CN104391311A

CN104391311A - Satellite passive positioning method based on GPS broadcast data

Info

Publication number: CN104391311A
Application number: CN201410461830.4A
Authority: CN
Inventors: 许哲; 沈庆丰; 叶小舟; 陈占胜; 黄欣; 董泽政
Original assignee: Shanghai Institute of Satellite Engineering
Current assignee: Shanghai Institute of Satellite Engineering
Priority date: 2014-09-11
Filing date: 2014-09-11
Publication date: 2015-03-04
Anticipated expiration: 2034-09-11
Also published as: CN104391311B

Abstract

The invention provides a satellite passive positioning method based on GPS broadcast data. The method comprises: acquiring measuring data of a satellite; according to the measuring data, calculating a conversion matrix M from an earth-fixed coordinate system to a satellite local system; selecting an earth ellipsoidal model and obtaining corresponding earth geometrical parameters and an earth ellipsoidal surface equation; and according to the conversion matrix M, solving an object position equation, and solving object position coordinates by simultaneously solving the earth ellipsoidal surface equation. The measuring data comprises the GPS broadcast data including a satellite real-time position P<se><-> and a speed V<se><->, object measurement angles beta<x> and beta<y> and attitude angle data, i.e., a pitch angle phi, a yaw angle yaw angle gamma and a roll angle theta, wherein the GPS broadcast data is obtained through a satellite data bus, the beta<x> and the beta<y> are obtained through a load subsystem, and the attitude angle data is obtained through an attitude orbit control subsystem. According to the invention, the high-precision GPS broadcast data is taken as a system input quality, such that the precision of each parameter during a calculation process is ensured, accordingly, the precision of a positioning result is ensured, at the same time, an offline processing process is unnecessary, and the satellite autonomous calculating capability is enhanced.

Description

Satellite passive positioning method based on GPS broadcast data

Technical Field

The invention relates to the technical field of satellite passive positioning, in particular to an on-satellite passive positioning method based on GPS broadcast data.

Background

The outer space is shared by human beings, no national boundary exists, and the positioning of the target by using the satellite is not limited by soil, sea and air. The satellite passive positioning is to position the target by utilizing the self-radiated radio signal of the satellite receiving target, and has the characteristics of long action distance, good concealment and the like, thereby being widely applied. Satellite passive positioning technology has been one of the hot spots in the research of satellite application technology.

The satellite passive positioning process puts high demands on the following capabilities of the satellite:

(1) the positioning principle uses the position and attitude information of the satellite and the direction vector of the target relative to the satellite as input, thereby providing higher requirements for the orbit determination and attitude determination capabilities of the satellite;

(2) in order to adapt to actual use conditions, certain requirements on timeliness and autonomous capability of the positioning system are necessary from the viewpoints of system safety and use convenience.

In the traditional positioning process, an attitude and orbit control system is used for providing recursion position and speed data of a satellite, the positioning calculation precision is relatively low, or target angle measurement data and attitude determination data are downloaded and then processed under a line, and the autonomous ability is weak.

Disclosure of Invention

Aiming at the defects in the prior art, the invention provides a satellite passive positioning method based on GPS broadcast data, and the method can realize high-precision observation data acquisition and rapid satellite autonomous calculation process. The invention has the advantages of high positioning precision, less resource occupation, strong on-satellite autonomous capability and the like.

The satellite passive positioning method based on the GPS broadcast data provided by the invention comprises the following steps:

step 1: collecting measurement data of a satellite;

step 2: calculating a conversion matrix M from the earth-fixed coordinate system to the satellite body system according to the measurement data;

and step 3: selecting an earth ellipsoid model and obtaining corresponding earth geometric parameters and an earth ellipsoid equation;

and 4, step 4: and solving a target position equation according to the conversion matrix M and solving a target position coordinate by combining an earth ellipsoid equation.

Preferably, the measurement data comprises real-time position including satelliteAnd velocityGPS broadcast data, target mapping angle beta_xTarget angle beta_yAnd attitude angle data, i.e. pitch angleA yaw angle gamma, a roll angle theta; said GPS broadcast data is obtained via satellite data bus, beta_xAnd beta_yThe attitude angle data is obtained through a posture and orbit control subsystem.

Preferably, the GPS broadcast data, the target angle measurement data, and the attitude angle data are recursively combined into a combined data packet and added with a time identifier.

