CN115996461A - Underwater wireless sensor network node positioning method based on matrix completion - Google Patents
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Abstract
The invention discloses a matrix completion-based underwater wireless sensor network node positioning method, which comprises the following steps: s1: acquiring Euclidean distances between every two different sensor nodes by using a TOA ranging method, so as to obtain an Euclidean distance matrix of the underwater sensor network nodes; s2: restoring and complementing the Euclidean distance matrix by adopting a matrix complementing algorithm based on non-convex rank approximation to obtain a restored and complemented Euclidean distance matrix, thereby obtaining distance measurement among all nodes in a network; s3: and calculating the relative coordinates of all the nodes under water by using an MDS-MAP algorithm, converting the relative coordinates of the conventional nodes into absolute coordinates by using the positions of a part of anchor nodes, and outputting the coordinates of the conventional nodes. The invention not only can accurately recover and complement the missing distance matrix, but also can effectively process the influence caused by Gaussian noise and outlier noise, and has better robustness and accuracy. The relative position is converted into the absolute position by using the position information of a small amount of anchor nodes, so that the node positioning precision is high.
Description
Technical Field
The invention relates to the technical field of underwater wireless sensor networks, in particular to an underwater wireless sensor network node positioning method based on matrix completion.
Background
The underwater wireless sensor network (Underwater Wireless Sensor Networks, UWSNs) is used as one of the key technologies for water area monitoring, and has wide application prospects in the fields of ocean exploration, underwater environment monitoring, underwater navigation target monitoring and tracking, auxiliary navigation and the like. The positioning technology of sensor nodes takes an important place in the application of UWSNs. The data acquisition with effective node position information is the basis of data application, and to realize effective acquisition and analysis of underwater resources and environmental data, the data acquisition must be based on a high-precision underwater sensor node positioning technology. Therefore, how to accurately determine the position of the underwater sensor node is a research focus of current domestic and foreign scholars.
Due to the specificity of the underwater communication environment, only part of nodes can be accurately positioned, such as submarines, ocean surface buoys, underwater fixed beacons and the like. In the UWSNs, these nodes with known positions are called anchor nodes, and other conventional nodes perform communication positioning with the anchor nodes as positioning references. Due to the difficulty of deployment, the number of underwater anchor nodes is generally small, which increases the difficulty of locating conventional nodes. Meanwhile, the topology of the underwater sensor network may cause network holes due to problems such as water flow or node faults, and difficulty in positioning the underwater nodes is brought.
The node positioning method based on the MDS algorithm is called an MDS-MAP algorithm, and the algorithm is characterized in that the number of anchor nodes is small, and the positioning accuracy is higher than that of a general algorithm. The MDS-MAP algorithm only needs 3 anchor nodes in a two-dimensional plane and only needs 4 anchor nodes in a three-dimensional space to realize the positioning of all nodes in the network, accords with the application characteristics of the underwater sensor network, and has obvious advantages compared with other algorithms. Generally, this type of node location technique requires that sufficient distance data be collected, while requiring that the obtained distance data be accurate, otherwise the location performance of the algorithm will be severely affected. The transmission distance of the underwater sensor is limited, so that the network part is connected, and the measured distance among a plurality of nodes is lost. In addition, the complex underwater communication environment causes the distance measurement among the nodes to be interfered by various noises, thereby reducing the positioning accuracy of the nodes.
Due to the complex underwater environment, the deployment of the underwater nodes is more sparse than that of land nodes, the underwater network structure is uneven, and the nodes are difficult to recycle, so that a larger positioning error is brought to the underwater node positioning method based on the MDS-MAP algorithm. The positioning error of the method mainly comes from the following 2 aspects:
1) In the actual collection process of the distance data, the distance information between all the nodes cannot be collected due to the energy condition and the communication range of the nodes, so that the obtained Euclidean distance matrix is often incomplete, and the difficulty of positioning the nodes is further increased.
2) The distance measurement process is inevitably interfered by various noise factors, so that the obtained distance data is inaccurate, and larger node positioning errors are caused. Typically, gaussian noise and outlier noise are the main contributors to the distance measurement process.
Disclosure of Invention
The invention provides a matrix-complement-based underwater wireless sensor network node positioning method, which can effectively improve node positioning accuracy, and aims to solve the problems that a large amount of node distance information is needed in the traditional underwater node positioning method and node distance errors are caused by a complex underwater environment.
