Data processing: vehicles – navigation – and relative location – Navigation – Employing position determining equipment
Reexamination Certificate
2001-04-06
2002-10-29
Cuchlinski, Jr., William A. (Department: 3661)
Data processing: vehicles, navigation, and relative location
Navigation
Employing position determining equipment
C342S357490
Reexamination Certificate
active
06473694
ABSTRACT:
FIELD OF THE INVENTION
The present invention relates to time-of-arrival ranging receivers, and more specifically to global positioning system (GPS) receivers.
BACKGROUND OF THE INVENTION
A global positioning system (GPS) receiver measures the time of arrival of a satellite ranging code provided by a GPS satellite. Based on the measured time of arrival of the ranging code (which includes orbit parameters and a timestamp indicating when the spacecraft transmitted the ranging code), the receiver determines the time required for the signal to propagate from the satellite to the receiver, and then multiplies this time by the speed of light to obtain a so-called pseudorange; the term pseudorange is used because there are a number of errors involved in the above measurement, the primary error being the error in the time of arrival of the signal because of an offset in the receiver clock from true GPS system time. The true range is just the difference between the time of reception of a specific code phase (of the ranging code) by the GPS receiver and the time of transmission of the specific code phase by the satellite multiplied by the speed of light, i.e.
r
=(
T
u
−T
s
)
c,
(1)
where r is the (true) range, T
u
is the GPS system time at which a specific code phase is received by the GPS receiver (i.e. the user of the GPS), T
s
is the GPS system time at which the specific code phase was transmitted, and c is the speed of light. Because of an offset t
u
of the receiver clock from GPS system time, what is measured is not the true range r but a pseudorange &rgr;, related to the true range according to,
&rgr;=
c
[(
T
u
+t
u
)−T
s
]=r+ct
u
. (2)
The true range r depends on the sought-after user position {right arrow over (u)} (at the time of receipt of the specific code phase) as well as on the position {right arrow over (s)} of the satellite transmitting the specific code phase (i.e. the position of the satellite at the time of transmission of the specific code phase of the ranging code), according to simply,
r=∥{right arrow over (s)}
(
t
tx
)−
{right arrow over (u)}
(
t
rx
)∥, (3)
where the time of transmission t
tx
and the time of reception t
rx
are both according to the same clock, the GPS system clock. (
FIG. 1
illustrates the notation of equation (3).) Therefore, from equations (2) and (3), the (measured) pseudorange depends on the sought-after user position and receiver clock offset according to,
&rgr;=∥{right arrow over (s)}
(
t
tx
)−
{right arrow over (u)}
(
t
rx
)∥+
ct
u
, (4)
in which the satellite position {right arrow over (s)}(t
tx
) is known from the navigation data modulating the ranging code, since the navigation data provide the orbit parameters and a timestamp indicating the time at which the spacecraft transmitted the ranging code, and so indicating the location of the spacecraft along its orbit at the time of transmitting the (specific code phase of the) ranging code.
The problem of determining the offset of the receiver clock is called the time-recovery problem. The prior art teaches solving for the receiver clock offset, as part of the solution for the user position, using various techniques, including linearizing a system of equations for pseudoranges from four or more satellites (using least squares estimation in case of more than four satellites) as well as other techniques such as using a Kalman filter. In a typical solution, at least four satellites are needed because there are four unknowns being solved for all three coordinates of the user position and the receiver clock offset. The approach based on linearizing a system of four pseudorange equations begins with the pseudorange equation, i.e. equation (4), for each of four satellites,
&AutoLeftMatch;
ρ
i
=
&LeftDoubleBracketingBar;
s
⇀
i
(
t
tx
)
-
u
⇀
(
t
rx
)
&RightDoubleBracketingBar;
+
ct
u
=
(
x
i
-
x
u
)
2
+
(
y
i
-
y
u
)
2
+
(
z
i
-
z
u
)
2
+
ct
u
=
f
i
(
x
u
,
y
u
,
z
u
,
t
u
)
,
for
i
=
1
,
2
…
,
4.
