CN109889971B - Base station three-dimensional cooperative positioning method applied to large indoor environment - Google Patents

Base station three-dimensional cooperative positioning method applied to large indoor environment Download PDF

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CN109889971B
CN109889971B CN201910094117.3A CN201910094117A CN109889971B CN 109889971 B CN109889971 B CN 109889971B CN 201910094117 A CN201910094117 A CN 201910094117A CN 109889971 B CN109889971 B CN 109889971B
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CN109889971A (en
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王智
王晓轩
陈敏麟
王雅菲
嵇程
柴晴峰
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Zhejiang University ZJU
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Abstract

The invention discloses a base station three-dimensional cooperative positioning method applied to a large indoor environment, which comprises the following steps: (1) manually measuring three-dimensional coordinates of at least four anchor base stations, and estimating measurement distances among unknown base stations and between the unknown base stations and the anchor base stations by a TPSN (tire pressure sensor and sensor network) ranging principle; (2) representing the position coordinates of the base stations by using a matrix, and multiplying the position coordinates by using the matrix to obtain the actual distance between the base stations represented by the coordinates; (3) relaxing the coordinate matrix in the step (2) into a semi-definite matrix, and establishing an optimized constraint condition; (4) taking the square sum of the difference between the measured distance obtained in the step (1) and the actual distance obtained in the step (2) as an objective function in the optimization process, and adding weight factors to construct a semi-definite planning problem; (5) and (4) solving the semi-definite programming problem in the step (4), wherein the obtained optimal matrix is the three-dimensional position coordinate of the base station. The method avoids the situation that positioning cannot be carried out due to insufficient connection number with the anchor base station, and meanwhile, the positioning precision is improved.

Description

Base station three-dimensional cooperative positioning method applied to large indoor environment
Technical Field
The invention belongs to the field of indoor positioning, and particularly relates to a base station three-dimensional cooperative positioning method applied to a large-scale indoor environment.
Background
With the rise of various large buildings and the improvement of the living standard of residents, Location-Based services (LBS) are receiving more and more attention, and become an essential part for people to go out, shop, travel, lodge, etc. daily. However, due to the rapid attenuation of satellite signals caused by building occlusion and the influence of multipath effects caused by reflection among building groups, the positioning accuracy of the global navigation satellite system in the indoor environment drops rapidly. Meanwhile, the requirements of indoor location services such as medical aid, accurate marketing, intelligent warehousing and the like are increasing day by day, and various indoor positioning methods are brought forward, wherein the base station positioning can achieve higher precision.
In various technical implementations, a positioning mode of fingerprint identification is mostly adopted for WIFI positioning and Bluetooth positioning, the positioning precision is low, the system performance is easily influenced by environmental changes, the stability is poor, and the maintenance difficulty is high; the ultrasonic technology and the ultra-wideband technology can obtain higher position estimation precision, but the technologies need to customize professional module equipment, so the cost is high; positioning technology based on acoustic signals has been the subject of leading-edge research in recent years due to its characteristics of low cost, good mobile phone compatibility, high stability and estimation accuracy, but also has the problems of clock synchronization, multipath effect, etc.
The most widely used positioning method is a positioning method based on distance measurement, and is divided into cooperative positioning and non-cooperative positioning according to whether the distance between unknown base stations can be used. The non-cooperative positioning only depends on the distance between the unknown base station and the anchor base station, and is mostly used for positioning the target. Cooperative positioning utilizes the distance between unknown base stations in addition to the distance between the unknown base stations and the anchor base station, as shown in fig. 3, and is mostly used in network node positioning.
The base station positioning can achieve higher precision but the precise position of the base station needs to be known in advance, however, the calibration of the base station position needs to be carried out manually, errors are easy to introduce, and when the number of the base stations increases along with the increase of the indoor area, the process is time-consuming and labor-consuming. It is desirable to be able to calibrate all base station locations indoors with several anchor base stations of known locations. However, there are many occlusions in the indoor environment, and the distance between every two base stations cannot be accurately measured. In addition, in a large indoor space, the distance between base stations exceeding the measurable maximum distance will cause measurement value defect, so that the number of anchor base stations connected with unknown base stations is not enough to carry out independent three-dimensional positioning. Cooperative positioning solves this problem to some extent by interfacing with other unknown base stations. However, the current semi-definite planning algorithm is two-dimensional, in order to improve the three-dimensional positioning accuracy in a complex large indoor building, a base station is often required to be distributed at various heights and corners, and the two-dimensional positioning algorithm is far from sufficient.
