CN116106823A - TDOA-PDOA combined positioning method based on particle swarm optimization - Google Patents

TDOA-PDOA combined positioning method based on particle swarm optimization Download PDF

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Publication number
CN116106823A
CN116106823A CN202310131914.0A CN202310131914A CN116106823A CN 116106823 A CN116106823 A CN 116106823A CN 202310131914 A CN202310131914 A CN 202310131914A CN 116106823 A CN116106823 A CN 116106823A
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pdoa
tdoa
particle swarm
cost function
swarm optimization
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吴昊
郝翎钧
邱千钧
石章松
孙世岩
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Naval University of Engineering PLA
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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
    • G01S5/02—Position-fixing by co-ordinating two or more direction or position line determinations; Position-fixing by co-ordinating two or more distance determinations using radio waves
    • G01S5/0257—Hybrid positioning
    • Y—GENERAL TAGGING OF NEW TECHNOLOGICAL DEVELOPMENTS; GENERAL TAGGING OF CROSS-SECTIONAL TECHNOLOGIES SPANNING OVER SEVERAL SECTIONS OF THE IPC; TECHNICAL SUBJECTS COVERED BY FORMER USPC CROSS-REFERENCE ART COLLECTIONS [XRACs] AND DIGESTS
    • Y02—TECHNOLOGIES OR APPLICATIONS FOR MITIGATION OR ADAPTATION AGAINST CLIMATE CHANGE
    • Y02T—CLIMATE CHANGE MITIGATION TECHNOLOGIES RELATED TO TRANSPORTATION
    • Y02T10/00—Road transport of goods or passengers
    • Y02T10/10—Internal combustion engine [ICE] based vehicles
    • Y02T10/40—Engine management systems

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  • General Physics & Mathematics (AREA)
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  • Management, Administration, Business Operations System, And Electronic Commerce (AREA)
  • Position Fixing By Use Of Radio Waves (AREA)

Abstract

The invention relates to a TDOA-PDOA joint positioning method based on particle swarm optimization, which combines the TDOA and PDOA measurement to form a joint cost function utilizing all available information, combines the influence average of the TDOA and the PDOA, improves the positioning precision, solves the optimization problem through quasi-Newton optimization and initializes a PSO algorithm, defines the PSO search space, improves the possibility of converging the solution of the node position to a global optimal value, reduces the total calculation complexity, and increases the population scale, wherein the root mean square error of the algorithm tends to converge to the lower limit; and the simulation experiment result shows that the combination method provided by the inventor has obvious advantages in position estimation.

