Distributed kalman filter multi-node cooperative positioning method based on average consensus

Abstract

The application discloses a kind of distributed Kalman filtering multi-node cooperative tracking positioning method and system based on average consensus, it is related to networking positioning tracking technical field.The application innovatively proposes to the problem of detection packet loss and communication packet loss in wireless sensor network, the validity of measurement is checked, in eliminating, repeating, predicting, estimating four methods, suitable method is selected to carry out data processing;For asynchronous network, by setting global update cycle, local posteriori estimation of the same time point is obtained in each cycle, the confidence value of each node data is set according to the error characteristics of measurement data by average consensus algorithm, and the local posteriori estimation is fused, the problem of measurement asynchrony in asynchronous sensor network is solved, the information fusion step is optimized, and higher estimation accuracy is realized.

Description

TECHNICAL FIELD

[0001] The present application relates to the field of multi-node networking positioning technology, and particularly relates to a multi-node cooperative positioning method based on average consensus distributed Kalman filtering. BACKGROUND

[0002] At present, wireless sensor networks formed by multi-nodes in a self-organizing manner are widely applied. Wireless sensor networks mostly adopt a distributed control mode, that is, each node has independent routing and host functions, and there is no network center control point similar to a base station, and the positions of nodes are equal to each other, so that the wireless sensor network has strong robustness and invulnerability. However, in actual engineering, due to different initial sampling times, different sampling periods and information transmission time delays of nodes, the measurement sampling times of nodes are different, so that the measurement information reaching the nodes is asynchronous, and such a network is called an asynchronous sensor network. In actual application of the asynchronous wireless sensor network, due to network time delay, network congestion, sensor failure and the like, packet loss often occurs. Packet loss mainly includes two types: ① communication packet loss in the network node, that is, the data packet sent by a node in the network to a neighbor node is lost; and ② detection packet loss, that is, the measurement data of the network node is lost. The packet loss problem will lead to reduced estimation accuracy or even divergent estimation values. SUMMARY

[0003] The present application provides a multi-node cooperative positioning method based on average consensus distributed Kalman filtering to solve the following problems:

[0004] 1) Asynchronous multi-node target tracking problem

[0005] In the cooperative positioning of a wireless sensor network (WSN), due to different initial sampling times, different sampling periods and information transmission time delays of the detectors of nodes, the measurement sampling times of nodes are different, so that the measurement information reaching the nodes is asynchronous.

[0006] 2) Communication packet loss and detection packet loss problem

[0007] In actual application of the wireless sensor network, due to network time delay, network congestion, sensor failure and the like, packet loss often occurs. Packet loss mainly includes two types: ① communication packet loss in the network node, that is, the data packet sent by a node in the network to a neighbor node is lost; and ② detection packet loss, that is, the measurement data of the network node is lost. The packet loss problem will lead to reduced estimation accuracy or even divergent estimation values.

[0008] The technical solution of the present application is as follows: a multi-node cooperative positioning method based on average consensus distributed Kalman filtering, comprising the following steps:

[0009] Step 1: Multiple sensors form a WSN (Wireless Networking System) through a self-organizing network, with each sensor acting as a node. The WSN collaboratively senses, collects, and processes information within the coverage area in real time. In the WSN, node i at time t... k Obtain the target azimuth angle With latitude and longitude information node j at time t k Obtain the target azimuth angle With latitude and longitude information Then, through multi-node collaborative computation, the time of node i at time t is obtained. k The measured values, and node j at time t k The measured value is then used to proceed to step 2.

[0010] Step 2: Perform a validity check on the measurements obtained in Step 1. For invalid data, select one of the four methods—elimination, duplication, prediction, or estimation—for data processing to obtain valid measurements. Proceed to step 3.

[0011] Step 3: Based on the EKF state estimation rule, construct the motion equations and observation equations of the target state within node i; for node i: calculate the state of node i at time t using the motion equations. k Local prior estimation Then based on local prior estimation The observation equation is used to calculate the time of node i at time t. k Local posterior estimation Proceed to step 4.

[0012] Step 4: Estimate based on local posterior. Node i advances through a state transition function to align with the endpoint t of the GUP. E To obtain time-aligned local posterior estimates Proceed to step 5.

