Global Positioning System Reference
In-Depth Information
[
]
T
, ,
&
,
&
,
()
X
t
≡
a e M
,,
, ,
ω ΩΩ∆
i
i
n C
,
,
C
,
C
,
C
,
C
,
C
(3.9)
oe
0
0
0
uc
us
ic
is
rc
rs
with an associated ephemeris reference time,
t
oe
, and are generated using a nonlinear
weighted least squares fit.
For a given subframe, the orbital elements,
X
()
t
oe
, are chosen to minimize the
performance objective:
(
)
[
]
T
(
)
(
)
()
()
rtt
−
g t
,
X
t
W
t
sa
l
k
eph
l
oe
l
∑
l
E
(3.10)
(
[
]
)
(
)
(
)
()
rtt
−
g t
,
X
t
sa
l
k
eph
l
oe
E
where
g
eph
( ) is a nonlinear function mapping the orbital elements,
X
()
t
oe
, to an ECEF
satellite antenna phase center position (see Section 2.3.1, Table 2.3) and
W
()isa
weighting matrix.
As defined in (3.10), all position vectors and associated weighting matrices are
in ECEF coordinates. Since the CS error budget is defined relative to the user range
error (see Section 7.2), the weighting matrix is resolved into radial, along-track, and
cross-track (RAC) coordinates, with the radial given the largest weight. The weight-
ing matrix of (3.10) has the form:
()
()
()
()
T
(3.11)
WM WM
t
=
t
⋅
t
⋅
t
l
E
←
RAC
l
RAC
l
E
←
RAC
l
() is a coordinate transformation from RAC to ECEF coordinates,
and
W
RAC
is a diagonal RAC weighting matrix.
For the orbital elements in (3.9), the performance objective in (3.10) can become
ill conditioned for small eccentricity,
e
. An alternative orbital set is introduced to
remove such ill conditioning; specifically, three auxiliary elements defined as
follows:
where
M
E
⋅
←
RAC
α
=
e
cos
ω
,
β
=
e
sin
ω
,
γ
=
M
+
ω
(3.12)
0
Thus, the ob
jec
tive function in (3.10) is minimized relative to the alternative
orbital elements,
X
()
having the form:
⋅
[
]
T
, ,
&
,
&
,
()
X
t
≡
a
,,,,
αβγΩΩ∆
00
i
i
n C
,
,
C
,
C
,
C
,
C
,
C
(3.13)
oe
uc
us
ic
is
rc
r
s
The three orbital elements (
e
,
M
0
,
ω
) are related to the auxiliary elements, (
α
,
β
,
γ
) by the inverse mapping
()
2
2
−
1
e
=
αβω
+
,
=
tan
βα
,
M
=
γω
−
(3.14)
0
The advantage of minimizing (3.10) with respect to
X
()
⋅
in (3.13) versus
X
()
⋅
in
(3.9) is that the auxiliary orbital elements are well defined for small eccentricity.
The minimization problem in (3.10) and (3.1
4)
is simplified by linearizing
g
eph
()
about a nominal orbital element set, denoted by
X
nom
t
()such that
oe
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