CN104197954B - Method for estimating precision of drop points of inertial navigation system in three-dimensional space - Google Patents

Abstract

The invention discloses a method for estimating the precision of drop points of an inertial navigation system in the three-dimensional space. According to the method, after standard deviations of a longitudinal deviation, a lateral deviation and a height deviation of the inertial navigation system are known, a space rectangular coordinate system OXYZ is established, an ellipsoid can be obtained by virtue of the standard deviations, and 50% of drop points of the inertial navigation system in the three-dimensional space are within the range of the ellipsoid; compared with a traditional sphere error probability (SEP) description method, the method has the advantages that an expression formula is accurate, and the characteristics of the drop points are accurately described.

Description

A kind of inertial navigation system three dimensions impact accuracy method of estimation

Technical field

The present invention relates to a kind of impact accuracy method of estimation, belong to inertial navigation system Accuracy extimate field.

Background technology

In inertial navigation system, the important indicator evaluating its three-dimensional impact accuracy is based on three-dimensional ball probability by mistake Difference sep (sphere error probablity).This Accuracy Measure method is to provide one for ballistic missile point of impact positional precision Plant simple tolerance, sep is defined as comprising the radius of 50% ball of guided missile warhead point of impact around realistic objective.

The theoretical calculation method of sep is as follows:

Define coordinate system oxyz, point centered on wherein o point in three dimensions, be located at point in this coordinate system coordinate (x, Y, z) obey three-dimensional normal distribution and separate, note

$\mathrm{\δp}=\left[\begin{array}{c}x\\ y\\ z\end{array}\right]$

Its average is

$m=\left[\begin{array}{c}{m}_x\\ m_y\\ m_z\end{array}\right]$

Its standard error is

$\mathrm{\σ}=\left[\begin{array}{c}{\mathrm{\σ}}_x\\ {\mathrm{\σ}}_y\\ {\mathrm{\σ}}_z\end{array}\right]$

This point is in initial point as the center of circle, the probability in the ball as r for the radius by error component Gauss distribution triple integral Be given.For x, y and z of Gauss distribution, zero-mean, its probability is

$p(x^2+y^2+z^2<r^2)=\frac{1}{{\left(2\mathrm{\&pi;}\right)}^{\frac{3}{2}}{\mathrm{\&sigma;}}_x{\mathrm{\&sigma;}}_y{\mathrm{\&sigma;}}_z}\&centerdot;{\&integral;}_{-r}^r{\&integral;}_{-\sqrt{r^2-x^2}}^{\sqrt{r^2-x^2}}{\&integral;}_{-\sqrt{r^2-x^2-y^2}}^{\sqrt{r^2-x^2-y^2}}\exp [-\frac{1}{2}(\frac{x^2}{{\mathrm{\&sigma;}}_x^2}+\frac{y^2}{{\mathrm{\&sigma;}}_y^2}+\frac{z^2}{{\mathrm{\&sigma;}}_z^2})]\mathrm{dzdydx}---\left(1\right)$

Above formula is the function of variable r, i.e. this point is in the size of the probability in ball, relevant with the size of radius r.If With polar coordinate, triple integral is just calculated in a suitable form.Polar parameter is made to be θ and r.Then

Using triple integral polar coordinate transform formula

Probability integral becomes

Using above formula, can in known probability p its corresponding r value of approximate calculation, but solution procedure is very loaded down with trivial details.

If σ_x=10m, σ_y=30m, σ_z=40m, then p (x²+y²+z²<r²) as shown in Figure 1 with the Changing Pattern of r.From figure In can obtain, in r=42.32, p (x²+y²+z²<r²)=0.5.

By above standard deviation simulation practice shooting 1000 times distribution as shown in Figure 2, in figure ball be sep ball.

From accompanying drawing 2 as can be seen that point of impact is substantially both less than the radius of this ball in a direction, but have in another two direction A large amount of discrete points are all in the outside of this ball.In addition, only the size according to sep value not can determine that the closeness in which direction is bigger. Therefore, sep such as is suitable at precision analysis during standard error, lacks directivity.Meanwhile, in the product of probability asking formula (1) and formula (3) Timesharing, does not have accurate expression formula, leads to gained sep value for an approximation, rather than precise results.

For navigation drop point, actual falling point the plane of riee the distance between subpoint and impact point referred to as longitudinally by mistake Difference；Distance between drop point and the plane of riee is referred to as lateral error；Deviation perpendicular to horizontal plane is referred to as height tolerance.

Content of the invention

The technology solve problem of the present invention is: overcomes the deficiencies in the prior art, provides a kind of inertial navigation system three-dimensional space Between impact accuracy method of estimation, be both given inertial navigation three-dimensional impact accuracy estimate relatively precisely expression formula, can sentence simultaneously Break the distribution character point of impact in longitudinal bias direction, lateral deviation direction and height tolerance direction.