In order to achieve the above purpose of the present invention, the following technical scheme is adopted:
a method for positioning an underwater wireless sensor network node based on matrix completion comprises the following steps:
s1: acquiring Euclidean distances between every two different sensor nodes by using a TOA ranging method, so as to obtain an Euclidean distance matrix of the underwater sensor network nodes;
s2: restoring and complementing the Euclidean distance matrix by adopting a matrix complementing algorithm based on non-convex rank approximation to obtain a restored and complemented Euclidean distance matrix, thereby obtaining distance measurement among all nodes in a network;
s3: and calculating the relative coordinates of all the nodes under water by using an MDS-MAP algorithm, converting the relative coordinates of the conventional nodes into absolute coordinates by using the positions of a part of anchor nodes, and outputting the coordinates of the conventional nodes.
Preferably, S1 is specifically as follows:
acquiring the Euclidean distance from a sensor node i to a sensor node j by using a TOA ranging method, wherein the node i receives a broadcast data packet of the node j in a positioning period T;
the node i calculates the Euclidean distance between the node i and the node j according to the sending time of the received data packet and the time of the received data packet, and the calculation formula is as follows:
d ij ＝V×(t 2 -t 1 ) (1)
wherein ,dij Is the Euclidean distance between node i and node j; v is the acoustic signal propagation velocity; t is t 1 Is the time at which the data packet is sent; t is t 2 Is the time of receipt of the data packet;
after node i calculates the Euclidean distance to node j, the Euclidean distance d ij Transmitting the data to a base station on the bank through buoy nodes, so as to obtain a Euclidean distance matrix of the underwater sensor network nodes;
wherein n represents the number of nodes in the underwater wireless sensor network, and EDM represents the Euclidean distance matrix.
Further, the data packet comprises a node label, a positioning period label and a sending time;
the node labels are arranged with a sequence before the network node deployment;
the positioning period is used for judging whether the data packet is the data packet of the positioning period, if not, discarding the data packet of the positioning period, continuing monitoring, otherwise, updating the data packet by the node i;
the sending time, the time of sending the data packet by the node j.
Preferably, S2, a matrix complement algorithm based on non-convex rank approximation is used to perform restoration and complement on the euclidean distance matrix, which is specifically as follows:
decomposing the Euclidean distance matrix to obtain a low-rank matrix comprising distance data and a sparse matrix comprising outlier noise; the low rank matrix is approximated by a kernel norm, and the sparse matrix is approximated by l 1 Approximating the norm;
the restoration complement of the euclidean distance matrix can be modeled as:
wherein D represents a complete Euclidean distance matrix, E represents a noise matrix, S represents an observation distance matrix, Ω represents a set of subscripts of known elements in the observation distance matrix S, and P Ω Representing projection operators II * The core norms of the matrix are represented, lambda represents Lagrangian multiplier 1 L representing matrix 1 A norm;
secondly, introducing a non-convex rank function to replace a kernel norm to approximate an original rank function, and obtaining a non-convex rank function approximation model;
then, solving the proposed non-convex rank function approximation model by using an alternate direction multiplier method to obtain a complete Euclidean distance matrix without noise pollution.
Further, the introduction of a non-convex rank function to replace the kernel norm to approximate the original rank function, and the non-convex rank function approximation model is obtained, which is specifically as follows:
relaxation is carried out on the nuclear norms in the formula (3) by adopting a non-convex regularization term based on a Laplace function, and the formula (3) is converted into a smaller-scale optimization problem by utilizing the idea of matrix decomposition:
wherein ,‖V‖γ Is a non-convex function; u and V both represent an n x r matrix, r representing the rank of matrix D, when U is satisfied T When u=i, d=uv;
considering equation (4) as a variable separable convex programming problem, the augmented lagrangian function form of equation (4) is:
where Λ represents a linear constraint Lagrangian operator,<·,·>represents the inner product operation of matrix, mu>0 represents penalty parameter,
Representing the F-norm of the matrix.
Still further, the proposed non-convex rank function approximation model is solved by using an alternate direction multiplier method, specifically as follows:
carrying out iterative solution on the formula (5) by using an alternate direction multiplier method, wherein the basic iterative formula is as follows:
μ k+1 ＝min(ρμ k ,μ max ) (10)
wherein ρ >1 is a constant, k represents the current iteration number;
step D1: to update U, save V, E, Λ unchanged, convert equation (6) to its equivalent:
wherein p=s+Λ k /ρ k The method comprises the steps of carrying out a first treatment on the surface of the Hypothesis (P-E) k )V＝AΣB T A and B are matrices (P-E k ) V left and right singular value vectors, then U's solution is:
U k+1 ＝AB T (12)
step D2: solving for V according to equation (13):
step D3: solving for E according to equation (14):
E k+1 ＝sgn(W k )·max{|W k |-λ/μ k ,0} (14)
wherein ,
sgn (·) is a sign function; />
Step D4: updating the lagrangian multiplier Λ according to (9) k+1 ；
Step D5: updating the parameter mu according to (10) k+1 ；
Step D6: judging
Whether or not it is true, if so, output
Restoring the full Euclidean distance matrix for the final; otherwise, returning to the step D1; where ε represents a tolerable error.