Disclosure of Invention
In view of the above disadvantages, the present invention provides a three-dimensional cooperative positioning method for a base station applied to a large-scale indoor environment, which solves the three-dimensional calibration problem of the base station in a large-scale complex indoor environment, converts the positioning problem into a matrix optimization problem, and finds a matrix that minimizes the sum of squares of the difference between the measured distance and the actual distance. Even if the number of the base stations is huge, the positions of the base stations can be calibrated at the same time in a short time, and the problem that each unknown base station cannot be independently positioned due to measurement distance limitation and complex indoor environment obstruction is solved.
The technical scheme adopted by the invention is as follows: a base station three-dimensional cooperative positioning method applied to a large indoor environment comprises the following steps:
(1) manually measuring three-dimensional coordinates of at least four anchor base stations, and estimating measurement distances among unknown base stations and between the unknown base stations and the anchor base stations by a TPSN (tire pressure sensor and sensor network) ranging principle;
(2) representing the position coordinates of unknown base stations by using a matrix, obtaining the actual distance represented by the position coordinates by using matrix multiplication, and constructing three-dimensional distance constraint between the base stations;
(3) relaxing the position coordinate matrix in the step (2) into a semi-definite matrix, and establishing an optimized constraint condition;
(4) taking the square sum of the difference between the measured distances between the unknown base stations, the unknown base stations and the anchor base stations obtained in the step (1) and the actual distance obtained in the step (2) as an objective function in the optimization process, and adding a weight factor into the objective function to construct a semi-definite planning problem;
(5) and (4) solving the semi-definite programming problem constructed in the step (4), wherein the obtained optimal matrix is the three-dimensional position coordinate of the base station.
Further, the TPSN (timing-sync protocol for sensor networks) ranging principle includes the following steps:
(1) by between every two base stationsTPSN ranging obtains the distance measurement value between the two base stations, and the base station A is made to send the actual time t of the signalA0The time of receiving signal of base station A and base station B is tA1，tB1(ii) a The actual time at which base station B transmits a signal is tB0The time of receiving signal of base station A and base station B is tA2，tB2(ii) a Will tA1And tB2Respectively approximating the real transmission time of the base station A and the base station B;
(2) in the case where the clocks of base station a and base station B are synchronized in advance, the distance between the two base stations is represented by D ═ t (t)B1-tA1)v，D＝(tA2-tB2) v, adding and finishing the two formulas to obtain:
Figure GDA0002490847130000021
and v is the sound velocity, and the distance measurement value can be obtained by directly estimating the time delay on the clock of the two base stations in the signals acquired by the two base stations and then carrying out difference, so that the measurement distances among the unknown base stations and between the unknown base stations and the anchor base station are obtained.
Further, the construction of the three-dimensional distance constraint in step 2 specifically includes:
using D as Euclidean distance between anchor base station k and unknown base station jjkIndicating that the Euclidean distance between the unknown base station i and the unknown base station j is represented by dijExpressed in a matrix of 3 × m
Figure GDA0002490847130000022
To represent the location of the anchor base station, the three-dimensional vector of each column represents the three-dimensional location coordinates of one anchor base station, and the 3 × n matrix
Figure GDA0002490847130000023
Figure GDA0002490847130000031
To represent the location of an unknown base station, where the three-dimensional vector of each column represents the three-dimensional location coordinates of an unknown base station, and Y ═ XTX∈Rn×nThis matrix satisfies the following distance information:
(ei-ej)TY(ei-ej)＝dij 2
Figure GDA0002490847130000032
wherein e isiIs an n-dimensional unit vector having an ith value of 1 and the remainder of 0, ejIs an n-dimensional unit vector with the j value of 1 and the rest of 0, and I is an identity matrix of 3 × 3, (a)xk，ayk，azk) Three-dimensional position coordinates, R, representing the kth anchor base stationn×nA matrix of real numbers of n × n.