(
5
)
The method the Taylor-series expands each of the equations (5) about an estimated position {right arrow over ({circumflex over (u)})} (related to the sought-after user position {right arrow over (u)} according to {right arrow over (u)}={right arrow over ({circumflex over (u)})}+&Dgr;{right arrow over (u)}) and an estimated offset {circumflex over (t)}
u
related to the sought-after receiver offset t
u
according to t
u
={circumflex over (t)}
u
+&Dgr;t
u
), and neglects second-order and higher terms in the expansion, leaving,
ρ
i
=
f
i
(
u
⇀
^
,
t
^
u
)
+
▽
u
f
i
(
u
⇀
^
,
t
^
u
)
·
Δ
u
⇀
+
∂
f
i
(
u
⇀
^
,
t
^
u
)
∂
t
u
Δ
t
u
,
for
i
=
1
,
2
…
,
4
(
6
)
where the notations ∇
u
ƒ
i
({right arrow over ({circumflex over (u)})},{circumflex over (t)}
u
) and ∂ƒ
i
({right arrow over ({circumflex over (u)})},{circumflex over (t)}
u
)/∂t
u
indicate that the function ƒ is to be treated as a function of {right arrow over (u)} and t
u
for the indicated differentiation, and the result evaluated at {right arrow over ({circumflex over (u)})} and {circumflex over (t)}
u
. Using equations (5), the equations (6) become,
ρ
i
=
ρ
^
i
-
s
⇀
i
-
u
⇀
^
r
^
i
Δ
u
⇀
+
c
Δ
t
u
,
for
i
=
1
,
2
…
,
4
,
(
7
)
with {circumflex over (ƒ)}
i
being used in place of ƒ
i
({right arrow over ({circumflex over (u)})},{circumflex over (t)}
u
) in eq. (6). Using
a
⇀
i
=
s
⇀
^
i
-
u
⇀
^
r
^
i
and &Dgr;ƒ
i
={circumflex over (ƒ)}
i
−ƒ
i
, equations (7) can be written as,
&Dgr;ƒ
i
={right arrow over (a)}
i
·&Dgr;{right arrow over (u)}+c&Dgr;t
u
, for i=1, 2 . . . , 4, (8)
and by constructing the vectors and matrix,
Δ
ρ
⇀
≡
(
Δ
ρ
1
Δ
ρ
2
Δ
ρ
3
Δ
ρ
4
)
H
=
(
a
x1
a
y1
a
z1
1
a
x2
a
y2
a
z2
1
a
x3
a
y3
a
z3
1
a
x4
a
y4
a
z4
1
)
and
Δ
x
=
(
Δ
x
u
Δ
y
u
Δ
z
u
-
c
t
u
)
,
(
9
)
equations (8) can be written as,
&Dgr;{right arrow over (&rgr;)}=H&Dgr;{right arrow over (x)}
(10)
which has the solution,
&Dgr;{right arrow over (x)}=H
−1
&Dgr;{right arrow over (&rgr;)}. (11)
When navigating by GPS in weak signal conditions such that the navigation component of the navigation signal cannot be decoded, the time when the signal is received is not precisely known. That time, when expressed according to the GPS clock, is called here simply GPS time at arrival and is indicated by the notation &tgr;. It differs from the time at which the signal is transmitted by a satellite (i.e. t
tx
in equation (5)) by the time of flight t
ƒ
of the signal from the satellite to the GPS receiver. In terms of the GPS time at arrival &tgr; and the time of flight t
ƒ
, the pseudorange given by equation (2) can be written as,
&rgr;=c
└(
T
u
+t
u
)−(&tgr;−
t
ƒ
)┘=
c
(
T
u
+t
u
)−
c
(
&tgr;−t
ƒ
)=
c
(
t
u
+t
ƒ
), (2′)
since, by definition, T
u
=&tgr;.
In normal circumstances (i.e. strong signal conditions), the time at which the ranging code was sent from the spacecraft, &tgr;&
Akopian David
Syrjarinne Jari
Nguyen Thu
Nokia Corporation
Ware, Freesola, Van Der Sluys & Adolphson LLP
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