U.S. patent application number 16/088518 was filed with the patent office on 2021-06-17 for using sdp relaxation for optimization of the satellites set chosen for positioning.
The applicant listed for this patent is Topcon Positioning Systems, Inc.. Invention is credited to LEV BORISOVICH RAPOPORT.
Application Number | 20210181358 16/088518 |
Document ID | / |
Family ID | 1000005623302 |
Filed Date | 2021-06-17 |
United States Patent
Application |
20210181358 |
Kind Code |
A1 |
RAPOPORT; LEV BORISOVICH |
June 17, 2021 |
Using SDP Relaxation for Optimization of the Satellites Set Chosen
for Positioning
Abstract
A method of determining coordinates, including receiving GNSS
(global navigation satellite system) signals from at least five
satellites, wherein at least two of the five satellites belong to
one constellation, and the remaining satellites belong to at least
one other constellation; processing the GNSS signals to measure
code and phase measurements for each of the satellites and each of
the GNSS signals; selecting a subset of the GNSS signals as an
optimal set for coordinate calculation, where the selecting is
based on Semi-Definite Programming (SDP) relaxation as applied to
an optimization of a PDOP (positional dilution of precision)
criterion; calculating coordinates of a receiver based on the code
and phase measurements of the selected subset; and outputting the
calculated coordinates. The total number of signals in the optimal
set should not exceed the predefined number of m signals.
Inventors: |
RAPOPORT; LEV BORISOVICH;
(Moscow, RU) |
|
Applicant: |
Name |
City |
State |
Country |
Type |
Topcon Positioning Systems, Inc. |
Livermore |
|
CA |
|
|
Family ID: |
1000005623302 |
Appl. No.: |
16/088518 |
Filed: |
April 18, 2018 |
PCT Filed: |
April 18, 2018 |
PCT NO: |
PCT/RU2018/000241 |
371 Date: |
September 26, 2018 |
Current U.S.
Class: |
1/1 |
Current CPC
Class: |
G01S 19/258 20130101;
G01S 19/29 20130101; G01S 19/30 20130101 |
International
Class: |
G01S 19/25 20060101
G01S019/25; G01S 19/29 20060101 G01S019/29; G01S 19/30 20060101
G01S019/30 |
Claims
1. A method of determining coordinates, comprising: receiving GNSS
(global navigation satellite system) signals from at least five
satellites, wherein at least two of the five satellites belong to
one constellation, and the remaining satellites belong to at least
one other constellation; processing the GNSS signals to measure
code and phase measurements for each of the satellites and each of
the GNSS signals; selecting a subset of the GNSS signals as an
optimal set for coordinate calculation, wherein the selecting is
based on Semi-Definite Programming (SDP) relaxation as applied to
an optimization of a PDOP (positional dilution of precision)
criterion; calculating coordinates of a receiver based on the code
and phase measurements of the selected subset; and outputting the
calculated coordinates.
2. The method of claim 1, wherein the receiver is a standalone
receiver.
3. The method of claim 1, wherein the receiver operates in a
smoothed standalone mode.
4. The method of claim 1, wherein the receiver operates in a
differential mode.
5. The method of claim 1, wherein the SDP relaxation minimizes a
linear criterion min S , X t r ( S ) ##EQU00011## subject to linear
matrix inequality constraint P ( S , X ) = ( A T X A I p I p S ) 0
, ##EQU00012## and algebraic inequality constraints i = 1 M x i
.ltoreq. m , 0 .ltoreq. x i .ltoreq. 1 , ##EQU00013## where p is a
number of navigation parameters, X = ( x 1 0 0 0 x 2 0 0 0 x M )
##EQU00014## is a diagonal matrix composed of x.sub.i, S is a
symmetric auxiliary matrix, m is the maximum number of signals
allowed in the optimal set, I.sub.p is an identity matrix, and
stands for positive semidefiniteness.
6. The method of claim 5, wherein the x.sub.i are non-binary, and
are rounded to 0 or 1.
7. The method of claim 5, wherein the SDP relaxation is used for
computation of a lower bound by using a Branch and Bound
technique.