[0013] Step 5: In WSN, calculate the state estimation fusion weight W by combining the distance between the target and the node, the number of the node's neighbors, and the measurement time point. ij W ij This is the fusion weight of node j relative to node i. Proceed to step 6.

[0014] Step 6: Node i broadcasts the current time t. k Local posterior estimation with time alignment Send it to neighboring nodes and estimate the fusion weight W based on the state. ij Time t is calculated based on the average consensus algorithm. k Global posterior estimate X k|k This enables multi-node collaborative positioning in WSN.

[0015] Compared with the prior art, the present application has the following advantages:

[0016] 1) In the local filtering stage, the present application processes data in the case of packet loss or misidentification, effectively suppresses the positioning error diffusion problem caused by observation loss or packet loss, and improves the stability of the system.

[0017] 2) After obtaining the local posterior estimation, the present application back-propagates the state estimation of each node to the end point of the GUP, thereby realizing time-aligned local estimation. It is suitable for periodic update monitoring systems and effectively solves the measurement asynchronous problem.

[0018] 3) In the fusion filter stage, the present application uses the average consensus algorithm to optimize the information fusion process, sets the confidence value of each node data according to the precision characteristics of the measurement data, integrates the time-aligned local posterior estimation of each node, forms a consistent global posterior estimation, and realizes higher estimation precision. BRIEF DESCRIPTION OF DRAWINGS

[0019] Figure 1 The flowchart of the present application based on average consensus distributed Kalman filter multi-node cooperative positioning method.

[0020] Figure 2 The single-node positioning error diagram of the present application.

[0021] Figure 3 The multi-node networking positioning error diagram of the present application. DETAILED DESCRIPTION

[0022] In order to make the above-mentioned purposes, features and advantages of the present application more obvious and easy to understand, the specific embodiments of the present application will be described in detail below with reference to the accompanying drawings.

[0023] In combination Figure 1 , the present application discloses a kind of based on average consensus distributed Kalman filter multi-node cooperative positioning method, including the following steps:

[0024] Step 1, multiple sensors are formed into WSN by self-organizing network, each sensor is regarded as a node, WSN in a cooperative manner real-time sensing, acquisition, processing information in coverage area;In WSN, node i at time t k Obtain target azimuth With latitude and longitude information Node j at time t k Obtain target azimuth With latitude and longitude information Again through multi-node cooperative calculation, the measurement value of node i at time t k , and the measurement value of node j at time t kThe measurement value of the multi-node cooperative calculation is as follows:

[0025] Step 11, the node i at time t k From the coordinates At an angle of A ray is drawn, and the node j at time t k From the coordinates At an angle of A ray is drawn, and the ray equation is established:

[0026]

[0027] Wherein, l represents the extension of the ray along the direction vector (cos(r)+sin(r)), l≥0, (N i (l), E i (l)) represents the ray equation established by the node i; (N j (l), E j (l)) represents the ray equation established by the node j; Indicates that the node i at time t k Obtains the target azimuth angle; Indicates that the node j at time t k Obtains the target azimuth angle; Indicates that the node i at time t k Obtains the latitude and longitude information; Indicates that the node j at time t k Obtains the latitude and longitude information.

[0028] Step 12, the intersection of the ray equation is solved, and the intersection coordinates (u m , v m ) are obtained, and then the coordinates of the intersection are averaged to obtain the measurement value, and the formula is as follows:

[0029]

[0030] Wherein, m represents a total of m intersection points; u m represents the horizontal coordinate of the intersection point m; v m represents the vertical coordinate of the intersection point m; represents the measurement value.

[0031] Go to step 2.

[0032] Step 2, the validity of the measurement value obtained in step 1 is checked, and the specific steps are as follows:

[0033] In practical applications, although the target's motion pattern cannot be predicted, there are certain limitations on the known target's velocity and acceleration. The target's velocity and acceleration are checked at this moment. If the target's velocity and acceleration exceed the known limits, or if the measurement value is lost, one of four methods—elimination, duplication, prediction, or estimation—is used to process the invalid data and obtain valid measurements. The four methods are detailed below:

[0034] a) Elimination: If node i is at time t k If the l-th measurement is invalid and the number of neighboring nodes of node i is less than 3, then this invalid measurement value will not be used for network positioning.

