Global Positioning System Reference
In-Depth Information
The
a
xj
,
a
yj
, and
a
zj
terms in (2.31) denote the direction cosines of the unit vector
pointing from the approximate user position to the
j
th satellite. For the
j
th satellite,
this unit vector is defined as
(
)
a
j
=
aaa
,
,
j
yj
zj
Equation (2.30) can be rewritten more simply as
∆
ρ
j
=
ax ay az ct
∆
+
∆
+
∆
−
∆
xj
u
yj
u
zj
u
u
t
u
, which can be solved for by
making ranging measurements to four satellites. The unknown quantities can be
determined by solving the set of linear equations that follow:
We now have four unknowns:
∆
x
u
,
∆
y
u
,
∆
z
u
, and
∆
∆
ρ
ρ
=
ax ay az ct
ax aya
∆
+
∆
+
∆
−
∆
1
x
1
u
y
1
u
z
1
u
u
∆
=
zct
ax ay az ct
ax a
∆
+
∆
+
∆
−
∆
2
x
2
u
y
2
u
z
2
u
u
(2.32)
∆
ρ
ρ
=
∆
+
∆
+
∆
−
∆
3
x
3
u
y
3
u
z
3
u
u
∆
=
∆
+
∆
yazct
u
+
∆
−
∆
4
x
4
u
y
4
z
4
u
u
These equations can be put in matrix form by making the definitions
∆
∆
∆
∆
ρ
ρ
ρ
ρ
aaa
aaa
a
1
1
∆
∆
∆
x
y
z
ct
1
x
1
y
1
z
1
u
2
x
2
y
2
z
2
u
∆
=
H
=
∆
x
=
a a
aaa
1
1
x
3
y
3
z
3
u
3
−
∆
x
4
y
4
z
4
4
u
One obtains, finally,
∆
=
Hx
∆
(2.33)
which has the solution
∆
xH
=
−
1
∆
(2.34)
Once the unknowns are computed, the user's coordinates
x
u
,
y
u
,
z
u
and the
receiver clock offset
t
u
are then calculated using (2.26). This linearization scheme
will work well as long as the displacement (
z
u
) is within close proximity of
the linearization point. The acceptable displacement is dictated by the user's accu-
racy requirements. If the displacement does exceed the acceptable value, this pro-
cess is reiterated with
∆
x
u
,
∆
y
u
,
∆
ρ
being replaced by a new estimate of pseudorange based on
the calculated point coordinates
x
u
,
y
u
, and
z
u
. In actuality, the true user-to-satellite
measurements are corrupted by uncommon (i.e., independent) errors, such as mea-
surement noise, deviation of the satellite path from the reported ephemeris, and
multipath. These errors translate to errors in the components of vector
∆
x
, as shown
here:
−
1
=
H
(2.35)
x
meas
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