8. A GNSS (global navigation satellite system) navigation receiver,
comprising: a plurality of radio frequency tracts receiving GNSS
signals from at least five satellites, wherein at least two of the
five satellites belong to one constellation, and the remaining
satellites belong to at least one other constellation; a digital
section processing the GNSS signals to measure code and phase
measurements for each of the satellites and each of the GNSS
signals; the digital section selecting a subset of the GNSS signals
as an optimal set for coordinate calculation, wherein the selecting
is based on Semi-Definite Programming (SDP) relaxation as applied
to an optimization of a PDOP (positional dilution of precision)
criterion; the digital section calculating coordinates of a
receiver based on the code and phase measurements of the selected
subset; and the digital section outputting the calculated
coordinates.
9. The receiver of claim 8, wherein the receiver is a standalone
receiver.
10. The receiver of claim 8, wherein the receiver operates in a
smoothed standalone mode.
11. The receiver of claim 8, wherein the receiver operates in a
differential mode.
12. The receiver of claim 8, wherein the SDP relaxation minimizes a
linear criterion min S , X t r ( S ) ##EQU00015## subject to linear
matrix inequality constraint P ( S , X ) = ( A T X A I p I p S ) 0
, ##EQU00016## and algebraic inequality constraints i = 1 M x i
.ltoreq. m , 0 .ltoreq. x i .ltoreq. 1 , ##EQU00017## where p is a
number of navigation parameters, X = ( x 1 0 0 0 x 2 0 0 0 x M )
##EQU00018## is a diagonal matrix composed of x.sub.i, S is a
symmetric auxiliary matrix, m is the maximal number of signals
allowed in the optimal set, I.sub.p is an identity matrix, and
stands for positive semidefiniteness.
13. The receiver of claim 12, wherein the x.sub.i are non-binary,
and are rounded to 0 or 1.
14. The receiver of claim 12, wherein the SDP relaxation is used
for computation of a lower bound by using a Branch and Bound
technique.
15. The receiver of claim 8, wherein the radio frequency tracts are
connected to a single antenna.
16. The receiver of claim 8, wherein the radio frequency tracts are
connected to multiple antennas.
Description
BACKGROUND OF THE INVENTION
Field of the Invention
[0001] The present invention relates to satellite navigation
systems, and, more particularly, to determination of a subset of
satellites, out of many available ones, that are to be used for
accurate calculation of position.
Description of the Related Art
[0002] When processing multiple navigation satellite systems,
including GPS, GLONASS, Galileo, Beidou, QZSS, the overall number
of the range and carrier phase signals can exceed several tens
(close to 100 today, and more are expected in the future). On the
other hand, a much smaller number of them is necessary for
processing to meet sufficient precision of positioning. Some parts
of RTK algorithms, like carrier phase ambiguity resolution, are
very sensitive to the dimension of the integer search problem. If
trying to process multiple satellite signals, it is very important
to correctly choose the subset of carrier phase ambiguities that
are fixed first, secondarily, and so on.
[0003] Assume that the receiver connected to each antenna or
multi-antenna receiver connected to plurality of antennas is
capable of receiving multiple GNSS signals of multiple GNSS
satellites, including (but not limited by this set) [0004] GPS L1,
L2, L5; [0005] Galileo L1, L2, E5a, E5b; [0006] QZSS L1, L2, L5,
E6; [0007] GLONASS L1, L2, L3; [0008] BEIDOU B1, B2, B3.
[0009] The number of signals is actually larger than the number of
satellites, because multiple frequency bands can be used for each
satellite. The total number of signals can exceed dozens and even
reach a value of a hundred.
[0010] The following fundamental set of observables is used:
P k , b s ( t ) = .rho. k s ( t ) + c d t k ( t ) - c d t s ( t ) +
( f L 1 s f b s ) 2 I k , L 1 s ( t ) + T k s ( t ) + d k , b , P +
M k , b , P s - D b , P s + k , b , P s ( t ) ( 1 ) .PHI. k , b s (
t ) = f b s c .rho. k s ( t ) + f b s d t k ( t ) - f b s d t s ( t
) + N k , b s ( t C S , k , b s ) - 1 c ( f L 1 s ) 2 f b s I k , L
1 s ( t ) + f b s c T k s ( t ) + d k , b , .PHI. s + M k , b ,
.PHI. s - D b , .PHI. s + k , b , .PHI. s ( 2 ) ##EQU00001##
[0011] where the following notations are used (see [3, Chapter 6,
7]): [0012] k is the index of the receiver stations; [0013] s is
the number of the satellite (out of S total satellites from which
signals are received). It is assumed that all satellites available
for tracking by the antenna k are ordered and this ordering number
includes the satellite system. For example, satellites with numbers
from 1 to 32 are GPS, satellites with numbers from 33 to 56 are
GLONASS, satellites with numbers from 57 to 88 are BEIDOU, etc.
