Cascade passive positioning method based on pure angle measurement

Abstract

The invention discloses a cascade passive positioning method based on pure angle measurement, and belongs to the technical field of passive positioning. Comprising the steps of S1, defining a coordinate system; s2, defining a sight angle of the foundation observation station to the unmanned aerial vehicle; s3, establishing a mathematical model of a cascade passive positioning algorithm; s4, I-level positioning: calculating an optimal estimation coordinate of the space-based observation station at the moment k through a double-station cooperative positioning algorithm based on EKF; and S5, II-level positioning: calculating an optimal estimation coordinate of the target at the moment k through a double-station cooperative positioning algorithm based on EKF. According to the invention, the foundation fixed station and the air-based mobile station are cascaded and based on pure angle measurement, so that high-precision positioning of a target is realized. The air-based observation station does not depend on GPS and inertial navigation, the self load can be effectively reduced, the cruising ability can be improved, the flexibility can be further enhanced, and the coverage can be further expanded. The method perfectly integrates the advantages of high positioning precision of a foundation observation station and the advantages of maneuverability, flexibility and wide coverage of an air-based observation station, and effectively avoids the respective defects of two single systems.

Description

Technical Field

[0001] The present invention belongs to the technical field of passive positioning, and particularly relates to a cascaded passive positioning method based on pure angle measurement. Background Art

[0002] As an important detection means, passive positioning technology has broad application prospects in many fields such as military and civilian. Compared with traditional active positioning technology, passive positioning technology does not actively emit signals and has advantages such as strong concealment and good anti-interference ability. After years of research by scholars from various countries, passive positioning technology has made remarkable progress in algorithms, signal sources, positioning accuracy, equipment development, etc., but still faces many challenges.

[0003] There are various algorithms for passive positioning technology, mainly including methods based on angle of arrival (AOA), time difference of arrival (TDOA), frequency difference of arrival (FDOA), etc., as well as hybrid algorithms of various algorithm mechanisms. Each of the various algorithms has its own advantages and disadvantages, but the passive positioning algorithm based on angle of arrival is still the earliest studied, most mature, and most stable passive positioning algorithm.

[0004] In terms of equipment, passive positioning technology has been widely applied in fields such as ground-to-air, air-to-air, and air-to-ground, but it also faces some difficulties in practical applications. The ground-based passive positioning system has high self-positioning accuracy, small signal interference from the environment, and is easier to perform high-precision positioning. However, due to terrain occlusion and fixed deployment, the flexibility and line-of-sight of the system are severely limited. The outstanding features of the space-based passive positioning system are its mobility, flexibility, and wide coverage, but multiple factors such as its own positioning accuracy, endurance, load capacity, GPS signal, and inertial navigation system accuracy will seriously affect the positioning accuracy. Summary of the Invention

[0005] In order to solve the above problems, the present invention further provides a cascaded passive positioning method based on pure angle measurement;

[0006] The technical solution adopted by the present invention is as follows:

[0007] A cascaded passive positioning method based on pure angle measurement includes the following steps:

[0008] S1. Define a coordinate system with the northeast celestial coordinate system as the reference;

[0009] S2. Define the line-of-sight angle of the ground-based observation station to the unmanned aerial vehicle;

[0010] S3. Establish a mathematical model of the cascaded passive positioning algorithm;

[0011] S4. Level-I positioning, based on S3, calculate the optimal estimated coordinates of the space-based observation station at time k through a dual-station cooperative positioning algorithm based on EKF;

[0012] S5. Level II positioning, calculating the optimal estimated coordinates of the target at time k through a dual-station collaborative positioning algorithm based on EKF.

[0013] The present invention has the following beneficial effects compared with the prior art:

[0014] The present invention realizes high-precision positioning of the target based on pure angle measurement through a cascaded method by a ground-based fixed station and an airborne mobile station. Compared with the existing single-stage passive positioning system, the present invention has the following significant advantages:

[0015] 1. Realize positioning based on pure angle measurement. In a complex strong electromagnetic interference environment, the line-of-sight angle parameter is still the easiest signal to obtain and the most stable one, and there is no need for strict time synchronization and frame rate unification.

