A three-dimensional positioning method with no constraints on the position and attitude of a linear array

Abstract

The present invention discloses a three-dimensional positioning method without constraints on the position and attitude of linear arrays. A plurality of linear arrays are arranged in space, and the reference rectangular coordinate system of the space, the number of linear arrays, the spatial position coordinates of the linear arrays relative to the reference rectangular coordinate system, and the attitude of the linear arrays relative to the reference rectangular coordinate system are initialized and determined. The spatial angle between the linear array and the direction of the target incoming wave is measured. According to the spatial position coordinates of the linear array, the attitude of the linear array, and the spatial angle between the linear array and the direction of the target incoming wave, a cost function of a convex optimization problem regarding the target position and the distances from the target to each linear array is established. A constraint relationship between the target position and the distances from the target to each linear array is established, so as to determine a constrained convex optimization problem, and the rough estimate of the target position is solved by the interior point method. A linear equation system of the rough estimate of the target position and the error quantity is established, and the least squares solution of the linear equation system is calculated. The rough estimate of the target position is corrected by using the obtained error quantity to obtain the accurate position of the target.

Description

Technical Field

[0001] The present invention relates to the field of electronic information technology, and particularly to a three-dimensional positioning method with unconstrained linear array position and attitude. Background Art

[0002] Location-dependent applications are very extensive in commercial services and defense activities, which has led to great attention to the development of positioning technologies. The fifth-generation communication system (5G) widely uses array antennas to improve the throughput of data transmission, and at the same time provides the possibility for the measurement of the angle of arrival (AOA). By using the AOAs measured from different antenna arrays and combining the spatial positions of the antenna arrays themselves, the position of the terminal can be calculated.

[0003] Generally, three-dimensional positioning in a three-dimensional space requires two-dimensional AOA measurements, namely azimuth angle and elevation angle, so a planar array needs to be used. Compared with a linear array, constructing a planar array usually requires more array elements and a more complex structure. The calibration of the element position error and amplitude-phase error is also more difficult than that of a linear array, and the space occupied is larger than that of a linear array, which is restricted when installed on small unmanned aerial vehicles, ships, vehicles, etc. Usually, a linear array with omnidirectional antennas is only used to measure one-dimensional AOA in a two-dimensional space, so only the two-dimensional position of the target is estimated. In view of the advantages of the linear array in terms of structure and space occupancy, the latest research and patents (such as CN201710333182.8, CN201910232128.3, CN201910232216.3) have disclosed methods for determining the position of a target in a three-dimensional space using one-dimensional direction finding. By establishing a non-linear equation system using the spatial angles (one-dimensional AOA in a three-dimensional space) measured by multiple linear arrays and the spatial positions of the linear arrays, the three-dimensional position of the target can be solved.

[0004] The patent with the application number CN201710333182.8 and the publication date of May 12, 2017 discloses a three-dimensional positioning method based on one-dimensional direction finding, which uses the pseudo-linear least squares method to determine the three-dimensional coordinate estimation of the target position. However, this method ignores the non-linear constraint relationship between the auxiliary variables, requires at least 6 linear arrays, and has a low positioning accuracy. The patent with the application number CN201910232128.3 and the publication date of June 14, 2019 discloses a method for quickly determining the three-dimensional coordinates of a target using one-dimensional direction finding, which uses an alternating iteration method to improve the positioning accuracy. The patent with the application number CN201910232216.3 and the publication date of June 14, 2019 discloses a convex optimization method for determining the three-dimensional coordinates of a target using one-dimensional direction finding. It is an improvement on the patent CN201710333182.8, uses the constraint relationship of the auxiliary variables, and improves the positioning accuracy in some scenarios based on the convex optimization method. However, all of these methods require all antenna arrays to be horizontally arranged, which not only increases the difficulty of actual operation but also loses the spatial flexibility brought by the linear array, limits the usage scenarios of the linear array three-dimensional positioning, and the positioning accuracy is still insufficient. Summary of the Invention

[0005] To solve the problems of limited usage scenarios and insufficient positioning accuracy in three-dimensional positioning using spatial angle measurement of linear arrays in the prior art, the present invention provides a three-dimensional positioning method with no constraints on the position and attitude of the linear array. This method realizes the three-dimensional positioning of the target through a new convex optimization method based on spatial angle measurement, the attitude of the linear array itself, and the spatial position of the linear array, overcoming the scene limitations and improving the positioning accuracy at the same time.