Preferably, the step 2 comprises the steps of:

step 2.1: calculating unit vectors of the satellite to target directions under a satellite body coordinate system, wherein specifically, two target measurement angles under the body coordinate system are respectively beta_x、β_yThen the unit vector L of the satellite to the target direction under the satellite system at this time₁Expressed as:

wherein l₁，m₁，n₁Respectively projecting the target sight direction under the x axis, the y axis and the z axis of the system of the satellite;

step 2.2: unit vector L for directing satellite to target direction₁Converted to the orbital coordinate system, expressed as:

is provided withThen L is₂＝M₁·L₁，

Wherein l₂，m₂，n₂Respectively projecting the target sight line direction under the x axis, the y axis and the z axis of the orbit system;

step 2.3: a unit vector L in an orbit coordinate system₂Converting into a ground-fixed coordinate system, specifically: taking the velocity vector of the satellite as an X axis, the reverse vector of the position vector of the satellite as a Z axis, and the cross multiplication of the Z axis and the X axis as a Y axis, then the orbit coordinate system is obtainedUnit vector L of₂Converting into a coordinate system of a ground-fixed coordinate system, and expressing as follows:

v is to be_siIs expressed as [ v ] in terms of component_x,v_y,v_z]^TThe three unit vectors are obtained as follows:

[\begin{matrix} c_{1} \\ c_{2} \\ c_{3} \end{matrix}] = [\begin{matrix} - x / \sqrt{x^{2} + y^{2} + z^{2}} \\ - y / \sqrt{x^{2} + y^{2} + z^{2}} \\ - z / \sqrt{x^{2} + y^{2} + z^{2}} \end{matrix}]

according to

[\begin{matrix} b_{1} \\ b_{2} \\ b_{3} \end{matrix}], [\begin{matrix} c_{1} \\ c_{2} \\ c_{3} \end{matrix}]

To obtain

Wherein v is_siThe representation of the inertial velocity of the satellite in the earth-fixed coordinate system is shown, and x, y and z are coordinates of the satellite in the earth-fixed coordinate system respectively; l₃，m₃，n₃The projections of the target sight line direction under the x axis, the y axis and the z axis of the earth fixation system are respectively. [ a ] A₁,a₂,a₃]^T，[b₁,b₂,b₃]^T，[c₁,c₂,c₃]^TRespectively projection of an x axis, a y axis and a z axis of the orbit system under the earth fixation system;

step 2.4: obtaining a conversion matrix M from a ground-fixed coordinate system to a satellite body coordinate system:

M＝M₂ ^T·M₁ ^T。

preferably, the earth geometric parameters include:

major radius a: a is 6378137 +/-2 m;

product of gravity and mass GM: GM 3.98600441500 x 10⁵km³/s²；

Rotational angular velocity ω of the earth: omega 7292115 x 10^-11rad/s±0.150×10^-11rad/s。

Preferably, the step 4 comprises the steps of:

step 4.1: calculating a target position equation x in a satellite body coordinate system_T,b＝Μ(x_T,e-x_S,e)；

Wherein,

x_T,bis the target position in the satellite body coordinate system, and the component form [ x_T,e,y_T,e,z_T,e]^T，x_T,eIs a target position in the earth's fixation system, x_S,eFor anchoring to the groundA satellite position of;

step 4.2: the longitude and latitude (L) of any point on the earth_T，B_T) And geographic elevation (H)_T) And (3) converting the data into a ground-fixed coordinate system to obtain an equivalent equation:

\{\begin{matrix} x_{T, e} = (N + H_{T}) \cos L_{T} \cos B_{T} \\ y_{T, e} (N + H_{T}) \cos L_{T} \sin B_{T} \\ z_{T, e} = (N - (1 - e^{2}) + H_{T}) \sin B_{T} \end{matrix}

wherein,the curvature radius of the local unitary fourth of twelve earthly branches, the earth long radius a is 6378137m, and the square e of the first eccentricity ratio²＝0.00669437999013；H_TIs the geographic elevation of any point on the earth, L_TIs the latitude of any point on the earth, B_TLongitude of any point on the earth;

step 4.3: x is to be_T,b、x_T,eAnd x_S,eSubstituting into an earth ellipsoid equation;

step 4.3: and simultaneously establishing an objective position equation, an equivalent equation and an earth ellipsoid equation to obtain the objective position.

