CN110658490B - RSS (really simple syndication) and AOA (automatic optical inspection) based three-dimensional wireless sensor network non-cooperative positioning method - Google Patents

RSS (really simple syndication) and AOA (automatic optical inspection) based three-dimensional wireless sensor network non-cooperative positioning method Download PDF

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CN110658490B
CN110658490B CN201910783420.4A CN201910783420A CN110658490B CN 110658490 B CN110658490 B CN 110658490B CN 201910783420 A CN201910783420 A CN 201910783420A CN 110658490 B CN110658490 B CN 110658490B
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李有明
戚勤科
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Ningbo University Science Park Development Co ltd
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    • G—PHYSICS
    • G01—MEASURING; TESTING
    • G01S—RADIO DIRECTION-FINDING; RADIO NAVIGATION; DETERMINING DISTANCE OR VELOCITY BY USE OF RADIO WAVES; LOCATING OR PRESENCE-DETECTING BY USE OF THE REFLECTION OR RERADIATION OF RADIO WAVES; ANALOGOUS ARRANGEMENTS USING OTHER WAVES
    • G01S5/00—Position-fixing by co-ordinating two or more direction or position line determinations; Position-fixing by co-ordinating two or more distance determinations
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    • G01S5/0257—Hybrid positioning
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Abstract

The invention discloses a three-dimensional wireless sensor network non-cooperative positioning method based on RSS and AOA, which comprises the steps of establishing a measurement model of an RSS measurement value obtained by each anchor node and respective measurement models of an azimuth angle measurement value and an elevation angle measurement value in an AOA measurement value obtained by each anchor node; obtaining a non-convex positioning problem according to respective approximate expressions of the three measurement models and a least square criterion; obtaining the equivalence problem of the non-convex positioning problem by introducing an auxiliary variable and a relaxation variable; a second-order cone relaxation method and a semi-positive definite relaxation method are adopted to relax constraint conditions in the equivalence problem of the non-convex positioning problem, so as to obtain a mixed semi-positive definite/second-order cone planning problem; solving the mixed semi-positive definite/second-order cone programming problem by adopting an interior point method to obtain a global optimal solution of the target source, namely a position estimation value of the target source in a reference coordinate system; the method has the advantages that the positioning accuracy is improved by utilizing the mixed measurement value, the positioning accuracy is high, and the influence of noise power is small.

Description

RSS (really simple syndication) and AOA (automatic optical inspection) based three-dimensional wireless sensor network non-cooperative positioning method
Technical Field
The invention relates to a target positioning method, in particular to a three-dimensional wireless sensor network non-cooperative positioning method based on RSS and AOA.
Background
A Wireless Sensor Network (WSN) is a distributed Sensor Network, which generally comprises a large number of sensors distributed in an inspection area to cooperatively sense, collect, process and transmit information of sensed objects in a geographical area covered by the WSN. To keep the implementation costs low, only some of the sensors in the wireless sensor network are equipped with a Global Positioning System (GPS), referred to as anchor nodes, and the remaining sensors use the known locations of the anchor nodes to determine their locations by using some kind of positioning scheme, referred to as target nodes, i.e., target sources. In many practical applications, the data collected by the sensors is meaningful only if the data carries corresponding position information, and therefore, the position estimation of the sensors is one of the key technologies of the wireless sensor network. Object localization schemes typically rely on distance measurements, which can be extracted from different transmitted signal characteristics, such as time of arrival (TOA), time difference of arrival (TDOA), angle of arrival (AOA), Received Signal Strength (RSS), by using different hardware devices. Recently, to further improve the positioning accuracy, more and more target positioning schemes start to adopt the mixed measurement values, because more information can be extracted from the mixed measurement values. Slavisa Tomic et al respectively adopt approximate expressions of RSS and AOA models in achievements published by IEEE Wireless Communications Letters (institute of Electrical and electronics Engineers (IEEE) Wireless communication Kuck newspaper), then convert norms into a vector dot product form, and obtain a closed solution expression of a positioning problem by a weighted least square method, but experiments show that the positioning performance of the uncooperative positioning method is rapidly deteriorated with the enhancement of noise power of mixed measurement values. Slavisa Tomic et al, in the work published by IEEE Transactions on Vehicular Technology, translated the positioning problem into a generalized confidence domain problem and solved by dichotomy, however, experiments showed that the positioning performance of the uncooperative positioning method has room for improvement.