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

[0007] A three-dimensional positioning method with no constraints on the position and attitude of the linear array, comprising the following steps:

[0008] Arrange multiple linear arrays in space, and initialize and determine the reference rectangular coordinate system of the space, the number of linear arrays, the spatial position coordinates of each linear array relative to the reference rectangular coordinate system, and the attitude of each linear array relative to the reference rectangular coordinate system, that is, the angle between the projection of each linear array on the x-y plane of the reference rectangular coordinate system and the x-axis of the reference rectangular coordinate system, and the angle between each linear array and the x-y plane of the reference rectangular coordinate system;

[0009] Measure the spatial angle between each linear array and the direction of the target incoming wave;

[0010] Based on the spatial position coordinates of each linear array, the angle between the projection of each linear array on the x-y plane of the reference rectangular coordinate system and the x-axis of the reference rectangular coordinate system, the angle between each linear array and the x-y plane of the reference rectangular coordinate system, and the spatial angle between each linear array and the target incoming wave direction, establish a convex optimization problem cost function for the target position and the distances from the target to each linear array;

[0011] Establish the constraint relationship between the target position and the distances from the target to each linear array according to the geometric relationship, so as to determine the constrained convex optimization problem, and solve the rough estimate of the target position by the interior point method;

[0012] Establish a linear equation system for the rough estimate of the target position and the error quantity according to the geometric relationship, calculate the least squares solution of the linear equation system, and then use the obtained error quantity to correct the rough estimate of the target position to finally obtain the accurate position of the target.

[0013] Preferably, set the number of linear arrays to M, and the spatial position coordinates of the m-th linear array to s m , m = 1, 2,..., M, the angle between the projection of the m-th linear array on the x-y plane of the reference rectangular coordinate system and the x-axis of the reference rectangular coordinate system is α m , m = 1, 2,..., M, the angle between the m-th linear array and the x-y plane of the reference rectangular coordinate system is β m , m = 1, 2,..., M, the attitude vector of the m-th linear array is γ m , m = 1, 2,..., M, the spatial angle between the m-th linear array and the target incoming wave direction is θ m , m = 1, 2,..., M; the convex optimization problem cost function is:

[0014] tr(XH),

[0015] where, U = uu T , u = [x, y, z, γ1, γ2,..., γ m T is the unknown,

[0016]

[0017]

[0018]

[0019] W is the weighting matrix, which is determined by the following formula:

[0020] W = (TQT) -1 ,

[0021]

[0022] ​Q is the covariance matrix of the spatial angles.

[0023] Preferably, the constraint relationship is:

[0024]

[0025]

[0026] u(i + 3) > 0,

[0027] i, j = 1, 2,..., M, and i ≠ j;

[0028] The above constraint relationship and the cost function of the convex optimization problem form a constrained convex optimization problem. The rough estimate of the target three-dimensional position can be obtained by the interior point method as where λ and e are respectively the largest eigenvalue of U and the corresponding eigenvector.

[0029] Preferably, the linear equation system is:

[0030] a - Bu + BFδ = Tn,

[0031] where n is the noise vector,

[0032]

[0033] I3 is the identity matrix,

[0034] Then the error quantity is:

[0035] δ = -(F T B T WBF) -1 F T B T W(a - Bu),

[0036] The exact position of the target is:

[0037] p2 = p1 - δ.

[0038] The beneficial effects of the present invention are as follows: By using an observation station equipped with a linear array to measure the spatial angles of the target, three-dimensional positioning of the target is achieved through a combination of convex optimization and error correction. On the one hand, the limitations of the existing methods on the deployment attitude and station positions are overcome. On the other hand, the positioning accuracy is greatly improved. Specific embodiments

[0039] The present invention will be described in detail below in conjunction with the embodiments.

[0040] Embodiment

[0041] In this embodiment, five observation stations equipped with linear arrays at known positions and one target to be located in three-dimensional space are taken as examples for illustration. The position coordinates of each observation station are (300, 0, 94), (187, 235, 122), (-67, 292, -112), (-270, 130, 124), (-270, -130, 39) (unit: meter) respectively. Each observation station has a 4-element uniform linear array with an element spacing of 0.0625 meters. The angles between each linear array and the x-axis of the reference rectangular coordinate system are {0, 1.2566, 2.5133, 3.7699, 5.0265} (unit: radian) respectively, and the angles between each linear array and the x-y plane of the reference rectangular coordinate system are {0, 0.8130, 0.2280, 2.4600, 2.2700} (unit: radian) respectively. The signal center frequency is 2.4 GHz, and the true three-dimensional coordinates of the target position are (424, 519, 375) (unit: meter).