Preferably, the relative coordinates of all nodes under water are calculated by using an MDS-MAP algorithm, and the specific steps are as follows:
firstly, calculating a double-centering similarity matrix G for recovering the full Euclidean distance matrix:
wherein ,
j represents a centralized matrix, I represents an identity matrix, and n represents the number of all nodes in the underwater wireless sensor network; d represents a restored and completed Euclidean distance matrix;
and then carrying out singular value decomposition on the double-centering similarity matrix G:
[A,Σ,B]＝svd(G) (16)
wherein Σ represents a diagonal matrix obtained by singular value decomposition of the matrix G, svd () represents a singular value decomposition operation, and a and B represent two orthogonal matrices obtained by singular value decomposition of the matrix G;
and finally, calculating a relative coordinate matrix R of the nodes:
wherein ,rn Representing the relative coordinates of the nth node;
d represents the dimension of the node coordinates and T represents the transpose operation of the matrix.
Further, the relative coordinates of the regular node are converted into absolute coordinates by the positions of a part of the anchor nodes, and the coordinates of the regular node are output, specifically as follows:
firstly, calculating a coordinate transformation matrix Q of an anchor node:
wherein ,Q1 Representing a rotation matrix, Q 2 Represents a mirror matrix, m represents the number of anchor nodes, t m Representing absolute coordinates of the mth node;
and then converting the relative coordinates of the conventional nodes into absolute coordinates:
t i ＝Q(r i -r 1 )+t 1 ,i＝m+1,…,n (19)
finally outputting the coordinates { t } of the conventional nodes i |t m+1 ,t m+2 ,...,t n }。
A computer device comprising a memory, a processor and a computer program stored on the memory and executable on the processor, the processor implementing the steps of the matrix-completion-based underwater wireless sensor network node positioning method when executing the computer program.
A computer readable storage medium having stored thereon a computer program which, when executed by a processor, implements the steps of the matrix-completion based underwater wireless sensor network node positioning method.
The beneficial effects of the invention are as follows:
aiming at the problems that the measured distance data is missing and the distance data is polluted by noise in the underwater wireless sensor network, the invention adopts a matrix complement algorithm based on non-rank approximation to recover and complement the Euclidean distance matrix. The method not only can accurately recover and complement the missing distance matrix, but also can effectively process the influence caused by Gaussian noise and outlier noise, and has better robustness and accuracy.
Aiming at the problems of low node positioning accuracy caused by the small number and uneven distribution of anchor nodes and large iteration error in an underwater wireless sensor network, the invention adopts an MDS-MAP algorithm to calculate the relative position of a conventional node based on the complete Euclidean distance matrix after the recovery, and then uses the position information of a small amount of anchor nodes to convert the relative position into an absolute position, thereby having higher node positioning accuracy.
Drawings
Fig. 1 is a schematic diagram of an underwater wireless sensor network structure.
Fig. 2 is a flow chart of the steps of the method for positioning the nodes of the underwater wireless sensor network based on matrix completion.
Fig. 3 is a schematic diagram of a network topology.
Fig. 4 is a schematic diagram of EDM completion error.
Fig. 5 is a schematic diagram of a node positioning error.
Fig. 6 is a schematic diagram of the effect of actual positioning of nodes.
Detailed Description
The invention is described in detail below with reference to the drawings and the detailed description.
Example 1
The network structure of the underwater wireless sensor network (Underwater Wireless Sensor Networks, UWSNs) considered in this embodiment is shown in fig. 1, and mainly comprises buoy nodes, anchor nodes, regular nodes and the like. The buoy node is a node deployed on the water surface, is provided with a GPS receiver, can be positioned through GPS signals, and can also communicate with a base station on the shore to send related data. The anchor nodes are nodes with stronger computing power in the UWSNs, and the nodes can acquire the position information through direct communication with the buoy nodes, so that self-positioning is realized. The conventional nodes refer to underwater nodes with weaker computing power and lower energy consumption, the nodes cannot directly acquire own position information, and the position coordinates of the nodes need to be solved by a positioning algorithm.
