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114 Autonomous Mobile Robots

and L2. Users with “two frequency” receivers can obtain pseudorange, phase,
and Doppler measurements for each of the two frequencies.
The L1 and L2 code and carrier phase measurements from a given satellite
can be modeled as

f 2
˜ ρ L1 = R + b u + c t sv + I a + E cm + MP 1 + η 1
f 1
f 1
˜ ρ L2 = R + b u + c t sv + I a + E cm + MP 2 + η 2
f 2
f 2
˜ φ L1 λ 1 + N 1 λ 1 = R + b u + c t sv − I a + E cm + mp 1 + n 1
f 1
f 1
˜ φ L2 λ 2 + N 2 λ 2 = R + b u + c t sv − I a + E cm + mp 2 + n 2
f 2
where R = X sv − X u is the geometric distance between the satellite position
X sv and receiver antenna position X u , b u is the receiver clock bias, and c t sv is
the satellite clock bias. The satellite clock bias can be partially corrected by eph-
emeris data. E cm represents common errors other than dispersive effects such as
ionosphere and I a represents ionospheric error. The symbols η and n represent
receiver measurement noise. The symbols mp and MP represent errors due to
multipath. Note that the receiver clock bias is identical across satellites for all
simultaneous pseudorange and phase measurements. Since the receiver phase
lock loops can only track changes in the signal phase and the initial number of
carrier wavelengths at the time of signal lock is not known, each phase signal is
biased by an unknown constant integer number of carrier cycles represented by
N 1 and N 2 . Use of the phase measurements as pseudorange signals for position
estimation also requires estimation of these unknown integers [28–32]. Use of
the change in the phase over a known period of time to estimate the receiver
velocity does not require estimation of these integers, since the integers are
canceled in the differencing operation [33,34]. The standard GPS texts [34,35]
include entire sections or chapters devoted to the physical aspects of the various
quantities that have been briefly defined in this section.
Note that only R and b u contain the position and receiver clock information
that we wish to estimate. The symbols c t sv , I a , E cm , MP, mp, η, and n all
represent errors that decrease the accuracy of the estimated quantities. There
are many techniques to reduce these measurement errors prior to the navigation
solution. Dualfrequencyreceiverscantakeadvantageofthecodemeasurements
from L1 and L2 to estimate the ionospheric delay error I a as

f 1 f 2
ρ
ρ
I a = 2 2 ( ˜ L1 −˜ L2 )
f − f
2 1

© 2006 by Taylor & Francis Group, LLC

FRANKL: “dk6033_c003” — 2006/3/31 — 16:42 — page 114 — #16