Further, the step 3 specifically includes:
changing Y to XTThe relaxation of X is that Y is more than or equal to XTX, setting up the boundary of feasible region, converting the problem into finding a matrix
Figure GDA0002490847130000033
Figure GDA0002490847130000034
And it satisfies the constraints:
(0；0；0；ei-ej)TZ(0；0；0；ei-ej)＝dij 2
(axk；ayk；azk；-ej)TZ(axk；ayk；azk；-ej)＝djk 2
Figure GDA0002490847130000035
further, the step 4 specifically includes:
assume anchor base station location akWith unknown base station location xjHas a distance error ofjkObedience mean 0 and variance σ1 2Normal distribution of (2); another unknown base station location is xi，xiAnd xjHas a distance error ofijCompliance withMean 0 and variance σ2 2Normal distribution of (e)jkAnd eijAre independent of each other; adding the weight information into the optimization target to obtain the optimization target:
Figure GDA0002490847130000036
Xmlrepresents the optimal solution of the matrix X, NaRepresents the distance space between the anchor base station and the unknown base station, Na＝{(j，k)：1≤j≤n，1≤k≤m}；d(xj，αk) Indicating anchor base station location akWith unknown base station location xjThe measurement distance therebetween; n is a radical ofxRepresenting the distance space between unknown base stations, Nx＝{(i，j)：1≤i，j≤n}；d(xi，xj) Representing another unknown base station xiWith unknown base station xjThe measurement distance therebetween;
by combining the relaxation technology, a semi-definite programming problem can be obtained:
Figure GDA0002490847130000041
Figure GDA0002490847130000042
Figure GDA0002490847130000043
(0；0；0；ei-ej)TZ(0；0；0；ei-ej)＝dij 2
(axk；ayk；azk；-ej)TZ(axk；ayk；azk；-ej)＝djk 2
Figure GDA0002490847130000044
further, a semi-definite programming problem is solved by utilizing a CVX tool kit of matlab.
The invention has the following beneficial effects: the invention avoids the error accumulation, the labor and time cost consumption caused by a large amount of manual measurement during the calibration of the base station. Compared with the traditional target positioning method, the distance between unknown base stations is considered in cooperative positioning, the situation that positioning cannot be performed due to the fact that the number of the base stations connected with the anchor base stations is insufficient is avoided, and positioning accuracy is improved; for node positioning, the distribution of the base stations is not limited to the same plane any more, and the base stations with different heights can effectively improve the precision of the system in positioning the three-dimensional target.
The method uses three-dimensional coordinates to represent an entity object, converts a positioning problem into a matrix optimization problem, uses measurement distance information and three-dimensional position coordinates of an anchor base station as constraints, uses the square sum of the difference between a measurement distance with weight and an actual distance as an optimization target, introduces a relaxation technology to relax a target matrix into a semi-definite matrix, and uses a convex optimization tool box to obtain the solution of the three-dimensional coordinate matrix. The invention can still maintain the accuracy of the decimeter grade when the distance measurement information is less than 50 percent of the total amount.
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FIG. 1 is a flow chart of an algorithm;
FIG. 2 is a schematic diagram of distance measurement;
FIG. 3 is a schematic view of cooperative positioning;
FIG. 4 is a three-dimensional representation of the positioning result;
FIG. 5 is a diagram of the relationship between positioning mean square error and distance measurement range;
figure 6 is a graph comparing SDP localization accuracy to the least squares algorithm (LLS).
Detailed Description
The present invention will be described in further detail with reference to the accompanying drawings and examples.
The method requires knowing the three-dimensional positions of four or more anchor base stations, the distance between unknown base stations, and the distance between an unknown base station and an anchor base station. The distance between base stations is estimated through the TPSN principle, and due to the fact that distance limitation exists in signal ranging and a plurality of obstacles exist in a complex indoor environment, only part of the distance between the base stations can be obtained.
We specify some codes in the positioning process for convenience of description. Assuming R in three dimensions3In, there are m anchor base stations akK is 1, 2,. multidot.m; n unknown base stations xj1, 2, n, the euclidean distance between the kth anchor base and the jth unknown base station is djkThe Euclidean distance between the ith unknown base station and the jth unknown base station is represented by dijAnd (4) showing. With NaRepresents the distance space between the anchor base station and the unknown base station, Na{ (j, k): j is more than or equal to 1 and less than or equal to n, and k is more than or equal to 1 and less than or equal to m }; with NxRepresenting the distance space between unknown base stations, Nx{ (i, j): i is more than or equal to 1 and j is less than or equal to n. Assuming that the maximum measurement distance of the signal is R, when the distance between two base stations exceeds R, the measurement distance cannot be obtained; when the distance between two base stations is smaller than R, the measured distance is considered to be true and effective, and the distance space is expressed as: n is a radical ofa＝{(j，k)：||xj-ak||2≤R，1≤j≤n，1≤k≤m}，Nx＝{(i，j)：||xi-xj||2≤R，1≤i，j≤n}。
In the following, the steps of three-dimensional cooperative positioning are introduced:
step 1: anchor base station position measurements and inter-base station distance estimates.