The technical solution of the present invention: a kind of inertial navigation system three dimensions impact accuracy method of estimation, step is such as Under:

(1) the n positional information that after the n navigation of collection inertial navigation system, carrier stops, n position composition position Group, n is at least 6；

(2) the n positional information being obtained according to step (1), obtains the coordinate at set of locations center, then measures each position With respect to the longitudinal bias at set of locations center, lateral deviation and height tolerance, it is calculated all longitudinal bias on this basis Standard deviation sigma₁, the standard deviation sigma of lateral deviation₂Standard deviation sigma with height tolerance₃；

(3) with set of locations center for initial point o, set up rectangular coordinate system in space oxyz, so that the direction of longitudinal bias is fallen in x-axis Or on one of axle of y-axis, the direction of height tolerance falls in z-axis；

(4) standard deviation sigma of the longitudinal bias being obtained according to step (2)₁, the standard deviation sigma of lateral deviation₂With height tolerance Standard deviation sigma₃, setting up an ellipsoid centered on initial point o in the coordinate system of step (3), make this ellipsoid cover n/2 carrier Stop position；

(5) this inertial navigation system three-dimensional impact accuracy esep=(a, b, c) is obtained according to the ellipsoid that step (4) is set up, Wherein a, b, c are respectively the ellipsoid set up of step (4) the half of longitudinal bias direction, lateral deviation direction and height tolerance direction Shaft length, thus complete the estimation of inertial navigation system three-dimensional impact accuracy.

The ellipsoid set up in described step (4) meets following form:

Ellipsoid meets equation in the half shaft length a in longitudinal bias direction

$\frac{1}{{\left(2\mathrm{\&pi;}\right)}^{3/2}{{\mathrm{\&sigma;}}_1}^3}{\&integral;}_0^a{r}^2\text{exp}(-\frac{1}{{{2\mathrm{\&sigma;}}_1}^2}{r}^2){\&integral;}_0^{2\mathrm{\&pi;}}{\&integral;}_0^{\mathrm{\&pi;}}\sin \mathrm{\&beta;d\&theta;d\&beta;dr}=\frac{1}{2}$

Ellipsoid meets equation in the half shaft length b in lateral deviation direction

$\frac{1}{{\left(2\mathrm{\&pi;}\right)}^{3/2}{{\mathrm{\&sigma;}}_2}^3}{\&integral;}_0^b{r}^2\text{exp}(-\frac{1}{{{2\mathrm{\&sigma;}}_2}^2}{r}^2){\&integral;}_0^{2\mathrm{\&pi;}}{\&integral;}_0^{\mathrm{\&pi;}}\sin \mathrm{\&beta;d\&theta;d\&beta;dr}=\frac{1}{2}$

Ellipsoid meets equation in the half shaft length c in height tolerance direction

$\frac{1}{{\left(2\mathrm{\&pi;}\right)}^{3/2}{{\mathrm{\&sigma;}}_3}^3}{\&integral;}_0^c{r}^2\text{exp}(-\frac{1}{{{2\mathrm{\&sigma;}}_3}^2}{r}^2){\&integral;}_0^{2\mathrm{\&pi;}}{\&integral;}_0^{\mathrm{\&pi;}}\sin \mathrm{\&beta;d\&theta;d\&beta;dr}=\frac{1}{2}$

WhereinRepresent that the n position that carrier stops falls vertical Probability in in bias direction [- a ,+a]；Represent that carrier stops The probability that falls on lateral deviation direction [- b ,+b] of n position； Represent the probability that the n position that carrier stops falls on height tolerance direction [- c ,+c]；

Obtain after solution

A=1.539 σ₁

B=1.539 σ₂

C=1.539 σ₃

Wherein θ, β and r are respectively polar parameter.

Present invention advantage compared with the conventional method is:

(1) because calculating standard error σ in three directions in the present invention respectively₁、σ₂、σ₃, and calculate the three of ellipsoid with it Individual axial length, can directly obtain the three-dimensional impact accuracy of inertial navigation system, and result has precise forms, non-approximated result；

(2) because employing three parameters to describe the distribution character of drop point, that is, using esep=(1.539 σ₁,1.539 σ₂,1.539σ₃) as index, can more accurately understand inertial navigation result distribution character spatially, it is navigation error analysis More accurately foundation is provided.

(3) this method calculates simplicity, fast it is not necessary to carry out troublesome calculation can obtain inertial navigation system performance.