The underwater node positioning method based on matrix completion mainly comprises three stages: the method for positioning the nodes of the underwater wireless sensor network based on the matrix completion comprises the following steps of:
s1: acquiring Euclidean distances between every two different sensor nodes by using a TOA ranging method, so as to obtain an Euclidean distance matrix of the underwater sensor network nodes;
s2: restoring and complementing the Euclidean distance matrix by adopting a matrix complementing algorithm based on non-convex rank approximation to obtain a restored and complemented Euclidean distance matrix, thereby obtaining distance measurement among all nodes in a network;
s3: and calculating the relative coordinates of all the nodes under water by using an MDS-MAP algorithm, converting the relative coordinates of the conventional nodes into absolute coordinates by using the positions of a part of anchor nodes, and outputting the coordinates of the conventional nodes.
In a specific embodiment, S1 is specifically as follows:
acquiring the Euclidean distance from a sensor node i to a sensor node j by using a TOA ranging method, wherein the node i receives a broadcast data packet of the node j in a positioning period T;
the node i calculates the Euclidean distance between the node i and the node j according to the sending time of the received data packet and the time of the received data packet, and the calculation formula is as follows:
d ij ＝V×(t 2 -t 1 ) (1)
wherein ,dij Is the Euclidean distance between node i and node j; v is the acoustic signal propagation velocity; t is t 1 Is the time at which the data packet is sent; t is t 2 Is the time of receipt of the data packet;
after node i calculates the Euclidean distance to node j, the Euclidean distance d ij Transmitting the data to a base station on the bank through buoy nodes, so as to obtain a Euclidean distance matrix of the underwater sensor network nodes;
wherein n represents the number of nodes in the underwater wireless sensor network, and EDM represents the Euclidean distance matrix.
In this embodiment, the data packet includes a node identifier, a positioning period identifier, and a transmission time;
see in particular table 1:
table 1 data packet of underwater sensor node
| Node label (j) | Positioning periodic marker (T) n ) | Time t of transmission 1 |
The node labels are arranged in order before the network nodes are deployed, namely, the nodes are arranged in order before the network nodes are deployed;
the positioning period is used for judging whether the data packet is the data packet of the positioning period, if not, discarding the data packet of the positioning period, continuing monitoring, otherwise, updating the data packet by the node i;
the sending time, the time of sending the data packet by the node j.
In a specific embodiment, S2, a matrix complement algorithm based on non-convex rank approximation is used to perform restoration and complement on the euclidean distance matrix, which is specifically as follows:
based on the low rank nature of the Euclidean distance matrix EDM, consider decomposing the original EDM into a low rank matrix and a sparse matrix. Decomposing the Euclidean distance matrix to obtain a low-rank matrix comprising distance data and a sparse matrix comprising outlier noise; the low rank matrix is approximated by a kernel norm, and the sparse matrix is approximated by l 1 Approximating the norm;
the restoration complement of the euclidean distance matrix can be modeled as:
wherein D represents a complete Euclidean distance matrix, E represents a noise matrix, S represents an observation distance matrix, Ω represents a set of subscripts of known elements in the observation distance matrix S, and P Ω Representing a projection operator; II * The core norms of the matrix are represented, lambda represents Lagrangian multiplier 1 L representing matrix 1 A norm;
secondly, introducing a non-convex rank function to replace a kernel norm to approximate an original rank function, and obtaining a non-convex rank function approximation model;
then, solving the proposed non-convex rank function approximation model by using an alternate direction multiplier method to obtain a complete Euclidean distance matrix without noise pollution.
In a specific embodiment, a non-convex rank function is introduced to replace a kernel norm to approximate an original rank function, so as to obtain a non-convex rank function approximation model, which is specifically as follows:
since the kernel norms in equation (3) are not the best approximation of the matrix rank, it is difficult to obtain an accurate solution to the model. Therefore, the present embodiment proposes a matrix reconstruction model based on non-convex rank approximation considering that a non-convex rank function is adopted to replace the kernel norm. The model adopts a non-convex regularization term (called gamma function) based on a Laplace function to relax the nuclear norms in the formula (3), and converts the formula (3) into a smaller-scale optimization problem by utilizing the idea of matrix decomposition:
wherein ,‖V‖γ Is a non-convex function; u and V both represent an n x r matrix, r representing the rank of matrix D, when U is satisfied T When u=i, d=uv;
this is because the rank of matrix D in equation (3) is r, and the size of matrix D is n×n, and the sizes of matrices U and V are n×r, respectively. When full ofFoot U T When u=i, d=uv T And r is in turn much smaller than n. The solution can be performed by decomposing the matrix D into two smaller-scale U and V by matrix decomposition.