[0016] The self-positioning of the airborne observation station is realized by the ground-based observation station, and it only needs to focus on obtaining the line-of-sight angle information. Based on this, the airborne observation station does not rely on GPS and inertial navigation, can effectively reduce its own load and improve the endurance ability, can achieve low cost, high endurance, miniaturization, and further enhance flexibility and expand the coverage area.

[0017] 2. Less signal transmission and strong anti-interference ability. For scenarios with real-time positioning requirements, only the arrival angle signal needs to be transmitted to realize positioning. For fixed targets such as land and sea where real-time positioning is not required, the target can be located after the airborne observation station detects and returns.

[0018] 3. The present invention perfectly combines the advantages of high positioning accuracy of the ground-based observation station and the mobility, flexibility, and wide coverage of the airborne observation station, and effectively avoids the disadvantages of the two single systems;

[0019] Without satellite navigation systems such as GPS and inertial navigation systems, which could not be achieved in previous airborne observation stations. Therefore, the application scenarios of the cascaded passive positioning system proposed by the present invention will far exceed those of single-system airborne observation stations. BRIEF DESCRIPTION OF THE DRAWINGS

[0020] Figure 1 is a simulation diagram of the online positioning scenario of the target by the present invention (in the figure, the white signal line represents the effective signal measurement, and the orange signal line represents the out-of-line-of-sight or invalid signal measurement);

[0021] Figure 2 is a simulation diagram of the offline positioning scenario of the target by the present invention (in the figure, the white signal line represents the effective signal measurement, and the orange signal line represents the out-of-line-of-sight or invalid signal measurement);

[0022] Figure 3 is the northeast celestial right-angle coordinate system;

[0023] Figure 4 is a schematic diagram of the line-of-sight angle of the present invention;

[0024] Figure 5 is the flow chart of the cascaded passive positioning algorithm of the present invention;

[0025] Figure 6 is the geometric relationship of the positions of the carrier aircraft A, B and the target M in space;

[0026] Figure 7 is the three-dimensional view of the scene simulation;

[0027] Figure 8 is the top view of the scene simulation;

[0028] Figure 9 is the three-dimensional view of the real trajectory and the filtered trajectory;

[0029] Figure 10 is the top view of the real trajectory and the filtered trajectory;

[0030] Figure 11 is the positioning accuracy RMSE of the space-based observation station 1;

[0031] Figure 12 is the positioning accuracy ADPE of the space-based observation station 1;

[0032] Figure 13 is the positioning accuracy RMSE of the space-based observation station 2;

[0033] Figure 14 is the positioning accuracy ADEP of the space-based observation station 2;

[0034] Figure 15 is the positioning accuracy RMSE of the target;

[0035] Figure 16 is the positioning accuracy ADEP of the target. Detailed implementation manners

[0036] In order to better understand the purpose, structure and function of the present invention, the following further describes the present invention in detail with reference to the accompanying drawings.

[0037] Usage scenario

[0038] The usage scenario of the cascaded passive positioning method based on pure angle measurement proposed by the present invention is as Figure 1 , Figure 2 shown.

[0039] Situation 1: The line-of-sight angle information of the own space-based observation station is obtained by the ground-based observation station, and the line-of-sight angle information of the target is obtained by the space-based observation station. The real-time state estimation and trajectory tracking of the target are realized online through the cascaded passive positioning algorithm based on the angle of arrival. ( Figure 1 )

[0040] Case 2: The line-of-sight angle information of its own space-based observation station is obtained and saved by the ground-based observation station, and the line-of-sight angle information of the target is obtained and saved by its own space-based observation station. After completing the observation and reconnaissance mission of the enemy target and returning, the positioning of the target is realized offline through the cascaded passive positioning algorithm based on the angle of arrival.( Figure 2 )

[0041] Case 2 mainly targets fixed targets and targets that do not require real-time positioning. Since information is collected first and then calculated offline, real-time trajectory information cannot be obtained. If it is a fixed target, it is to locate the coordinates of the target. If it is a moving target, it is to locate the trajectory at the time of being observed.