Disclosure of Invention
The technical problem to be solved by the invention is to provide a three-dimensional wireless sensor network non-cooperative positioning method based on RSS and AOA, which utilizes mixed measurement values to improve positioning accuracy, has high positioning accuracy and is less influenced by noise power of the mixed measurement values.
The technical scheme adopted by the invention for solving the technical problems is as follows: a three-dimensional wireless sensor network non-cooperative positioning method based on RSS and AOA is characterized by comprising the following steps:
the method comprises the following steps: establishing a space rectangular coordinate system as a reference coordinate system in a three-dimensional wireless sensing network environment, setting N anchor nodes and a target source in the three-dimensional wireless sensing network environment, and recording the coordinate position of the ith anchor node in the reference coordinate system as siThe coordinate position of the target source in the reference coordinate system is recorded as x, si＝(si1,si2,si3)，x＝(x1,x2,x3) (ii) a Wherein N is a positive integer, N represents the total number of anchor nodes in the three-dimensional wireless sensor network environment, N is more than or equal to 4, i is a positive integer, i is more than or equal to 1 and less than or equal to N, si1、si2、si3Corresponding representation si1 st, 2 nd, 3 rd coordinate component, x1、x2、x3Correspondingly representing the 1 st coordinate component, the 2 nd coordinate component and the 3 rd coordinate component of x;
step two: transmitting a measuring signal by a target source in a three-dimensional wireless sensing network environment, and obtaining an RSS measuring value and an AOA measuring value after each anchor node receives the measuring signal transmitted by the target source; then, a measurement model of the RSS measurement value obtained by each anchor node is established, and the measurement model of the RSS measurement value obtained by the ith anchor node is described as follows:
Figure BDA0002177258970000031
and establishing respective measurement models of an azimuth angle measurement value and an elevation angle measurement value in the AOA measurement values obtained by each anchor node, and describing the measurement model of the azimuth angle measurement value in the AOA measurement values obtained by the ith anchor node as follows:
Figure BDA0002177258970000032
a measurement model of elevation measurements among AOA measurements obtained by the ith anchor node is described as:
Figure BDA0002177258970000033
wherein L isiRepresents the RSS measurement obtained by the ith anchor node, i.e. the path loss, L, existing on the transmission path experienced by the measurement signal transmitted from the target source to the ith anchor node0Representing the path loss existing on a transmission path from the transmission of a measurement signal from a target source to the reception of a reference point, gamma representing the path loss coefficient of the transmission distance from the transmission of the measurement signal from the target source to the reception of each anchor node, the value range of gamma being 2.2-2.8, the symbol "| | |" being the symbol of Euclidean norm, d0Representing the distance of a reference point to a target source in a reference coordinate system, niRepresents LiOf the measurement noise, niObeying a Gaussian distribution
Figure BDA0002177258970000034
Figure BDA0002177258970000035
Represents niPower of phiiRepresenting azimuthal measurements, m, in AOA measurements obtained by the ith anchor nodeiIs indicative of phiiMeasurement noise present in, miObeying a Gaussian distribution
Figure BDA0002177258970000036
Figure BDA0002177258970000037
Represents miPower of alphaiRepresenting elevation measurements, v, of AOA measurements obtained at the ith anchor nodeiDenotes alphaiOf the measurement noise, viObeying a Gaussian distribution
Figure BDA0002177258970000038
Figure BDA0002177258970000039
Denotes viThe power of (d);