Brief description

Fig. 1 is ball probability integral schematic diagram；

Fig. 2 is that simulation is practiced shooting the distribution of 1000 times and sep schematic diagram；

Fig. 3 is the inventive method flow chart；

Fig. 4 is that simulation is practiced shooting the distribution of 1000 times and ellipsoid schematic diagram.

Specific embodiment

A kind of inertial navigation system three dimensions impact accuracy method of estimation proposed by the present invention, flow chart as shown in figure 3, Specifically comprise the following steps that

(1) the n positional information that after using gps to gather n navigation of inertial navigation system, carrier stops, n position letter Breath constitutes position group information, and n is at least 6；

(2) the n positional information being obtained according to step (1), obtains the coordinate at set of locations center, then measures each position With respect to the longitudinal bias at set of locations center, lateral deviation and height tolerance, on this basis by statistical computation obtain n vertical Standard deviation sigma to deviation₁, the standard deviation sigma of n lateral deviation₂Standard deviation sigma with n height tolerance₃.

For example, test through 10 times, collected with longitude, latitude, 10 final positions highly representing using gps Point, by being averaged to three position coordinateses respectively, it is possible to obtain the longitude at set of locations center, latitude, height coordinate, Ran Houtong Cross longitudinal error, lateral error and the height error asking for measurement point and set of locations center, obtain n longitudinal error, n horizontal Error and n height error, can obtain corresponding longitudinal bias standard deviation, lateral deviation standard deviation and height based on this three class value Degree deviation standard is poor.

(3) with set of locations center for initial point o, set up rectangular coordinate system in space oxyz, so that the direction of longitudinal bias is fallen in x-axis Or on one of axle of y-axis, the direction of height tolerance falls in z-axis；

(4) standard deviation sigma being obtained according to step (2)₁、σ₂And σ₃, being built centered on initial point o in the coordinate system of step (3) A vertical ellipsoid, makes this ellipsoid cover n/2 carrier stop position；

Wherein, ellipsoid meets equation in the half shaft length a in longitudinal bias direction

$\frac{1}{{\left(2\mathrm{\&pi;}\right)}^{3/2}{{\mathrm{\&sigma;}}_1}^3}{\&integral;}_0^a{r}^2\text{exp}(-\frac{1}{{{2\mathrm{\&sigma;}}_1}^2}{r}^2){\&integral;}_0^{2\mathrm{\&pi;}}{\&integral;}_0^{\mathrm{\&pi;}}\sin \mathrm{\&beta;d\&theta;d\&beta;dr}=\frac{1}{2}$

Ellipsoid meets equation in the half shaft length b in lateral deviation direction

$\frac{1}{{\left(2\mathrm{\&pi;}\right)}^{3/2}{{\mathrm{\&sigma;}}_2}^3}{\&integral;}_0^b{r}^2\text{exp}(-\frac{1}{{{2\mathrm{\&sigma;}}_2}^2}{r}^2){\&integral;}_0^{2\mathrm{\&pi;}}{\&integral;}_0^{\mathrm{\&pi;}}\sin \mathrm{\&beta;d\&theta;d\&beta;dr}=\frac{1}{2}$

Ellipsoid meets equation in the half shaft length c in height tolerance direction

$\frac{1}{{\left(2\mathrm{\&pi;}\right)}^{3/2}{{\mathrm{\&sigma;}}_3}^3}{\&integral;}_0^c{r}^2\text{exp}(-\frac{1}{{{2\mathrm{\&sigma;}}_3}^2}{r}^2){\&integral;}_0^{2\mathrm{\&pi;}}{\&integral;}_0^{\mathrm{\&pi;}}\sin \mathrm{\&beta;d\&theta;d\&beta;dr}=\frac{1}{2}$

WhereinRepresent that the n position that carrier stops falls in longitudinal direction Probability in [- a ,+a] in bias direction；Represent that n carrier stops The probability that falls on lateral deviation direction [- b ,+b] of n position； Represent the probability that the n position that carrier stops falls on height tolerance direction [- c ,+c].

Obtain after solution

A=1.539 σ₁

B=1.539 σ₂

C=1.539 σ₃

Wherein θ, β and r are respectively polar parameter.

(5) this inertial navigation system three-dimensional impact accuracy esep=(a, b, c) is obtained according to the ellipsoid that step (4) is set up, Wherein a, b, c be respectively the ellipsoid set up of step (4) longitudinally, laterally, the half shaft length in height tolerance direction.