Preferably, when the earth-fixed coordinate system is approximately regarded as a coordinate system of fixed axis rotation, thenWhereinIs the angular velocity vector of the rotation of the earth,

<math><mrow> <msub> <mover> <mi>ω</mi> <mo>&RightArrow;</mo> </mover> <mi>e</mi> </msub> <mo>=</mo> <mfenced open='[' close=']'> <mtable> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>7.292115</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>-</mo> <mn>5</mn> </mrow> </msup> </mtd> </mtr> </mtable> </mfenced> <mo>;</mo> </mrow></math>

the position of the satellite in the earth-fixed coordinate system, namely the GPS position given by the satellite data bus;the speed of the satellite in the earth-fixed coordinate system, namely the GPS speed given by the satellite data bus.

Preferably, the step 4 further comprises the following steps:

assuming the target is at a distance r from the star, β is measured_x，β_yAfter the angle, the azimuth cosine angle measured in the satellite body coordinate system can be decomposed into positions:

x_T,b＝ru

thus is provided with

x_T,e＝x_S,e+Μ^-1x_T,b＝x_S,e+rΜ^-1u

Substituting into an earth ellipsoid equation to obtain a unitary quadratic equation about r; and solving the quadratic equation to obtain two roots, wherein the minimum solution is the distance solution of the target position, and further the position solution of the target is obtained, and u is a unit vector of the satellite in the satellite body system from the satellite to the target direction.

Compared with the prior art, the invention has the following beneficial effects:

1. the invention takes the high-precision GPS broadcast data as the input quantity of the system, ensures the precision of each parameter in the calculation process, further ensures the precision of the positioning result, simultaneously does not need the off-line processing process, and enhances the on-board autonomous calculation capability;

2. the invention obtains the satellite inertia speed in the earth-fixed coordinate system by a simple conversion mode, immediately and directly obtains the conversion matrix between the orbit coordinate system and the earth-fixed coordinate system, avoids the complex calculation required by the conversion between the inertia system and the earth-fixed coordinate system, ensures the conversion precision and greatly reduces the occupancy rate of the calculation resources;

3. the invention constructs an earth ellipsoid model based on international standard, which is more accurate than flat earth and spherical earth models.

Drawings

Other features, objects and advantages of the invention will become more apparent upon reading of the detailed description of non-limiting embodiments with reference to the following drawings:

FIG. 1 is a block diagram of a positioning method of the present invention;

FIG. 2 is a schematic view of the angle of measurement definition of the load system of the present invention;

FIG. 3 is a diagram illustrating the relationship between the velocity of the satellite relative to the earth-fixed coordinate system and the inertial velocity of the satellite.

Detailed Description

The present invention will be described in detail with reference to specific examples. The following examples will assist those skilled in the art in further understanding the invention, but are not intended to limit the invention in any way. It should be noted that variations and modifications can be made by persons skilled in the art without departing from the spirit of the invention. All falling within the scope of the present invention.

The satellite positioning method provided by the invention can meet the target positioning precision provided by a user. The method is reasonable and reliable from the test condition in the satellite development process. With the rapid development of the aerospace technology, the satellite plays a more important role in information support combat and support disaster relief, and the method provides reference and basis for a future low-orbit passive electronic investigation satellite positioning method.

In this embodiment, as shown in fig. 1, fig. 2, and fig. 3, the method for passive positioning on a satellite based on GPS broadcast data provided by the present invention includes the following steps:

step 1: acquiring measurement data of a satellite, in particular by acquiring real time data comprising the satellite on a satellite data bus

Position ofAnd velocityCombined with target angle beta of the load subsystem_x、β_yPosture and orbit control

Attitude angle data of the subsystem, i.e. pitch angleThe yaw angle gamma and the rolling angle theta are combined into the same angle after recursion treatment

And combining the data packets at a moment and adding a time identifier.

Step 2: calculating a conversion matrix M from the earth-fixed coordinate system to a satellite body system according to the measurement data, and comprising the following steps:

step 2.1: the unit vector of the satellite to target direction in the satellite body coordinate system is calculated, specifically, as shown in fig. 2, two target measurement angles in the satellite body coordinate system are β respectively_x、β_yThe unit vector of the satellite to target direction under the satellite system can be expressed as:

wherein l₁，m₁，n₁The projection of the target sight line direction under the x-axis, the y-axis and the z-axis of the system in the satellite is respectively.

Step 2.2: unit vector L for directing satellite to target direction₁The conversion to the orbital coordinate system can be expressed as:

is provided withThen L is₂＝M₁·L₁。

Wherein l₂，m₂，n₂Respectively are the projection of the target sight line direction under the x-axis, the y-axis and the z-axis of the orbital system.

