A three-dimensional angle of arrival positioning method and system based on geometric center intersection positioning

Abstract

This invention discloses a three-dimensional angle-of-arrival (AOA) positioning method and system based on geometric center intersection positioning. The method first establishes a three-dimensional AOA positioning system model based on geometric center intersection, expressing the azimuth and elevation angles as nonlinear functions related to the target distance. Second, it establishes three-dimensional AOA positioning rules based on the inscribed sphere center. At each measurement node, two planes are extended based on the two angles, and the distance from any point in space to these planes is calculated. Finally, based on the definition of the inscribed sphere of a three-dimensional tetrahedron, the inscribed sphere center of the geometric body enclosed by these geometric planes is obtained as the estimated target position. Simultaneously, based on the radius of the inscribed sphere obtained at the same time, the noise variance of the current angle measurement is calculated. The system includes a system model construction module, a plane extension module, and a target position estimation module. This invention achieves high performance with extremely low complexity, is suitable for three-dimensional AOA positioning scenarios, and has a wide range of applications.

Description

Technical Field

[0001] This invention relates to the field of wireless communication technology, and in particular to a three-dimensional (3D) angle of arrival (AOA) positioning method and system based on geometric center cross-positioning. Background Technology

[0002] AOA positioning is a wireless positioning method, which, along with fingerprint positioning, received energy intensity (RSS) positioning, and time of arrival (TOA) positioning, are among the more popular positioning methods today.

[0003] Fingerprint positioning relies on the varying channel state information at different locations, requiring pre-training. Therefore, it's unsuitable for outdoor scenarios where channel characteristics change rapidly or pre-training is impossible. RSS-based positioning is simple in structure, requiring only a single antenna for the receiver, but it's susceptible to uneven antenna patterns in practice. TOA-based positioning is more commonly used; current GPS systems are based on TOA technology. However, due to the high speed of light, time synchronization and calculation between different nodes can easily introduce errors.

[0004] AOA positioning uses angle information to locate a target, and is mainly affected by the number of antennas. However, with the development of fifth-generation communication technology (5G) and massive MIMO, most devices are equipped with more and more antennas, which has accelerated the popularization of AOA positioning. For example, new Bluetooth protocols now use AOA positioning for position estimation.

[0005] However, most current methods are based on 2D scenes, with few studies on 3D localization. Furthermore, the addition of pitch angles makes many 2D methods unsuitable for direct application in 3D scenes. Therefore, a low-complexity, high-performance 3D AOA localization method is needed. Summary of the Invention

[0006] The purpose of this invention is to provide a three-dimensional angle of arrival (AOA) estimation method and system based on geometric center cross-positioning, which achieves high three-dimensional AOA estimation performance with extremely low complexity.

[0007] The technical solution to achieve the purpose of this invention is: a three-dimensional arrival angle positioning method based on geometric center intersection positioning, the steps of which are as follows:

[0008] Step 1: Based on the azimuth and elevation angles measured by the measurement nodes, establish a three-dimensional angle-of-arrival positioning system model based on the intersection of geometric centers, and write the azimuth and elevation angles as nonlinear functions related to the target distance;

[0009] Step 2: Based on the system model, establish a three-dimensional arrival angle positioning rule based on the center of the inscribed sphere. At each measurement node, extend two planes based on two angles and calculate the distance from any point in space to these planes.

[0010] Step 3: Based on the definition of the inscribed sphere of a tetrahedron in three-dimensional space, find the center of the inscribed sphere of the geometric body enclosed by these geometric planes, and use it as the estimated target position. At the same time, based on the radius of the inscribed sphere obtained simultaneously, calculate the noise variance of the current angle measurement.

[0011] A three-dimensional angle-of-arrival (AOA) positioning system based on geometric center intersection positioning is disclosed. This system implements the aforementioned three-dimensional AOA positioning method based on geometric center intersection positioning. The system includes a system model construction module, a plane expansion module, and a target position estimation module, wherein:

[0012] The system model construction module establishes a three-dimensional angle-of-arrival positioning system model based on the azimuth and elevation angles measured by the measurement nodes, and writes the azimuth and elevation angles as nonlinear functions related to the target distance.

