System and method for determining the position error of a satellite
localization receiver
5 The invention concerns the field of satellite positioning. In
particular it concerns the determination of a positioning error that a receiver
produces when using a satellite localization system. These systems are for
I example implemented in systems for improving the accuracy of the
localization supplied by the GPS (Global Positioning System). These systems
10 are also referred to as localization augmentation systems. They are also
known by the acronym SBAS for Satellite-Based Augmentation Systems. The
system EGNOS, an acronym for European Geostationary Navigation Overlay
Service, is another known GPS augmentation system.
15 Knowing this positioning error of the receiver makes it possible to
determine a volume for which the probability of presence of the receiver is
above a threshold set by a standard. Knowing this volume thus makes it
possible, for example, to determine the minimum distance at which two
aircraft must not come closer to each other. Knowing this information is in
20 particular necessary for integrity services. The term integrity service refers to
the capability of a system to supply an alert to the pilot when the navigation
system can no longer be used with the requisite performance in terms of user
risk.
25 These error determination systems may be used in aircraft but
also in ground vehicles or ships, for example.
Systems are known in the prior art in which information
representing the positions of the satellites, and times of passage through the
30 ionosphere as well as information representing the error on these positions
and these times of passage are sent to the various receivers. Knowing this
information makes it possible to determine the position error of the receiver,
also referred to as the integrity of the position of the object located by the
receiver.
35
It is known in the prior art that the items of information
representing errors are marginal standard deviations (ai)of the distributions
of the errors committed. Thus the error distribution is modelled by a centred
Gaussian law of the form N(0,ai2).H owever, the modelling of the error
5 distribution in the form of a centred Gaussian law is too coarse, and leads to
the need for a safety margin to be applied, which is in some cases too large.
This is particularly the case when the combination of the marginal standard
deviations (ai)o,f the position error of each satellite used, must be small to
allow sufficient accuracy for the performance of the manoeuvres to be
10 executed. In such cases, the margin, chosen to cover the lack of fit of a
model based on centred Gaussians, too frequently makes the service
unavailable. It has also been possible to demonstrate that the use of centred
Gaussians is only correct from a mathematical point of view if one can posit
the hypothesis that the errors are distributed in a unimodal and symmetrical
15 way, which is in no way guaranteed as a general rule.
The present invention aims to remedy these problems by providing
a system for determining the position error of a receiver whose accuracy is
increased in relation to the accuracy of systems based on the exchange of
20 marginal standard deviations of the distribution of the error on the position of
the satellites.
According to an aspect of the invention, a system is suggested for
determining a distribution of a position error of a receiver of localization
25 signals, the signals being sent by at least one satellite. The system includes;
0 the receiver, one position of which is known as first position and is
affected by an error, known as first error, having a distribution, known as
first distribution;
a first device for determining at least one position of the or each or at
30 least one satellite, known as second position,
a device for transmitting the or each or at least one second position of
the first determination device to the receiver.
The system is characterized in that
0 the first distribution is defined by at least one first cumulant, of higher-
35 than-second order,
the first device is furthermore adapted for determining at least one
second cumulant, of higher-than-second order, representing a second
distribution of a second error on the second position,
0 the transmission device is furthermore adapted for transmitting the or
5 each or at least one second cumulant of the first determination device to
the receiver,
0 and in that the receiver includes:
o a second device for determining the or each or at least one first
cumulant, on the basis of the second position, the second cumulant and
10 a model for determining the first position of the receiver from distances
between the receiver and the satellite(s) and
o a third device for determining the first distribution, from said first
cumulant.
15 The first and second cumulants of higher-than-second order can
also be referred to by the expression "series of cumulants" or "set of
cumulants".
The cumulants K, of the random variable X are defined by the
20 cumulant generating function g ( t ) :
m
(it)"
g(t) = ln((eiXt),) = 1 -- K" n!
n=O
In this equation ( ), represents the mathematical expectation of
the random variable X and K, the nth order cumulant.
25
The system therefore allows the transmission to the receivers of
satellite signals of an item of information representing the distribution of the
position error of the satellites. In addition, the use of this information makes it
possible to determine the positioning error that the receiver produces by
30 using the satellite localization system.
In addition, the use of the cumulants makes it possible to transmit
to the users an item of information representing the distribution of the position
error of the satellite that is more reliable than in the case of the systems
known in the prior art. The use of these cumulants then obviates the need for
applying so many margins in the position, for two reasons:
5 - It makes it possible to describe the distributions of
positioning/synchronization error of the positioning signal sources in more
detail, and due to this fact makes it possible to improve the accuracy of the
localization information computed by the receiver. The region of tolerance to
the integrity risk can be smaller than in current solutions.