step three: in that
Figure BDA00021772589700000310
mi＜＜1、viWhen < 1, i.e. under the condition of high signal-to-noise ratio, for
Figure BDA00021772589700000311
Making approximate transformation to obtain niThe approximate expression of (c), described as:
Figure BDA00021772589700000312
and to
Figure BDA00021772589700000313
Making approximate transformation to obtain miThe weighted approximate expression is described as:
Figure BDA0002177258970000041
to pair
Figure BDA0002177258970000042
Making approximate transformation to obtain viThe approximate expression of (c), described as:
Figure BDA0002177258970000043
wherein the content of the first and second substances,
Figure BDA0002177258970000044
Figure BDA0002177258970000045
Figure BDA0002177258970000046
represents miWeighted value, ci＝[-sin(φi),cos(φi),0]TThe term "[ 2 ]]"is a vector representing a symbol, k ═ 0,0,1]TThe symbol "T" is transposed symbol, β, μ, λi、ciK is an introduced intermediate variable;
step four: according to
Figure BDA0002177258970000047
Figure BDA0002177258970000048
And
Figure BDA0002177258970000049
and combining the least square criterion to obtain a non-convex positioning problem for solving x, which is described as:
Figure BDA00021772589700000410
wherein, min () is a minimum function;
step five: introducing auxiliary variables f, h, r and z and relaxation variables e, g and t into the description of solving the non-convex positioning problem of x to obtain an equivalent problem of solving the non-convex positioning problem of x, wherein the description is as follows:
Figure BDA0002177258970000051
wherein e isiDenotes the i-th component in e, giDenotes the ith component in g, "s.t." means "constrained to … …", hiDenotes the ith component in h, hi＝||x-si||2，I3Represents a 3-dimensional identity matrix, riDenotes the i-th component in r, ri＝||x-si||，fiThe i-th component in f is represented,
Figure BDA0002177258970000052
step six: using a second order cone relaxation method, r in the description of the equivalence problem of the non-convex localization problem of x will be solvedi＝||x-siRelaxation of | to ri≥||x-siL; and a semi-positive definite relaxation method is adopted to relax the z-xTx in the description of solving the equivalence problem of the non-convex positioning problem of the x into a form of linear matrix inequality
Figure BDA0002177258970000053
Then according to ri≥||x-siI and
Figure BDA0002177258970000054
and obtaining a mixed semi-positive definite/second order cone programming problem for solving x, wherein the description is as follows:
Figure BDA0002177258970000061
wherein the content of the first and second substances,
Figure BDA0002177258970000062
to represent
Figure BDA0002177258970000063
Is a semi-positive definite matrix;
step seven: and solving the mixed semi-positive definite/second-order cone programming problem of x by adopting an interior point method to obtain the global optimal solution of x, wherein the global optimal solution is used as the position estimation value of the target source in the reference coordinate system.
Compared with the prior art, the invention has the advantages that:
1) the method of the invention approximates the existing minimization problem based on the maximum likelihood criterion to the least square problem based on the least square criterion, namely, solves the non-convex positioning problem of x, and relaxes the non-convex positioning problem of x to the mixed semi-positive/second-order cone planning problem of x by combining the second-order cone relaxation technology and the semi-positive relaxation technology, thereby ensuring to obtain the global optimal solution of the position of the target source in the reference coordinate system and improving the positioning precision.
2) The method further improves the positioning accuracy by utilizing the RSS measurement value and the AOA measurement value, thereby more accurately estimating the position of the target source.
3) Experiments prove that the method of the invention has stable performance under the condition of large measurement noise power.
