Global Positioning System Reference
In-Depth Information
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P
pq
km,
IF
P
pq
km,I
β
f
P
pq
km,
1
− γ
f
P
pq
km,
2
P
pq
P
pq
km,
2
km,
1
−
≡
=
pq
km,
IF
Φ
pq
km,
1
− γ
f
Φ
pq
km,
2
Φ
β
f
Φ
pq
km,I
pq
km,
1
− Φ
pq
km,
2
Φ
(7.41)
pq
km
+
T
pq
km
d
pq
km,
IF
,P
d
pq
km,I,P
d
pq
km,
IF
,
ρ
1
0
0
0
I
pq
km,
1
,P
N
pq
km,
1
N
pq
km,
2
− α
01
0
0
f
+
1
0
β
f
λ
1
−γ
f
λ
2
Φ
01
− α
f
λ
1
−λ
2
d
pq
km,I,
Φ
The receiver and satellite hardware delay and signal multipath terms have been suit-
ably transformed (note the subscripts of the
[25
δ
terms).
Lin
—
-0.
——
Lon
PgE
7.4 POINT POSITIONING
According to the first equation in the geometry-free model (7.19), we estimate the
sum
every epoch, using sequential least-squares or Kalman filtering. The
topocentric distance
ρ + ∆
, of course, is a function of the receiver antenna position
x
k
and the satellite position
x
p
. By using
x
k
and
x
p
explicitly in (7.19) we introduce the
dependency on the receiver-satellite geometry. We further replace the auxiliary quan-
tity
ρ
with the original definition (7.20), thus introducing the receiver and satellite
clock errors explicitly.
Point positioning refers to the estimation of receiver antenna coordinates
x
k
and the
receiver clock error
d
t
k
using pseudorange observables. The role of carrier phases is
limited to smoothing the pseudoranges, if used at all. More specifically, the term point
positioning as used here implies several simplifying assumptions. The satellite posi-
tions
x
p
at transmission times are assumed known and available from the broadcast
ephemeris. While we estimate the receiver clock error at every epoch, we neglect the
residual satellite clock errors
d
∆
[25
t
p
. Of course, the satellite clock broadcast correction
must be applied following (5.38). The satellite clocks are constantly monitored by
the control center, which models the clock offsets by polynomials in time. The latter
are part of the navigation message. The ionospheric and tropospheric delays are also
computed from models, as explained in Chapter 6. Hardware delays and multipath
are neglected.
The four unknowns
x
k
and
d
t
k
can be computed using four pseudoranges measured
simultaneously to four satellites. Using the simplifying assumptions made above we
can write four equations of the type
¯
cT
GD
=
x
p
x
k
−
P
k
p
k
−
−
cd
t
k
= ρ
−
cd
t
k
,
(7.42)
one for each satellite (superscript
p
varies). The effect of the earth's rotation during
the signal travel time must be incorporated in (7.42) following Section 5.3.2. Since the
receiver clock error
d
t
k
is solved together with the position of the receiver's antenna
