Global Positioning System Reference
In-Depth Information
(
)
Geometric range,
rcT T
=
−
=
ct
∆
u
s
[
]
(
)
(
)
Pseudorange
,
ρ
=
c
T
+
t
−
T
+
δ
t
u
u
S
(
)
(
)
=
cT
−
T
+
ct
−
δ
t
u
S
u
(
)
=+
rct
−
δ
t
u
Therefore, (2.15) can be rewritten as:
(
)
ρ
−
ct
−
δ
t
=
su
−
u
where
t
u
represents the advance of the receiver clock with respect to system time,
t
represents the advance of the satellite clock with respect to system time, and
c
is the
speed of light.
The satellite clock offset from system time,
δ
t
, is composed of bias and drift con-
tributions. The GPS ground-monitoring network determines corrections for these
offset contributions and transmits the corrections to the satellites for rebroadcast to
the users in the navigation message. These corrections are applied within the user
receiver to synchronize the transmission of each ranging signal to system time.
Therefore, we assume that this offset is compensated for and no longer consider
δ
t
an unknown. (There is some residual offset, which is treated in Section 7.2.1, but in
the context of this discussion we assume that this is negligible.) Hence, the preceding
equation can be expressed as
δ
ρ −=−
ct
u
su
(2.18)
2.4.2 Calculation of User Position
In order to determine user position in three dimensions (
x
u
,y
u
,z
u
) and the offset
t
u
,
pseudorange measurements are made to four satellites resulting in the system of
equations
ρ
j
=−+
s u
ct
(2.19)
j
u
where
j
ranges from 1 to 4 and references the satellites. Equation (2.19) can be
expanded into the following set of equations in the unknowns
x
u
,
y
u
,
z
u
, and
t
u
:
(
)
2
(
)
2
(
)
2
ρ
1
=
xx
−
+−
yy
+−
zz
+
t
(2.20)
1
u
1
u
1
u
u
(
)
2
(
)
2
(
)
2
ρ
2
=
xx
−
+− +−
yy
z
z
+
ct
(2.21)
2
u
2
u
2
u
u
(
)
(
)
(
)
2
2
2
ρ
3
=
xx
−
+−
yy
+−
z
z
+
ct
(2.22)
3
u
3
u
3
u
u
(
)
2
(
)
2
(
)
2
ρ
4
=
xx
−
+−
yy
+−
z
z
+
ct
(2.23)
u
u
u
u
4
4
4
where
x
j
,
y
j
, and
z
j
denote the
j
th satellite's position in three dimensions.
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