[0035] b) Repeat: If node i is at time t k If the l-th measurement is invalid, it is replaced with time t. k-1 Measured values

[0036] c) Prediction: If node i is at time t k If the l-th measurement is invalid and the number of neighbors of the node is less than 3, then time t is used. k-1 Global posterior estimate X k-1|k-1 As a valid measurement value

[0037] d) Estimation: If node i is at time t k If the l-th measurement is invalid and the number of neighbors of the node is less than 3, then time t is used. k-1 Global posterior estimate X k-1|k-1 Calculate time t using Kalman filtering k The local state estimate is used as an effective measurement.

[0038] In practical wireless sensor network applications, packet loss frequently occurs due to network latency, network congestion, and sensor malfunctions. Whether it's communication packet loss or detection packet loss, it leads to reduced accuracy or even divergence in the estimated values. This invention innovatively proposes a method for checking the validity of measured values. Based on the known limitations of the target's velocity and acceleration, the method checks the target's velocity and acceleration. For invalid data, one of four methods—elimination, duplication, prediction, or estimation—is used for data processing to obtain valid measured values. This reduces the impact of packet loss on estimation accuracy, improves positioning performance, and increases the accuracy of acquiring target information.

[0039] Proceed to step 3.

[0040] Step 3: According to the EKF state estimation rule, construct the motion equation and observation equation of the target state within node i. For node i: calculate the motion equation and observation equation of node i at time t.k Local prior estimation Then through local prior estimation Calculate node i at time t k Local posterior estimation Specifically as follows:

[0041] Step 31: Based on node i at time t k Valid measurement value Calculate the nonlinear function h(x) at time t k Jacobian matrix The nonlinear function f(x) at time t k Jacobian matrix

[0042] Step 32: Construct the motion equations and measurement equations for the target state, and calculate the state prediction and error covariance prediction. The calculation equations are as follows:

[0043]

[0044] Where f(·) represents the state transition function, X k-1|k-1 Represents time t k-1 Global posterior estimate; This indicates that node i at time t k Process noise; This indicates that node i at time t k Local prior estimates of P; k-1|k-1 Represents time t k-1 The global error covariance; This indicates that node i at time t k The prediction error covariance; This indicates that node i at time t k The process noise covariance matrix; This represents the nonlinear function f(x) at time t. k The Jacobian matrix; T denotes the transpose.

[0045] Step 33: Calculate the Kalman gain, update the state estimate and error covariance. The calculation formula is as follows:

[0046]

[0047]

[0048] in, This indicates that node i at time t k Kalman gain; This indicates that node i at time t k The prediction error covariance; This indicates that node i at time tk the local prior estimate of node i at time t denotes the local state estimate of node i at time t k the local prior estimate of node i at time t denotes the effective measurement value; h() denotes a nonlinear function; is the update error covariance of node i at time t k the local prior estimate of node i at time t denotes the Jacobian matrix of the nonlinear function h() of node i at time t k the identity matrix.

[0049] Go to Step 4.

[0050] Step 4, the local posterior estimate obtained in Step 3 is advanced by the state transition function to align to the end time t E of the GUP, obtaining the time-aligned local posterior estimate The calculation formula is as follows:

[0051]

[0052] where f E (·) denotes the state transition function for advancing the local posterior estimate to the end time t E of the GUP; Δt denotes the time interval; denotes the local posterior estimate of node i at time t k denotes the time-aligned local posterior estimate.

[0053] In actual engineering, due to different initial sampling times, different sampling periods and information transmission delays of nodes, the measurement values arriving at each node are asynchronous. In such an asynchronous wireless sensor network, the application innovatively introduces a global update period method. After obtaining the local posterior estimate, each node advances the local posterior estimate to the end time t E of the GUP, thereby realizing the time-aligned local posterior estimate. This method is suitable for periodic update monitoring systems and effectively solves the measurement asynchronous problem.

[0054] Go to Step 5.

[0055] In Step 5, the state estimation fusion weight W ij is calculated in combination with the distance between the target and the node, the number of neighbors of the node and the measurement time point. The specific steps are as follows:

[0056] Step 51, calculate the initial confidence value of the node:

[0057] ​Since the measurement value is related to the distance between the target and the node, the number of neighbors of the node and the measurement time, the initial confidence value of the node i is calculated as follows:

[0058] C i = t' x w t + n' x w n + d' x w d

[0059] where t', n' and d' are the normalized values of the measurement time, the number of neighbors of the node and the distance between the target and the node; w t , w n and w d are the corresponding weights; C i represents the initial confidence value of the node i.