[0042] Implementation solution

[0043] (1) Define the coordinate system;

[0044] In the present invention, in order to facilitate the accurate analysis and description of the observed quantity and the target motion state, the northeast celestial coordinate system is uniformly adopted as the research benchmark. As Figure 3 This coordinate system is a kind of Cartesian rectangular coordinate system. Its three coordinate axes point to the geographical east, north, and sky directions respectively, are perpendicular to each other in pairs and intersect at the origin, forming a standard rectangular coordinate system. Specifically, the positive direction of the X-axis points to the east direction in the geographical coordinate system, the positive direction of the Y-axis points to the north direction in the geographical coordinate system, and the positive direction of the Z-axis points to the sky, that is, the sky direction of the geographical coordinate system, perpendicular to the earth's surface and upward.

[0045] (2) Define the line-of-sight angle

[0046] Taking the line-of-sight angle of the ground-based observation station to the unmanned aerial vehicle as an example, as Figure 4 shown, let the coordinate of the ground-based observation station be the origin O of the coordinate system, the coordinate of the unmanned aerial vehicle in the three-dimensional space be U, and the projection of U on the xoy plane be U'. Define the yaw angle θ as the angle between OU' and the x-axis, and define the pitch angle as the angle between OU and the OU' axis.

[0047] (3) Abbreviation and symbol description

[0048]

[0049]

[0050] (4) Implementation process

[0051] The cascaded passive positioning system consists of two ground-based observation stations D a , ground-based observation station D b and two space-based observation stations U a , space-based observation station U bIt consists of, with the target being T. The motion state and trajectory of target T are located by a space-based observation station. When positioning systems such as satellite navigation and inertial navigation positioning systems cannot be used, the coordinates of the space-based observation station are unknown, and thus the target cannot be located. The cascade passive positioning algorithm process is as follows Figure 5 As shown, the overall process can be divided into level-I positioning and level-II positioning. The filtering algorithms relied on for passive positioning are shown in Table 1. Under the framework of the cascade passive positioning algorithm proposed in the present invention, different filtering algorithms can be replaced as needed to meet the requirements. Among them, the EKF technology is mature and most widely used. Moreover, in the application scenario of passive positioning, the target is not aware that it is being locked and located. Therefore, it can be assumed that the motion state of the target is basically stable, and within a very short time interval of one sampling period, the target motion mode can be approximated by a single motion model. Therefore, all passive positioning filtering algorithms in the present invention are implemented taking the EKF as an example.

[0052] Table 1 Technical characteristics and usage scenarios of different filtering algorithms

[0053]

[0054] Level-I positioning: Ground-based observation station D a and ground-based observation station D b The coordinates are known. At time k, the line-of-sight angles of ground-based observation station D a and ground-based observation station D b to the space-based observation station U a are respectively The optimal estimated coordinates of the space-based observation station U a are Using the optimal estimated coordinates of the space-based observation station U a at time k-1 to obtain the predicted coordinates at time k Using and Through the dual-station cooperative positioning algorithm based on EKF to calculate the optimal estimated coordinates of U a at time k Similarly, calculate the optimal estimated coordinates of U b at time k

[0055] Level-II positioning: At time k, the line-of-sight angles of the space-based observation station U a and the space-based observation station U b to the target T are respectively Using the optimal estimated coordinates of the target T at time k-1 to obtain the predicted coordinates at time k Using and Through the dual-station cooperative positioning algorithm based on EKF to calculate the optimal estimated coordinates of the target T at time k

[0056] (5) Establish a mathematical model

[0057] 1. Bistatic passive localization principle based on pure angle measurement

