Distributed antenna cooperative positioning method based on particle flow filtering

Abstract

The invention discloses a distributed antenna cooperative positioning method based on particle flow filtering, and relates to the field of wireless sensor network distributed positioning. The method comprises the following steps: setting initial state distribution for each antenna node, and generating a particle set with corresponding uniform weight; in each discrete moment, the antenna node updates the speed and position of the particle according to the INS data, updates the position of the particle according to the GNSS data, and iteratively updates the position of the particle according to the distance measurement of the adjacent node; and introducing a logarithmic homotopy function to evolve a particle state from prior distribution obtained by INS data to posterior distribution predicted by GNSS data and distance measurement, and iteratively updating the particle state to realize cooperative positioning of distributed antennas at each moment. According to the method, the particle degeneration phenomenon can be effectively relieved, the resampling overhead is reduced, the positioning precision is high, the operation time is short, the influence of noise is small, and the method can better adapt to nonlinearity and high-dimensional measurement uncertainty in a distributed antenna cooperative positioning scene.

Description

Technical Field

[0001] This invention relates to a distributed antenna cooperative localization method based on particle flow filtering, belonging to the field of distributed localization in wireless sensor networks. Background Technology

[0002] The accuracy of the spatial coordinates of each antenna node in a distributed array antenna is crucial for ensuring coherent signal synthesis across multiple nodes. Cooperative positioning has been widely applied in wireless sensor networks, vehicle-to-everything (V2X) systems, and other fields, overcoming the limitations of single sensor capabilities and variable environments to improve positioning accuracy, system robustness, and coverage. In cooperative positioning with distributed array antennas, each node exchanges and fuses its measured relative distance information. This not only effectively reduces reliance on GPS accuracy but also suppresses the accumulation and amplification of various measurement errors through information sharing among multiple nodes.

[0003] Existing cooperative localization algorithms can be divided into non-Bayesian and Bayesian methods. In non-Bayesian algorithms, node positions are treated as unknown and deterministic parameters, and node positions can only be estimated through observation, ignoring prior information. Bayesian algorithms treat node positions as realizations of random variables, estimating their posterior distribution based on Bayes' theorem, thus fully utilizing prior information to estimate position uncertainty. From an implementation perspective, cooperative localization can also be divided into centralized and distributed methods. Centralized implementations require a central processor to uniformly process large-scale observation data, typically offering higher accuracy but with poor scalability and real-time performance. Distributed implementations involve each node exchanging local information with its neighbors and calculating its own position, effectively reducing computational and communication overhead while maintaining high accuracy, and exhibiting superior scalability.

[0004] Bayesian cooperative localization often employs a message-passing framework based on a probabilistic graphical model, where each antenna can be considered a node in a graph network, and the ranging links between antenna nodes correspond to the connections in the network. In this graphical model, calculating the spatial coordinates of an antenna node corresponds to solving the marginal probability distribution of that node. However, as the network size increases and nonlinear measurements become more complex, Bayesian inference requires numerical approximation of high-dimensional integrals, leading to a sharp increase in computational complexity. Belief propagation methods utilize local node messages to iterate through the graph multiple times to approximate the posterior distribution, reducing direct calculation of high-dimensional integrals. Their computation time is typically proportional to the number of connections in the network. In distributed antenna cooperative localization applications, due to the nonlinearity and non-Gaussian uncertainty of the ranging model, standard belief propagation algorithms often require further approximation. Nonparametric belief propagation methods introduce the idea of ​​particle filtering to perform Monte Carlo sampling approximation of the belief propagation messages, avoiding overly strict prior assumptions about the posterior distribution. Nonparametric belief propagation methods use particles to represent the probability distribution and continuously resample or weight the messages during updates to adapt to complex ranging error models and noise interference. However, as the number of iterations increases, particle filtering in nonlinear observations for cooperative positioning will suffer from particle degradation problems such as low sampling efficiency and unstable estimation accuracy, which in turn affects the cooperative positioning accuracy of distributed antennas. Summary of the Invention

[0005] To avoid particle degradation problems such as low sampling efficiency and unstable estimation accuracy in nonlinear observations of distributed antenna cooperative localization (DAL), and to improve the accuracy of DDL, this invention proposes a DDL cooperative localization method based on particle flow filtering. In the Bayesian filtering framework, by constructing a homotopy relationship between the prior and posterior distributions and introducing a "pseudo-time" parameter, particles can gradually move towards regions with higher posterior probabilities. This effectively approximates the posterior distribution and alleviates particle degradation, better adapting to the nonlinearity and high-dimensional measurement uncertainties in DDL cooperative localization scenarios.