[0013] The planar expansion module establishes a three-dimensional arrival angle positioning rule based on the inscribed sphere center according to the system model. At each measurement node, two planes are expanded based on two angles, and the distance from any point in space to these planes is calculated.

[0014] The target position estimation module, based on the definition of the inscribed sphere of a tetrahedron in three-dimensional space, obtains the inscribed center of the geometric body enclosed by these geometric planes, which is used as the estimated target position. At the same time, based on the radius of the inscribed sphere obtained simultaneously, the noise variance of the current angle measurement is calculated.

[0015] A three-dimensional angle-of-arrival positioning device based on geometric center cross-positioning, the device comprising: a processor, and a non-transient computer-readable storage medium connected to the processor via a bus; the non-transient computer-readable storage medium storing one or more computer programs executable by the processor; the processor executing the one or more computer programs implements the steps of the three-dimensional angle-of-arrival positioning method based on geometric center cross-positioning.

[0016] A non-transitory computer-readable storage medium storing instructions that, when executed by a processor, cause the processor to perform steps in the three-dimensional angle-of-arrival positioning method based on geometric center cross-positioning.

[0017] Compared with the prior art, the significant advantages of this invention are: (1) The azimuth and elevation angles are extended into two planes respectively. These planes in space will form a polyhedron around the target position. The inscribed center of the geometry is obtained. The inscribed center is the estimated geometric position of the target. It achieves high performance with extremely low complexity. The complexity is close to that of the traditional LS method, but the performance is much higher than that of LS. It can approach the performance of maximum likelihood estimation. At the same time, it can also estimate the noise variance of the current angle measurement; (2) It is applicable to three-dimensional AOA positioning scenarios. It has a wide range of applications and can be used for indoor and outdoor positioning.

[0018] Additional aspects and advantages of the invention will be set forth in part in the description which follows, and will become apparent from the description or may be learned by practice of the invention. Attached Figure Description

[0019] Figure 1 This is a block diagram of a 3D AOA positioning system with M nodes.

[0020] Figure 2 This is a flowchart of a three-dimensional angle-of-arrival positioning method based on geometric center intersection positioning.

[0021] Figure 3 This is a schematic diagram of the inscribed center of a tetrahedron.

[0022] Figure 4 These are the mean square error performance graphs of different methods under different measurement noise levels.

[0023] Figure 5 This is a graph showing the mean square error performance of different methods under different numbers of nodes. Detailed Implementation

[0024] This invention provides a three-dimensional angle-of-arrival positioning method based on geometric center intersection positioning, the steps of which are as follows:

[0025] Step 1: Based on the azimuth and elevation angles measured by the measurement nodes, establish a three-dimensional angle-of-arrival positioning system model based on the intersection of geometric centers, and write the azimuth and elevation angles as nonlinear functions related to the target distance;

[0026] Step 2: Based on the system model, establish a three-dimensional arrival angle positioning rule based on the center of the inscribed sphere. At each measurement node, extend two planes based on two angles and calculate the distance from any point in space to these planes.

[0027] Step 3: Based on the definition of the inscribed sphere of a tetrahedron in three-dimensional space, find the center of the inscribed sphere of the geometric body enclosed by these geometric planes, and use it as the estimated target position. At the same time, based on the radius of the inscribed sphere obtained simultaneously, calculate the noise variance of the current angle measurement.

[0028] As a specific example, in step 1, a three-dimensional angle-of-arrival positioning system model based on the intersection of geometric centers is established, and the azimuth and pitch angles are written as nonlinear functions related to the target distance, as follows:

[0029] Assume there are M devices measuring azimuth and elevation angles, and the positions of the target and the devices are denoted as u0 = [x0, y0, z0], respectively. T and u m =[x m y m , z m ] T m = 1, 2, ..., M; x m y m , z m Let θ represent the three-dimensional coordinates of the m-th device in space, and let θ represent the pitch and azimuth angles measured by the m-th device. m and φ m ;

[0030] Based on geometric principles, the relationship between the azimuth and elevation angles measured at the m-th node and the target is as follows:

[0031]

[0032]