[0058] Define the coordinates of two ground-based observation stations as D a = [x Da , y Da , z Da T , D b = [x Db , y Db , z Db T , and the coordinates of two space-based observation stations are U a = [x Ua , y Ua , z Ua T , U b = [x Ub , y Ub , z Ub T , and the line-of-sight angles of the ground-based observation stations to the space-based observation stations are respectively The line-of-sight angles of the two space-based observation stations to the target are The coordinates of the target are T = [x T , y T , z T T ,

[0060] According to the line-of-sight angle definition in subsection (2), the relationship between the line-of-sight angle and the coordinates is expressed as follows:

[0061] The ground-based observation station D a , the ground-based observation station D b locates the space-based observation station U a :

[0062]

[0063] In the above equations, the unknowns are the coordinates [x a , y Ua , z Ua of the space-based observation station U Ua . The equations are overdetermined equations and have a unique solution, so the coordinates [x a , y Ua , z Ua of the space-based observation station U Ua can be determined.

[0064] Similarly, using D a = [x Da , y​​​​​Da , z Da T , D b = [x Db , y Db , z Db T 、 The coordinates [x b , y Ub , z Ub of U can be determined. Ub

[0066] Using U a = [x Ua , y Ua , z Ua T 、U b = [x Ub , y Ub , z Ub T 、 The coordinates [x T , y T , z T of the target T can be uniquely determined.

[0067] 2. Motion Model

[0068] Assume that the sampling period of the system is dt, and the state of the moving object (the moving object is an airborne observation station or a target. In level-I positioning, the airborne observation station can be regarded as a moving object with unknown coordinates. In level-II positioning, the target is regarded as a moving object) at time k is Within a sampling period, the motion model of the moving object is approximated as uniform motion. Then the state of the moving object at time k + 1 is X k+1 It can be expressed as:

[0069]

[0070] 3. Flow of Passive Positioning Algorithm Based on EKF

[0071] (1) Prediction Step

[0072]

[0073] Among them, F is the state transition matrix, P is the state covariance matrix, and Q is the covariance matrix of the process noise.

[0074] (2) Update Step

[0075] Calculate the observation residual:

[0076] ​​​​

[0077] Calculate the Kalman gain:

[0078]

[0079] Update the state and covariance:

[0080]

[0081]

[0082] where h(X) is the observation function, which is a non-linear function of X, and H k is the Jacobian matrix of h(X) at time k, and R is the measurement error covariance matrix.

[0083] 4. Level-I positioning algorithm

[0084] Ground-based observation stations: The positions are known, which are D1 = [x1, y1, z1] T , D2 = [x2, y2, z2] T ;

[0085] Space-based observation stations: The positions are unknown. Assume that the states at time k are respectively

[0086] X 1,k = [x U1,k , y U1,k , z U1,k , v xU1,k , v yU1,k , v zU1,k T , X 2,k = [x U2,k , y U2,k , z U2,k , v xU2,k , v yU2,k , v zU2,k T The observable quantity of the ground-based observation station to the space-based observation station at time k is:

[0087]

[0088] Locate the space-based observation station U1:

[0089] The state transition matrix of the space-based observation station U1 is:

[0090] X 1,k = FX k-1 + w k (9)

[0091] where, according to formula (2), w k is the process noise, and its covariance matrix is Q;​​

[0092] According to formula (1), the observation equation is:

[0093]

[0094] where v k is the measurement noise, and its covariance matrix is R; in the formula, h(X 1,k ) is a system of non-linear equations. After linearizing h(X 1,k ), the obtained Jacobian matrix can be expressed as:

[0095]

[0096] where

[0097] i ∈ (1, 2), j ∈ (1, 2).

[0098] Then, according to formulas (3)-(8), the state X 1,k of U1 at time k is obtained.

[0099] Similarly, the state X 2,k of U2 at time k can be obtained.