[0006] This invention provides a distributed antenna cooperative localization method based on particle flow filtering, comprising the following four steps:

[0007] Step 1: Set an initial state distribution for each antenna node and generate a set of particles with corresponding uniform weights. Each particle state represents an antenna node state, which includes the node's position and velocity.

[0008] Step 2: At each discrete moment, the antenna node receives INS data and updates the velocity and position of the particle at the previous moment according to the INS motion model.

[0009] Step 3: The antenna node receives Global Navigation Satellite System (GNSS) measurement information and updates the particle position. Based on the Daum-Huang particle flow, a logarithmic homotopy function is introduced to transform the particle state from a prior distribution updated from INS data to a posterior distribution predicted by GNSS data and ranging. This yields a distribution function for the particle state. After receiving the current position coordinates from the GNSS measurement, the antenna first calculates the mean and variance of the node's state, updates the parameter matrix of the distribution function, and then updates the particle position.

[0010] Step 4: Antenna nodes receive ranging measurements from neighboring nodes, calculate the mean and variance of the current node state, and each particle updates the parameter matrix of the distribution function using the ranging measurements, thus updating the particle coordinates. At each discrete time step, each antenna node synchronously updates the particle coordinates based on the ranging measurements, performing an iterative update each time a new neighbor ranging measurement is received, until the maximum number of iterations is reached or the current discrete time step ends, at which point the iteration stops.

[0011] At each discrete time step, steps 2-4 are executed to update the particle state of each antenna node; each antenna node estimates the radar position and velocity at the current discrete time step based on the final updated particle state at the current discrete time step.

[0012] In step 1, for each antenna node, a set of K particles is generated according to the set initial state distribution, with each particle having an initial weight of 1 / K. If there is no prior observation, particles are randomly generated in the cooperative region. Let the initially generated particle state set be... , It is an antenna node The state of the k-th particle; the distribution of each radar node is approximated by a Gaussian approximation, described by weighted mean and weighted covariance.

[0013] In step 3, a pseudo-time parameter is introduced. Constructing antenna nodes The logarithmic homotopy form is as follows:

[0014] ;

[0015] in, Represents particle flux density, It is a node The prior distribution of the state, It is a node The posterior distribution of the state, Is it following Adjusted normalization constant; ,exist The process from 0 to 1 transforms the prior distribution of particle flux density corresponding to the particle state into a posterior distribution. Assuming that the prior distribution and likelihood function of the antenna node state in cooperative localization are both Gaussian distributions, we obtain the distribution function. as follows:

[0016] ;

[0017] in, and For parameter matrices;

[0018] Suppose the antenna node is observed by GNSS. exist The position of the moment is , ,in, To measure the noise, it follows a mean of 0 and a covariance matrix of... Gaussian distribution; observation matrix , used to extract position information from radar state vector;

[0019] Calculate antenna nodes The particles in Weighted mean at time points with weighted covariance Then calculate and as follows:

[0020] ;

[0021] Then update the antenna nodes. The states of each particle are as follows:

[0022] ;

[0023] in, For pseudo time step, and These are the particle's state before and after the update, respectively. The particle state after step 2 update. The particle state is updated using GNSS observation data.

[0024] In step 4, the antenna node Upon receiving information from neighboring nodes Distance measurement Then, a particle state iteration update is performed, including:

[0025] Antenna Node Calculate the Jacobian matrix for the k-th particle , and Represent the radar position of the k-th particle and its neighboring nodes, respectively. The positional deviation is projected onto the x-axis and y-axis;

[0026] Antenna Node Calculate the mean of the states based on the current states of all particles at this node. and variance Calculate the particle flow update parameter matrix and as follows:

[0027] ;

[0028] Among them, the observation matrix Distance measurement The error follows a mean of 0 and a covariance of . Gaussian distribution; The current antenna node The state of the kth particle;

[0029] Then update using the distribution function. Antenna node at time The state of the k-th particle is as follows:

[0030] ;

[0031] in, This is the state of the particles before the update. If it's the first iteration, then this... This is the state after updating using GNSS measurement data in step 3; otherwise, this... This represents the state updated using a certain distance measurement in the previous iteration; It is the updated state using the ranging data from the current iteration.