Drawings
FIG. 1 is a block diagram of an overall implementation of the method of the present invention;
FIG. 2 is a schematic diagram of positions of a target source and an ith anchor node in a reference coordinate system in an uncooperative three-dimensional wireless sensor network environment;
FIG. 3 is a graph illustrating the variation of the root mean square error with the number of anchor nodes for the method of the present invention, the existing generalized confidence sub-domain method, and the existing weighted least squares method, given a standard deviation of the measurement noise in the path loss of 6dB, given a standard deviation of the measurement noise in the azimuth measurements, and given a standard deviation of the measurement noise in the elevation measurements of 5 degrees;
FIG. 4 is a graphical representation of the root mean square error of the method of the present invention versus the existing generalized confidence sub-domain method, the existing weighted least squares method as a function of the standard deviation of the measurement noise in a given path loss, for a given standard deviation of the measurement noise present in the azimuth measurements and for a given standard deviation of the measurement noise present in the elevation measurements of both 5 degrees and 6 anchor nodes;
FIG. 5 is a graphical representation of the RMS error of the method of the present invention versus the existing generalized confidence sub-domain method, the existing weighted least squares method as a function of the standard deviation of the measured noise present in a given azimuth measurement for a given standard deviation of the measured noise in path loss of 6dB, a given standard deviation of the measured noise present in an elevation measurement of 5 degrees, and a number of anchor nodes of 6;
fig. 6 is a graph illustrating the variation of the root mean square error of the method of the present invention with the standard deviation of the measurement noise present in the given elevation measurement, compared to the existing generalized confidence sub-domain method and the existing weighted least squares method, with the standard deviation of the measurement noise present in the given path loss being 6dB, the standard deviation of the measurement noise present in the given azimuth measurement being 5 degrees, and the number of anchor nodes being 6.
Detailed Description
The invention is described in further detail below with reference to the accompanying examples.
The general implementation block diagram of the three-dimensional wireless sensor network non-cooperative positioning method based on RSS and AOA provided by the invention is shown in FIG. 1, and the method comprises the following steps:
the method comprises the following steps: establishing a space rectangular coordinate system as a reference coordinate system in a three-dimensional wireless sensing network environment, setting N anchor nodes and a target source in the three-dimensional wireless sensing network environment, and recording the coordinate position of the ith anchor node in the reference coordinate system as siThe coordinate position of the target source in the reference coordinate system is recorded as x, si＝(si1,si2,si3)，x＝(x1,x2,x3) (ii) a N is a positive integer, N represents the total number of anchor nodes in the three-dimensional wireless sensor network environment, N is greater than or equal to 4, in this embodiment, N is 6, i is a positive integer, i is greater than or equal to 1 and less than or equal to N, and s isi1、si2、si3Corresponding representation si1 st, 2 nd, 3 rd coordinate component, x1、x2、x3Corresponding to the 1 st, 2 nd and 3 rd coordinate components representing x.
Fig. 2 is a schematic diagram showing positions of a target source and an ith anchor node in a reference coordinate system in an uncooperative three-dimensional wireless sensor network environment.