Considering equation (4) as a variable separable convex programming problem, the augmented lagrangian function form of equation (4) is:
where Λ represents a linear constraint Lagrangian operator,<·,·>represents the inner product operation of matrix, mu>0 represents penalty parameter,
Representing the F-norm of the matrix.
The "approximation" in the present embodiment means "approximation" or "replacement", and the original rank function is one l 0 Norms are difficult to solve. The conventional idea is to replace l with a kernel norm 0 The norm calculates the approximate solution of the problem, and in this embodiment, the non-convex rank function is used in equation (4) γ Replace l 0 Norms are used to calculate an approximate solution to the problem. The non-convex function approximation model described in this embodiment is shown in equation (5).
In a specific embodiment, the proposed non-convex function approximation model is solved by using an alternate direction multiplier method, which is specifically as follows:
carrying out iterative solution on the formula (5) by using an alternate direction multiplier method, wherein the basic iterative formula is as follows:
μ k+1 ＝min(ρμ k ,μ max ) (10)
wherein ρ >1 is a constant, k represents the current iteration number;
step D1: to update U, save V, E, Λ unchanged, convert equation (6) to its equivalent:
wherein p=s+Λ k /ρ k The method comprises the steps of carrying out a first treatment on the surface of the Hypothesis (P-E) k )V＝AΣB T A and B are matrices (P-E k ) V left and right singular value vectors, then U's solution is:
U k+1 ＝AB T (12)
step D2: solving for V according to equation (13):
step D3: solving for E according to equation (14):
E k+1 ＝sgn(W k )·max{|W k |-λ/μ k ,0} (14)
wherein ,
sgn (·) is a sign function;
in this embodiment, equation (14) may be used to obtain the optimal solution by classical orthogonal Procrusters techniques.
Step D4: updating the lagrangian multiplier Λ according to (9) k+1 ；
Step D5: updating the parameter mu according to (10) k+1 ；
Step D6: judging
Whether or not it is true, if so, output
D1, recovering the full Euclidean distance matrix for the final result, otherwise, returning to the step D1; where ε represents a tolerable error.
The complete Euclidean distance matrix EDM obtained by recovering the matrix completion method based on the non-convex rank approximation can obtain distance measurement among all nodes in a network, and at the moment, the relative coordinates of all the nodes under water can be calculated by using an MDS-MAP algorithm, and the method comprises the following specific steps:
firstly, calculating a double-centering similarity matrix G for recovering the full Euclidean distance matrix:
wherein ,
j represents a centralized matrix, I represents an identity matrix, and n represents the number of all nodes in the underwater wireless sensor network; d represents a restored and completed Euclidean distance matrix;
and then carrying out singular value decomposition on the double-centering similarity matrix G:
[A,Σ,B]＝svd(G) (16)
wherein Σ represents a diagonal matrix obtained by singular value decomposition of the matrix G, svd () represents a singular value decomposition operation, and a and B represent two orthogonal matrices obtained by singular value decomposition of the matrix G;
and finally, calculating a relative coordinate matrix R of the nodes:
wherein ,rn Representing the relative coordinates of the nth node;
d represents the dimension of the node coordinates and T represents the transpose operation of the matrix.
In a specific embodiment, the relative coordinates of the regular node are converted into absolute coordinates by the position of a part of the anchor nodes, and the coordinates of the regular node are output, specifically as follows:
firstly, calculating a coordinate transformation matrix Q of an anchor node:
wherein ,Q1 Representing a rotation matrix, Q 2 Represents a mirror matrix, m represents the number of anchor nodes, t m Representing absolute coordinates of the mth node;
and then converting the relative coordinates of the conventional nodes into absolute coordinates:
t i ＝Q(r i -r 1 )+t 1 ,i＝m+1,…,n (19)
finally outputting the coordinates { t } of the conventional nodes i |t m+1 ,t m+2 ,...,t n }。
Aiming at the problems that measured distance data is missing and the distance data is polluted by noise in an underwater wireless sensor network, the embodiment provides a low-rank matrix complement model based on non-convex rank approximation. The model not only can accurately recover and complement the missing distance matrix, but also can effectively process the influence caused by Gaussian noise and outlier noise, and has good robustness and accuracy.
Aiming at the problems of low node positioning accuracy caused by the small number and uneven distribution of anchor nodes and large iteration error in an underwater wireless sensor network, the embodiment of the invention adopts an MDS-MAP algorithm to calculate the relative position of a conventional node based on the full Euclidean distance matrix after the recovery, and then uses the position information of a small amount of anchor nodes to convert the relative position into an absolute position, thereby having higher node positioning accuracy.