10
- It makes it possible to avoid having to form a hypothesis with a
false mathematical basis, and which consists in modelling the errors
committed for the position of each satellite by a combination of centred
Gaussians N(0,0i2), whereas the distribution is not unimodal or symmetrical.
15
In other words, the form of the distribution allowed by the invention
is much more accurate than a centred Gaussian distribution. However, this
generic form can be described by a few parameters only, the cumulants. It is
therefore possible to transmit much more detailed information on the
20 probability distribution of the second positioning errors of the satellites.
This accurate information then makes it possible to model the first
position errors of the receiver in as reliable a manner as possible. Thus, it is
not necessary to apply large safety margins to ensure safety, and it is
25 possible to use the localization system to perform manoeuvres in which the
demand for accuracy is high.
According to a technical feature the second determination device
is adapted for applying the model to the second cumulant(s).
30
According to a technical feature the first determination device is
furthermore adapted for determining the second cumulants of first to fifth
order, and in which the second determination device is furthermore adapted
for determining the first cumulants of first to fifth order.
35
The larger the number of cumulants the more accurate the
modelling will be, but a larger number of cumulants are more complicated to
measure experimentally.
5 According to a technical feature the third device is furthermore
adapted for using an Edgeworth expansion.
The invention also concerns a method for determining a distribution of a
position error of a receiver of localization signals, the signals being sent by at
10 least one satellite. The method includes;
0 a step of reception. by a receiver, of the satellite localization signals, one
position of the receiver being known as first position and is affected by an
error, known as first error, having a distribution, known as first
distribution,
15 0 a first step of determination, by a first determination device, of at least
one position of the or each or at least one satellite, known as second
positions,
a step of transmission, by a transmission device, of the or each or at
least one second position of the first determination device to the receiver.
20 The method is characterized in that
the first distribution is defined by at least one first cumulant, of higherthan-
second order,
the first determination step is furthermore adapted for determining at
least one second cumulant, of higher-than-second order, representing a
25 second distribution of a second error on the second position,
the transmission step is furthermore adapted for transmitting the or each
or at least one second cumulant, associated with said second position of
the first determination device to the receiver,
in addition the method includes:
30 o a second step of determination, by a second determination device of
said receiver, of the or each or at least one first cumulant, on the basis of
the second position, the second cumulant and a model for determining
the first position of the receiver from distances between the receiver and
the satellite(s) and
o a third step of determination, by a third determination device of the
receiver, of the first distribution, from the first cumulant.
According to a technical feature the second determination step is adapted for
5 applying the model to the second cumulant(s).
According to a technical feature the first determination step is furthermore
adapted for determining the second cumulants of first to fifth order, and in
which the second determination step is furthermore adapted for determining
10 the first cumulants of first to fifth order.
According to a technical feature the third determination step is furthermore
adapted for using an Edgeworth expansion.
15 The invention will be better understood, and other advantages will
become apparent, upon reading the detailed description, given by way of
non-limiting example. This detailed description is made using the following
figures:
Figure 1 shows a first embodiment of the system shown in this
20 invention.
Figure 1 shows the system including a satellite 101 and a receiver
of satellite signals 102. The system allows the receiver to determine the
distribution of a first error associated with a first position of the receiver. This
25 first distribution of the first error is modelled by at least one first set of
cumulants of higher-than-second order.
In order to perform this determination, the system allows the
transmission (via a transmission device 103) of second cumulants that
30 represent a second distribution representing a second error associated with
the second position of a satellite. The determination of these elements is
performed by a first determination device 104.
The first cumulants are determined by the receiver using a second
35 determination device 105.
Finally a third device 106 enables the determination of the first
distribution, from the first cumulants.
5 This modelling is based on the use of the Edgeworth expansion of
the probability density of the error associated with the position of a satellite.
The cumulants of a random variable X distributed according to a
probability density f (note that X- f) are determined by introducing the
10 function ~ ( t=) ( eixt)x.
e e represents the exponential function
m ( ), represents the mean on the values of X
i being the imaginary unit (iZ = -1).
15
It will be noted that the expansion of this function, as a function of
the powers of the exponent, is a series that involves the nth-order moments of
f: kn = (X").
m
(it)"
~ ( t=) C --ilnn!
It is also possible to carry out the expansion of the function
1n((eixt),), in which case a set of coefficients K, is obtained, which are
defined in the following manner
m
(it)"
~n((e~~'),) = 2 - s,
n!
Each K, thus defined is the nth-order cumulant of the distribution f.