[0033] Where, θ m ∈(-π, π) and θ m Let ∈(-π / 2, π / 2) be the azimuth and elevation angles respectively, and u0 = [x0, y0, z0]. T Let u be the three-dimensional coordinates of the target. m =[x m y m , z m ] T , m = 1, 2, ..., M represent the three-dimensional coordinates of the m-th measurement node; sin() and cos() represent the sine and cosine functions respectively, and arctan() represents the arctangent function;

[0034] Considering measurement noise, the actual measured angle is expressed as:

[0035] θ r =θ+n θ (3)

[0036] φ r =φ+n φ (4) Among them, θ = [θ1, θ2,…, θ M ] T and φ=[φ1, φ2, …φ M ] TA vector composed of the true values ​​of angles; and n φ =[n φ n φ ,…,n φ ] T It is additative Gaussian white noise. and I M It is an M×M identity matrix. Represents n θ The covariance matrix is It follows a normal distribution.

[0037] As a specific example, in step 2, based on the system model, a three-dimensional arrival angle positioning rule is established based on the center of the inscribed sphere. At each measurement node, two planes are extended based on two angles, and the distance from any point in space to these planes is calculated, as follows:

[0038] For the azimuth angle of the m-th node, extend the coordinates of that node along the azimuth direction perpendicular to the plane to form the first plane. For any point [x, y, z] in the first plane... T Satisfy the following relationship

[0039] tanθ m xy-tanθ m x m +y m =0 (5)

[0040] Where tan() represents the tangent function, θ m Let x represent the azimuth angle measured by the m-th measurement node. m y m , z m Represents the three-dimensional coordinates of the m-th measurement node;

[0041] Then, passing through this node, along the pitch direction, the angle between the pitch direction and the horizontal plane is also φ. m Extending to a second plane, the set of points in the second plane is represented as...

[0042] tanφ m cosθ m x+-tanφ m sinθ m yz-tanφ m cosθ m x m -tanφ m sinθ m y m +z m =0 (6)

[0043] Where sin() and cos() represent the sine and cosine functions respectively, tan() represents the tangent function, and x m y m , z m Let θ represent the three-dimensional coordinates of the m-th measurement node. m φ represents the azimuth angle measured by the m-th measuring node. m This represents the pitch angle measured by the m-th measurement node;

[0044] Assume any point in space is Calculate the Euclidean distances from the point to the first and second planes, respectively. and

[0045] As a specific example, in step 3, based on the definition of the inscribed sphere of a tetrahedron in three-dimensional space, the inscribed center of the geometric body enclosed by these geometric planes is obtained as the estimated target position. At the same time, based on the radius of the inscribed sphere obtained simultaneously, the noise variance of the current angle measurement is calculated, as follows:

[0046] According to the definition of the incenter of a tetrahedron, the incenter of a tetrahedron is equidistant from the four planes, and equal to the radius of the sphere, i.e., d. m，θ =d m，φ = r, where r is the radius of the inscribed sphere;

[0047] To find the inscribed sphere of a multiplane, the objective function is written as:

[0048]

[0049] Where u is the coordinate of the center of the inscribed sphere, and d m，θ and d m，φ Represent the θ measured from u to the m-th node. m and φ m The distance between the formed planes;

[0050] The SFP method for calibration points is used to solve for the coordinates of the sphere's center and the radius of the inscribed sphere. Based on the obtained radius r of the inscribed sphere, the variance of the current angle measurement is calculated.

[0051]

[0052] Where cos() represents the cosine function, M represents the number of measurement nodes, and d m This represents the estimated distance from the center of the sphere to the plane corresponding to the m-th node.

[0053] This invention also provides a three-dimensional angle-of-arrival positioning system based on geometric center intersection positioning. This system is used to implement the aforementioned three-dimensional angle-of-arrival positioning method based on geometric center intersection positioning. The system includes a system model construction module, a plane expansion module, and a target position estimation module, wherein:

[0054] The system model construction module establishes a three-dimensional angle-of-arrival positioning system model based on the azimuth and elevation angles measured by the measurement nodes, and writes the azimuth and elevation angles as nonlinear functions related to the target distance.

[0055] The planar expansion module establishes a three-dimensional arrival angle positioning rule based on the inscribed sphere center according to the system model. At each measurement node, two planes are expanded based on two angles, and the distance from any point in space to these planes is calculated.