Because the number of anchor base stations is small, manual measurement can be carried out. Obtaining the distance measurement value between two base stations by performing TPSN ranging between the two base stations, as shown in FIG. 2, making the actual time t of the base station A transmitting signalA0The time of receiving signal of base station A and base station B is tA1，tB1(ii) a The actual time at which base station B transmits a signal is tB0The time of receiving signal of base station A and base station B is tA2，tB2(ii) a Because the distance between the base station signal receiving and transmitting module and the distance between two base stations can be ignored, when in actual use, t is usedA1And tB2Approximating the real transmission time of base station a and base station B, respectively.
In the case of base station a and base station B clocks synchronized in advance, the distance between the two base stations can be expressed as D ═ (t)B1-tA1)v，D＝(tA2-tB2) v, adding and finishing the two formulas to obtain:
Figure GDA0002490847130000051
and v is the sound velocity, usually measured at 340m/s, and the distance measurement can be obtained by directly estimating the time delay on the clock of the two base stations in the signals acquired by the two base stations and then carrying out difference, so that the method can successfully avoid the clock synchronization problem in the distance measurement process. And the base station transmits the distance information to a server for the next position calculation.
Step 2: and constructing a distance application basis.
Anchor base station position 3 × m matrix
Figure GDA0002490847130000052
To indicate that the three-dimensional vector of each column represents the three-dimensional position of one anchor base station, using a 3 × n matrix
Figure GDA0002490847130000053
To represent the position of a position base station, the three-dimensional vector of each column represents the three-dimensional position coordinates of an unknown base station, and the problem is to find a matrix X, where Y is equal to XTX∈Rn×nThe following distance information is satisfied:
(ei-ej)TY(ei-ej)＝dij 2
Figure GDA0002490847130000061
wherein e isiIs a unit vector of length n with the ith bit being 1 and the other bits being 0, ejIs an n-dimensional unit vector with the j value of 1 and the rest of 0, and I is an identity matrix of 3 × 3, (a)xk，ayk，azk) Three-dimensional position coordinates, R, representing the kth anchor base stationn×nA matrix of real numbers of n × n.
And step 3: and a relaxation technology is introduced to construct a convex optimization problem.
In order to convert the problem into a convex optimization problemProblem, by relaxation technique, Y is equal to XTThe relaxation of X is that Y is more than or equal to XTX, setting up feasible region boundary, converting the problem into finding a matrix
Figure GDA0002490847130000062
And it satisfies the constraints:
(0；0；0；ei-ej)TZ(0；0；0；ei-ej)＝dij 2
(axk；ayk；azk；-ej)TZ(axk；ayk；azk；-ej)＝djk 2
Figure GDA0002490847130000063
and 4, step 4: and comprehensively considering the distance measurement error.
In a real scene, due to clock synchronization, multipath effect, noise interference and other reasons, an error between a distance ak measured by using a TPSN technology and an unknown base station position xj is jk, and the error obeys normal distribution with an average value of 0 and a variance of sigma 12; another unknown base station location is xi，xiAnd xjWith an error ofijObedience mean 0 and variance σ2 2Normal distribution of (e)jkAnd eijAre independent of each other; namely:
Figure GDA0002490847130000064
Figure GDA0002490847130000065
taking the longer distance, the larger measurement error as a criterion, and taking the reciprocal of the variance as weight to obtain an optimization target:
Figure GDA0002490847130000066
Xmlrepresents the optimal solution of the matrix X, NaRepresents the distance space between the anchor base station and the unknown base station, Na＝{(j，k)：1≤j≤n，1≤k≤m}；d(xj，αk) Representing anchor base station akWith unknown base station xjThe measurement distance therebetween; n is a radical ofxRepresenting the distance space between unknown base stations, Nx＝{(i，j)：1≤i，j≤n}；d(xi，xj) Representing another unknown base station xiWith unknown base station xjThe measurement distance therebetween;
by combining the relaxation technology, the semi-definite programming problem in the optimization problem can be obtained:
Figure GDA0002490847130000067
s.t.Na＝{(j，k)：||xj-ak||2≤R，1≤j≤n，1≤k≤m}
Nx＝{(i，j)：||xi-xj||2≤R，1≤i，j≤n}
Figure GDA0002490847130000071
Figure GDA0002490847130000072
(0；0；0；ei-ej)TZ(0；0；0；ei-ej)＝dij 2
(axk；ayk；azk；-ej)TZ(axk；ayk；azk；-ej)＝djk 2
Figure GDA0002490847130000073
and 5: and (5) solving the semi-definite programming problem.