In order to verify the effectiveness of the underwater wireless sensor node positioning method (NRAM) based on matrix completion, provided by the invention, all sensor nodes in a network model are assumed to be provided with pressure sensors, and depth information of the underwater nodes is acquired through the pressure sensors, so that the node positioning problem of an underwater three-dimensional space is solved by utilizing a two-dimensional plane technology. The present invention contemplates a network topology as shown in fig. 3. As shown in fig. 3, 100 sensor nodes are uniformly distributed in a 100×100m 2 Wherein 5 anchor nodes are respectively arranged at four vertex angles and the center position of the monitoring area, and the rest 95 conventional nodes are uniformly and randomly distributed in the square monitoring area.
In order to research the performance of the algorithm under the noise interference condition, the invention examines the performance of the algorithm under the influence of Gaussian noise and outlier noise at the same time. Meanwhile, the experimental results are compared with several existing latest methods, including a positioning algorithm based on online robust matrix completion (Online Robust Matrix Completion, ORMC), a positioning algorithm based on truncated kernel norm and sparse regularization term matrix decomposition (matrix decomposition via truncated nuclear norm and a sparse regularizer, TNNSR) and a positioning algorithm based on semi-definite programming (Semidefinite Relaxation Localization, SDRL) based on a positioning algorithm with column outliers and sparse noise matrix completion (Matrix Completion with column Outliers and Sparse noise, MCOS). The simulation experiment platform is configured as follows: windows 10 64-bit operating system, interR Core (TM) i7-7700k 4.20GHz CPU,16GB RAM, simulation software MATLAB R2022a. All simulations were run repeatedly 100 times, taking their average as the final result.
In order to effectively evaluate the performance of the algorithm, two evaluation indexes selected by the invention are defined as follows:
(1) Average completion error (Average Completion Error, ACE):
wherein D represents the actual complete EDM,
indicating EDM after restoration of completion.
(2) Average positioning error (Average Localization Error, ALE):
wherein X represents the actual node coordinate matrix,
representing the estimated node coordinate matrix.
The invention assumes that the distance information is simultaneously interfered by Gaussian noise and outlier noise in the process of collecting the distance information. In the experiment, gaussian noise with the mean value of 0 and the variance of 100 and outlier noise with the value of 500-1000 are added at the same time, wherein the outlier noise proportion is set to be 1%. Fig. 4 and 5 show EDM completion errors and node positioning errors for the five algorithms under the influence of the mixed noise, respectively.
As shown in fig. 4, both NRAM and ORMC algorithms can show a good matrix complement effect under the influence of mixed noise, and not only smooth gaussian noise, but also effectively cope with interference caused by abnormal value noise. When the observation rate reaches 0.3, the matrix complement errors of the ORMC algorithm and the NRAM algorithm can be stabilized at 0.02m. In contrast, the SDRL algorithm achieves a matrix-complement error of 0.02m when the observation rate reaches 0.7. The MCOS algorithm achieves a complement error of 0.02m when the observation rate reaches 0.9. The TNNSR algorithm still has larger complement error when the observation rate reaches 0.9. While the ORMC algorithm can achieve nearly the same matrix completion accuracy as the NRAM algorithm when the observation rate is greater than 0.3, the completion performance of the ORMC algorithm is significantly inferior to the NRAM algorithm when the observation rate is less than 0.2. Experimental results show that the NRAM algorithm provided by the invention not only can effectively process the influence of mixed noise, but also has higher matrix complement precision under the condition of lower observation rate.
As can be seen from fig. 5, the results of the node positioning errors and matrix completion errors for the five algorithms are similar. Under the influence of mixed noise, the node positioning error of the NRAM algorithm provided by the invention can reach the positioning error of 1.5m when the observation rate is 0.1. Under the same conditions, the positioning errors of the TNNSR, ORMC, MCOS and SDRL algorithms are respectively 4.8m, 3.6m, 4.5m and 3.5m. Experimental results show that the NRAM algorithm provided by the invention can effectively process the interference of Gaussian noise and outlier noise, and can still obtain a good matrix complement effect under the condition of low data observation rate, so that higher node positioning accuracy is obtained.
Fig. 6 shows the final positioning results for 100 sensor nodes. Wherein the solid triangle represents the actual position of the anchor node, the open circle represents the actual position of the conventional node, and the solid point represents the position of the conventional node calculated by the positioning method provided by the invention. In this experiment, the observation rate of the distance data was 0.3, the gaussian noise mean was 0, the variance was 100, and the proportion of outlier noise was 1%. As shown in FIG. 6, the solid dots can fall substantially within the open circles more accurately. The result shows that the positioning method based on matrix completion can realize accurate positioning of the sensor node under the conditions of partial sampling data missing and noise pollution.