Step two: transmitting a measuring signal by a target source in a three-dimensional wireless sensing network environment, and obtaining an RSS measuring value and an AOA measuring value after each anchor node receives the measuring signal transmitted by the target source; then, a measuring model of RSS measured values obtained by each anchor node is established, and the ith anchor node is connected with the RSS measuring moduleThe measurement model of the point-obtained RSS measurements is described as:
Figure BDA0002177258970000081
and establishing respective measurement models of an azimuth angle measurement value and an elevation angle measurement value in the AOA measurement values obtained by each anchor node, and describing the measurement model of the azimuth angle measurement value in the AOA measurement values obtained by the ith anchor node as follows:
Figure BDA0002177258970000082
a measurement model of elevation measurements among AOA measurements obtained by the ith anchor node is described as:
Figure BDA0002177258970000091
wherein L isiRepresents the RSS measurement obtained by the ith anchor node, i.e. the path loss, L, existing on the transmission path experienced by the measurement signal transmitted from the target source to the ith anchor node0Representing the path loss present on the transmission path experienced by the measurement signal transmitted from the target source to the reference point, taken in the experiment L0The value of gamma is 40dB, gamma represents the path loss coefficient of the transmission distance of the measurement signal transmitted from the target source to each anchor node, the value range of gamma is 2.2-2.8, if gamma is 2.6, the symbol "| | | |" is the symbol for solving the Euclidean norm, d0Representing the distance from the reference point to the target source in the reference coordinate system, and taking d in the experiment0Is 1 m, niRepresents LiOf the measurement noise, niObeying a Gaussian distribution
Figure BDA0002177258970000092
Figure BDA0002177258970000093
Represents niPower of phiiRepresenting azimuthal measurements, m, in AOA measurements obtained by the ith anchor nodeiIs indicative of phiiMeasurement noise present in, miObeying a Gaussian distribution
Figure BDA0002177258970000094
Figure BDA0002177258970000095
Represents miPower of alphaiRepresenting elevation measurements, v, of AOA measurements obtained at the ith anchor nodeiDenotes alphaiOf the measurement noise, viObeying a Gaussian distribution
Figure BDA0002177258970000096
Figure BDA0002177258970000097
Denotes viOf the power of (c).
Step three: in that
Figure BDA0002177258970000098
mi＜＜1、viWhen < 1, i.e. under the condition of high signal-to-noise ratio, for
Figure BDA0002177258970000099
Making approximate transformation to obtain niThe approximate expression of (c), described as:
Figure BDA00021772589700000910
and to
Figure BDA00021772589700000911
Making approximate transformation to obtain miThe weighted approximate expression is described as:
Figure BDA00021772589700000912
to pair
Figure BDA00021772589700000913
Making approximate transformation to obtain viThe approximate expression of (c), described as:
Figure BDA00021772589700000914
wherein the content of the first and second substances,
Figure BDA00021772589700000915
Figure BDA00021772589700000916
Figure BDA00021772589700000917
represents miWeighted value, ci＝[-sin(φi),cos(φi),0]TThe term "[ 2 ]]"is a vector representing a symbol, k ═ 0,0,1]TThe symbol "T" is transposed symbol, β, μ, λi、ciAnd k are all introduced intermediate variables.
Step four: according to
Figure BDA00021772589700000918
Figure BDA00021772589700000919
And
Figure BDA0002177258970000101
and combining the least square criterion to obtain a non-convex positioning problem for solving x, which is described as:
Figure BDA0002177258970000102
wherein min () is a minimum function.
Step five: introducing auxiliary variables f, h, r and z and relaxation variables e, g and t into the description of solving the non-convex positioning problem of x to obtain an equivalent problem of solving the non-convex positioning problem of x, wherein the description is as follows:
Figure BDA0002177258970000103
wherein e isiDenotes the i-th component in e, giDenotes the ith component in g, "s.t." means "constrained to … …", hiDenotes the ith component in h, hi＝||x-si||2，I3Represents a 3-dimensional identity matrix, riDenotes the i-th component in r, ri＝||x-si||，fiThe i-th component in f is represented,
Figure BDA0002177258970000104
step six: using a second order cone relaxation method, r in the description of the equivalence problem of the non-convex localization problem of x will be solvedi＝||x-siRelaxation of | to ri≥||x-siL; and adopting a semi-positive definite relaxation method to solve the problem that x is not the equal value of x in the description of the problem of the convex positioningTx relaxation is in the form of a Linear Matrix Inequality (LMI)
Figure BDA0002177258970000111
Then according to ri≥||x-siI and
Figure BDA0002177258970000112
and obtaining a mixed semi-positive definite/second order cone programming problem for solving x, wherein the description is as follows:
Figure BDA0002177258970000113
wherein the content of the first and second substances,
Figure BDA0002177258970000114
to represent
Figure BDA0002177258970000115
Is a semi-positive definite matrix.