Description

TDOA-PDOA combined positioning method based on particle swarm optimization
Technical Field
The invention relates to the technical field of positioning methods, in particular to a TDOA-PDOA combined positioning method based on particle swarm optimization.
Background
Along with the rapid development of Wireless Sensor Network (WSN) technology and a positioning method thereof, positioning becomes one of basic services for WSN data acquisition. Positioning accuracy generally depends on the accuracy of the distance estimation. Due to limitations in the size, power consumption and cost of sensor nodes, research into efficient positioning algorithms meeting the basic accuracy requirements of WSNs has met new challenges.
Disclosure of Invention
The invention provides a TDOA-PDOA combined positioning method based on particle swarm optimization for effectively solving the problems.
In order to solve the technical problems, the invention adopts the following technical scheme:
a TDOA-PDOA combined positioning method based on particle swarm optimization comprises the following steps of;
step 1, setting p as a target to be positioned, wherein N anchor nodes are arranged in a space where p is located, R1, R2 and R.the. RN are respectively arranged, and the coordinates of the anchor nodes Ri are R i I=1,..
Figure SMS_1
；
Step 2, defining a lower limit of the search space
Figure SMS_2
And upper limit->
Figure SMS_3
:
Figure SMS_4
Wherein,,
Figure SMS_6
、/>
Figure SMS_9
is the lower and upper limit of the space covered by the anchor point,/->
Figure SMS_11
Is in combination with->
Figure SMS_7
Vectors of the same dimension;
Figure SMS_8
，/>
Figure SMS_10
is the standard deviation of TDOA error, +.>
Figure SMS_12
Is an all 1 vector, +.>
Figure SMS_5
Is a scaling factor;
step 3, calculating PDOA cost function
Figure SMS_13
And TDOA cost function->
Figure SMS_14
；
Step 4, defining a composite function
Figure SMS_15
Solving to obtain the estimated position of p by using PSO particle swarm optimization algorithm>
Figure SMS_16
：
Figure SMS_17
Further, the PDOA cost function
Figure SMS_18
The calculation formula of (2) is as follows:
Figure SMS_19
；
given signal frequency
Figure SMS_20
,/>
Figure SMS_21
Is at the frequency +.>
Figure SMS_22
PDOA between the signals received at anchors Ri and Rj;
Figure SMS_23
，/>
Figure SMS_24
， />
Figure SMS_25
v is the signal propagation velocity;
Figure SMS_26
wherein->
Figure SMS_27
And->
Figure SMS_28
The variance of the TDOA and PDOA estimation errors in square seconds, respectively; />
Figure SMS_29
Is the PDOA variance in square radians.
Further, the cost function
Figure SMS_30
The calculation formula of (2) is as follows:
Figure SMS_31
，
Figure SMS_32
。
after the technical scheme is adopted, compared with the prior art, the invention has the following advantages:
the invention combines the TDOA and PDOA measurement to form a combined cost function utilizing all available information, combines the influence average of the TDOA and the PDOA, improves the positioning precision, solves the optimization problem through quasi-Newton optimization, initializes the PSO algorithm, defines the PSO search space, improves the possibility of converging the solution of the node position to the global optimal value, reduces the total calculation complexity, and increases the population scale, the root mean square error of the algorithm tends to converge to the lower bound; and the simulation experiment result shows that the combination method provided by the inventor has obvious advantages in position estimation.
The invention will now be described in detail with reference to the drawings and examples.
Drawings
FIG. 1 is a schematic diagram of a conventional gas turbine
Figure SMS_33
And->
Figure SMS_34
A relation diagram of the performance and the number s of groups;
FIG. 2 is the standard deviation of RMSE performance from TDOA error
Figure SMS_35
Relation diagram (++)>
Figure SMS_36
，/>
Figure SMS_37
）；
FIG. 3 is the standard deviation of RMSE performance and PDOA error
Figure SMS_38
Relation diagram (++)>
Figure SMS_39
）。
Description of the embodiments
The principles and features of the present invention are described below with reference to the drawings, the examples are illustrated for the purpose of illustrating the invention and are not to be construed as limiting the scope of the invention.
1. TDOA method
Consider a 2D or 3D space with N anchors R1, R2, an. P is the target to be located, and for any signal the difference in arrival time between anchor Ri and anchor Rj at point p is given by:
Figure SMS_40
（1）