although this numbering scheme is more for convenience than due to
any fundamental physical limitation; [0014] b is the frequency band
(for example L1, LP2, L2C, L5, E6, B1, and so on); [0015] (s,b) the
pair of indices indicating the signal of the satellites at the
frequency band b; [0016] t is the current time; the term "epoch" is
also used to denote the current discrete time instant; [0017] c is
the speed of light; [0018] f.sub.b.sup.s is the frequency of the
signal corresponding to the satellite s and the frequency band b;
[0019] dt.sub.k(t) is the current clock bias of the station k;
[0020] dt.sup.s(t) is the current clock bias of the satellite s;
[0021] I.sub.k,b.sup.s(t) is the ionospheric delay affecting the
signal {s,b} received by the station k. Thus I.sub.k,L1.sup.s(t) is
related to the L1 band. Basically, the ionospheric delay depends on
the position of the station, position of the satellite, frequency
of the signal, and the total electronic content (TEC) corresponding
to the time instant t; [0022] T.sub.k.sup.s(t) is the tropospheric
delay. In contrast to the ionospheric delay it doesn't depend on
the signal frequency and is called "non-dispersive" delay. [0023]
P.sub.k,b.sup.s(t) and .phi..sub.k,b.sup.s(t) are pseudorange and
phase measurements respectively; [0024] M.sub.k,b,P.sup.s and
M.sub.k,b,.phi..sup.s are code and phase multipath, affecting
pseudorange and carrier phase measurements respectively; [0025]
N.sub.k,b.sup.s(t.sub.CS,k,b.sup.s) is carrier phase ambiguity
corresponding to the signal {s, b} received by the station k. Note
that it corresponds to the last detected cycle slip and therefore
it explicitly depends on the time instant t.sub.CS,k,b.sup.s, when
the cycle slip was detected. The carrier phase ambiguity remains
unchanged until the cycle slip occurs. [0026] .rho..sub.k.sup.s(t)
is the true topocentric range between the satellite and the
station. The measurement equations (1) and (2) relate this quantity
with pseudorange and carrier phase observables, respectively;
[0027] Quantities D.sub.b,.phi..sup.s, D.sub.b,P.sup.s,
.epsilon..sub.k,b,P.sup.s(t) and .epsilon..sub.k,b,.phi..sup.s
denote hardware biases and noise. D stands for satellite-related
bias, while d denotes the receiver-related bias. Biases reflect a
systematic hardware component of the measurement error. Due to
their physical nature biases are constant or slow varying, in
contrast to the noise component of the error having the stochastic
nature. Satellite position error is part of the slow varying bias.
It is also called "the ephemerides" error.
[0028] Thus, the receiver position is measured by the pseudorange
and carrier phase observables for the plurality of satellites.
Error components, including biases and noise, affecting the
observable equations (1) and (2), prevent a direct solution for the
receiver antenna position.
[0029] Carrier phase measurements are much more precise, compared
to the pseudorange measurements, since the carrier phase noise has
standard deviation in the centimeter or even millimeter range,
while the standard deviation of the pseudorange measurements is
usually of the meter or decimeter range. On the other hand, the
carrier phase measurement is affected by the carrier phase
ambiguity, which is an unknown integer valued quantity.
[0030] Thus, elimination of measurement errors is necessary for
precise positioning. To achieve a high precision in position
determination, different methods of errors mitigation are applied.
For example, tropospheric errors can be precisely modeled and
compensated in observables of equations (1) and (2). Ionospheric
errors can be estimated along with other unknowns. Noise is easily
filtered.