Step 2.3: a unit vector L in an orbit coordinate system₂And converting into a ground-fixed coordinate system.

And obtaining the position coordinates (x, y, z) of the satellite in the WGS-84 coordinate system, namely the earth-fixed coordinate system according to the combined data packet, wherein the speed of the satellite relative to the WGS-84 coordinate system is (vx, vy, vz), so that the inertial speed of the satellite in the WGS-84 coordinate system can be obtained.

On the premise that the precision requirement can meet the engineering realization, a ground-fixed coordinate system can be regarded as a coordinate system with fixed axis rotation, and the actual velocity vector is formed by superposing the velocity of a particle under the coordinate system relative to the coordinate system, namely the velocity given by a GPS system.

As shown in fig. 3, v_siRepresenting the direction of flight of the satellite in inertial space, i.e. X in the orbital coordinate system_oDirection in which the axis points, v_eRepresenting the speed of involvement, v, provided by the earth-fixed coordinate system (rotating coordinate system of fixed axis) for the location of the satellite_seRepresenting the speed provided by the GPS system, i.e. the speed of the earth-fixed coordinate system relative to the rotation of the satellites.

This conversion relationship can be expressed as follows:

<math><mrow> <mover> <msub> <mi>V</mi> <mi>si</mi> </msub> <mo>&RightArrow;</mo> </mover> <mo>=</mo> <mover> <msub> <mi>ω</mi> <mi>e</mi> </msub> <mo>&RightArrow;</mo> </mover> <mo>×</mo> <msub> <mover> <mi>P</mi> <mo>&RightArrow;</mo> </mover> <mi>se</mi> </msub> <mo>+</mo> <msub> <mover> <mi>V</mi> <mo>&RightArrow;</mo> </mover> <mi>se</mi> </msub> </mrow></math>

wherein,is the angular velocity vector of the rotation of the earth,

the position of the satellite in the earth-fixed coordinate system, namely the GPS position given by the satellite data bus;the velocity of the satellite in the earth-fixed coordinate system, i.e. the GPS velocity given by the bus.

The velocity vector of the satellite is taken as an X axis, the reverse vector of the position vector of the satellite is taken as a Z axis, and the difference product of the X axis and the Z axis is taken as a Y axis. Then the unit vector L in the orbital coordinate system is assigned₂Converted to the WGS-84 coordinate system, which can be expressed as:

wherein,

[\begin{matrix} a_{1} \\ a_{2} \\ a_{3} \end{matrix}], [\begin{matrix} b_{1} \\ b_{2} \\ b_{3} \end{matrix}], [\begin{matrix} c_{1} \\ c_{2} \\ c_{3} \end{matrix}]

the unit vectors of the X axis, the Y axis and the Z axis of the orbit coordinate system are respectively in the WGS84 coordinate system. V is to be_siIs expressed as [ vx, vy, vz]^TThe three unit vectors can be found as follows:

[\begin{matrix} c_{1} \\ c_{2} \\ c_{3} \end{matrix}] = [\begin{matrix} - x / \sqrt{x^{2} + y^{2} + z^{2}} \\ - y / \sqrt{x^{2} + y^{2} + z^{2}} \\ - z / \sqrt{x^{2} + y^{2} + z^{2}} \end{matrix}]

according to

[\begin{matrix} b_{1} \\ b_{2} \\ b_{3} \end{matrix}], [\begin{matrix} c_{1} \\ c_{2} \\ c_{3} \end{matrix}]

To obtain

Wherein v is_siX, y and z are respectively the representation of the inertial velocity of the satellite in the earth-fixed coordinate systemCoordinates of the lower part; l₃，m₃，n₃The projections of the target sight line direction under the x axis, the y axis and the z axis of the earth fixation system are respectively. [ a ] A₁,a₂,a₃]^T，[b₁,b₂,b₃]^T，[c₁,c₂,c₃]^TThe projections of the x-axis, the y-axis and the z-axis of the orbit system under the earth fixation system are respectively.