The two first cumulants are the mean and the variance of the distribution
In addition, if X and Y are two random variables distributed
30 according to f and g respectively, and whose nth-order cumulants are ~ , [ f ]
and ~ ~ [ rges]pe ctively, then the nth-order cumulants of the distribution 5
associated with the random variable Z = pX + qY, are given by:
Kn[hI = pnKn[fI + qnKn[gI
In addition it is known that any distribution that results from the
5 combination of m random variables can be represented by an expansion,
known as the Edgeworth expansion and having the following form:
In this equation the variables are as follows:
- Y(x) is a reference function according to choice (Gaussian for
10 example)
- K, is the p'h-order cumulant of the distribution of the orbit and/or
clock errors
- P, is a polynomial of order 3 j in x, which involves the j first K,S in
its coefficients, and the expression of which depends on the choice of Y(x).
15 - n represents the number of variables combined to obtain x
In addition, it is known that this expansion converges as n approaches m
Based on the mathematical concepts above, the invention
20 provides the determination of the first position error of the receiver as follows:
a transmission device 103 supplies the information on the distribution
of the position and synchronization errors of the satellites in the form
of cumulants of higher-than-second order of this distribution. This
25 transmission is performed for each of the Ns positioning signal sources
(for example satellites emitting a signal observing the GPS standard)
the receiver determines its position and a reference time using a linear
combination of measurements of (pseudo-)distances pj made
between its antenna and the Ns signal sources used for the positioning
the receiver determines the m first cumulants (K,) of the first
distribution of the error associated with its position, from the
transmitted cumulants rcijj, using the following relationship:
5 With n = l;..,m representing the order of the cumulant, j the
satellite, M,,, the coefficient p,j of the matrix that makes it possible to
determine the second position of the receiver on the basis of the distances
between the receiver and the satellites, p represents the direction (x, y or z)
for which the cumulant is determined.
10 In one embodiment, it is possible to use the least squares method
to determine the matrix Mp,j. In this embodiment the vector of the distances
between the receiver and the satellites is modelled as follows:
p= e x +
In this equation p = [PI ... P N ~i]s the vector of the distances
between the receiver and the satellites, E = [ E l ... EN^] is the vector of the
15 errors of the distances between the receiver and the satellites and X =
[x,y,z,Atusr] is the vector of the second position and of the clock shift of the
receiver and
exel eyel
is the observation matrix of the problem. In the matrix &, c represents the
20 speed of light and exej is the cosine of the angle between the vector in the
direction x and the vector towards the satellite j.
Using the least squares method, the relationship between the second
position of the receiver and the distances between the receiver and the
satellites can be written X,,, = ( e t f - l e)-l G t t - l P.
25 In this relationship f =< E E >~is the error correlation matrix. In this
embod.im ent it can then be determined that M = ( C t t - l e1-l etf-I. Using the Edgeworth expansion truncated to the mth order,
which involves the first cumulants K,, the first distribution F that
approaches the distribution of the first error associated with the
positioning of the receiver is determined.
Finally, this distribution of the first error can be used to determine the
dimension of a trust region (i.e. a region in which the probability of finding the
receiver is greater than or equal to a determined threshold). It is possible to
find this trust region by solving the following equation.
- R ~ m / F(x)dx + / F(x)dx = PnMI
-m RP
5 This determination must be carried out for each direction in space (vertical,
horizontal). P, represents the tolerated probability of non-integrity, in order
to ensure that these R,s are smaller than the dimensions of the tolerance
region (for example the alert radii used in civil navigation).
10 It is also possible to directly find the risk of being found outside the requisite
tolerance region (R,), and for this the following equation can be used:
The system as presented in this invention necessitates certain
I 15 pre-requisites before being used. In particular, it is necessary that:
I
! the order of the expansion used to determine the first distribution
must be known in advance by the receiver and the satellite(s)
* the computation of the second cumulants must be done in such a
way that the level of trust of their estimate is consistent with the probability of
20 non-integrity demand required for the overall system. It is also necessary that
the resulting approximation remain conservative, i.e. that one is sure that the
cumulants have not been undervalued.
finally, it is necessary that the reference function Y ( x ) be also
known in advance by the satellite and the receiver.
25
In another embodiment of the system the latter uses the
knowledge of the cumulants up to the fourth or fifth order, associated with the
position error of each satellite, andlor with the error on the time of passage of
the signal of each satellite through the ionospheric layer.
This computation of the error distribution is based on a
combination of the statistical calibrations that are carried out over a long
period and on contributions arriving within a short period. The latter are more
reactive and are based for example on the observation of the
5 position/synchronization/ionospheric delay computations.
The broadcasting of the cumulants to the receivers is performed
with an alert device and/or by re-updating the values of the cumulants in the
case where they have turned out to be poorly fitted to the integrity
10 requirements following a change in the state of the system.
