Global Positioning System Reference
In-Depth Information
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the time argument for the broadcast and precise ephemerides. Because the satellite
transmissions are steered by the nominal time of the individual satellite (satellite
time), it is important to know the differences between GPS time and the individual
satellite time. In the notation and sign convention as used by the interface control
document, the time correction to the nominal space vehicle time
t
SV
is
t
oc
)
2
∆
t
SV
=
a
f
0
+
a
f
1
(t
SV
−
t
oc
)
+
a
f
2
(t
SV
−
+ ∆
t
R
(5.38)
with
t
GPS
=
t
SV
− ∆
t
SV
(5.39)
and
[17
2
2
c
2
c
2
√
a
X
∆
t
R
=−
µ
e
sin
E
=−
X
·
(5.40)
Lin
—
1.7
——
Nor
*PgE
Th
e polynomial coefficients are transmitted in units of sec, sec/sec, and sec/sec
2
;
th
e clock data reference time
t
oc
is also broadcast in seconds in subframe 1 of the
na
vigation message. As is required when using the ephemeris expressions, the value
of
t
SV
must account for the beginning or end-of-week crossovers. That is, if
(t
SV
−
t
oc
)
is
greater than 302,400, subtract 604,800 from
t
SV
.If
(t
SV
−
t
oc
)
is less than
−
302,400,
ad
d 604,800 to
t
SV
.
The second part of (5.40) follows from (3.61).
t
R
is a small relativistic clock
co
rrection caused by the orbital eccentricity
e
. The symbol
∆
denotes the gravitational
co
nstant,
a
is the semimajor axis of the orbit, and
E
is the eccentric anomaly. See
Ch
apter 3 for details on these elements. Using
a
µ
[17
≈
26,600 km we have
∆
t
R
[
µ
sec]
≈−
2
e
sin
E
(5.41)
5.
3.2 Topocentric Range
The pseudorange equation (5.7) and the carrier phase equation (5.10) require that the
topocentric distance
p
k
ρ
be computed. In the inertial coordinate system
(
X
)
, this is
simply accomplished by
=
X
k
(t
k
)
X
p
(t
p
)
p
k
ρ
−
(5.42)
In the inertial coordinate system, the receiver coordinates are a function of time due
to the earth's rotation. If the receiver antenna and satellite ephemeris are available
in the terrestrial coordinate system, we must take the earth's rotation explicitly into
account. Neglecting polar motion, the Greenwich apparent sidereal time relates the
terrestrial coordinate system
(
x
)
to the inertial system
(
X
)
by (2.34). Let
p
θ
k
denote the Greenwich apparent sidereal times for transmission and reception of the
signal, then
θ
and