[0060] Step 52, fusion weight calculation:

[0061] The initial confidence value C i of a single node is used to dynamically adjust the fusion weight, so as to more accurately reflect the importance of each node and the strength of the adjacent relationship.

[0062]

[0063] where C i represents the initial confidence value of the node i; C j represents the initial confidence value of the node j, C n represents the initial confidence value of the neighbor node of the node i; n represents the neighbor node of the node i, W ii represents the fusion weight of the node i itself; W ij represents the fusion weight of the node j relative to the node i; β represents the proportion of W ii and W ij ; N i represents the neighbor node set of the node i, and A represents all nodes in the WSN.

[0064] Since the measurement value is related to the distance between the target and the node, the number of neighbors of the node and the measurement time in actual engineering, the measurement value is more accurate when the target is closer to the node, the measurement value is more accurate when the number of neighbors of the node is larger, and the measurement value is more accurate when the measurement time is newer, the application innovatively proposes a method of introducing a fusion weight when information is fused, and according to the characteristics of the measurement value, the fusion weight of the more accurate local posterior estimate is increased when the average consensus algorithm is used to fuse information, the local posterior estimates of each node after time alignment are integrated, and higher positioning accuracy is achieved.

[0065] Go to step 6.

[0066] Step 6, according to the state estimation fusion weight Wij , the global posterior estimation X k at time t k|k is calculated based on the average consensus algorithm, and the specific steps are as follows:

[0067] Step 61, information exchange:

[0068] In the WSN, the node i sends the locally posterior estimation X k time-aligned to the neighbor nodes by broadcasting the current time t ;

[0069] Step 62, state update:

[0070] The update rule is shown in the following formula:

[0071]

[0072] wherein, denotes the consensus state variable of the node i at time t k of the (ξ+1)th iteration;

[0073] denotes the consensus state variable of the node i at time t k of the ξth iteration; denotes the consensus state variable of the node j at time t k of the ξth iteration; denotes the consensus error covariance matrix of the node i at time t k of the (ξ+1)th iteration; denotes the consensus error covariance matrix of the node i at time t k of the ξth iteration; denotes the consensus error covariance matrix of the node j at time t k of the ξth iteration; ξ denotes the ξth iteration of the GUP of the average consensus algorithm at time t k ; W ii denotes the fusion weight of the node i itself; W ij denotes the fusion weight of the node j relative to the node i; N i denotes the neighbor node set of the node i;

[0074] Step 63, convergence inspection:

[0075] When the absolute value of the difference between the (ξ+1)th iteration and the ξth iteration is less than a set value ε, it is considered that the state estimation converges, and the global posterior estimation X k at time t k|k is obtained, and the calculation is shown in the following formula:

[0076]

[0077] wherein, Xi(t) represents the local posterior estimation of node i at time t k consensus state variable of the (ξ+1)th iteration;

[0078] Xi(t) represents the local posterior estimation of node i at time t k consensus state variable of the ξth iteration; ξ represents the ξth iteration of the average consensus algorithm at time t k during the GUP; ε represents the convergence tolerance; X k|k represents the global posterior estimation of node i at t k .

[0079] Finally, the global posterior estimation X k at time t k|k is obtained, realizing the multi-node cooperative positioning in the WSN.

[0080] Embodiment 1

[0081] Considering that the communication range of each node and the detection range of the acoustic array are limited (both the communication distance and the detection distance are within 300 m), five nodes are selected to conduct the test in an area of 450 m x 200 m. Among them, the target travels along the set straight path at an average speed of 36 km / h, and the experimental scene is as shown in Figure 3 . In order to simulate the situation that the number of nodes detecting the target is uncertain, the motion trajectory of the target is set to be not detected by the five nodes throughout the journey, that is, there will be a case of less than five nodes for networking positioning, and the number of neighbors of each node is different.

[0082] The local posterior estimation results of the five observation stations are as shown in Figure 2 , wherein the blue line represents the actual motion trajectory of the target, and the dotted line represents the local posterior estimation of each node for each step of the target. As can be seen from the figure, when the target enters the detection range of each node, each node begins to perform local state estimation, and at the same time, when the detection distance is far, the error is large due to signal attenuation and environmental interference. In addition, due to the larger number of neighbors of node 3, the positioning accuracy is higher, verifying the influence of the number of neighbors of the node on the confidence value of the measurement value.