The distribution of the first errors is produced using the broadcast
cumulants and modelling the reference function Y as a Gaussian centred on
the first first-order cumulant and of the width of the first second-order
cumulant.
15 Next it is possible to evaluate the availability of the service and
therefore the region in which probability of the receiver being present
exceeds a threshold using the preceding equations.
The first device 104 for determining the positions of the satellite(s)
20 can be located on the ground or in one of the satellites.
The various determination devices described in this invention can
be computers or processors programmed in such a way as to produce the
various operations performed by the devices. It is also possible to use
25 dedicated components, programmable logic circuits, programmable logic
networks (also known by the acronym FPGA for Field-Programmable Gate
Array) or integrated circuits specific to one application (also known by the
acronym ASIC for Application-Specific Integrated Circuit) programmed in
such a way as to produce the various operations performed by the devices.
30
The present invention can also be implemented from hardware
and software elements. It can be available as a computer program product
on a computer-readable medium. The medium can be electronic, magnetic,
optical, electro-magnetic or be an infra-red type broadcasting medium. Such
35 media are, for example, semi-conductor memories (Random Access Memory
RAM, Read-Only Memory ROM), tapes, diskettes or magnetic or optical
disks (Compact Disk - Read Only Memory (CD-ROM), Compact Disk -
ReadNVrite (CD-RNV) and DVD).
CLAIMS
1. System for determining a distribution of a position error of a receiver (1 02)
of localization signals, said signals being sent by at least one satellite, said
system including;
said receiver (102), a position of which is called first position and is
affected by an error, called first error, having a distribution, called first
distribution;
a first determination device (104) for determining at least one position of
said or at least one said or each said satellite, known as second position,
a transmission device (1 03) for transmitting said or said at least one said
or each said second position from said first determination device to said
receiver,
said system being characterized in that
e said first distribution is defined by at least one first cumulant, of higherthan-
second order,
said first device is furthermore adapted for determining at least one
second cumulant, of higher-than-second order, representing a second
distribution of a second error on said second position,
said transmission device (103) is furthermore adapted for transmitting
said or at least one said or each said second cumulant from said first
determination device to said receiver,
e and in that said receiver includes:
o a second determination device (105) for determining said or at least
one said or each said first cumulant, on the basis of said second position,
of said second cumulant and of a model for determining said first position
of the receiver from distances between said receiver and said satellite or
satellites, and
o a third determining device (106) for determining said first distribution,
from said or at least one said or each said first cumulant.
2. System according to Claim 1, in which said second determination device
(105) is adapted for applying said model to said second cumulant or
35 cumulants.
3. System according to Claim 1 or 2, in which said first determination device
is furthermore adapted for determining said second cumulants of first to fifth
order, and in which said second determination device is furthermore adapted
11 for determining said first cumulants of first to fifth order.
'I
: I 5
:I 4. System according to one of Claims 1 to 3, in which said third device is
furthermore adapted for using an Edgeworth expansion.
5. Method for determining a distribution of a position error of a receiver (102)
10 of localization signals, said signals being sent by at least one satellite, said
method including;
a a reception step comprising receiving, by a receiver (102), said satellite
localization signals, a position of said receiver being called first position
and being affected by an error, called first error, having a distribution,
15 called first distribution,
a a first determination step comprising determining, by a first determination
device (104), at least one position of said or at least one said or each
said satellite, called second position,
a a transmission step comprising transmitting, by a transmission device
20 (103), said or at least one said or each said second position from said
first determination device to said receiver,
said method being characterized in that
a said first distribution is defined by at least one first cumulant, of higherthan-
second order,
25 e the first determination step also comprises determining at least one
second cumulant, of higher-than-second order, representing a second
distribution of a second error on said second position,
said transmission step also comprises transmitting said or at least one
said or each said second cumulant, associated with said second position
30 from said first determination device to said receiver,
said method also including:
o a second determination step comprising determining, by a second
determination device of said receiver, said or at least one said or each
said first cumulant, on the basis of said second position, of said second
35 cumulant and of a model for determining said first position of said
receiver from distances between said receiver and said satellite or
satellites and -
o a third determination step comprising determin~ng, by a third
determination device of said receiver, said first distribution, from said or
5 at least one said or each said first cumulant.
6. Method according to Claim 5, in which said secoind determination step
comprises applying said model to said second cumulant or cumulants.
7. Method according to Claim 5 or 6, in which said first determination step
comprises determining said second cumulants of first to fifth order, and in
which said second determination step comprises determining said first
cumulants of first to fifth order.
8. Method according to one of Claims 5 to 7, in which said third determination
1 step comprises using an Edgeworth expansion.