wherein I II 2 the L2 norm is represented by the number, and v is the signal propagation velocity. This formula gives the exact (true) TDOA. In practice, we have to rely on the estimated value τij of TDOA. With time synchronization maintained between the receivers, two methods can be used to estimate TDOA. By transferring the received signals to a central processor, a cross-correlation based method may be used to obtain the TDOA estimate. Another approach is to make TDOA estimates based on independently estimating the time of arrival of the signal at each anchor, thereby calculating the time difference.
Using the TDOA information acquired by any one of the above or other methods, the target position may be estimated. In the present invention, we use the maximum likelihood method, which can be expressed as follows:
Figure SMS_41
（2）
to understand the function in (2)
Figure SMS_42
We use (1) to develop its expression to yield
Figure SMS_43
（3）
From equation (3), we can see that for a particular
Figure SMS_44
，/>
Figure SMS_45
May be a convex function. However, in general, the function +.>
Figure SMS_46
Must be handled in a non-convex optimization framework. For this reason, convex relaxation is a method representing an effective way.
PDOA method
Given signal frequency
Figure SMS_47
The PDOA is related to TDOA by
Figure SMS_48
（4）
Here the number of the elements is the number,
Figure SMS_49
is the PDOA between the signals received at anchors Ri and Rj at frequency fm. Since PDOA is a conventional estimate, it is given by:
Figure SMS_50
（5）
wherein the method comprises the steps of
Figure SMS_51
E [ -pi, pi) is the covered PDOA, (-pi)>
Figure SMS_52
The remainder of a divided by b is returned.
For a given estimated PDOA, we create the following cost function:
Figure SMS_53
（6）
in equation (6), the covered PDOA can be calculated using (5)
Figure SMS_54
. Scalar->
Figure SMS_55
Is a weighting factor given by:
Figure SMS_56
（7）
wherein the method comprises the steps of
Figure SMS_57
And->
Figure SMS_58
Is the variance of the TDOA and PDOA estimation errors in square seconds; />
Figure SMS_59
Is the PDOA variance in square radians. A practical way to estimate the error variance of TDOA and PDOA is to use the signal-to-noise ratio.
(6) The PDOA cost function set forth in (2) may be related to the TDOA cost function. From the definition of the wrap function, it is easy to see
Figure SMS_60
（8）
Based on equation (8), equation (6) can be said to represent another form of TDOA location where the residual is subject to a nonlinear transformation by a convolution function. This transformation is not unique, which makes the cost function difficult to optimize.
3. TDOA-FPOA joint cost function
It is contemplated that the TDOA and PDOA measurements may be combined to form a joint cost function that utilizes all available information. We propose a simple way to achieve this by using a complex function given by the following formula
Figure SMS_61
（9）
Wherein the method comprises the steps of
Figure SMS_62
And->
Figure SMS_63
Are defined in equations (3) and (6), respectively. It is important to note here that the cost function
Figure SMS_64
The effects of TDOA and PDOA are not combined on average. Retrospective (6) and formula (7), we observe +.>
Figure SMS_65
Has been variance +.>
Figure SMS_66
And (5) weighting. Furthermore, different frequency pairs->
Figure SMS_67
The influence of (2) is weighted by the inverse of the corresponding PDOA error variance.
4. Optimization
In the present invention we consider a positioning based on cost functions (6) and (9). The target position can be estimated by solving for the global minimum:
Figure SMS_68
（10）
or (b)
Figure SMS_69
（11）
To avoid having local minima as the final result, PSO particle swarm optimization is used. PSO surrounds an initial point
Figure SMS_70
A set of candidate solutions (swarm size s) is initialized. However, it is not always guaranteed that the PSO finds a global minimum, as the colony size and search area determine performance.
Two key processes to achieve good performance are: 1) The initial point of the PSO algorithm is set judiciously, 2) the boundary of the search space is set.
In order to provide a good starting point for PSO, we rely on solving the classical TDOA problem, equation (2), as a means to obtain a good starting point. To accomplish this task, there are a number of methods with well-known performance characteristics. I.e. using quasi-newton optimization to solve the optimization problem, equation (2), and initialize the PSO algorithm. We define the search space of the PSO as:
Figure SMS_71
（12）
wherein,,
Figure SMS_73
、/>
Figure SMS_76
is the lower and upper limit of the space covered by the anchor point,/->
Figure SMS_78
Is in combination with->
Figure SMS_74
Vectors of the same dimension;
Figure SMS_75
，/>
Figure SMS_77
is the standard deviation of TDOA error, +.>
Figure SMS_79
Is an all 1 vector, +.>
Figure SMS_72
Is a scaling factor.
We used in all tests
Figure SMS_80