M＝M₂ ^T·M₁ ^T

and step 3: the earth model is set up and the earth model,

step 3.1: selecting a reference ellipsoid model, specifically, selecting a WGS-84 model as the reference ellipsoid model, wherein four basic geometric parameters are as follows:

long radius: a is 6378137 +/-2 m

Product of earth's gravity and earth's mass:

GM＝3.98600441500×10⁵km³/s²

rotation angular velocity of the earth: omega 7292115 x 10^-11rad/s±0.150×10^-11rad/s

And 4, step 4: the coordinates of the target position are resolved,

the target position equation in the satellite body coordinate system is

x_T,b＝Μ(x_T,e-x_S,e)

In the formula, M is a transformation matrix from the earth-fixed coordinate system to the satellite body coordinate system.

Wherein x is_T,bIs the target position in the satellite body coordinate system, and the component form [ x_T,e,y_T,e,z_T,e]^T，x_T,eFor target position in earth fixation systemX is arranged_S,eIs the satellite position in the earth's fixed system;

assume that the average earth ellipsoid WGS-84 model is used, with the target located on the earth's surface, the geographic latitude and longitude (L) of any point of the earth_T，B_T) And geographic elevation (H)_T) With its three-dimensional rectangular coordinate X of ground_T,b＝[x_T,e y_T,e z_T,e]^TIs in the equivalent form of

\{\begin{matrix} x_{T, e} = (N + H_{T}) \cos L_{T} \cos B_{T} \\ y_{T, e} (N + H_{T}) \cos L_{T} \sin B_{T} \\ z_{T, e} = (N - (1 - e^{2}) + H_{T}) \sin B_{T} \end{matrix}

Wherein,the curvature radius of the local unitary fourth of twelve earthly branches, the earth long radius a is 6378137m, and the square e of the first eccentricity ratio²＝0.00669437999013。

The above formula can be written as another form, namely the earth ellipsoid equation under the WGS84 series

\frac{{x_{T, e}}^{2} + {y_{T, e}}^{2}}{{(N + H_{T})}^{2}} + \frac{{z_{T, e}}^{2}}{{[N (1 - e^{2}) + H_{T}]}^{2}} = 1 .

The problem of direction finding positioning can be regarded as the problem of how to solve the target position by using the target position equation and the earth ellipsoid equation under the WGS-84 system. In the above-described system of nonlinear equations, there are a total of three equations and three unknowns. Geometrically, if a positioning solution exists, two intersection points can be obtained by intersecting a general direction-finding line with an ellipsoid of the earth, and in the equation solving process, two solutions are necessarily generated, so that a solution with a larger distance from the other end of the earth needs to be removed, and the position of the radiation source can be solved. The form of the equation used covers the consideration of the height of the target and does not address the situation where the target is located on the surface of the earth as in conventional solutions.

If the distance of the target from the star is assumed to be r, beta is obtained in the measurement_x，β_yAfter the angle, the azimuth cosine angle measured in the satellite body coordinate system can be decomposed into positions:

x_T,b＝ru

thus is provided with

x_T,e＝x_S,e+Μ^-1x_T,b＝x_S,e+rΜ^-1u

Substituting into the earth ellipsoid equation can obtain a quadratic equation of one unit about r. Solving this quadratic equation of one unit can get two roots, the minimum solution is the distance dissociation of the target position. And further obtain a position solution of the target.

However, in the earth ellipse equation, the curvature radius N of the target local unitary fourth-element circle is unknown, and therefore an iterative method can be adopted for solving. That is, considering the closer N values of the satellite and the target, the satellite N is used first₀＝N_SatThe values are substituted into the earth's ellipse equation,

wherein N is₀Comprises the following steps: an initial value of the radius of the target local unitary fourth of twelve earthly branches; n is a radical of_SatComprises the following steps: and (5) the radius of the satellite lower point-fourth-prime unitary circle.

Calculating the ith iteration distance r_iRecalculating the targetCalculating to obtain the curvature radius N of the ith iteration estimated target current-fourth prime unitary ring_iAnd circulating the steps until the target distance converges in the following formula.

|r_i-r_i-1|≤_r

Wherein_rIs a set distance error threshold.

The invention designs a set of method for positioning a passive target on the ground or at a specified height based on position and speed data of a satellite-borne GPS (global positioning system) aiming at the characteristic that the precision requirement of satellite passive direction finding positioning on measurement data is high, and the method comprises the steps of measurement data acquisition, coordinate system transformation matrix calculation, elliptical earth model setting, target position calculation and the like, wherein the whole positioning flow chart is shown in figure 1. The invention exerts the characteristic of high precision of satellite-borne GPS for measuring the position and speed data of the satellite, can accurately obtain the transformation matrix of the system of the satellite and the earth-fixed coordinate system, and uses the characteristic in cooperation with the high-precision target angle measurement and attitude determination technology, thereby ensuring the positioning precision of the satellite to the target, enhancing the autonomous calculation capability on the satellite, and improving the emergency efficiency of the satellite in the application fields of information support, disaster monitoring and the like. The measurement precision of the current practical engineering can ensure that the theoretical positioning precision of the method is superior to 3 km.