[0056] The target position estimation module, based on the definition of the inscribed sphere of a tetrahedron in three-dimensional space, obtains the inscribed center of the geometric body enclosed by these geometric planes, which is used as the estimated target position. At the same time, based on the radius of the inscribed sphere obtained simultaneously, the noise variance of the current angle measurement is calculated.

[0057] The present invention also provides a three-dimensional angle-of-arrival positioning device based on geometric center cross-positioning. The device includes: a processor and a non-transient computer-readable storage medium connected to the processor via a bus; the non-transient computer-readable storage medium stores one or more computer programs that can be executed by the processor; when the processor executes the one or more computer programs, it implements the steps in the three-dimensional angle-of-arrival positioning method based on geometric center cross-positioning.

[0058] The present invention also provides a non-transitory computer-readable storage medium storing instructions that, when executed by a processor, cause the processor to perform the steps in the three-dimensional angle-of-arrival positioning method based on geometric center cross-positioning.

[0059] The present invention will be further illustrated below with reference to the accompanying drawings and specific examples. It should be understood that these examples are for illustrative purposes only and are not intended to limit the scope of the invention. After reading this invention, any modifications of the invention in various equivalent forms by those skilled in the art will fall within the scope defined by the appended claims.

[0060] Example

[0061] Combination Figures 1-2 The three-dimensional angle-of-arrival positioning method based on geometric center intersection positioning of the present invention includes the following steps:

[0062] Step 1: Establish a 3D AOA positioning system model based on the intersection of geometric centers, such as... Figure 1As shown, assume there are a total of M devices that can measure azimuth and elevation angles.

[0063] The locations of the target and the equipment are denoted as follows:

[0064] u0 = [x0, y0, z0] T

[0065] u m =[x m y m , z m ] T m = 1, 2, ..., M (9)

[0066] The pitch and azimuth angles measured by the m-th device are denoted as θ. m and φ m .according to Figure 1 The geometric relationship shown indicates that

[0067]

[0068]

[0069] Where, θ m ∈(-π, π) and θ m ∈(-π / 2, π / 2).

[0070] In actual measurements, angle measurements will always have a certain degree of error; therefore, the received angle information can be written as...

[0071] θ r =θ+nθ (12)

[0072] φ r =φ+n φ (13)

[0073] where θ=[θ1, θ2,…,θ M ] T and φ=[φ1, φ2, …φ M ] T It is a true value. andn φ =[n φ n φ ,…,n φ ] T For addable Gaussian white noise, the following distribution relationship must be satisfied: and

[0074] Now, two planes are defined based on the two angles measured by each device. The first plane contains the line u0u m It is also perpendicular to the xoy plane. Therefore, this plane can be written as

[0075] tanθ m xy-tanθ m x m +y m =0 (14)

[0076] Where, [x, y, z] T Let be any point on the plane. The second plane also contains the line u0u. m At the same time, the angle between it and the xoy plane is φ. m This plane can be written as

[0077] tanφ m cosθ m x+tanφ m sinθ m yz-tanφ m cosθ m x m -tanφ m sinθ m y m +z m =0 (15)

[0078] Now, suppose any fixed point in space is The Euclidean distance between this point and the two planes defined above can be written as:

[0079]

[0080]

[0081] Then, to simplify the expression of the two equations above, we define the following vectors and scalars.

[0082] a m，θ =[sinθ m -cosθ m ,0] T (18)

[0083] b m，θ =sinθ m x m -cosθ m y m (19)

[0084] a m，φ =[sinφ m cosθ m sinφ m sinθ m -cosφ m ] T(20)

[0085] b m，φ =sinφ m cosθ m x m +sinφ m sinθ m y m -cosφ m z m (twenty one)

[0086] Then, substituting (18)-(21) into (16) and (17), we can obtain

[0087]

[0088]

[0089] This completes the system modeling for 3DAOA positioning based on geometric center intersection.