[0031] Errors common to two receivers, like clock and hardware
biases of the satellite, can be compensated in a difference between
two receivers. Usually one of receivers occupies a known position,
while an antenna of another receiver is attached to the object to
be located. The first receiver is called "the base" while another
receiver is called "the rover". The processing mode involving
calculation of the across-receiver difference (also called the
"first difference") is referred to as differential GNSS processing
or DGNSS. The DGNSS processing is performed in real time and
includes not only pseudoranges but also carrier phase observables,
and is referred to as real time kinematic (RTK) processing.
[0032] Another sources of errors partially eliminated by
across-receiver differences are ionospheric delay and ephemerides
error. The closer the rover is to the base, the better is
compensation of the ionospheric and ephemerides error.
[0033] For two stations k and l the across-receiver differences of
pseudorange and carrier phase measurements can be written as
P _ kl , b s ( t ) = .rho. k s ( t ) - .rho. l s ( t ) + c d t k l
( t ) + ( f L 1 s f b s ) 2 I kl , L 1 s ( t ) + d kl , b , P + _
kl , b , P s ( t ) , ( 3 ) .PHI. _ kl , b s ( t ) = 1 .lamda. b s (
.rho. k s ( t ) - .rho. l s ( t ) ) + f b s d t k l ( t ) + N kl ,
b s ( t C S , kl , b s ) - 1 .lamda. b s ( f L 1 s f b s ) 2 I kl ,
L 1 s ( t ) + d kl , b , .PHI. s + _ kl , b , .PHI. s , ( 4 )
##EQU00002##
[0034] Another way for error mitigation includes using a precise
satellite clock and precise ephemerides. They are available through
a dedicated real time service. Precise point positioning (PPP)
allows to achieve the centimeter level position with only one rover
receiver, provided satellite clock and precise satellite position
are available. The base station is not necessary in this case.
[0035] Finally, if neither base station, nor precise clock and
ephemerides are available, the quality of the standalone position
can be improved if carrier phase ambiguity and ionospheric delay
are estimated, along with position, using broadcast ephemerides.
The corresponding processing mode is equivalent smoothing of
pseudoranges using carrier phase measurements or, in short, carrier
phase smoothing of code pseudoranges, see [2]. Whatever processing
mode is used, the linearization and filtering algorithms are used
for recursive estimation of unknown position, carrier phase
ambiguity, and ionospheric delay.
[0036] The general form of the linearized navigation model has the
following form (see [2, Chapt. 7])
b.sub.P(t)=Adx(t)+e.xi.(t)+.GAMMA.i(t)+d.sub.P (5)
b.sub..phi.(t)=.LAMBDA..sup.-1Adx(t)+.LAMBDA..sup.-1e.xi.(t)+n-.LAMBDA..-
sup.-1.GAMMA.i(t)+d.sub..phi. (6)
[0037] Two last quantities are undifferences for carrier phase
smoothing and PPP processing modes. For DGNSS and RTK processing
modes the carrier phase ambiguity and ionospheric delay are
across-receiver differenced.
[0038] Let M be total number of satellite signals, including
different satellite systems, different satellites, different
frequency bands.
[0039] In the following description, all vectors are represented by
columns, and the superscript symbol T denotes the matrix transpose.
R.sup.N is the N--dimensional Euclidean space. Given a
linearization point x.sub.0(t).di-elect cons.R.sup.3, notations
used in equations (5) and (6) are as follows: [0040]
b.sub.P(t).di-elect cons.R.sup.M is the M--dimensional vector of
pseudorange residuals calculated at the linearization point; [0041]
b.sub..phi.(t).di-elect cons.R.sup.M is the M--dimensional vector
of carrier phase residuals calculated at the linearization point;
[0042] e=(1, 1, . . . , 1).sup.T.di-elect cons.R.sup.M is the
vector consisting of all "ones"; [0043] d x(t).di-elect
cons.R.sup.3 is the correction to the linearization point. Thus,
the corrected position is calculated as x(t)=x.sub.0(t)+d x(t);
[0044] .xi.(t) is the arbitrary varying rover clock shift; it is
undifferenced for the standalone and PPP processing modes, and it
is across-receiver differenced in the DGNSS and RTK processing
modes; [0045] .LAMBDA. is the M-dimensional diagonal matrix with
wavelengths .lamda..sub.b.sup.s=c/f.sub.b.sup.s in the main
diagonal. Each wavelength corresponds to the specific signal (s,b);
[0046] A is the M.times.3 matrix of directional cosines; [0047]
.GAMMA. is the M.times.M diagonal matrix with quantities
(f.sub.L1.sup.s/f.sub.b.sup.s).sup.2 in the main diagonal; [0048]
i(t).di-elect cons.R.sup.S is the vector of ionospheric delays
related to the L1 frequency band.
