System and method for determining the position error of a satellite localization receiver

Abstract

A system and method are provided for determining a distribution of a position error of a receiver of localization signals, the signals being sent by at least one satellite. The system includes 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 positions of the satellite(s), known as second positions, a device for transmitting the second position of the first determination device to the receiver, and the first distribution is defined by at least one first cumulant, of higher-than-second order.

Claims

1. A system for determining a distribution of a position error of a receiver of localization signals, said signals being sent by at least one satellite, said system comprising: said receiver, having a first position and being affected by a first error, having a first distribution; a first determination device for determining at least one position of said at least one satellite, being a second position, a transmission device for transmitting said second position from said first determination device to said receiver, said first distribution being defined by at least one first cumulant which is of higher-than-second order, said first determination device being configured for determining at least one second cumulant which is of higher-than-second order, representing a second distribution of a second error on said second position, said transmission device being configured for transmitting said at least one second cumulant from said first determination device to said receiver, wherein said receiver includes: a second determination device for determining said at least one 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 at least one satellite, and a third determining device for determining said first distribution, from said at least one first cumulant.

2. The system of claim 1, wherein said second determination device is adapted for applying said model to said at least one second cumulant.

3. The system of claim 1, wherein said first determination device is furthermore adapted for determining cumulants of first to fifth order, and in which said second determination device is furthermore adapted for determining cumulants of first to fifth order.

4. The system of claim 1, wherein said third device is furthermore adapted for using an Edgeworth expansion.

5. A method for determining a distribution of a position error of a receiver of localization signals, said signals being sent by at least one satellite, said method including; a reception step comprising receiving, by a receiver, said satellite localization signals, a position of said receiver being a first position and being affected by a first error, having a first distribution, a first determination step comprising determining, by a first determination device, at least one position of said at least one satellite, being a second position, a transmission step comprising transmitting, by a transmission device, said second position from said first determination device to said receiver, said first distribution being defined by at least one first cumulant which is of higher-than-second order, wherein the first determination step also comprises determining at least one second cumulant which is of higher-than-second order, representing a second distribution of a second error on said second position, wherein said transmission step also comprises transmitting said at least one second cumulant, associated with said second position from said first determination device to said receiver, said method also including: a second determination step comprising determining, by a second determination device of said receiver, said at least one first cumulant, on the basis of said second position, of said second cumulant and of a model for determining said first position of said receiver from distances between said receiver and said at least one satellite, and a third determination step comprising determining, by a third determination device of said receiver, said first distribution, from said at least one first cumulant.

6. The method of claim 5, wherein said second determination step comprises applying said model to said at least one second cumulant.

7. The method of claim 5, wherein said first determination step comprises determining cumulants of first to fifth order, and in which said second determination step comprises determining cumulants of first to fifth order.

8. The method of claim 5, wherein said third determination step comprises using an Edgeworth expansion.

Description

BRIEF DESCRIPTION OF THE DRAWINGS

(1) 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:

(2) FIG. 1 shows a first embodiment of the system shown in this invention.

DETAILED DESCRIPTION

(3) FIG. 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 first distribution of the first error is modelled by at least one first set of cumulants of higher-than-second order.

(4) In order to perform this determination, the system allows the transmission (via a transmission device 103) of second cumulants that 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.

(5) The first cumulants are determined by the receiver using a second determination device 105.

(6) Finally a third device 106 enables the determination of the first distribution, from the first cumulants.

(7) 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.

(8) The cumulants of a random variable X distributed according to a probability density f (note that X˜f) are determined by introducing the function φ(t)=e.sup.iXt.sub.X. e represents the exponential function .sub.X represents the mean on the values of X i being the imaginary unit (i.sup.2=−1).

(9) 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 n.sup.th-order moments of f: μ.sub.n=X.sup.n.

(10) $\phi \left(t\right)=\underset{n=0}{\overset{\infty }{.Math.}}\frac{{\left(\mathrm{it}\right)}^n}{n!}{\mu }_n$

(11) It is also possible to carry out the expansion of the function ln(e.sup.iXt.sub.X), in which case a set of coefficients κ.sub.n is obtained, which are defined in the following manner:

(12) $\ln \left({.Math.e^{i\mathrm{Xt}}.Math.}_X\right)=\underset{n=0}{\overset{\infty }{.Math.}}\frac{{\left(\mathrm{it}\right)}^n}{n!}{\kappa }_n$

(13) Each κ.sub.n thus defined is the n.sup.th-order cumulant of the distribution f. The two first cumulants are the mean and the variance of the distribution.