And solving the semi-definite programming problem through a CVX tool box of the matlab to obtain the coordinates of the positions of the unknown base stations when the signal ranging sensing radius is R.
Description of the implementation:
the algorithm flow is shown in fig. 1, and the positioning error is related to various factors, including the number and layout of anchor base stations, the signal distance measurement range, the distance measurement error distribution, and the like. The more accurate the distance measurement, the smaller the positioning error. However, the distance measurement may be based on increasing the signal bandwidth and increasing the transmission power, the number of signals that can be simultaneously transmitted in the former may be reduced, and the latter may generate noise, and the improvement of the positioning accuracy is limited in this respect by the time cost and the influence of noise. With the increase of the number of the anchor base stations, the positioning error is gradually reduced, but when the occupation ratio of the anchor base stations reaches a certain degree, the precision is slowly improved. The determination of the positions of the anchor base stations is time-consuming and labor-consuming, so the selection of the number of anchor base stations is a compromise between accuracy requirements and labor cost. The larger the ranging range is, the larger the number of connections between base stations is, which is also beneficial to improving the positioning accuracy, as shown in fig. 5, but also needs to improve the power to bring about the influence of noise.
The invention avoids the error accumulation, the labor and time cost consumption caused by a large amount of manual measurement during the calibration of the base station. Compared with the traditional target positioning method, the cooperative positioning method considers the distance between unknown base stations, avoids the situation that positioning cannot be performed due to insufficient connection number with anchor base stations, and improves the positioning accuracy, as shown in a comparison diagram of fig. 6; for node positioning, the distribution of base stations is no longer limited to the same plane, and as shown in fig. 4, the base stations with different heights will also effectively improve the accuracy of the system in positioning the three-dimensional target.
The application environment is as follows:
the positioning system applied to the complex large-scale indoor environment needs a large number of base stations with known positions, is time-consuming and labor-consuming in manual calibration, and can introduce uncertain errors. More seriously, in some large ancient buildings, the anchor base station positioned in some special positions in the air is difficult to measure, and manual calibration cannot be carried out. Meanwhile, due to the attenuation of signals in the process of propagation, the measurement distance is limited, namely, the sensing radius exists, and the distance between the base stations exceeds the sensing radius and cannot be communicated. For an unknown base station in a large indoor space, the number of connections between the unknown base station and the known base station is insufficient, and the unknown base station cannot be calibrated independently through an anchor base station. In addition, large-scale aerial robot formation, underwater submarine formation and the like similarly need a three-dimensional cooperation algorithm, and position information of all machines is obtained through coordinates with known few positions and distance information among the coordinates.

Claims (5)

1. A three-dimensional cooperative positioning method of a base station applied to a large indoor environment is characterized by comprising the following steps:
(1) manually measuring three-dimensional coordinates of at least four anchor base stations, and estimating measurement distances among unknown base stations and between the unknown base stations and the anchor base stations by a TPSN (tire pressure sensor and sensor network) ranging principle;
(2) representing the position coordinates of unknown base stations by using a matrix, obtaining the actual distance represented by the position coordinates by using matrix multiplication, and constructing three-dimensional distance constraint between the base stations;
(3) relaxing the position coordinate matrix in the step (2) into a semi-definite matrix, and establishing an optimized constraint condition;
(4) taking the square sum of the difference between the measured distances between the unknown base stations, the unknown base stations and the anchor base stations obtained in the step (1) and the actual distance obtained in the step (2) as an objective function in the optimization process, and adding a weight factor into the objective function to construct a semi-definite planning problem; the step (4) is specifically as follows:
assume anchor base station location akWith unknown base station location xjHas a distance error ofjkObedience mean 0 and variance σ1 2Normal distribution of (2); another unknown base station location is xi，xiAnd xjHas a distance error ofijObedience mean 0 and variance σ2 2Normal distribution of (e)jkAnd eijAre independent of each other; adding the weight information into the optimization target to obtain the optimization target:
Figure FDA0002505909600000011
Xmlrepresents the optimal solution of the matrix X, NaRepresents the distance space between the anchor base station and the unknown base station, Na＝{(j，k)：1≤j≤n，1≤k≤m}；d(xj，αk) Indicating anchor base station location akWith unknown base station location xjThe measurement distance therebetween; n is a radical ofxRepresenting the distance space between unknown base stations, Nx＝{(i，j)：1≤i，j≤n}；d(xi，xj) Representing another unknown base station location xiWith unknown base station location xjThe measurement distance therebetween; djkRepresenting the Euclidean distance between the kth anchor base and the jth unknown base station, dijRepresenting the Euclidean distance between the ith unknown base station and the jth unknown base station;
by combining the relaxation technology, a semi-definite programming problem can be obtained:
Figure FDA0002505909600000012
Figure FDA0002505909600000013
Figure FDA0002505909600000014
(0；0；0；ei-ej)TZ(0；0；0；ei-ej)＝dij 2
(axk；ayk；azk；-ej)TZ(axk；ayk；azk；-ej)＝djk 2
Figure FDA0002505909600000021
wherein e isiIs an n-dimensional unit vector with the ith value of 1 and the rest of 0; e.g. of the typejIs that the jth value is 1 and the rest is 0n-dimensional unit vector, I is a unit matrix of 3 × 3 (a)xk，ayk，azk) Representing three-dimensional position coordinates of a kth anchor base station; x represents a position matrix of an unknown base station; y ═ XTX, representing the three-dimensional position coordinates of an unknown base station;
(5) and (4) solving the semi-definite programming problem constructed in the step (4), wherein the obtained optimal matrix is the three-dimensional position coordinate of the base station.
2. The method as claimed in claim 1, wherein the TPSN (timing-sync protocol for sensor networks) ranging principle comprises the following steps:
(1) obtaining the distance measurement value between two base stations by TPSN ranging between every two base stations, and enabling the base station A to transmit the actual time t of the signalA0The time of receiving signal of base station A and base station B is tA1，tB1(ii) a The actual time at which base station B transmits a signal is tB0The time of receiving signal of base station A and base station B is tA2，tB2(ii) a Will tA1And tB2Respectively approximating the real transmission time of the base station A and the base station B;
(2) in the case where the clocks of base station a and base station B are synchronized in advance, the distance between the two base stations is represented by D ═ t (t)B1-tA1)v，D＝(tA2-tB2) v, adding and finishing the two formulas to obtain:
Figure FDA0002505909600000022
and v is the sound velocity, and the distance measurement value can be obtained by directly estimating the time delay on the clock of the two base stations in the signals acquired by the two base stations and then carrying out difference, so that the measurement distances among the unknown base stations and between the unknown base stations and the anchor base station are obtained.
3. The method for three-dimensional cooperative positioning of a base station applied to a large-scale indoor environment according to claim 1, wherein the construction of the three-dimensional distance constraint in the step (2) is specifically as follows:
using D as Euclidean distance between anchor base station k and unknown base station jjkIndicating that the Euclidean distance between the unknown base station i and the unknown base station j is represented by dijExpressed in a matrix of 3 × m
Figure FDA0002505909600000023
To represent the location of the anchor base station, the three-dimensional vector of each column represents the three-dimensional location coordinates of one anchor base station, and the 3 × n matrix
Figure FDA0002505909600000024
Figure FDA0002505909600000025
To represent the location of an unknown base station, where the three-dimensional vector of each column represents the three-dimensional location coordinates of an unknown base station, and Y ═ XTX∈Rn×nThis matrix satisfies the following distance information:
(ei-ej)TY(ei-ej)＝dij 2
Figure FDA0002505909600000031
wherein e isiIs an n-dimensional unit vector having an ith value of 1 and the remainder of 0, ejIs an n-dimensional unit vector with the j value of 1 and the rest of 0, and I is an identity matrix of 3 × 3, (a)xk，ayk，azk) Three-dimensional position coordinates, R, representing the kth anchor base stationn×nA matrix of real numbers of n × n.
4. The method for three-dimensional cooperative positioning of a base station applied to a large indoor environment according to claim 3, wherein the step (3) is specifically as follows:
changing Y to XTThe relaxation of X is that Y is more than or equal to XTX, setting up the boundary of feasible region, converting the problem into finding a matrix
Figure FDA0002505909600000032
Figure FDA0002505909600000033
And it satisfies the constraints:
(0；0；0；ei-ej)TZ(0；0；0；ei-ej)＝dij 2
(axk；ayk；azk；-ej)TZ(axk；ayk；azk；-ej)＝djk 2
Figure FDA0002505909600000034
5. the method according to claim 1, wherein the semi-definite programming problem is solved by using a CVX toolkit of matlab.
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