Global Positioning System Reference
In-Depth Information
9.2.4.2 Transition Matrix
To formulate the state transition matrix, one can write down the transition equa-
tions for position, velocity, clock bias, and clock drift as follows:
&
δ
x
=
δ
x
+
δ
x
t
n
n
1
n
1
δ
x
&
=
=
δ
x
&
n
n
1
&
δ
t
δ
t
+
δ
t
t
u
u
u
n
n
1
n
1
&
&
δ
t
=
δ
t
u
u
n
n
1
From these equations, the state transition matrix can be formulated as
1 0 0
0100
001
0001
t
(
)
(
)
tt
,
=
tt
,
=
n
n
1
1
0
t
For this example, we will denote
Φ
( t 1 , t 0 ) as
Φ
.
9.2.4.3 Measurement Matrix
The elements in the measurement matrix H ( t 1 ) relate observations—in this case,
range error and range rate error—to the state vector. To accomplish this, as each
measurement is received, we create a LOS unit vector from the user's inertial-based
position to the satellite's position. This unit vector is then placed in the measurement
matrix to decompose the range error and range rate error into its x -dimension com-
ponent. The navigation processor computes the unit vector from the user to the sat-
ellite by subtracting the user's inertial position ( x ui , y ui , z ui ) from the satellite position
( x j , y j , z j ) (where j denotes the j th satellite), generating an estimated range vector to
the satellite. This is then normalized to a unit vector by dividing the range vector by
its scalar range. Normally each element of the unit vector is placed in the measure-
ment matrix to convert the range error and range rate error, but in our case we only
need to incorporate the x -axis component of its xyz components. The errors in the
clock bias and drift states are simply set to 1, since the LOS pseudorange and
pseudorange rate errors directly map into the clock bias and clock drift states.
Let
~
~
~
r
=−
=−
=−
xx
x
j
ui
r
yy
y
j
ui
r
zz
(9.5)
z
j
ui
~ ~~~
2
2
2
r
=
r
+
r
+
r
x
y
z
~
~
r
r
x
010
()
H
t
=
~
~
1
r
r
x
0
0
1
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