The foregoing description of specific embodiments of the present invention has been presented. It is to be understood that the present invention is not limited to the specific embodiments described above, and that various changes and modifications may be made by one skilled in the art within the scope of the appended claims without departing from the spirit of the invention.

Claims

1. A satellite passive positioning method based on GPS broadcast data is characterized by comprising the following steps:

step 1: collecting measurement data of a satellite;

2. The method of claim 1, wherein the measurement data comprises a real-time position of the satelliteAnd velocityGPS broadcast data, target angle measurement beta_xTarget angle beta_yAnd attitude angle data, i.e. pitch angleA yaw angle gamma, a roll angle theta; said GPS broadcast data is obtained via satellite data bus, beta_xAnd beta_yThe attitude angle data is obtained through a posture and orbit control subsystem.

3. The method of claim 2, wherein the GPS broadcast data, the target angle measurement data and the attitude angle data are recursively combined into a combined data packet and added with a time identifier.

4. The method for passive positioning on satellite based on GPS broadcast data according to claim 2, wherein the step 2 comprises the following steps:

is provided withThen L is₂＝M₁·L₁，

step 2.3: a unit vector L in an orbit coordinate system₂Converting into a ground-fixed coordinate system, specifically: taking the velocity vector of the satellite as an X axis, the reverse vector of the position vector of the satellite as a Z axis, and the cross multiplication of the Z axis and the X axis as a Y axis, then a unit vector L in an orbit coordinate system is obtained₂Converting into a coordinate system of a ground-fixed coordinate system, and expressing as follows:

[\begin{matrix} c_{1} \\ c_{2} \\ c_{3} \end{matrix}] = [\begin{matrix} - x / \sqrt{x^{2} + y^{2} + z^{2}} \\ - y / \sqrt{x^{2} + y^{2} + z^{2}} \\ - z / \sqrt{x^{2} + y^{2} + z^{2}} \end{matrix}]

according to

[\begin{matrix} b_{1} \\ b_{2} \\ b_{3} \end{matrix}], [\begin{matrix} c_{1} \\ c_{2} \\ c_{3} \end{matrix}]

To obtain

M＝M₂ ^T·M₁ ^T。

5. the method of claim 4, wherein the geometric parameters of the earth comprise:

major radius a: a is 6378137 +/-2 m;

product of gravity and mass GM: GM 3.98600441500 x 10⁵km³/s²；

6. The method for passive positioning on satellite based on GPS broadcast data according to claim 5, wherein the step 4 comprises the following steps:

Wherein x is_T,bIs the target position in the satellite body coordinate system, and the component form [ x_T,e,y_T,e,z_T,e]^T，x_T,eIs a target position in the earth's fixation system, x_S,eIs the satellite position in the earth's fixed system;

\{\begin{matrix} x_{T, e} = (N + H_{T}) \cos L_{T} \cos B_{T} \\ y_{T, e} (N + H_{T}) \cos L_{T} \sin B_{T} \\ z_{T, e} = (N - (1 - e^{2}) + H_{T}) \sin B_{T} \end{matrix}

step (ii) of4.3: x is to be_T,b、x_T,eAnd x_S,eSubstituting into an earth ellipsoid equation;

7. The method of claim 4, wherein the earth-fixed coordinate system is approximately the coordinate system of fixed axis rotationWhereinIs the angular velocity vector of the rotation of the earth,

<math> <mrow> <msub> <mover> <mi>ω</mi> <mo>&RightArrow;</mo> </mover> <mi>e</mi> </msub> <mo>=</mo> <mfenced open='[' close=']'> <mtable> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>7.292115</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>-</mo> <mn>5</mn> </mrow> </msup> </mtd> </mtr> </mtable> </mfenced> <mo>;</mo> </mrow> </math>

8. The method for passive positioning on satellite based on GPS broadcast data according to claim 6, characterized in that, the step 4 is followed by the following steps:

x_T,b＝ru

thus is provided with

x_T,e＝x_S,e+Μ^-1x_T,b＝x_S,e+rΜ^-1u