[0049] Ionospheric delays are undifferenced for the standalone and
PPP processing modes, and they are across-receiver differenced in
the DGNSS and RTK processing modes; [0050] n.di-elect cons.R.sup.M
is the vector of carrier phase ambiguities related to the L1
frequency band. Ambiguities are undifferenced for the standalone
and PPP processing modes, and they are across-receiver differenced
in the DGNSS and RTK processing modes; [0051] d.sub.P.di-elect
cons.R.sup.M and d.sub..phi..di-elect cons.R.sup.M are vectors of
pseudorange and carrier phase receiver hardware biases.
[0052] Consideration of pseudorange hardware biases leads to a
necessity to consider the plurality of signals the receiver is able
to track. In the case of a multi-frequency and multi-system
receiver supporting the following bands:
[0053] L1, L2 and L5 bands for GPS,
[0054] L1 and L2 GLONASS,
[0055] L1, E5a, E5b and E6 Galileo,
[0056] L1, L2, L5 and E6 QZSS,
[0057] L1 an, L5 SBAS,
[0058] B1, B2, and B3 Beidou,
[0059] the signals (L1 GPS, L1 Galileo, L1 SBAS, L1 QZSS), (L2 GPS,
L2 QZSS), (L5 GPS, E5a Galileo, L5 SBAS, L5 QZSS), (E6 Galileo, E6
QZSS), respectively, can share the same hardware channel and
therefore will be affected by the same hardware bias, as noted in
[2, Chapter 7]. Note that the biases vector d.sub.P and the clock
shift variable .xi.(t) appear as a sum in equation (5). This means
that one of the biases, say d.sub.L.sub.1.sub.,G/E/S/Q,P, can be
combined with j(t), while others can be replaced with their
difference with d.sub.L.sub.1.sub.,G/E/S/Q,P. Thus, new bias
variables appear:
.eta..sub.1=d.sub.L.sub.2.sub.,G/Q,P-d.sub.L.sub.1.sub.,G/E/S/Q,P,.eta..-
sub.2=d.sub.L.sub.1.sub.,R,P-d.sub.L.sub.1.sub.,G/E/S/Q,P,.eta..sub.3=d.su-
b.L.sub.2.sub.,R,P-d.sub.L.sub.1.sub.,G/E/S/Q,P
.eta..sub.4=d.sub.L.sub.5.sub.,G/Q,P-d.sub.L.sub.1.sub.,G/E/S/Q,P,.eta..-
sub.5=d.sub.E.sub.56.sub.,E,P-d.sub.L.sub.1.sub.,G/E/S/Q,P,.eta..sub.6=d.s-
ub.E.sub.6.sub.,E/Q,P-d.sub.L.sub.1.sub.,G/E/S/Q,P
.eta..sub.7=d.sub.B.sub.1.sub.,B,P-d.sub.L.sub.1.sub.,G/E/S/Q,P,.eta..su-
b.8=d.sub.B.sub.2.sub.,B,P-d.sub.L.sub.1.sub.,G/E/S/Q,P,.eta..sub.9=d.sub.-
B.sub.3.sub.,B,P-d.sub.L.sub.1.sub.,G/E/S/Q,P (7)
[0060] This representation can be referred to as establishing a
bias datum.
[0061] In one possible embodiment, the linearized equations (5) can
now be expressed in the form
b.sub.P(t)=Adx(t)+e.xi.(t)+.GAMMA.i(t)+W.sub..eta..eta. (8)
[0062] The bias vector .eta. has the appropriate dimension
m.sub..eta.. Note again that in this exemplary embodiment we follow
notations introduced in [2], which is incorporated herein by
reference in its entirety.
