Global Positioning System Reference
In-Depth Information
(
)
−
1
T
T
SIHHHH
≡−
(7.71)
n
where
I
n
is the
n
×
n
unit matrix. Then, the
n
×
1 range residual vector,
w
, is given as
w
S
(used in the simulations). The range residual vec-
tor,
w
, could be used as a measure of consistency. This is not ideal, however, because
there are four constraints (associated with the four unknown components of the vec-
tor
x
) among the
n
elements of
w
, which obscure some of the aspects of the inconsis-
tency that are of interest. Therefore, it is useful to perform a transformation that
eliminates the constraints and transforms the information contained in
w
into
another vector known as the parity vector,
p
.
Performing a transformation on
y
,
p
=
Sy
(used in practice) or
w
=
=
Py
, where the parity transformation
matrix
P
is defined as an (
n
n
matrix, which can be obtained by QR
factorization of the
H
matrix [44]. The rows of
P
are mutually orthogonal, unity in
magnitude, and mutually orthogonal to the columns of
H
. Due to these defining
properties, the resultant
p
has special properties, especially with respect to the noise
[43]. If
−
4)
×
has independent random elements that are all N(0,
σ
2
), then
pPw
=
(7.72a)
pP
=
(7.72b)
T
T
pp ww
=
(7.72c)
These equations state that the same transformation matrix
P
that takes
y
into the
parity vector,
p
, also takes either
w
or into
p
. The sum of the squared residuals is
the same in both range space and parity space. In performing failure detection, it is
much easier to work with
p
than with
w
.
Using a case of six visible satellites as an example, the following analysis demon-
strates how the parity transformation affects a deterministic error in one of the
range measurements. Suppose there is a range bias error,
b
, in satellite 3. From
(7.72b),
0
0
PPP
L
L
P
b
11
12
13
16
p
=
or
PPP
P
0
0
0
21
22
23
26
(
)
p
=×
b
3
rd column of
P
The third column of
P
defines a line in parity space called the characteristic bias
line associated with satellite 3. Each satellite has its own characteristic bias line. The
magnitude of the parity bias vector induced by the range bias
b
is given by
|parity bias vector|
=
b
· norm |[
P
13
P
23
]
T
|, (bias on satellite 3, assuming
b
>
0)
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