[0032] Compared with the prior art, the advantages and positive effects of the present invention are as follows:

[0033] (1) The method of the present invention is based on the Daum-Huang particle flow idea, which smoothly transforms the prior distribution of particle state of antenna nodes into a posterior distribution, effectively alleviates particle degradation and reduces resampling overhead, and improves the accuracy of distributed antenna cooperative positioning.

[0034] (2) The method of the present invention has been experimentally verified to have superior positioning fastening under conditions such as nonlinearity and ranging noise. Compared with the existing methods, the method of the present invention has the advantages of high positioning accuracy, short running time and less impact from noise, and can well meet the high precision requirements of collaborative positioning. Attached Figure Description

[0035] Figure 1This is an overall flowchart of the distributed antenna cooperative localization method based on particle flow filtering according to an embodiment of the present invention;

[0036] Figure 2 Comparison of the collaborative localization results of the method of this invention and the particle filter algorithm for 100 randomly deployed stationary nodes on a 100m×100m two-dimensional plane;

[0037] Figure 3 The image shows a comparison of the collaborative localization results of the method of this invention and the existing particle filter algorithm for six moving nodes on a 50m×50m two-dimensional plane. Detailed Implementation

[0038] The present invention will be further described in detail below with reference to the accompanying drawings and embodiments.

[0039] In the distributed antenna cooperative localization task scenario of this invention embodiment, it is assumed that N distributed antenna nodes move in a two-dimensional plane, and the continuous movement time is divided into multiple equally spaced discrete moments. , Let n be the total number of discrete moments, and n be the number of discrete moments. Each antenna node The time is located at the coordinate At that point, with speed Move, in and Represents the position coordinates on the x-axis and y-axis. and Represents the velocity components along the x and y axes, with the superscript T indicating transpose. Each antenna receives information from the inertial navigation system (INS) to update the velocity vector. Receive GNSS information and update the position vector .definition Let be the set of all antenna nodes, and any antenna node The set of neighboring nodes is denoted as Antenna nodes within ranging range With antenna node The distance information is represented as Antenna node At discrete time state vector Cooperative localization using Bayesian methods faces exponentially increasing computational complexity as the number of antenna nodes grows, particularly when directly marginalizing high-dimensional posterior distributions. To overcome this challenge, this invention decomposes the high-dimensional optimization problem of multivariate coupling in a factor graph-based belief transfer algorithm into local belief exchange between factor nodes and variable nodes. This transforms the original global optimization problem into a parallelizable local inference process, enabling cooperative localization of distributed multi-antenna nodes with near-linear computational complexity. Within this approximate inference framework, due to the nonlinearity of the ranging function and the high dimensionality of the state vector to be estimated, the integration of neighboring variable nodes is complex. Therefore, a particle flow filtering algorithm is proposed to address this type of nonlinear and non-Gaussian ranging problem. This algorithm uses a set of weighted random particles to represent the posterior probability and estimates the system state from noisy or incomplete observation sequences.

[0040] like Figure 1 As shown in the figure, a distributed antenna cooperative localization method based on particle flow filtering according to an embodiment of the present invention mainly includes the following four steps.

[0041] Step 1: Set the initial state distribution for each antenna node And generate a set of particles with corresponding uniform weights, where the initial weight of particle k is... K is the total number of particles generated for each node. If there is no prior observation, particles are randomly generated in the cooperative region. Let the initial set of generated particles be... , Let be the k-th particle of antenna node i, and each particle is the state vector of an antenna node. A Gaussian approximation is applied to the distribution of each node, that is, its center and diffusion are described by the weighted mean and weighted covariance.

[0042] In the In the next iteration, the node The mean and weighted covariance matrix are represented by particles as follows:

[0043] ; ;

[0044] in, It performs a normalization correction on the covariance matrix to prevent the covariance from being too small when the particle weights are concentrated.

[0045] Step 2: At each discrete time... In the process, the antenna node receives data from the inertial navigation system and first updates the velocity and position of the particle at the previous moment according to the motion model of the INS.

[0046] The motion of each antenna node is described by a first-order Markov process, and its state transition process is as follows:

[0047] ;

[0048] Among them, control input This represents the acceleration vector measured by INS. and Represents the acceleration components along the x-axis and y-axis; The acceleration measurement error and process noise are combined and modeled as zero-mean Gaussian white noise, i.e. , Its covariance matrix; state transition matrix The matrix represents the update of the antenna node's position with respect to velocity and the velocity with respect to acceleration. The acceleration transformation into a state transition is represented as follows:

[0049] ;

[0050] in, The sampling period is a discrete period with equal intervals. and They represent The identity matrix and the all-zero matrix.