[0090] Step 2 proposes a 3DAOA localization method based on the center of the inscribed sphere. For example... Figure 3 As shown, the center of an inscribed sphere of a tetrahedron is equidistant from the four planes, and the distances are equal to the radius of the sphere. That is, d m，θ =d m，φ = r, where r is the radius of the inscribed sphere. Therefore, to find the inscribed sphere of a multiplane, the objective function can be written as:

[0091]

[0092] It can be seen that this optimization function is a non-convex function with a rough surface, making it difficult to solve using methods for convex functions. However, a minimization method, the fixed-point method, can be used to solve it. First, calculate the partial derivatives of this function with respect to u and r.

[0093]

[0094]

[0095] Then, the result can be solved iteratively using the following two formulas.

[0096]

[0097]

[0098] in

[0099]

[0100]

[0101]

[0102] Next, let's look at the calculated radius r. Assuming the angle measurement error is small, we can obtain the following approximation.

[0103]

[0104] Substituting it into (27) and combining it with geometric relationships, we can obtain

[0105]

[0106] When the angular noise is equal, the following condition is met: At that time, we can estimate the variance of the angle noise based on r.

[0107]

[0108] This completes the derivation of 3DAOA based on the inscribed ball.

[0109] Step 3 proposes a 3DAOA localization method based on minimum mean square distance. Observing (24), it can be found that the introduction of radius r leads to the objective function being non-smooth and non-convex. At the same time, considering that in practical applications, it is not necessarily necessary to estimate the noise variance of the angle, we can subtract r and transform the optimization objective into minimum mean square distance, which can be written as

[0110]

[0111] Redefining

[0112]

[0113]

[0114] in

[0115]

[0116] Substituting it into (35), we can get

[0117]

[0118] Clearly, this can be solved using the least squares method, and we can obtain...

[0119]

[0120] Therefore, this method has low computational complexity. However, since A contains noise, more accurate solutions can be obtained using full least squares or weighted least squares.

[0121] This completes the derivation of the minimum mean square distance 3DAOA localization.

[0122] exist Figure 4 In the diagram, we present the mean square error (MSE) of angle measurement errors using different methods. The target position is in the range [0, 0, 0]. T At this location, 20 devices are randomly distributed within a 50×50×50 cubic space, with its center at [0, 0, 0]. T All parameters and results are displayed in dB. σ 2 The maximum likelihood estimation (MLE) is almost identical to CRLB, which aligns with common sense. Furthermore, our proposed inscribed sphere center method and minimum mean square distance method can approach CRLB. Moreover, both proposed methods offer approximately 8 dB of gain compared to LS. More importantly, our proposed method has a significantly lower MLE.

[0123] Figure 5 The relationship between MSE and the number of devices is shown when using different 3D AOA methods. The devices are randomly distributed in space. It can be seen that both proposed methods have a gain of approximately 8 dB compared to LS. Furthermore, the performance differences of all these methods are similar. Notably, when the number of devices decreases, all methods approach CRLB. In summary, the proposed method exhibits satisfactory performance and low computational complexity.