Step seven: and solving the mixed semi-positive definite/second-order cone programming problem of x by adopting an interior point method to obtain the global optimal solution of x, wherein the global optimal solution is used as the position estimation value of the target source in the reference coordinate system.
The feasibility, effectiveness and positioning performance of the method are verified through simulation experiments.
Setting that N is 6 anchor nodes exist in the three-dimensional wireless sensing network environment, and the coordinate positions of the target source and the 6 anchor nodes in the reference coordinate system are randomly selected in a cube of 15 x 15 cubic meters. The power of the measurement noise in the path loss present on the transmission path experienced by the measurement signal transmitted from the target source to any anchor node reception is assumed to be the same, i.e.
Figure BDA0002177258970000116
The power of the measurement noise present in the azimuth measurements in the AOA measurements obtained by any anchor node is the same, i.e.
Figure BDA0002177258970000121
The power of the measurement noise present in the elevation measurements in the AOA measurements obtained by any anchor node is the same, i.e.
Figure BDA0002177258970000122
Wherein the content of the first and second substances,
Figure BDA0002177258970000123
corresponding to the power representing the measurement noise in the path loss existing on the transmission path experienced by the measurement signal transmitted from the target source to the 1 st anchor node reception, the power representing the measurement noise in the path loss existing on the transmission path experienced by the measurement signal transmitted from the target source to the nth anchor node reception,
Figure BDA0002177258970000124
representing the power of the measurement noise in a given path loss,
Figure BDA0002177258970000125
representing the standard deviation of the measurement noise in a given path loss,
Figure BDA0002177258970000126
corresponding to the power representing the measurement noise present in the azimuth angle measurements in the AOA measurements obtained by the 1 st anchor node, the power of the measurement noise present in the azimuth angle measurements in the AOA measurements obtained by the Nth anchor node,
Figure BDA0002177258970000127
representing the power of measurement noise present in a given azimuth measurement,
Figure BDA0002177258970000128
to representThe standard deviation of the measurement noise present in a given azimuthal measurement,
Figure BDA0002177258970000129
corresponding to a power representing measurement noise present in elevation measurements in AOA measurements obtained by the 1 st anchor node, a power of measurement noise present in elevation measurements in AOA measurements obtained by the Nth anchor node,
Figure BDA00021772589700001210
represents the power of measurement noise present in a given elevation measurement,
Figure BDA00021772589700001211
representing the standard deviation of the measurement noise present in a given elevation measurement.
The performance of the method of the invention is tested along with the change situation of the increase of the number of the anchor nodes.
Fig. 3 is a graph showing that the root mean square error of the method of the present invention, the existing generalized confidence sub-domain method and the existing weighted least square method varies with the number of anchor nodes under the condition that the standard deviation of the measurement noise in the given path loss is 6dB, the standard deviation of the measurement noise in the given azimuth measurement value and the standard deviation of the measurement noise in the given elevation measurement value are both 5 degrees. It can be observed from fig. 3 that the Root Mean Square Error (RMSE) of the method of the present invention is smaller than the other two methods and closer to the cramer-Circle (CRLB), and the difference in RMSE values between the method of the present invention and the two existing methods increases as the number of anchor nodes increases, which indicates the superior performance of the method of the present invention in terms of positioning accuracy.
The performance of the method of the invention was tested as a function of increasing standard deviation of the measured noise.