[0051] In this embodiment of the invention, an iteration is performed at each discrete time step. In this iteration, the particle state is updated in this step according to the above process. Defineable time The set of state vectors for all antenna nodes is: .

[0052] Step 3: The node receives Global Navigation Satellite System (GNSS) measurement information. Based on the Daum–Huang particle flow approach, the prior distribution predicted by INS is smoothly evolved into a posterior distribution that fuses GNSS and RS ranging information. A log-homotopy function is introduced at the node. The form of logarithmic homotopy is defined as follows:

[0053] ;

[0054] in, Represents particle flux density, It is a node The prior distribution of the state, It is a node The posterior distribution of the state, Is it following Adjusted normalization constant. It is the "pseudo-time" parameter that constructs this distribution function transition. When hour, The observation information for the corresponding particles is a priori distribution. When At that time, according to Bayes' theorem, we know Corresponding to the posterior distribution, in The process from 0 to 1 transforms the particle flux density from a priori to a posteriori. To achieve this homotopy transformation, assume the state vector... Follow The evolution satisfies the following stochastic differential equation:

[0055] ;

[0056] in, For the Wiener process, It is the coefficient matrix of the diffused noise; This is the drift term, used to determine the form of the flow distribution function. When the prior distribution of the antenna node state and the likelihood function both belong to the exponential family distribution, it can be... and Written in the form of sufficient statistics, it can be derived that The precise form of . Assume that the prior distribution and likelihood function of the antenna node states in cooperative localization are both Gaussian distributions, and for After matching the coefficients of the linear and quadratic terms, we can obtain The format is as follows:

[0057] ;

[0058] Wherein, parameter matrix and The update equations are usually given by the Daum-Huang particle flow.

[0059] In cooperative localization, for particles At each pseudo time step Discrete updates are performed. When a particle receives new measurement information, the current node is first calculated. State mean and variance Updated again and .

[0060] GNSS provides antenna nodes with position coordinates in an absolute coordinate system. Let the node... exist The current position coordinates are obtained by measuring the time. , represented as:

[0061] ;

[0062] in, For antenna nodes exist The actual location coordinates at any given moment; To measure the noise, we set it to zero-mean Gaussian white noise, i.e. , The covariance matrix; GNSS only observes the position components of nodes, the observation matrix Used to extract two-dimensional position information from the state vector. Define time. Set of antenna node locations for all GNSS measurements When node particles Received GNSS observation information Then, the particle flow update matrix is ​​calculated as follows:

[0063] ;

[0064] ;

[0065] Update status as follows:

[0066] ;

[0067] in, and They are Antenna node at time The states of the k-th particle before and after the update, where the particle state before the update is shown below. The updated particle state is the state after step 2 using INS data. This is the updated state using GNSS measurement data.

[0068] Step 4: During cooperative positioning, each antenna node measures distance to its neighboring nodes within its communication range. Let's assume... Time Node and nodes Direct connection, the distance measured between two nodes. Represented as:

[0069] ;

[0070] in, Represents Euclidean distance. Distance measurement error is used to describe the impact of conditions such as noise and multipath on measurements. Under the assumption of satisfying the Line of Sight (LOS), the distance measurement error... It follows a zero-mean Gaussian distribution, i.e. , For covariance.

[0071] Set at time The mutual ranging information between all antenna nodes is When node particles Received from neighboring nodes RS (Focus Rangefinder) Distance Measurement Then, the Jacobian matrix is ​​calculated as follows: ,in and They represent particles respectively with neighboring nodes The positional deviation is projected onto the x-axis and y-axis, and the particle flow update parameter matrix is ​​calculated:

[0072] ;

[0073] ;

[0074] in, This represents the state of the k-th particle at the current antenna node i.

[0075] Calculate the current antenna node mean of the state and variance Update according to the formula above and Then, the particle coordinates are updated using the distribution function, as follows:

[0076] ;

[0077] in, and These are the particle's state before and after the update, respectively. Here, the state before the update is... This refers to the state updated using GNSS measurement data in step 3, or the state updated using distance measurement data from neighbors in the previous iteration. It is the updated state using the neighbor ranging data in the current iteration.