[0083] Each node calculates the global posterior estimation through the average consensus algorithm, and the result is as shown in Figure 3 , wherein the blue line represents the actual motion trajectory of the target, and the red star marks the global posterior estimation of each node after iteration. By comparing Figure 2 and Figure 3 , it can be seen that each node significantly improves the estimation accuracy of the target position by combining the local posterior information of the adjacent nodes through iterative calculation.

Claims

1. A distributed Kalman filter multi-node cooperative localization method based on average consensus, characterized in that, Includes the following steps: Step 1: Multiple sensors form a WSN through a self-organizing network, with each sensor acting as a node. The WSN collaboratively senses, collects, and processes information in the coverage area in real time. In WSN, node i at time t k Obtain the target azimuth angle with latitude and longitude information node j at time t k Obtain the target azimuth angle with latitude and longitude information Then, through multi-node collaborative computation, the time of node i at time t is obtained. k The measured values, and node j at time t k The measured value is then transferred to step 2; Step 2: Perform a validity check on the measurements obtained in Step 1. For invalid data, select one of the four methods—elimination, duplication, prediction, or estimation—for data processing to obtain valid measurements. Proceed to step 3; Step 3: Based on the EKF state estimation rule, construct the motion equations and observation equations of the target state within node i; for node i: calculate the state of node i at time t using the motion equations. k Local prior estimation Then based on local prior estimation The observation equation is used to calculate the time of node i at time t. k Local posterior estimation Proceed to step 4; Step 4: Estimate based on local posterior. Node i advances through a state transition function to align with the endpoint t of the GUP. E To obtain time-aligned local posterior estimates The calculation formula is as follows: Among them, f E (·) indicates that the local posterior estimate is used. Advance to the GUP finish line E The state transition function; Δt represents the time interval; This indicates that node i at time t k Local posterior estimation; Represents the time-aligned local posterior estimate; Proceed to step 5; Step 5: In WSN, calculate the state estimation fusion weight W by combining the distance between the target and the node, the number of the node's neighbors, and the measurement time point. ij W ij This is the fusion weight of node j relative to node i. Proceed to step 6. Step 6: Node i broadcasts the current time t. k Local posterior estimation with time alignment Send it to neighboring nodes and estimate the fusion weight W based on the state. ij Time t is calculated based on the average consensus algorithm. k Global posterior estimate X k|k This enables multi-node collaborative positioning in WSN.

2. The distributed Kalman filter multi-node cooperative localization method based on average consensus as described in claim 1, characterized in that, In step 2, the measurement values ​​obtained in step 1 are checked for validity. For invalid data, one of four methods—elimination, duplication, prediction, or estimation—is used for data processing to obtain valid measurement values. Specifically as follows: a) Elimination: If node i is at time t k If the l-th measurement is invalid and the number of neighboring nodes of node i is less than 3, then this invalid measurement value will not be used for network positioning. b) Repeat: If node i is at time t k If the l-th measurement is invalid, it is replaced with time t. k-1 Measured values c) Prediction: If node i is at time t k If the l-th measurement is invalid and the number of neighbors of the node is less than 3, then time t is used. k-1 Global posterior estimate X k-1|k-1 As a valid measurement value d) Estimation: If node i is at time t k If the l-th measurement is invalid and the number of neighbors of the node is less than 3, then time t is used. k-1 Global posterior estimate X k-1|k-1 Calculate time t using Kalman filtering k The local state estimate is used as an effective measurement.