Fig. 4 is a graph showing the variation of the root mean square error of the method of the present invention with the standard deviation of the measurement noise in the given path loss, compared with the existing generalized confidence sub-domain method and the existing weighted least square method, when the standard deviation of the measurement noise in the given azimuth measurement value and the standard deviation of the measurement noise in the given elevation measurement value are both 5 degrees and the number of anchor nodes is 6. Fig. 5 is a graph showing the variation of the root mean square error of the method of the present invention with the standard deviation of the measurement noise present in the given azimuth measurement value, which is 6dB, the standard deviation of the measurement noise present in the given elevation measurement value, which is 5 degrees, and the number of anchor nodes, which is 6, compared with the existing generalized confidence sub-domain method and the existing weighted least-squares method, with the standard deviation of the measurement noise present in the given azimuth measurement value. Fig. 6 shows a graphical representation of the root mean square error of the method of the invention versus the existing generalized confidence sub-domain method, the existing weighted least squares method as a function of the standard deviation of the measurement noise present in a given elevation measurement, given a standard deviation of the measurement noise in the path loss of 6dB, given a standard deviation of the measurement noise present in the azimuth measurement of 5 degrees, and given a number of anchor nodes of 6. It can be observed from fig. 4, 5 and 6 that the method of the present invention has more accurate positioning accuracy and is closer to the lower limit of cramer than the two existing methods within the standard deviation variation range of the measurement noise considered, and the positioning performance is more stable.
The first prior art method in fig. 3 to 6 is a 3-D Target Localization in Wireless Sensor Network Using RSS and AOA Measurements (three-dimensional object Localization based on RSS and AOA Measurements in Wireless Sensor networks), which is disclosed in IEEE Transactions on Vehicular Technology (institute of electrical and electronics engineers (IEEE) vehicle technologies) by slave Tomic et al, which is referred to as the generalized trust sub-domain method for short; the second existing method is a closed-form Solution for RSS/AOA Target Localization by spatial Coordinates mapping (RSS/AOA Target Localization based on Spherical coordinate transformation), which is disclosed in IEEE Wireless Communications Letters (institute of electrical and electronics engineers (IEEE) Wireless communication prompter) by slave Tomic et al, and is referred to as a weighted least square method for short.
The simulation result shows that the method has good positioning performance, can well meet the requirement of high positioning precision, and is less influenced by measurement noise.

Claims (1)

1. A three-dimensional wireless sensor network non-cooperative positioning method based on RSS and AOA is characterized by comprising the following steps:
the method comprises the following steps: establishing a space rectangular coordinate system as a reference coordinate system in a three-dimensional wireless sensing network environment, setting N anchor nodes and a target source in the three-dimensional wireless sensing network environment, and recording the coordinate position of the ith anchor node in the reference coordinate system as siThe coordinate position of the target source in the reference coordinate system is recorded as x, si＝(si1,si2,si3)，x＝(x1,x2,x3) (ii) a Wherein N is a positive integer, N represents the total number of anchor nodes in the three-dimensional wireless sensor network environment, N is more than or equal to 4, i is a positive integer, i is more than or equal to 1 and less than or equal to N, si1、si2、si3Corresponding representation si1 st, 2 nd, 3 rd coordinate component, x1、x2、x3Correspondingly representing the 1 st coordinate component, the 2 nd coordinate component and the 3 rd coordinate component of x;
step two: transmitting a measuring signal by a target source in a three-dimensional wireless sensing network environment, and obtaining an RSS measuring value and an AOA measuring value after each anchor node receives the measuring signal transmitted by the target source; then, a measurement model of the RSS measurement value obtained by each anchor node is established, and the measurement model of the RSS measurement value obtained by the ith anchor node is described as follows:
Figure FDA0002177258960000011
