Global Positioning System Reference
In-Depth Information
From (38), it can be proved that
R
is also piecewise linear. In order to shape the SCF into an
ideal triangle, one have to make all of the ending points of lines in
R
be zero except the
central one.
To ensure the triangular shape, SCF must satisfy the following request:
()
(
⎧
R
RkT
00
/
≠
⎨
(41)
)
(
)
M
=
0,
k
≠
0
⎩
c
where the first term is equivalent to
r
=
0
(42)
0
and using (38) and (40), the second term can be simplified as
rr
−
≤
0
(43)
kk
for
k
≠ .
0
The constraints (42) and (43) are necessary but not sufficient, since the absolute-magnitude
operation introduces additional endpoints at the zero crossing points in
R
and
R
. If
R
(
)
⎦
,
(
)
has a zero crossing point
τ within the interval
⎡
kT
/,
M
k
+
1 /
T
M
⎤
k
>
0
, easily
⎣
c
c
proved, it must be
r
kT T
Mr rM
k
τ
=+
+
c
c
(44)
0
k
k
+
1
From (43) we know that
R
must have a zero crossing point within the same interval, which
is
kT T
Mr r M
r
−
k
τ
′ =+
+
c
c
(45)
0
−
k
− −
k
1
In order to eliminate the inclined lines on both sides of the zero crossing point, we must
have
τ τ= , which can be simplified as
0
0
rr
=
r r
(46)
kk
− −
1
−
kk
+
1
for
k
> .
0
6.3 Step 3 - UVS establishment
The necessary and sufficient conditions for SCF being triangular are (40), (42), (43), and (46).
From (40), we obtain that
d
and
d
are mirror images of each other, i.e.
dd
− −
′ =
(47)
k
Mk
1
where
d
and
d
′ are the entries of
d
and
d
, respectively. When
M
= , by using the
2
k
relationship (20), the UVS can be represented as
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