[0063] The bias vector .eta. is three-dimensional (m.sub..eta.=3)
for dual-band and dual-system GPS/GLONASS receivers, as only biases
.eta..sub.1, .eta..sub.2, .eta..sub.3 are presented among all
possible biases listed in (7). In the case of the multi-band,
multi-system receiver, the dimension of the vector .eta. can be
large. It is one-dimensional in the case of dual-band GPS-only
receivers or single band GPS/GLONASS receivers.
[0064] The W.sub..eta. is referred to as bias allocation matrix and
has dimensions n.times.m.sub..eta.. It allocates a single bias, or
none, to a certain signal. No bias is allocated to the signal
corresponding to the GPS, Galileo, SBAS, or QZSS systems and b=L1
because we combined the bias d.sub.L.sub.1.sub.,G/E/S/Q,P with the
clock bias .xi.(t). In this case, the row of W.sub..eta. consists
of zeroes.
[0065] Consider, for example, a dual-band GPS/GLONASS receiver.
Suppose it tracks six GPS satellites and six GLONASS satellites.
The total number of dual-band signals is 24. Let the signals be
ordered in the following way: six GPS L1, six GPS L2, six GLONASS
L1, and six GLONASS L2 signals. The biaseallocation matrix
W.sub..eta. presented in the linearized single difference
pseudorange equation (8) takes the following form:
W .eta. T = [ 0 0 0 | 1 1 1 | 0 0 0 | 0 0 0 0 0 0 | 0 0 0 | 1 1 1 |
0 0 0 0 0 0 | 0 0 0 | 0 0 0 | 1 1 1 ] ( 9 ) ##EQU00003##
[0066] Further, the real-valued carrier phase ambiguities (also
called float ambiguities) are combined with biases d.sub..phi.,
while pseudorange biases are considered as a real valued constant
unknown parameter. Thus, after combination of carrier phase
ambiguities with carrier phase biases, the equation (6) takes the
form
b.sub..phi.=.LAMBDA..sup.-1Adx(t)+.LAMBDA..sup.-1e.xi.(t)+n-.LAMBDA..sup-
.-1.GAMMA.i(t). (10)
[0067] Note that the noise component is omitted in equations (8),
(10) for the sake of brevity.
[0068] The ambiguity vector n and the bias vector .eta. are further
recursively estimated based on successively processed set of data
(8) and (10) for successive time epochs t. Ambiguity vector is then
fixed at the integer valued-vector called fixed ambiguity n*. Both
recursive estimation and especially integer search are
computationally consuming. Computational cost of recursive
estimation cubically depends on M. Even more computationally
excessive (actually near exponential) is the integer search. On the
other hand not all signals are necessary for accurate position
determination.
[0069] Accordingly, there is a need in the art for a technique for
selecting a set of GNSS signals for use in all modes of
positioning.
SUMMARY OF THE INVENTION
[0070] Accordingly, the present invention is directed to optimal
selection of satellites to be used for accurate precision of
positioning in the following processing modes:
[0071] In one aspect, there is provide a method of determining
coordinates, including receiving GNSS (global navigation satellite
system) signals from at least five satellites, wherein at least two
of the five satellites belong to one constellation (e.g., GPS), and
the remaining satellites belong to at least one other constellation
(e.g., to Galileo); processing the GNSS signals to measure code and
phase measurements for each of the satellites and each of the GNSS
signals; selecting a subset of the GNSS signals as an optimal set
for coordinate calculation, where the selecting is based on
Semi-Definite Programming (SDP) relaxation as applied to an
optimization of a PDOP (positional dilution of precision)
criterion; calculating coordinates of a receiver based on the code
and phase measurements of the selected subset; and outputting the
calculated coordinates. The total number of signals in the optimal
set should not exceed the predefined number of m signals.