and establishing respective measurement models of an azimuth angle measurement value and an elevation angle measurement value in the AOA measurement values obtained by each anchor node, and describing the measurement model of the azimuth angle measurement value in the AOA measurement values obtained by the ith anchor node as follows:
Figure FDA0002177258960000012
a measurement model of elevation measurements among AOA measurements obtained by the ith anchor node is described as:
Figure FDA0002177258960000013
wherein L isiRepresents the RSS measurement obtained by the ith anchor node, i.e. the path loss, L, existing on the transmission path experienced by the measurement signal transmitted from the target source to the ith anchor node0Representing the path loss existing on a transmission path from the transmission of a measurement signal from a target source to the reception of a reference point, gamma representing the path loss coefficient of the transmission distance from the transmission of the measurement signal from the target source to the reception of each anchor node, the value range of gamma being 2.2-2.8, the symbol "| | |" being the symbol of Euclidean norm, d0Representing the distance of a reference point to a target source in a reference coordinate system, niRepresents LiOf the measurement noise, niObeying a Gaussian distribution
Figure FDA0002177258960000021
Figure FDA0002177258960000022
Represents niPower of phiiRepresenting azimuthal measurements, m, in AOA measurements obtained by the ith anchor nodeiIs indicative of phiiMeasurement noise present in, miObeying a Gaussian distribution
Figure FDA0002177258960000023
Figure FDA0002177258960000024
Represents miPower of alphaiRepresenting elevation measurements, v, of AOA measurements obtained at the ith anchor nodeiDenotes alphaiOf the measurement noise, viObeying a Gaussian distribution
Figure FDA0002177258960000025
Figure FDA0002177258960000026
Denotes viThe power of (d);
step three: in that
Figure FDA0002177258960000027
mi＜＜1、viWhen < 1, i.e. under the condition of high signal-to-noise ratio, for
Figure FDA0002177258960000028
Making approximate transformation to obtain niThe approximate expression of (c), described as:
Figure FDA0002177258960000029
and to
Figure FDA00021772589600000210
Making approximate transformation to obtain miThe weighted approximate expression is described as:
Figure FDA00021772589600000211
to pair
Figure FDA00021772589600000212
Making approximate transformation to obtain viThe approximate expression of (c), described as:
Figure FDA00021772589600000213
wherein the content of the first and second substances,
Figure FDA00021772589600000214
Figure FDA00021772589600000215
Figure FDA00021772589600000216
represents miWeighted value, ci＝[-sin(φi),cos(φi),0]TThe term "[ 2 ]]"is a vector representing a symbol, k ═ 0,0,1]TThe symbol "T" is transposed symbol, β, μ, λi、ciK is an introduced intermediate variable;
step four: according to
Figure FDA00021772589600000217
And
Figure FDA00021772589600000218
and combining the least square criterion to obtain a non-convex positioning problem for solving x, which is described as:
Figure FDA00021772589600000219
wherein, min () is a minimum function;
step five: introducing auxiliary variables f, h, r and z and relaxation variables e, g and t into the description of solving the non-convex positioning problem of x to obtain an equivalent problem of solving the non-convex positioning problem of x, wherein the description is as follows:
Figure FDA0002177258960000031
wherein e isiDenotes the i-th component in e, giDenotes the ith component in g, "s.t." means "constrained to … …", hiDenotes the ith component in h, hi＝||x-si||2，I3Represents a 3-dimensional identity matrix, riDenotes the i-th component in r, ri＝||x-si||，fiThe i-th component in f is represented,
Figure FDA0002177258960000032
step six: using a second order cone relaxation method, r in the description of the equivalence problem of the non-convex localization problem of x will be solvedi＝||x-siRelaxation of | to ri≥||x-siL; and adopting a semi-positive definite relaxation method to solve the problem that x is not the equal value of x in the description of the problem of the convex positioningTx relaxation is in the form of a linear matrix inequality
Figure FDA0002177258960000033
Then according to ri≥||x-siI and
Figure FDA0002177258960000034
and obtaining a mixed semi-positive definite/second order cone programming problem for solving x, wherein the description is as follows:
Figure FDA0002177258960000041
wherein the content of the first and second substances,
Figure FDA0002177258960000042
to represent
Figure FDA0002177258960000043
Is a semi-positive definite matrix;
step seven: and solving the mixed semi-positive definite/second-order cone programming problem of x by adopting an interior point method to obtain the global optimal solution of x, wherein the global optimal solution is used as the position estimation value of the target source in the reference coordinate system.
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