[0100] Level-II positioning algorithm

[0101] Space-based observation stations: The motion states are known, which are U 1,k = [x U1,k , y U1,k , z U1,k , v xU1,k , v yU1,k , v zU1,k T ,

[0102] U 2,k = [x U2,k , y U2,k , z U2,k , v xU2,k , v yU2,k , v zU2,k T ;

[0103] Target: The motion information is unknown. Let the motion state at time k be

[0104] X T,k = [x T,k , y T,k , z T,k , v T,k , v T,k , v T,k T ,

[0105] The observation value of the space-based observation station for the target at time k is as follows: According to the principle of the level-I positioning algorithm, the state X of T at time k is obtained T,k .

[0106] It should be noted that a good initial filtering value helps to avoid the divergence of the EKF filter and accelerate the convergence speed.

[0107] Therefore, the present invention selects the initial filtering value by obtaining the analytical solution of the target, and the specific derivation is as follows:

[0108] The positional relationship among the observation station A, the observation station B, and the target M in space is described as Figure 6 . The triangle formed by the positions of the observation station A, the observation station B, and the target M in space may be an obtuse triangle, as shown in Figure 6 (a), or it may be an acute triangle, as shown in Figure 6 (b).

[0109] In the case of Figure (a):

[0110] MH = -AHtanα = BHtanβ = (AB + AH)tanβ (2-2)

[0111]

[0112] In the case of Figure (b):

[0113] MH = AHtanα = BHtanβ = (AB + AH)tanβ (2-5)

[0114]

[0115] As can be seen from equations (2-4) and (2-7), the expressions of AM in the two cases are the same. AB can be obtained from the positioning information of A and the positioning information of B, while α and β are unknowns. It is not difficult to find from the figure that α is the angle between the vector and the vector , and β is the angle between the vector and the vector . Through the line-of-sight angles of M from A and B, the unit vector of the vector is:

[0116]

[0117] The unit vector of the vector is:

[0118]

[0119] The vector Both it and the modulus L of the vector can be obtained from the positioning information of A and the positioning information of B. Then, the expressions for α and β are as follows:

[0120]

[0121] At this time, all variables on the right side of the equation in Equation (2-4) are known quantities, and thus AM can be obtained. Then, the formula for obtaining the position state of M relative to A is:

[0122]

[0123] Note: The above is the detailed derivation of the process in Case 1, and the positioning of the target can be completed in real time online. Case 2 is for ground-fixed targets. The algorithm principle is the same. Since it is a ground-fixed target, there is no need to solve the height parameter, and only the x and y coordinate information needs to be solved. Therefore, the settlement can be completed by a single station, that is, two ground-based observation stations locate an air-based observation station, and an air-based observation station locates a ground target. It can be positioned in real time, or the data can be collected and then solved offline. The advantage of offline solution is that the air-based observation station does not need to communicate with the ground at all because there is no need to transmit data in real time, which can further increase the adaptability to complex electromagnetic environments.

[0124]

[0125]

[0126] Simulation Results

[0127] Simulation scenario settings: Set the simulation duration to 100 s and the sampling period to dt = 0.1 s;

[0128] Ground-based observation station: D1 = [1000, 500, 0] T , D2 = [500, 1000, 0] T ;

[0129] Air-based observation station 1: Initial coordinates X 1,0 = [1000, 2000, 2000] T , Initial velocity

[0130] V 1,0 = [30 / s, 0 / s, 0 / s], Initial velocity direction angle α 1,0 = [0°, 0°] T , Direction angle change rate

[0131] V α1,0 = [0.6° / s, 0.1° / s] T , At the k-th moment

[0132]

[0133] Space-based Observation Station 2: Initial Coordinate X 2,0 = [3000, 2000, 2000] T , Initial velocity

[0134] V 2,0 = [30 / s, 0 / s, 0 / s] Initial velocity direction angle α 2,0 = [0°, 0°] T , Direction angle change rate

[0135] V α2,0 = [0.6° / s, 0.1° / s] T , At time k

[0136]

[0137] Target: Initial Coordinate X T,0 = [500, 6000, 3000] T , Initial velocity V 2,0 = [90 / s, 30 / s, 5 / s], At time k

[0138]

[0139] Note: The trajectory of the aerial vehicle is affected by interference and cannot be a smooth curve. Adding random numbers is to make the simulation closer to the real situation of the vehicle flying in the air.