In summary, the underwater node positioning algorithm based on matrix completion provided by the invention not only can effectively process noise influence in an underwater complex environment, but also can still store higher positioning precision and better robustness under the conditions of random arrangement of underwater nodes, uneven network structure, fewer anchor nodes and larger network area.
Example 2
A computer device comprising a memory, a processor and a computer program stored on the memory and executable on the processor, the processor implementing the steps of the matrix-complement based underwater wireless sensor network node positioning method of embodiment 1 when the computer program is executed.
Example 3
A computer readable storage medium having stored thereon a computer program which, when executed by a processor, performs the steps of the matrix-complement based underwater wireless sensor network node positioning method as described in embodiment 3.
It is to be understood that the above examples of the present invention are provided by way of illustration only and not by way of limitation of the embodiments of the present invention. Any modification, equivalent replacement, improvement, etc. which come within the spirit and principles of the invention are desired to be protected by the following claims.
Claims (10)
1. A method for positioning an underwater wireless sensor network node based on matrix completion is characterized by comprising the following steps of: the method comprises the following steps:
s1: acquiring Euclidean distances between every two different sensor nodes by using a TOA ranging method, so as to obtain an Euclidean distance matrix of the underwater sensor network nodes;
s2: restoring and complementing the Euclidean distance matrix by adopting a matrix complementing algorithm based on non-convex rank approximation to obtain a restored and complemented Euclidean distance matrix, thereby obtaining distance measurement among all nodes in a network;
s3: and calculating the relative coordinates of all the nodes under water by using an MDS-MAP algorithm, converting the relative coordinates of the conventional nodes into absolute coordinates by using the positions of a part of anchor nodes, and outputting the coordinates of the conventional nodes.
2. The underwater wireless sensor network node positioning method based on matrix completion according to claim 1, wherein the method is characterized in that: s1, specifically, the following steps are adopted:
acquiring the Euclidean distance from a sensor node i to a sensor node j by using a TOA ranging method, wherein the node i receives a broadcast data packet of the node j in a positioning period T;
the node i calculates the Euclidean distance between the node i and the node j according to the sending time of the received data packet and the time of the received data packet, and the calculation formula is as follows:
d ij ＝V×(t 2 -t 1 ) (1)
wherein ,dij Is the Euclidean distance between node i and node j; v is the acoustic signal propagation velocity; t is t 1 Is the time at which the data packet is sent; t is t 2 Is the time of receipt of the data packet;
after node i calculates the Euclidean distance to node j, the Euclidean distance d ij Transmitting the data to a base station on the bank through buoy nodes, so as to obtain a Euclidean distance matrix of the underwater sensor network nodes;
wherein n represents the number of nodes in the underwater wireless sensor network, and EDM represents the Euclidean distance matrix.
3. The underwater wireless sensor network node positioning method based on matrix completion according to claim 2, wherein the method is characterized in that: the data packet comprises a node label, a positioning period label and a sending time;
the node labels are arranged with a sequence before the network node deployment;
the positioning period is used for judging whether the data packet is the data packet of the positioning period, if not, discarding the data packet of the positioning period, continuing monitoring, otherwise, updating the data packet by the node i;
the sending time, the time of sending the data packet by the node j.
4. The underwater wireless sensor network node positioning method based on matrix completion according to claim 1, wherein the method is characterized in that: s2, restoring and complementing the Euclidean distance matrix by adopting a matrix complementing algorithm based on non-convex rank approximation, wherein the method comprises the following steps of:
decomposing the Euclidean distance matrix to obtain a low rank matrix including distance data and a matrix including different distancesA sparse matrix of constant noise; the low rank matrix is approximated by a kernel norm, and the sparse matrix is approximated by l 1 Approximating the norm;
the restoration complement of the euclidean distance matrix can be modeled as:
wherein D represents a complete Euclidean distance matrix, E represents a noise matrix, S represents an observation distance matrix, Ω represents a set of subscripts of known elements in the observation distance matrix S, and P Ω Representing projection operators II * The core norms of the matrix are represented, lambda represents Lagrangian multiplier 1 L representing matrix 1 A norm;
secondly, introducing a non-convex rank function to replace a kernel norm to approximate an original rank function, and obtaining a non-convex rank function approximation model;
then, solving the proposed non-convex rank function approximation model by using an alternate direction multiplier method to obtain a complete Euclidean distance matrix without noise pollution.