The inventive method can be applied in the three dimensions impact accuracy of guided missile is estimated, for example, when being existed according to this method The impact accuracy of certain guided missile gathers the drop point site of 1000 guided missiles in estimating, be then passed through being calculated σ₁=30m, σ₂= 10m、σ₃=40m, sets up coordinate system oxyz according to said method, and the set of locations center making the drop point site of 1000 guided missiles is former Point o, the direction of longitudinal bias falls in x-axis, and the direction of height tolerance falls in z-axis, sets up ellipsoid, make this in this coordinate system Ellipsoid covers the drop point site of 500 guided missiles, and three semiaxis of the ellipsoid set up are respectively 46.17m, 15.39m, 61.56m, I.e. esep=(46.17m, 15.39m, 61.56m), wherein, 46.17m is the half shaft length in longitudinal bias direction for the ellipsoid, 15.39m is the half shaft length in lateral deviation direction for the ellipsoid, and 61.56m is the half shaft length in height tolerance direction for the ellipsoid.Mould The distribution that plan target practice is 1000 times is as shown in Figure 4.It can be seen that describing distribution with ellipsoid relatively compared with Fig. 2 Accurately, and sep assessment result can not accurate description drop point feature.

The non-detailed description of the present invention is known to the skilled person technology.

Claims (1)

1. a kind of inertial navigation system three dimensions impact accuracy method of estimation is it is characterised in that step is as follows:

(1) the n positional information that after the n navigation of collection inertial navigation system, carrier stops, n position constitutes set of locations, and n is extremely It is 6 less；

(2) the n positional information being obtained according to step (1), obtains the coordinate at set of locations center, then measures each position relatively In the longitudinal bias at set of locations center, lateral deviation and height tolerance, it is calculated the mark of all longitudinal bias on this basis Quasi- difference σ₁, the standard deviation sigma of lateral deviation₂Standard deviation sigma with height tolerance₃；

(3) with set of locations center for initial point o, set up rectangular coordinate system in space oxyz, so that the direction of longitudinal bias is fallen in x-axis or y On one of axle of axle, the direction of height tolerance falls in z-axis；

(4) standard deviation sigma of the longitudinal bias being obtained according to step (2)₁, the standard deviation sigma of lateral deviation₂Standard with height tolerance Difference σ₃, setting up an ellipsoid centered on initial point o in the coordinate system of step (3), make this ellipsoid cover n/2 carrier stopping Position；

The ellipsoid set up meets following form:

Ellipsoid meets equation in the half shaft length a in longitudinal bias direction

$\frac{1}{{\left(2\mathrm{\&pi;}\right)}^{3/2}{{\mathrm{\&sigma;}}_1}^3}{\&integral;}_0^a{r}^2\exp \left(-\frac{1}{2{{\mathrm{\&sigma;}}_1}^2}{r}^2\right){\&integral;}_0^{2\mathrm{\&pi;}}{\&integral;}_0^{\mathrm{\&pi;}}sin\mathrm{\&beta;}d\mathrm{\&theta;}d\mathrm{\&beta;}dr=\frac{1}{2}$

Ellipsoid meets equation in the half shaft length b in lateral deviation direction

$\frac{1}{{\left(2\mathrm{\&pi;}\right)}^{3/2}{{\mathrm{\&sigma;}}_2}^3}{\&integral;}_0^b{r}^2\exp \left(-\frac{1}{2{{\mathrm{\&sigma;}}_2}^2}{r}^2\right){\&integral;}_0^{2\mathrm{\&pi;}}{\&integral;}_0^{\mathrm{\&pi;}}sin\mathrm{\&beta;}d\mathrm{\&theta;}d\mathrm{\&beta;}dr=\frac{1}{2}$

Ellipsoid meets equation in the half shaft length c in height tolerance direction

$\frac{1}{{\left(2\mathrm{\&pi;}\right)}^{3/2}{{\mathrm{\&sigma;}}_3}^3}{\&integral;}_0^c{r}^2\exp \left(-\frac{1}{2{{\mathrm{\&sigma;}}_3}^2}{r}^2\right){\&integral;}_0^{2\mathrm{\&pi;}}{\&integral;}_0^{\mathrm{\&pi;}}sin\mathrm{\&beta;}d\mathrm{\&theta;}d\mathrm{\&beta;}dr=\frac{1}{2}$

WhereinRepresent that the n position that carrier stops falls in longitudinal direction Probability in [- a ,+a] in bias direction；Represent what carrier stopped The probability that n position falls on lateral deviation direction [- b ,+b]； Represent the probability that the n position that carrier stops falls on height tolerance direction [- c ,+c]；

Obtain after solution

A=1.539 σ₁

B=1.539 σ₂

C=1.539 σ₃

Wherein θ, β and r are respectively polar parameter；

(5) this inertial navigation system three-dimensional impact accuracy esep=(a, b, c) is obtained according to the ellipsoid that step (4) is set up, thus Complete the estimation of inertial navigation system three-dimensional impact accuracy.