[0078] like Figure 1 As shown, in the method of this invention, each antenna node detects whether it receives INS data, GNSS data, and ranging data with neighboring nodes within each discrete time step; when INS data is detected, step 2 is executed to update the particle state over time; when GNSS data is detected, each antenna node executes step 3 to calculate the mean and variance of the current particle state as shown in step 1, and updates the parameter matrix based on the Daum-Huang particle flow. and Then, the particle state is updated using the distribution function; when a distance to a neighboring node is detected, the antenna node executes step 4, calculates the mean and variance of the current particle state, and updates it based on the Daum-Huang particle flow. and The particle state is then updated using a distribution function. Specifically, GNSS data and ranging data are used to update the antenna node positions in the particle state. Within each discrete time step, each antenna node synchronously and continuously detects the ranging with its neighboring nodes. Upon receiving a new neighbor ranging measurement, the particle position is updated. Iteration stops when the number of iterations reaches a set upper limit or the current discrete time step ends. After iteration stops at the current discrete moment, each antenna node estimates its current state based on the particle state. During the particle state iteration process, a message-passing iteration method is used. In each iteration, the antenna node obtains the mean and variance of each particle state and calculates the mean and variance. Each particle then updates its parameter matrix based on the mean, variance, and ranging. and Then, the particle positions are updated. The method of this invention iteratively updates the state of the particles using the Daum-Huang particle flow formula, changing their prior distribution to a posterior distribution, thereby achieving cooperative localization of distributed antennas.

[0079] The technical effects of the method of the present invention will be further explained below in conjunction with experimental results.

[0080] Figure 2 The performance comparison chart shows the collaborative localization results of the proposed method and the particle filter algorithm for 100 randomly deployed stationary nodes on a 100m × 100m two-dimensional plane. The prior distribution of each node is represented by 300 particles, and the total number of iterations is set to 10. In the simulation scenario, the prior mean of each user node is set as its true coordinates, and the covariance is... Distance variance In the diagram, the black "□" represents the actual location of the node, and the red "□" represents the actual location of the node. The symbols “” and “○” represent the node position estimates based on the method of this invention (PFlowBP) and the existing particle filter algorithm (PBP), respectively. The maximum distance measurement range between any two nodes in the network is 20 meters. The experimental results show that the particle flow filter algorithm proposed in this invention has high positioning accuracy.

[0081] Figure 3 The image shows a comparison of the cooperative localization results of six moving nodes on a 50m × 50m two-dimensional plane using the method of this invention (PFlowBP) and the existing particle filter algorithm (PBP). Six antenna nodes are deployed within the area. The simulation covers 20 discrete time points, with each node moving along a preset trajectory. Nodes 1 to 4 move along a clockwise circle with a radius of 25 meters, while nodes 5 and 6 move along the x-axis and x-axis, respectively. y Linear motion along the axis. Each node acquires INS measurement data every second and fuses it with GNSS measurement information and RS ranging information from neighboring nodes. The RS ranging variance is... The variance of GNSS measurements is Distance measurement and information exchange can be achieved between any two nodes. The results show that the method of this invention has higher positioning accuracy and more closely approximates the actual trajectory.

[0082] Generally, the embodiments disclosed in this invention can be implemented in hardware or dedicated circuitry, software, firmware, logic, or any combination thereof. Some aspects can be implemented in hardware, while others can be implemented in firmware or software executed by a controller, microprocessor, or other computing device. When aspects of this embodiment are illustrated or described as block diagrams, flowcharts, or using some other graphical representation, it will be understood that the blocks, apparatuses, systems, techniques, or methods described herein can be implemented as non-limiting examples in hardware, software, firmware, dedicated circuitry or logic, general-purpose hardware or controllers or other computing devices, or some combination thereof.

[0083] The above description, in conjunction with specific preferred embodiments, provides a further detailed explanation of the present invention. It should not be construed that the specific implementation of the present invention is limited to these descriptions. For those skilled in the art, various simple deductions or substitutions can be made without departing from the inventive concept, and all such modifications and substitutions should be considered within the scope of protection of the present invention. Except for the technical features described in the specification, all other technologies are known to those skilled in the art. The present invention omits descriptions of well-known components and technologies to avoid redundancy and unnecessary limitation of the invention.