3. The distributed Kalman filter multi-node cooperative localization method based on average consensus as described in claim 2, characterized in that, In step 3, according to the EKF state estimation rule, the motion equations and observation equations of the target state are constructed within node i. For node i: calculate the motion equations and observation equations of node i at time t. k Local prior estimation Then through local prior estimation Calculate node i at time t k Local posterior estimation Specifically as follows: Step 31: Based on node i at time t k Valid measurement value Calculate the nonlinear function h(x) at time t k Jacobian matrix The nonlinear function f(x) at time t k Jacobian matrix Step 32: Construct the motion equations and measurement equations for the target state, and calculate the state prediction and error covariance prediction. The calculation equations are as follows: Where f(·) represents the state transition function, X k-1|k-1 Represents time t k-1 Global posterior estimate; This indicates that node i at time t k Process noise; This indicates that node i at time t k Local prior estimates of P; k-1|k-1 Represents time t k-1 The global error covariance; This indicates that node i at time t k The prediction error covariance; This indicates that node i at time t k The process noise covariance matrix; The nonlinear function f(x) at node i at time t k The Jacobian matrix; T denotes transpose; Step 33: Calculate the Kalman gain, update the state estimate and error covariance. The calculation formula is as follows: in, This indicates that node i at time t k Kalman gain; This indicates that node i at time t k The prediction error covariance; This indicates that node i at time t k Local prior estimates; This indicates that node i at time t k Local state estimation; Represents the valid measurement value; h() represents the nonlinear function; Is node i at time t k The update error covariance; The nonlinear function h() at node i at time t k The Jacobian matrix; I represents the identity matrix.

4. The distributed Kalman filter multi-node cooperative localization method based on average consensus as described in claim 3, characterized in that, In step 5, the state estimation fusion weight W is calculated by combining the distance between the target and the node, the number of the node's neighbors, and the measurement time point. ij The specific steps are as follows: Step 51: Calculate the initial confidence values ​​of the nodes: Since the measurement information is related to the distance between the target and the node, the number of the node's neighbors, and the measurement time, the initial confidence value of node i is calculated as follows: C i =t'×w t +n'×w n +d'×w d Where t', n', and d' are the standardized values ​​of measurement time, number of node neighbors, and distance between the target and the node, respectively, and w t w n and w d That is the corresponding weight; C i Indicates the initial confidence value; Step 52, Calculation of fusion weights: Using the initial confidence value C of node i i To dynamically adjust the fusion weights so as to more accurately reflect the importance of each node and the strength of its adjacency relationships; Among them, C i C represents the initial confidence value of node i; j C represents the initial confidence value of node j. n Let W represent the initial confidence values ​​of the neighboring nodes of node i; n represents the neighboring nodes of node i; W represents the initial confidence values ​​of the neighboring nodes of node i. ii W represents the fusion weight of node i itself; ij β represents the fusion weight of node j relative to node i; β represents the weight used to adjust W. ii With W ij Specific gravity; N i Let A represent the set of neighboring nodes of node i, and let A represent all nodes in WSN.

5. The distributed Kalman filter multi-node cooperative localization method based on average consensus as described in claim 4, characterized in that, In step 6, the fusion weights W are estimated based on the state obtained in step 5. ij t is calculated based on the average consensus algorithm. k Global posterior estimate X at time t k|k The specific steps are as follows: Step 61, Information Exchange: In WSN, node i broadcasts the current time t. k Local posterior estimation with time alignment Send to neighboring nodes; Step 62, Status Update: The update rules are shown in the following formula: in, This indicates that node i at time t k The consensus state variables in the (ξ+1)th iteration; This indicates that node i at time t k The consensus state variables in the ξth iteration; This indicates that node j at time t k The consensus state variables in the ξth iteration; This indicates that node i at time t k The consensus error covariance matrix of the (ξ+1)th iteration; This indicates that node i at time t k The consensus error covariance matrix of the ξth iteration; This indicates that node j at time t k The consensus error covariance matrix of the ξ-th iteration; ξ represents the average consensus algorithm at time t. k The ξth iteration during the GUP period; W ii W represents the fusion weight of node i itself; ij N represents the fusion weight of node j relative to node i; i Represents the set of neighboring nodes of node i; Step 63, Convergence Check: When the absolute value of the difference between the (ξ+1)th iteration and the ξth iteration is less than the set value ε, the state estimation is considered to have converged, and time t is obtained. k Global posterior estimate X k|k The calculation is shown in the following formula: in, This indicates that node i at time t k The consensus state variables in the (ξ+1)th iteration; This indicates that node i at time t k The consensus state variable in the ξ-th iteration; ξ represents the average consensus algorithm at time t. k The ξ-th iteration during the GUP period; ε represents the convergence tolerance; X k|k Indicates that node i is in t k Global posterior estimate at time step; Finally, time t is obtained. k Global posterior estimate X k|k This enables multi-node collaborative positioning in WSN.

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