[0042] The specific positioning process is as follows:

[0043] Step 1: Set the observation stations at arbitrary positions in three-dimensional space. Initialize and determine that the number of observation stations M = 5, determine the position coordinates of each observation station as (300, 0, 94), (187, 235, 122), (-67, 292, -112), (-270, 130, 124), (-270, -130, 39) (unit: meter) respectively, determine the attitude angles of each observation station's linear array (i.e., the angles between the linear array and the x-axis of the reference rectangular coordinate are {0, 1.2566, 2.5133, 3.7699, 5.0265} (unit: radian) respectively, and the angles between the linear array and the x-y plane of the reference rectangular coordinate system are {0, 0.8130, 0.2280, 2.4600, 2.2700} (unit: radian) respectively), and determine the spatial angle measurements between each linear array and the direction of the target incoming wave as {1.3876, 0.3809, 1.7628, 0.3989, 1.0064} (unit: radian);

[0044] Step 2: Establish a convex optimization problem cost function regarding the target position and the distances from the target to each observation station based on the coordinates of each observation station, the attitude angles of each linear array, and the spatial angle measurements of each linear array as follows:

[0045] tr(XH),

[0046] where,

[0047]

[0048]

[0049]

[0050]

[0051] Step 3: Establish the constraint relationship between the target position and the distances from the target to each observation station according to the geometric relationship as follows:

[0052]

[0053]

[0054] u(i + 3) > 0,

[0055] i, j = 1, 2,..., M, and i ≠ j;

[0056] The above constraint relationship and the cost function of the convex optimization problem form a constrained convex optimization problem. The rough estimate p1 of the target three-dimensional position can be obtained by the interior point method as (379.65, 473.43, 416.95) (unit: meter);

[0057] Step 4: Establish a linear equation system about the rough estimate p1 of the target position and the error quantity δ as follows:

[0058] a - Bu + BFδ = Tn,

[0059] where n is the noise vector,

[0060]

[0061]

[0062] Then the weighted least squares estimate of the error quantity is:

[0063] (-33.52, -47.53, 55.78) (unit: meter),

[0064] The corrected accurate estimate of the target is:

[0065] (413.18, 525.05, 354.87) (unit: meter),

[0066] Define the positioning error as the distance between the positioning position coordinates of the target and the actual position coordinates of the target. In this embodiment, the actual position coordinates of the target are (424, 519, 375) (unit: meter). It can be seen that the positioning error of implementing the method of the present invention is equal to 23.6427 meters. In particular, in this embodiment, the array is arranged in a three-dimensional space and the array arrangement is in any direction. The existing patents (CN201710333182.8, CN201910232128.3, CN201910232216.3) are not applicable to the scenario of this embodiment.

[0067] Embodiment 2

[0068] In this embodiment, five observation stations equipped with linear arrays with known positions and one target to be positioned in three-dimensional space are taken as examples for illustration. The position coordinates of each observation station are (3, -250, 0), (37, -114, 0), (-89, 213, 0), (-167, 105, 0), (193, -23, 0) (unit: meter) respectively. Each observation station has a uniform linear array with 4 array elements, and the element spacing is 0.0625 meter. The angles between each linear array and the x-axis of the reference rectangular coordinate system are {0, 1.2566, 2.5133, 3.7699, 5.0265} (unit: radian) respectively, and the angles between each linear array and the x-y plane of the reference rectangular coordinate system are all 0 (unit: radian). The signal center frequency is 2.4 GHz, and the true three-dimensional coordinates of the target position are (424, 519, 375) (unit: meter).

[0069] The specific positioning process is as follows:

[0070] Step 1: Set the observation stations at arbitrary positions in three-dimensional space, initialize and determine that the number of observation stations M = 5, determine that the position coordinates of each observation station are (3, -250, 0), (37, -114, 0), (-89, 213, 0), (-167, 105, 0), (193, -23, 0) (unit: meter) respectively, determine the attitude angles of each observation station's linear array (that is, the angles between the linear array and the x-axis of the reference rectangular coordinate are {0, 1.2566, 2.5133, 3.7699, 5.0265} (unit: radian) respectively, and the angles between the linear array and the x-y plane of the reference rectangular coordinate system are all 0 (unit: radian), and determine that the spatial angle measurements of each linear array with respect to the target incoming wave direction are {1.1374, 0.5425, 1.8995, 2.6713, 2.2335} (unit: radian);

[0071] Step 2: Establish a convex optimization problem cost function regarding the target position and the distances from the target to each observation station according to the coordinates of each observation station, the attitude angles of each linear array, and the spatial angle measurements of each linear array as follows:

[0072] tr(XH),

[0073] where,

[0074]

[0075]

[0076]

[0077]

[0078] Step 3: According to the geometric relationship, the constraint relationship between the target position and the distances from the target to each observation station is established as follows:

[0079]

[0080]

[0081] u(i + 3)>0,

[0082] i, j = 1, 2,..., M, and i ≠ j;