. By using equation (12) and equation (13), we can set the boundaries of the PSO's search space. Defining the boundary in this way improves the convergence of the solutions of equation (10) and equation (11) to globalThe likelihood of an optimal value, while reducing the overall computational complexity.
5. Summary of TDOA-PDOA positioning methods
1) Using the TDOA measurements, the initial position is calculated by solving equation (2) using quasi-newton optimization.
2) Defining a lower limit of the search space using equation (12) and equation (13)
Figure SMS_81
And upper limit->
Figure SMS_82
。
3) Comparative example: algorithm 1 (TDOA-PDOA-1) was used: executing the formula (1) and the formula (2), and then solving the formula (10) using PSO.
4) The invention comprises the following steps: algorithm 2 (TDOA-PDOA-2) was used: executing the formula (1) and the formula (2), and then solving the formula (11) using PSO.
Note that for algorithm 1, although TDOA is missing from the expression of the cost function, the use of TDOA information is still necessary for successful initialization of the algorithm.
Simulation of
To simplify the representation of the results, we tested the approach we proposed using one specific anchor configuration (n=8). The receiver is located at
Figure SMS_83
Where a is formed of
Figure SMS_84
m 2 Is a square of (c). In all simulations, d=2.5 m was used.
For simplicity we consider the problem of two-dimensional positioning of the target on the same plane as the anchor point. All simulation results given in the following subsections are for
Figure SMS_85
(all coordinates are given in m) target. For a single audio frequency, calculate using equation (1)TDOA, while the covered PDOA is calculated based on equation (5). />
Figure SMS_86
(e.g., m=1). However, the result can be easily extended to the case of multi-frequency/multi-carrier. TDOA and PDOA errors are modeled as zero mean and zero standard difference, respectively>
Figure SMS_87
And->
Figure SMS_88
Is a gaussian noise of (c). Is provided with->
Figure SMS_89
Root Mean Square Error (RMSE) was used as a performance metric. For each simulation, RMSE was calculated from 5000 trials.
The proposed method yields two algorithms, TDOAPDOA-1 and TDOA-PDOA-2, which are summarized in section 5. The performance of the proposed method is compared with the linear closed form method (TDOALinear), the SDP relaxation method (TDOA-SDP) [ iterative optimization method (TDOA-Iter). TDOA location-based CRLB (CRLB-T), PDOA location-based lower bound (LB-P), and joint TDOA-PDOA lower bound (LB-J) are also used to compare performance.
In fig. 1, we studied the effect of population numbers of particle swarm algorithm on the performance of both algorithms. The setting considers the TDOA error standard deviation
Figure SMS_90
(normalized by multiplying the signal propagation speed), PDOA error standard deviation
Figure SMS_91
And a size of +.>
Figure SMS_92
Is used for searching the space of the search. As population size increases, the root mean square error of the proposed algorithm tends to converge to a lower bound. However, in the case of TDOA informationAided by the fact that TDOA-PDOA-2 requires fewer clusters to reach the lower bound. The results in fig. 2 show the significant advantage of our proposed joint approach in position estimation.
FIG. 2 is a drawing of
Figure SMS_93
And->
Figure SMS_96
RMSE at individual time and TDOA error standard deviation +.>
Figure SMS_98
Is a relationship of (2). When->
Figure SMS_95
Less than->
Figure SMS_97
When the joint lower bound LB-J approaches TDOA CRLB. On the contrary, when->
Figure SMS_99
When larger, LB-J is closer to LB-P, but when
Figure SMS_100
When too large, the erroneous TDOA location is detrimental to the initialization of the algorithm. Further, for a sufficiently small +.>
Figure SMS_94
The proposed method produces results with RMSE close to the lower limit.
In FIG. 3, we change the PDOA error standard deviation
Figure SMS_101
Simultaneously fix->
Figure SMS_102
And->
Figure SMS_103
. When (when)
Figure SMS_104
When the initial position estimation is larger, the TDOA-PDOA-2 has performance close to CRLB-T under the condition of accurate initial position estimation. However, TDOA-PDOA-1 has poor performance because TDOA information is not utilized in the optimization process. TDOA-PDOA-2 is preferred over other methods, highlighting the advantage of the joint cost function (9).
The foregoing is illustrative of the best mode of carrying out the invention, and is not presented in any detail as is known to those of ordinary skill in the art. The protection scope of the invention is defined by the claims, and any equivalent transformation based on the technical teaching of the invention is also within the protection scope of the invention.

