Global Positioning System Reference
In-Depth Information
and the offset of the user's position from the linearization point is expressed as a lin-
ear function of . In the case of
n
> 4, the method of least squares can be used to
solve (7.30) for
x
(see Appendix A). The least square result can be obtained for-
mally by multiplying both sides of (7.30) on the left by the matrix transpose of
H
obtaining
H
T
H x
4 matrix,
and one can solve for
x
by multiplying both sides by the inverse, (
H
T
H
)
−1
. (The
matrix will be invertible provided the tips of the unit vectors
a
i
do not all lie in a
plane.) One obtains
=
H
T
. The matrix combination
H
T
H
is a square 4
×
(
)
−
1
T
T
xHHH
=
(7.33)
which is the least square formulation for
x
as a function of
. We observe that if
n
H
−1
(H
T
)
−1
and (7.33) reduces to (7.32).
The pseudorange measurements are not error-free and can be viewed as a linear
combination of three terms,
4, (
H
T
H)
−1
=
=
=−+
T
d
(7.34)
L
where
L
is the vector of
pseudorange values computed at the linearization point, and
d
represents the net
error in the pseudorange values. Similarly,
x
can be expressed as
T
is the vector of error-free pseudorange values,
xx x x
=−+
T
d
(7.35)
L
where
x
T
is the error-free position and time,
x
L
is the position and time defined as the
linearization point, and
d
x
is the error in the position and time estimate. Substitut-
ing (7.34) and (7.35) into (7.33) and using the relation
x
T
(
H
T
H
)
−1
H
T
(
T
x
L
=
−
L
)—this follows from the relation
H
(
x
T
−
x
L
)
=
(
T
−
L
), which is a restatement of
(7.30)—one obtains
(
)
−
1
T T
xHHH K
d
=
d
=
d
(7.36)
The matrix
K
is defined by the expression in brackets. Equation (7.36) gives the
functional relationship between the errors in the pseudorange values and the
induced errors in the computed position and time bias. It is valid provided that the
linearization point is sufficiently close to the user's location and that the
pseudorange errors are sufficiently small so that the error in performing the
linearization can be ignored.
Equation (7.36) is the fundamental relationship between pseudorange errors
and computed position and time bias errors. The matrix (
H
T
H
)
−1
H
T
, which is some-
times called the least-squares solution matrix, is a 4
n
matrix and depends only on
the relative geometry of the user and the satellites participating in the least square
solution computation. In many applications, the user/satellite geometry can be con-
sidered fixed, and (7.36) yields a linear relationship between the pseudorange errors
and the induced position and time bias errors.
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