[0140] The simulation trajectory is as Figure 7 、 Figure 8 shown:

[0141] Simulation algorithm settings: The observation error of the ground-based observation station for the space-based observation station is Gaussian white noise with a mean of 0 and σ = 0.1°. The observation error of the space-based observation station for the target is Gaussian white noise with a mean of 0 and σ = 0.2°. In the simulation, to eliminate the influence of accidental errors on the results, the RMSE and ADPE of 1000 times of MC simulation are used as indicators.

[0142]

[0143] where N is the number of MC times, respectively represent the state posterior estimate and the true value at the k-th time in the i-th simulation.

[0144]

[0145] where

[0146] respectively represent the coordinates of the two observation stations in the i-th MC experiment at the k-th time.

[0147] The simulation results are as Figures 9 to 16 shown below.

[0148] It can be understood that the present invention is described by means of some embodiments. Those skilled in the art will appreciate that, without departing from the spirit and scope of the present invention, various changes or equivalent replacements can be made to these features and embodiments. Additionally, under the teaching of the present invention, these features and embodiments can be modified to adapt to specific circumstances and materials without departing from the spirit and scope of the present invention. Therefore, the present invention is not limited by the specific embodiments disclosed herein, and all embodiments falling within the scope of the claims of this application belong to the scope protected by the present invention.

Claims

1. A cascaded passive positioning method based on pure angle measurement, characterized in that: It includes the following steps: S1. Define a coordinate system with the northeast celestial coordinate system as the reference; S2. Define the line-of-sight angle of the ground-based observation station to the UAV; S3. Establish a mathematical model of the cascaded passive positioning algorithm; S4. Level-I positioning: Based on S3, calculate the optimal estimated coordinates of the space-based observation station at time k through a dual-station cooperative positioning algorithm based on EKF; S5. Level-II positioning: Calculate the optimal estimated coordinates of the target at time k through a dual-station cooperative positioning algorithm based on EKF.

2. The cascade passive positioning method based on pure angle measurement according to claim 1, wherein: The cascaded passive positioning system used in the cascaded passive positioning method consists of two ground-based observation stations D a , ground-based observation station D b and two space-based observation stations U a , space-based observation station U b and is composed of them.

3. The cascade passive positioning method based on pure angle measurement according to claim 2, wherein: The line-of-sight angle defined in S2 is: Let the coordinates of the ground-based observation station be the origin O of the coordinate system, and the coordinates of the UAV in the three-dimensional space be U. The projection of U on the xoy plane is U'. Define the yaw angle θ as the angle between OU' and the x-axis, and define the pitch angle as the angle between OU and the OU' axis.

4. A cascaded passive positioning method based on pure angle measurement according to claim 3, characterized in that: The mathematical model established in S3 is: S31. Based on the principle of dual-station passive positioning with pure angle measurement, Define the coordinates of two ground-based observation stations as D a = [x Da , y Da , z Da T , D b = [x Db , y Db , z Db T , and the coordinates of two space-based observation stations are U a = [x Ua , y Ua , z Ua T , U b = [x Ub , y Ub , z Ub T , and the line-of-sight angles of the ground-based observation stations to the space-based observation stations are respectively The line-of-sight angles of the two space-based observation stations to the target are The coordinates of the target are T = [x T , y T , z T T ,​​​​​ According to the line-of-sight angle definition in S2, the relationship between the line-of-sight angle and coordinates is expressed as follows: Ground-based Observation Station D a and Ground-based Observation Station D b Locate Space-based Observation Station U a : In the above system of equations, the unknown is U a with coordinates [x Ua , y Ua , z Ua . The system of equations is an overdetermined system with a unique solution, so the coordinates [x a , y Ua , z Ua of U Ua can be determined; Similarly, using D a = [x Da , y Da , z Da T , D b = [x Db , y Db , z Db T , can determine the coordinates [x b , y Ub , z Ub , z Ub of U;​​ Using U a = [x Ua , y Ua , z Ua T 、U b = [x Ub , y Ub , z Ub T 、 The solid can uniquely determine the coordinates [x T , y T , z T of the target T;​​ S32. Motion model: Let the sampling period of the system be dt, and the state of the moving object at time k be Within one sampling period, approximate the motion model of the moving object as uniform motion. Then the state of the moving object at time k+1 is X k+1 Expressed as: S33. Flow of the passive positioning algorithm based on EKF: (1) Prediction step where F is the state transition matrix, P is the state covariance matrix, and Q is the covariance matrix of the process noise; (2) Update step Calculate the observation residual: Calculate the Kalman gain: Update the state and covariance: Among them, h(X) is the observation function, which is a non-linear function of X, and H k is the Jacobian matrix of h(X) at time k, and R is the measurement error covariance matrix.