(14) In addition, if X and Y are two random variables distributed according to f and g respectively, and whose n.sup.th-order cumulants are κ.sub.n[f] and κ.sub.n[g] respectively, then the n.sup.th-order cumulants of the distribution h associated with the random variable Z=pX+qY, are given by:
κ.sub.n[h]=p.sup.nκ.sub.n[f]+q.sup.nκ.sub.n[g]

(15) In addition it is known that any distribution that results from the combination of m random variables can be represented by an expansion, known as the Edgeworth expansion and having the following form:

(16) $F_n\left(x\right)=\left[1+\underset{j=1}{\overset{\infty }{.Math.}}\frac{1}{n^{j/2}}{P}_j\left({\kappa }_1,\mathrm{.Math.},{\kappa }_j,x\right)\right]\Psi \left(x\right)$

(17) In this equation the variables are as follows: Ψ(x) is a reference function according to choice (Gaussian for example) κ.sub.p is the p.sup.th-order cumulant of the distribution of the orbit and/or clock errors P.sub.j is a polynomial of order 3j in x, which involves the j first κ.sub.ps in its coefficients, and the expression of which depends on the choice of Ψ(x). n represents the number of variables combined to obtain x
In addition, it is known that this expansion converges as n approaches ∞.

(18) Based on the mathematical concepts above, the invention 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 transmission is performed for each of the N.sub.S 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 ρ.sub.j made between its antenna and the N.sub.S signal sources used for the positioning the receiver determines the m first cumulants (κ.sub.n) of the first distribution of the error associated with its position, from the transmitted cumulants κ.sub.i,j, using the following relationship:

(19) $K_{n,p}=\underset{j=1}{\overset{N_S}{.Math.}}{\left(M_{p,j}\right)}^n{\kappa }_{n,j}$
With n=1, . . . , m representing the order of the cumulant, j the satellite, M.sub.p,j 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.

(20) In one embodiment, it is possible to use the least squares method to determine the matrix M.sub.p,j. In this embodiment the vector of the distances between the receiver and the satellites is modelled as follows:
ρ=ĜX+ε

(21) In this equation ρ=[ρ.sub.1 . . . ρ.sub.N.sub.S] is the vector of the distances between the receiver and the satellites, ε=[ε.sub.1 . . . ε.sub.N.sub.S] is the vector of the errors of the distances between the receiver and the satellites and X=[x, y, z, Δtusr] is the vector of the second position and of the clock shift of the receiver and

(22) $G=\left[\begin{array}{cccc}{e}_x{e}_1& e_y{e}_1& e_z{e}_1& c\\ \mathrm{.Math.}& \mathrm{.Math.}& \mathrm{.Math.}& \mathrm{.Math.}\\ e_x{e}_{N_S}& e_y{e}_{N_S}& e_z{e}_{N_S}& c\end{array}\right]$
is the observation matrix of the problem. In the matrix Ĝ, c represents the speed of light and e.sub.xe.sub.j is the cosine of the angle between the vector in the direction x and the vector towards the satellite j.

(23) 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.sub.est=(Ĝ.sup.tĈ.sup.−1 Ĝ).sup.−1 Ĝ.sup.tĈ.sup.−1ρ.

(24) In this relationship Ĉ=<ε ε.sup.t> is the error correlation matrix. In this embodiment it can then be determined that M=(Ĝ.sup.tĈ.sup.−1 Ĝ).sup.−1 Ĝ.sup.tĈ.sup.−1. Using the Edgeworth expansion truncated to the m.sup.th order, which involves the first cumulants K.sub.n, 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.

(25) ${\int }_{-\infty }^{-R_p}F\left(x\right)dx+{\int }_{R_p}^{\infty }F\left(x\right)dx=P_{\mathrm{HMI}}$
This determination must be carried out for each direction in space (vertical, horizontal). P.sub.HMI represents the tolerated probability of non-integrity, in order to ensure that these R.sub.ps are smaller than the dimensions of the tolerance region (for example the alert radii used in civil navigation).

(26) It is also possible to directly find the risk of being found outside the requisite tolerance region (R.sub.a), and for this the following equation can be used:

(27) ${\int }_{-\infty }^{-R_a}F\left(x\right)dx+{\int }_{R_a}^{\infty }F\left(x\right)dx<P_{\mathrm{HMI}}$

(28) The system as presented in this invention necessitates certain pre-requisites before being used. In particular, it is necessary that:

(29) the order of the expansion used to determine the first distribution must be known in advance by the receiver and the satellite(s)

(30) 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 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.

(31) finally, it is necessary that the reference function Ψ(x) be also known in advance by the satellite and the receiver.

(32) 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, and/or with the error on the time of passage of the signal of each satellite through the ionospheric layer.

(33) 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 position/synchronization/ionospheric delay computations.

(34) 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 requirements following a change in the state of the system.

(35) The distribution of the first errors is produced using the broadcast cumulants and modelling the reference function Ψ as a Gaussian centred on the first first-order cumulant and of the width of the first second-order cumulant.

(36) 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.

(37) The first device 104 for determining the positions of the satellite(s) can be located on the ground or in one of the satellites.

(38) 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 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.

(39) 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 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-Read/Write (CD-R/W) and DVD).