[0072] Optionally, the receiver is a standalone receiver, or
operates in a smoothed standalone mode, or operates in a
differential mode. Optionally, the SDP relaxation minimizes a
linear criterion
min S , X t r ( S ) ##EQU00004## [0073] subject to linear matrix
inequality constraint
[0073] P ( S , X ) = ( A T X A I p I p S ) 0 , ##EQU00005## [0074]
and algebraic inequality constraints
[0074] i = 1 M x i .ltoreq. m , 0 .ltoreq. x i .ltoreq. 1 ,
##EQU00006## [0075] where p is a number of navigation
parameters,
[0075] X = ( x 1 0 0 0 x 2 0 0 0 x M ) ##EQU00007##
is a diagonal matrix composed of x.sub.i, [0076] S is a symmetric
auxiliary matrix, [0077] m is the maximum number of signals allowed
in the optimal set, [0078] I.sub.p is an identity matrix, and
[0079] stands for positive semidefiniteness.
[0080] Optionally, the x.sub.i are non-binary, and are rounded to 0
or 1.
[0081] Optionally, the SDP relaxation is used for computation of a
lower bound by using a Branch and Bound technique.
[0082] Thus, the optimal choice problem is reduced to the convex
optimization using semidefinite programming (SDP) relaxation
approach. Within this approach the linear objective function is
optimized over the cone of positive semidefinite matrices.
[0083] Additional features and advantages of the invention will be
set forth in the description that follows, and in part will be
apparent from the description, or may be learned by practice of the
invention. The advantages of the invention will be realized and
attained by the structure particularly pointed out in the written
description and claims hereof as well as the appended drawings.
[0084] It is to be understood that both the foregoing general
description and the following detailed description are exemplary
and explanatory and are intended to provide further explanation of
the invention as claimed.
BRIEF DESCRIPTION OF THE ATTACHED FIGURES
[0085] The accompanying drawings, which are included to provide a
further understanding of the invention and are incorporated in and
constitute a part of this specification, illustrate embodiments of
the invention and together with the description serve to explain
the principles of the invention.
[0086] In the drawings comparison of optimal and non-optimal
constellation choices is illustrated:
[0087] FIG. 1 illustrates an example of non-optimal selection of
constellations of 9 satellites from a total 19 satellites
available. The constellation is shown in the coordinate frame
`X,Y`, where the axis `X` is for the quantity `elevation
angle*sin(azimuth)`. The axis `Y` is for the quantity `elevation
angle*cos(azimuth)`. Both quantities are measured in degrees from
-90 to +90. Each satellite is denoted as a point in this coordinate
frame.
[0088] FIG. 2 illustrates the optimal choice for the same example
as in FIG. 1. The same notations are used as in the FIG. 1.
[0089] FIG. 3 shows a structure of an exemplary navigation
receiver.
DETAILED DESCRIPTION OF EMBODIMENTS OF THE INVENTION
[0090] Reference will now be made in detail to the embodiments of
the present invention.
[0091] To reduce the computational cost of the positioning
calculation, the optimal choice of signals should be used.
Semi-definite relaxation (SDP) approach to solution of this
optimization problem is proposed in this invention. Different cost
functions controlling precision of positioning, for example PDOP,
can be considered in the optimization problem. The constraint used
in optimization includes the total number of signals. We must
choose the subset whose total number does not exceed a given value
(e.g., 19, or 24) in such a way that PDOP takes minimal possible
value.
[0092] The optimization problem falls into class of binary
optimization problems which are NP hard and therefore their precise
solution assumes exponential computational complexity. To make
solution more computationally tractable, the convex real valued
relaxation is proposed. In the proposed approach, the problem is
reduced to the convex optimization using semidefinite programming
(SDP) relaxation approach. Within this approach the linear
objective function is optimized over the cone of positive
semidefinite matrices. This approach is attractive because of its
generality, although it has never been applied to this specific
application before.
[0093] Many problems of operation research, control and systems
theory can be efficiently reduced to SDP. For convex optimization
there are efficient fast computational algorithms with polynomial
complexity which can be implemented in real time, see [1]. Most
algorithms are based on interior point barrier functions method.
There are robust packages, like SeDuMi, SDPT3 and SDPA, some of
which are available in Linux as libraries, and which can be used in
real time.
[0094] The quantity subject to optimization is known as positioning
dilution of precision (PDOP) and is defined as
tr(A.sup.TA).sup.-1 (11)
[0095] where A stands for the matrix of the linearized system, for
example (10). For different problems the matrix A can be chosen
different ways.
