the sub-
T
sequent sections. In this section, the state vector is defined as x =[x, y, z, b u ]
where (x, y, z) is the receiver antenna position in earth centered earth fixed
(ECEF) coordinates and b u is the receiver clock bias.
Taylor series expansion of Equation (3.44) about the current state estimate
ˆ
ˆ x =[ˆx, ˆy, ˆz, b u ] yields

˜ ρ i (x) =ˆρ i (ˆ x) +[h i ,1]δx + h.o.t.s + ε i

where

ˆ
δx = x − ˆ x =[x −ˆx, y −ˆy, z −ˆz, b u − b u ] T

2 2 2 ˆ
˜ ρ i (ˆ x) = (X i −ˆx) + (Y i −ˆy) + (Z i −ˆz) + b u (3.45)

∂ρ i ∂ρ i ∂ρ i
h i = , ,
∂x ∂y ∂z
(ˆx,ˆy,ˆz)

© 2006 by Taylor & Francis Group, LLC

FRANKL: “dk6033_c003” — 2006/3/31 — 16:42 — page 115 — #17