[0083] The above constraint relationship and the cost function of the convex optimization problem form a constrained convex optimization problem. By using the interior point method, the rough estimate p1 of the target three-dimensional position can be obtained as (407.28, 504.05, 376.16) (unit: meter);

[0084] Step 4: Establish a linear equation system about the rough estimate p1 of the target position and the error quantity δ as follows:

[0085] a - Bu + BFδ = Tn,

[0086] where n is the noise vector,

[0087]

[0088]

[0089] Then the weighted least squares estimate of the error quantity is:

[0090] (-11.78, -14.59, -13.04) (unit: meter),

[0091] The corrected accurate estimate of the target is:

[0092] (419.06, 518.64, 389.20) (unit: meter),

[0093] The positioning error is defined as the distance between the positioning position coordinates of the target and the actual position coordinates of the target. In this embodiment, the actual position coordinates of the target are (424, 519, 375) (unit: meter), and the positioning error of implementing the method of the present invention is equal to 14.10 meters. The positioning errors of patents CN201710333182.8, CN201910232128.3, and CN201910232216.3 are 770.82 meters, 174.37 meters, and 701.79 meters respectively. It can be seen that the positioning error of the method of this patent is much smaller than the above-mentioned existing patents.

[0094] The above-described embodiments merely represent specific embodiments of the present invention. Their descriptions are relatively specific and detailed, but they should not be construed as limiting the scope of the patent for the present invention. It should be noted that for those of ordinary skill in the art, without departing from the concept of the present invention, several modifications and improvements can still be made, and these all fall within the protection scope of the present invention.

Claims

1. A three-dimensional positioning method without constraints on the position and attitude of a linear array, characterized in that, It includes the following steps: Arrange multiple linear arrays in space, and initialize to determine the reference rectangular coordinate system of the space, the number of linear arrays, the spatial position coordinates of each linear array relative to the reference rectangular coordinate system, and the attitude of each linear array relative to the reference rectangular coordinate system, that is, the angle between the projection of each linear array on the x-y plane of the reference rectangular coordinate system and the x-axis of the reference rectangular coordinate system, and the angle between each linear array and the x-y plane of the reference rectangular coordinate system; Measure the spatial angle between each linear array and the direction of the target incoming wave; Based on the spatial position coordinates of each linear array, the angle between the projection of each linear array on the x-y plane of the reference rectangular coordinate system and the x-axis of the reference rectangular coordinate system, the angle between each linear array and the x-y plane of the reference rectangular coordinate system, and the spatial angle between each linear array and the direction of the target incoming wave, establish a convex optimization problem cost function regarding the target position and the distances from the target to each linear array; Establish the constraint relationship between the target position and the distances from the target to each linear array according to the geometric relationship, so as to determine the constrained convex optimization problem, and solve the rough estimate of the target position by the interior point method; set the number of linear arrays as M, and the spatial position coordinates of the m-th linear array as s m , m = 1, 2, …, M, the angle between the projection of the m-th linear array on the x-y plane of the reference rectangular coordinate system and the x-axis of the reference rectangular coordinate system is α m , m = 1, 2, …, M, the angle between the m-th linear array and the x-y plane of the reference rectangular coordinate system is β m , m = 1, 2, …, M, the attitude vector of the m-th linear array is γ m , m = 1, 2, …, M, the spatial angle between the m-th linear array and the target incoming wave direction is θ m , m = 1, 2, …, M; the cost function of the convex optimization problem is as follows: tr(XH), wherein, U = uu T , u = [x, y, z, γ1, γ2, …, γ m T are unknowns,​ W is a weighting matrix, which is determined by the following formula: W = (TQT) -1 , Q is the covariance matrix of the spatial angle; Establish a linear equation system for the rough estimation of the target position and the error quantity according to the geometric relationship, calculate the least squares solution of the linear equation system, and then use the obtained error quantity to correct the rough estimation of the target position, and finally obtain the accurate position of the target.

2. The three-dimensional positioning method without constraints on the position and attitude of the linear array according to claim 1, characterized in that The constraint relationship is: u(i + 3)>0, i,j = 1,2,…,M, and i≠j; The above constraint relationships and the cost function of the convex optimization problem form a constrained convex optimization problem. Through the interior point method, a rough estimate of the target three-dimensional position can be obtained as where λ and e are the maximum eigenvalue of U and the corresponding eigenvector, respectively.

3. The three-dimensional positioning method without constraints on the position and attitude of the linear array according to claim 2, characterized in that, The linear equation system is: a - Bu + BFδ = Tn, where, n is the noise vector, I3 is the identity matrix, then the error quantity is: δ = -(F T B T WBF) -1 F T B T W(a - Bu), The accurate position of the target is: p2 = p1 - δ.

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