5. A cascaded passive positioning method based on pure angle measurement according to claim 4, characterized in that: The process of level-I positioning in S4 is: Ground-based Observation Station D a 、Ground-based Observation Station D b The coordinates are known. At time k, Ground-based Observation Station D a 、Ground-based Observation Station D b The line-of-sight angles to the space-based Observation Station U a are respectively Space-based Observation Station U a The optimal estimated coordinates at time k - 1 are Using the optimal estimated coordinates of the space-based Observation Station U a at time k - 1 to obtain the predicted coordinates at time k Using and Calculate the optimal estimated coordinates of U a at time k through the dual-station collaborative positioning algorithm based on EKF Similarly, calculate the optimal estimated coordinates of U b at time k 6. A cascade passive positioning method based on pure angle measurement according to claim 5, characterized in that: The algorithm of level-I positioning in S4 is: Ground-based observation stations: The positions are known and are D1 = [x1, y1, z1] T , D2 = [x2, y2, z2] T ; Space-based observation station: The position is unknown. Assume the states at time k are X 1,k = [x U1,k , y U1,k , z U1,k , v xU1,k , v yU1,k , v zU1,k T , X 2,k = [x U2,k , y U2,k , z U2,k , v xU2,k , v yU2,k , v zU2,k T ​​ The observable quantity of the ground-based observation station to the space-based observation station at time k is: Locate the space-based observation station U1: The state transition matrix of the space-based observation station U1 is: X 1,k = FX k-1 + w k (9) where, according to formula (2), w k is the process noise, and its covariance matrix is Q; According to formula (1), the observation equation is: where, v k is the measurement noise with covariance matrix R; in the formula, h(X 1,k ) is a system of nonlinear equations. After linearizing h(X 1,k ), the obtained Jacobian matrix is expressed as: Among them, Then, according to formulas (3)-(8), obtain the state X of U1 at time k 1,k , Similarly, the state X of U2 at time k can be obtained 2,k .

7. A cascaded passive positioning method based on pure angle measurement according to claim 6, characterized in that: The process of level-II positioning in S4 is: At time k, the space-based observation station U a and the space-based observation station U b have line-of-sight angles to the target T as Using the optimal estimated coordinates of the target T at time k-1 obtain the predicted coordinates at time k Using and Calculate the optimal estimated coordinates of the target T at time k through the dual-station cooperative positioning algorithm based on EKF 8. A cascade passive positioning method based on pure angle measurement according to claim 7, characterized in that: The algorithm of level-II positioning in S5 is: Space-based observation station: The motion states are known and are U respectively 1,k = [x U1,k , y U1,k , z U1,k , v xU1,k , v yU1,k , v zU1,k T ,​ U 2,k = [x U2,k , y U2,k , z U2,k , v xU2,k , v yU2,k , v zU2,k T ;​ Target: The motion information is unknown. Assume the motion state at time k is X T,k = [x T,k , y T,k , z T,k , v T,k , v T,k , v T,k T ,​ The observed quantity of the target by the space-based observation station at time k is as follows: According to the principle of the level-I positioning algorithm, obtain the state X of T at time k T,k .

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