Claims

1. A three-dimensional angle-of-arrival positioning method based on geometric center intersection positioning, characterized in that, The steps are as follows: Step 1: Based on the azimuth and elevation angles measured by the measurement nodes, establish a three-dimensional angle-of-arrival positioning system model based on the intersection of geometric centers, and write the azimuth and elevation angles as nonlinear functions related to the target distance; Step 2: Based on the system model, establish a three-dimensional arrival angle positioning rule based on the center of the inscribed sphere. At each measurement node, extend two planes based on two angles and calculate the distance from any point in space to these planes. Step 3: Based on the definition of the inscribed sphere of a tetrahedron in three-dimensional space, find the inscribed center of the geometric body enclosed by these geometric planes, which is used as the estimated target position. At the same time, based on the radius of the inscribed sphere obtained simultaneously, calculate the noise variance of the current angle measurement. In step 1, a three-dimensional angle-of-arrival positioning system model based on the intersection of geometric centers is established, and the azimuth and elevation angles are expressed as nonlinear functions related to the target distance, as follows: Assuming there are a total of A device for measuring azimuth and elevation angles; the positions of the target and the device are denoted as follows: and ; They represent the first The three-dimensional coordinates of the device in space, the first The pitch and azimuth angles measured by each device are denoted as follows: and ; According to geometric principles, the first The relationship between the azimuth and elevation angles measured by each node and the target is shown below. (1) (2) in, and These are the azimuth and elevation angles, respectively. The three-dimensional coordinates of the target Indicates the first The three-dimensional coordinates of each measurement node; and Let them represent the sine and cosine functions, respectively. Represents the arctangent function; Considering measurement noise, the actual measured angle is expressed as: (3) (4) in, and A vector composed of the true values ​​of angles; and It is additative Gaussian white noise. and ; for The identity matrix, express The covariance matrix is The normal distribution; In step 2, based on the system model, a three-dimensional arrival angle positioning rule is established based on the center of the inscribed sphere. At each measurement node, two planes are extended based on two angles, and the distance from any point in space to these planes is calculated, as follows: For the The azimuth angle of each node, and the coordinates of that node extending perpendicularly to the plane along the azimuth direction, form the first plane. Any point in the first plane... Satisfy the following relationship (5) in, Represents the tangent function. Indicates the first The azimuth angle measured by each measurement node. Indicates the first The three-dimensional coordinates of each measurement node; Then, passing through this node, along the pitch direction, the angle with the horizontal plane is also... Extending to a second plane, the set of points in the second plane is represented as... (6) in, and Let them represent the sine and cosine functions, respectively. Represents the tangent function. Indicates the first The three-dimensional coordinates of each measurement node Indicates the first The azimuth angle measured by each measurement node. Indicates the first The pitch angle measured by each measurement node; Assume any point in space is Calculate the Euclidean distances from the point to the first and second planes, respectively. and ; In step 3, based on the definition of the inscribed sphere of a tetrahedron in three-dimensional space, the inscribed center of the geometric body enclosed by these geometric planes is obtained, which serves as the estimated target position. Simultaneously, based on the radius of the inscribed sphere obtained at the same time, the noise variance of the current angle measurement is calculated, as follows: According to the definition of the incenter of a tetrahedron, the distance from the center of the incenter of the tetrahedron to each of the four planes is equal to the radius of the sphere. ,in Let be the radius of the inscribed sphere; To find the inscribed sphere of a multiplane, the objective function is written as: (7) in, To find the coordinates of the center of the inscribed sphere, and They represent To the Measured by each node and The distance between the formed planes; The SFP method of calibration points is used to solve for the coordinates of the sphere center and the radius of the inscribed sphere. Based on the obtained radius of the inscribed sphere... The noise variance of the current angle measurement is calculated as follows: (8) in, Let them represent the cosine function, Indicates the number of measurement nodes. Indicates the estimated center of the ball up to the 1st. The distance between the planes corresponding to each node.

2. A three-dimensional angle-of-arrival positioning system based on geometric center intersection positioning, characterized in that, This system is used to implement the three-dimensional angle-of-arrival positioning method based on geometric center intersection positioning as described in claim 1. The system includes a system model construction module, a plane expansion module, and a target position estimation module, wherein: The system model construction module establishes a three-dimensional angle-of-arrival positioning system model based on the azimuth and elevation angles measured by the measurement nodes, and writes the azimuth and elevation angles as nonlinear functions related to the target distance. The planar expansion module establishes a three-dimensional arrival angle positioning rule based on the inscribed sphere center according to the system model. At each measurement node, two planes are expanded based on two angles, and the distance from any point in space to these planes is calculated. The target position estimation module, based on the definition of the inscribed sphere of a tetrahedron in three-dimensional space, obtains the inscribed center of the geometric body enclosed by these geometric planes, which is used as the estimated target position. At the same time, based on the radius of the inscribed sphere obtained simultaneously, the noise variance of the current angle measurement is calculated.

3. A three-dimensional angle-of-arrival positioning device based on geometric center intersection positioning, characterized in that, The device includes: a processor and a non-transitory computer-readable storage medium connected to the processor via a bus; the non-transitory computer-readable storage medium stores one or more computer programs executable by the processor; when the processor executes the one or more computer programs, it implements the steps of the three-dimensional angle-of-arrival positioning method based on geometric center cross-positioning as described in claim 1.

4. A non-transitory computer-readable storage medium, wherein the non-transitory computer-readable storage medium stores instructions, characterized in that, When executed by the processor, the instructions cause the processor to perform the steps in the three-dimensional angle-of-arrival positioning method based on geometric center cross-positioning as described in claim 1.

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