5. The underwater wireless sensor network node positioning method based on matrix completion as claimed in claim 4, wherein the method comprises the following steps: the introduction of a non-convex rank function replaces a nuclear norm to approximate an original rank function, and a non-convex rank function approximation model is obtained, which is specifically as follows:
relaxation is carried out on the nuclear norms in the formula (3) by adopting a non-convex regularization term based on a Laplace function, and the formula (3) is converted into a smaller-scale optimization problem by utilizing the idea of matrix decomposition:
wherein ,‖V‖γ Is a non-convex function; u and V both represent an n x r matrix, r representing the rank of matrix D, when U is satisfied T When u=i, d=uv;
considering equation (4) as a variable separable convex programming problem, the augmented lagrangian function form of equation (4) is:
where Λ represents a linear constraint Lagrangian operator,<·,·>represents the inner product operation of the matrix, mu > 0 represents the penalty parameter,
Representing the F-norm of the matrix.
6. The underwater wireless sensor network node positioning method based on matrix completion according to claim 5, wherein the method is characterized in that: the proposed non-convex rank function approximation model is solved by using an alternate direction multiplier method, and the method is specifically as follows:
carrying out iterative solution on the formula (5) by using an alternate direction multiplier method, wherein the basic iterative formula is as follows:
μ k+1 ＝min(ρμ k ，μ max ) (10)
wherein ρ >1 is a constant, k represents the current iteration number;
step D1: to update U, save V, E, Λ unchanged, convert equation (6) to its equivalent:
wherein p=s+Λ k /ρ k The method comprises the steps of carrying out a first treatment on the surface of the Hypothesis (P-E) k )V＝AΣB T A and B are matrices (P-E k ) V left and right singular value vectors, then U's solution is:
U k+1 ＝AB T (12)
step D2: solving for V according to equation (13):
step D3: solving for E according to equation (14):
E k+1 ＝sgn(W k )·max{|W k |-λ/μ k ，0} (14)
wherein ,
sgn (·) is a sign function; />
Step D4: updating the lagrangian multiplier Λ according to (9) k+1 ；
Step D5: updating the parameter mu according to (10) k+1 ；
Step D6: judging
Whether or not it is true, if so, output
Restoring the full Euclidean distance matrix for the final; otherwise, returning to the step D1; where ε represents a tolerable error.
7. The underwater wireless sensor network node positioning method based on matrix completion according to claim 1, wherein the method is characterized in that: calculating the relative coordinates of all nodes under water by using an MDS-MAP algorithm, wherein the method comprises the following specific steps:
firstly, calculating a double-centering similarity matrix G for recovering the full Euclidean distance matrix:
wherein ,
j represents a centralized matrix, I represents an identity matrix, and n represents the number of all nodes in the underwater wireless sensor network; d represents a restored and completed Euclidean distance matrix;
and then carrying out singular value decomposition on the double-centering similarity matrix G:
[A，∑，B]＝svd(G) (16)
wherein, sigma represents a diagonal matrix obtained by singular value decomposition of the matrix G, svd () represents singular value decomposition operation, and A and B represent two orthogonal matrices obtained by singular value decomposition of the matrix G;
and finally, calculating a relative coordinate matrix R of the nodes:
wherein ,rn Representing the relative coordinates of the nth node;
d represents the dimension of the node coordinates and T represents the transpose operation of the matrix.
8. The matrix-complement-based underwater wireless sensor network node positioning method of claim 7, wherein the method is characterized by: the relative coordinates of the regular nodes are converted into absolute coordinates through the positions of a part of anchor nodes, and the coordinates of the regular nodes are output, specifically as follows:
firstly, calculating a coordinate transformation matrix Q of an anchor node:
wherein ,Q1 Representing a rotation matrix, Q 2 Represents a mirror matrix, m represents the number of anchor nodes, t m Representing absolute coordinates of the mth node;
and then converting the relative coordinates of the conventional nodes into absolute coordinates:
t i ＝Q(r i -r 1 )+t 1 ，i＝m+1，…，n (19)
finally outputting the coordinates { t } of the conventional nodes i |t m+1 ，t m+2 ，…，t n }。
9. A computer device comprising a memory, a processor and a computer program stored on the memory and executable on the processor, characterized in that the processor, when executing the computer program, carries out the steps of the matrix-complement based underwater wireless sensor network node positioning method as claimed in any of claims 1 to 8.
10. A computer-readable storage medium having stored thereon a computer program, characterized by: the computer program, when executed by a processor, implements the steps of the method for positioning an underwater wireless sensor network node based on matrix completion according to any one of claims 1 to 8.
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