[0096] Given the matrix A of directional cosines, with M being the
total number of signals (19 in the example presented in FIGS. 1,
2), m the number of signals to be processed (9 in the example),
x.sub.i the variable indicating if the signal is chosen, X=diag
(x.sub.1, . . . , x.sub.M), the optimization problem looks
like:
tr(A.sup.TXA).sup.-1.fwdarw.min (12)
[0097] subject to constraints
i = 1 M x i .ltoreq. m , x i = 0 , 1 . ( 13 ) ##EQU00008##
Here diag (x.sub.1, . . . , x.sub.M) means a diagonal matrix with
variables x.sub.i taking values 0 or 1. The maximum number of
signals allowed in the optimal set is denoted by m.
[0098] The optimization problem (12), (13) is non-convex and
binary. Its precise solution assumes exhaustive search, which is
computationally hard. In the present invention, convex relaxation
is proposed, which solves the linear cost function under
semi-definite constraints as suggested below:
min S , X t r ( S ) ( 14 ) ##EQU00009##
[0099] subject to constraints
S = S T , ( 15 ) P ( S , X ) = ( A T X A I p I p S ) 0 , ( 16 ) i =
1 M x i .ltoreq. m , 0 .ltoreq. x i .ltoreq. 1 ( 17 )
##EQU00010##
[0100] where p is the number of navigation parameters. In one
embodiment it can be 5 (x, y, z, .DELTA.t's). The notation I.sub.p
stands for identity matrix, the symbol stands for positive
semidefiniteness of matrices.
[0101] The inequality (16) is known as linear matrix inequality
(LMI) and establish convex constraints on the matrix which is
linearly dependent on unknowns X, S. Thus, we reduced the optimal
choice problem to the convex optimization problem for which there
are very efficient numerical algorithms, see [1].
[0102] Possible algorithms include: [0103] Ellipsoid method, [0104]
Path-following method for -log det (.) barrier functions.
[0105] These and others algorithms are available in the art for
engineers.
[0106] After relaxed problem (14)-(17) is solved, the real valued
estimates of binary variables x.sub.i* are obtained which are
subject to further rounding to nearest 0 or 1.
[0107] The relaxed problem can be used for calculation of the lower
estimates together with Branch and Bound algorithm [4] for precise
solution. This idea was illustrated by practical computations.
[0108] The SDP relaxation algorithm can be implemented and used for
both batch processing and real time for L1 only and
L1/L2/L5/G1/G2/B1/B2 multi-constellation multi-band signal
processing.
[0109] A structure of the navigation receiver that can be used in
the invention is shown in FIG. 3. The antenna (or multiple
antennas) is connected to the receiver consisting among others of
the radio frequency (RF) part. RF signals are convolved with
reference signals generated by the numerically controlled
oscillator and digitized by A/D converter. Result of convolution is
used for multiple signal tracking. Multiple signals corresponding
to plurality of satellites and multiple frequency bands are used
for coordinate determination. The processor runs signal tracking
and coordinate determination algorithms.
[0110] Having thus described the different embodiments of a system
and method, it should be apparent to those skilled in the art that
certain advantages of the described method and apparatus have been
achieved. It should also be appreciated that various modifications,
adaptations, and alternative embodiments thereof may be made within
the scope and spirit of the present invention. The invention is
further defined by the following claims.
REFERENCES (ALL INCORPORATED HEREIN BY REFERENCE IN THEIR
ENTIRETY)
[0111] 1. Ben-Tal A., Nemirovski A., Lectures on Modern Convex
Optimization: Analysis, Algorithms, and Engineering Applications,
MOS-SIAM Series on Optimization, 2001. [0112] 2. A. Leick, L.
Rapoport, D. Tatarnikov, GPS Satellite Surveying, Wiley & Sons,
2015. [0113] 3. Ning Wang, Liuqing Yang, Semidefinite Programming
for GPS, // Proceedings of 24th International Technical Meeting of
the Satellite Division of The Institute of Navigation, Portland,
Oreg., Sep. 19-23, 2011. [0114] 4. Wolsey L.A., Nemhauser G. L.,
Integer and Combinatorial Optimization, John Wiley & Sons,
2014.
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