Claims

1. A distributed antenna cooperative localization method based on particle flow filtering, characterized in that, Includes the following steps: Step 1: Set an initial state distribution for each antenna node and generate a set of particles with corresponding uniform weights. Each particle state represents an antenna node state, which includes the node's position and velocity. Step 2: The antenna node receives data from the inertial navigation system (INS) and updates the velocity and position of the particles from the previous moment based on the INS motion model. Step 3: The antenna node receives GNSS measurement data from the Global Navigation Satellite System and updates the particle position; Based on the Daum-Huang particle flow, a log-homotope function is introduced to transform the particle state from a prior distribution obtained from INS data updates to a posterior distribution predicted by GNSS data and range measurements, thus obtaining a distribution function for the particle state. After receiving the current position coordinates measured by GNSS, the antenna node calculates the mean and variance of the current node state, updates the parameter matrix of the distribution function, and then updates the particle state. Step 4: The antenna node receives ranging information from neighboring nodes, calculates the mean and variance of the current node state, and each particle updates the parameter matrix of the distribution function using ranging to update the particle position. In each discrete moment, each antenna node synchronously performs particle state updates based on ranging. Each time a new neighbor ranging is received, an iterative update is performed until the maximum number of iterations is reached or the current discrete moment ends, at which point the iteration stops. At each discrete time step, steps 2-4 are executed to update the particle state of each antenna node; each antenna node estimates its position and velocity at the current discrete time step based on the final updated particle state at the current discrete time step.

2. The method according to claim 1, characterized in that, In step 1, for each antenna node, a set of K particles is generated according to the set initial state distribution, with each particle having an initial weight of 1 / K. If there is no prior observation, particles are randomly generated in the cooperative region. Let the initially generated particle state set be... , It is an antenna node The state of the k-th particle; the distribution of each antenna node is approximated by a Gaussian approximation, described by the weighted mean and weighted covariance; calculate Within a given time, the current node weighted mean with weighted covariance as follows: ； ； in, yes Time Antenna Node The state of the k-th particle, This is the weight of the particle, and the superscript T indicates transpose.

3. The method according to claim 1, characterized in that, In step 2, the antenna nodes reach... At discrete time Obtain the acceleration components on the x and y axes measured by INS. and , the acceleration vector As the control input, the motion of the antenna node is described by a first-order Markov process, updating the particle state at each discrete time step. Let the discrete time step be... Update antenna nodes The state of the kth particle is: .

4. The method according to claim 1, characterized in that, In step 3, a pseudo-time parameter is introduced. Constructing antenna nodes The logarithmic homotopy form is as follows: ； in, Represents particle flux density, It is a node The prior distribution of the state, It is a node The posterior distribution of the state, Is it following Adjusted normalization constant; ,exist The process from 0 to 1 transforms the prior distribution of particle flux density corresponding to the particle state into a posterior distribution. Assuming that the prior distribution and likelihood function of the antenna node state in cooperative localization are both Gaussian distributions, we obtain the distribution function. as follows: ； in, and For parameter matrices; Suppose the antenna node is observed by GNSS. exist The position of the moment is , ,in, To measure the noise, it follows a mean of 0 and a covariance matrix of... Gaussian distribution; observation matrix This is used to extract position information from the antenna node state vector; Antenna Node First calculate the mean of the current particle state. With covariance ; then calculate and as follows: ； Then update Antenna node at time The state of the k-th particle is as follows: ； in, For pseudo time step, and These are the particle's state before and after the update, respectively. The particle state after step 2 update. The particle state is updated using GNSS observation data.

5. The method according to claim 1, characterized in that, In step 4, the antenna node Upon receiving information from neighboring nodes Distance measurement Then, a particle state iteration update is performed, including: Antenna Node Calculate the Jacobian matrix for the k-th particle , and These represent the node position of the k-th particle and its neighboring nodes, respectively. The positional deviation is projected onto the x-axis and y-axis; Antenna Node Calculate the mean of the states based on the current states of all particles at this node. and variance Calculate the particle flow update parameter matrix and as follows: ； Among them, the observation matrix Distance measurement The error follows a mean of 0 and a covariance of . Gaussian distribution; The current antenna node The state of the kth particle; Then update using the distribution function. Antenna node at time The state of the k-th particle is as follows: ； in, This is the state of the particles before the update. If it's the first iteration, then this... This is the state after updating using GNSS measurement data in step 3; otherwise, this... This represents the state updated using a certain distance measurement in the previous iteration; It is the updated state using the ranging data from the current iteration.

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