Claims (3)

1. The TDOA-PDOA combined positioning method based on particle swarm optimization is characterized by comprising the following steps of;
step 1, setting p as a target to be positioned, wherein N anchor nodes are arranged in a space where p is located, R1, R2 and R.the. RN are respectively arranged, and the coordinates of the anchor nodes Ri are R i I=1,..
Figure QLYQS_1
；
Step 2, defining a lower limit of the search space
Figure QLYQS_2
And upper limit->
Figure QLYQS_3
:
Figure QLYQS_5
Wherein (1)>
Figure QLYQS_8
、/>
Figure QLYQS_10
Is the lower and upper limit of the space covered by the anchor point,/->
Figure QLYQS_6
Is in combination with->
Figure QLYQS_9
Vectors of the same dimension; />
Figure QLYQS_11
，/>
Figure QLYQS_12
Is the standard deviation of TDOA error, +.>
Figure QLYQS_4
Is an all 1 vector, +.>
Figure QLYQS_7
Is a scaling factor;
step 3, calculating PDOA cost function
Figure QLYQS_13
And TDOA cost function->
Figure QLYQS_14
；
Step 4, defining a composite function
Figure QLYQS_15
Solving to obtain the estimated position of p by using PSO particle swarm optimization algorithm>
Figure QLYQS_16
：
Figure QLYQS_17
。
2. The particle swarm optimization-based TDOA-PDOA joint positioning method according to claim 1, wherein the PDOA cost function
Figure QLYQS_18
The calculation formula of (2) is as follows:
Figure QLYQS_19
；
given signal frequency
Figure QLYQS_20
,/>
Figure QLYQS_21
Is at the frequency +.>
Figure QLYQS_22
PDOA between the signals received at anchors Ri and Rj;
Figure QLYQS_23
，/>
Figure QLYQS_24
，
Figure QLYQS_25
v is the signal propagation velocity;
Figure QLYQS_26
wherein->
Figure QLYQS_27
And->
Figure QLYQS_28
The variance of the TDOA and PDOA estimation errors in square seconds, respectively;
Figure QLYQS_29
is the PDOA variance in square radians.
3. The particle swarm optimization-based TDOA-PDOA joint positioning method according to claim 1, wherein the cost function
Figure QLYQS_30
The calculation formula of (2) is as follows: />
Figure QLYQS_31
，
Figure QLYQS_32
。/>
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Citations (5)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN101730227A (en) * 2009-11-10 2010-06-09 大连理工大学 Multi-base station secondary positioning method based on toughness estimation and arrival time difference
CN106793087A (en) * 2017-03-16 2017-05-31 天津大学 A kind of array antenna indoor positioning algorithms based on AOA and PDOA
CN107770859A (en) * 2017-09-21 2018-03-06 天津大学 A kind of TDOA AOA localization methods for considering base station location error
CN115210596A (en) * 2020-01-30 2022-10-18 Idac控股公司 Reference signal design and device procedures for downlink-based positioning/ranging using multi-frequency phase difference of arrival
CN115379558A (en) * 2022-08-22 2022-11-22 温州理工学院 Mobile station position estimation method based on particle swarm optimization crow algorithm

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* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN101730227A (en) * 2009-11-10 2010-06-09 大连理工大学 Multi-base station secondary positioning method based on toughness estimation and arrival time difference
CN106793087A (en) * 2017-03-16 2017-05-31 天津大学 A kind of array antenna indoor positioning algorithms based on AOA and PDOA
CN107770859A (en) * 2017-09-21 2018-03-06 天津大学 A kind of TDOA AOA localization methods for considering base station location error
CN115210596A (en) * 2020-01-30 2022-10-18 Idac控股公司 Reference signal design and device procedures for downlink-based positioning/ranging using multi-frequency phase difference of arrival
CN115379558A (en) * 2022-08-22 2022-11-22 温州理工学院 Mobile station position estimation method based on particle swarm optimization crow algorithm

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Title
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胡骏;乐英高;蔡绍堂;曹莉;吴浩;: "基于改进变异粒子群算法的TDOA/AOA定位研究", 组合机床与自动化加工技术, no. 04 *

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