A method for dynamic monitoring and sway compensation of urban bridges using tower-mounted true aperture radar

Abstract

This invention discloses a method for dynamic monitoring and sway compensation of urban bridges using a tower-end true aperture radar. The method includes acquiring the radar coordinates and coordinates of each control point at the initial moment, and real-time acquisition of the line-of-sight distance from the radar to each control point. An extended Kalman filter radar-control point spatial pose coupling trajectory tracking model is constructed to obtain preliminary coordinate estimates of the radar and control points. Based on these preliminary coordinate estimates, a Levenberg-Marquardt optimization model is constructed to obtain high-precision sway trajectory coordinates. A sway compensation model is constructed based on the line-of-sight sway offset, and segmented compensation is performed on different distance units to obtain a time-series displacement signal of the bridge target point that eliminates the influence of sway. This invention employs a trajectory tracking method that combines extended Kalman filtering with a Levenberg-Marquardt algorithm for iterative optimization, along with a dynamic compensation model segmented by distance units. This solves the accuracy problems of traditional monitoring technologies due to obstruction and sway, achieving high-precision, long-term time-series monitoring of bridge dynamic deflection.

Description

Technical Field

[0001] This invention relates to the field of urban bridge dynamic monitoring data compensation technology, and in particular to a method for sway compensation of urban bridge dynamic monitoring using a tower-end true aperture radar. Background Technology

[0002] As core nodes in transportation networks, the structural health of urban bridges directly impacts public safety and urban operational efficiency. Aging bridges suffer structural degradation due to traffic loads and environmental erosion, leading to a continuous increase in structural safety risks. This necessitates efficient and reliable dynamic deflection monitoring technology for risk early warning. Existing monitoring methods, such as contact sensors (e.g., strain gauges, accelerometers), suffer from narrow measurement ranges, high installation and maintenance costs, and susceptibility to environmental interference. Non-contact technologies, such as total stations, GNSS, and terrestrial laser scanning (TLS), are limited by single-point monitoring capabilities, insufficient data sampling frequency, or poor environmental adaptability, making them unsuitable for the high-precision, high-frequency, and dynamic monitoring needs of urban bridges. While ground-based synthetic aperture radar (GBSAR) possesses high-precision, high-frequency sampling, and full-area dynamic monitoring capabilities, making it suitable for urban bridge dynamic deflection monitoring, the complex urban environment often limits its application. GBSAR is deployed using traditional ground-based tripods, which are susceptible to interference from building signal obstruction and multipath effects caused by complex ground reflections. In densely populated urban areas, ideal observation points are often lacking around bridges.

[0003] To address the pain points in urban scenarios, existing technologies can mount GBSAR onto communication towers for high-altitude observation ("tower-mounted radar"). Leveraging the 30-50 meter height of the towers, this overcomes urban obstruction limitations, expanding the monitoring range to 3-5 kilometers. Simultaneously, it shares the tower's power, communication, and wind and earthquake-resistant infrastructure, significantly reducing deployment costs and maintenance complexity. However, towers are susceptible to irregular, random swaying at the centimeter or even decimeter level due to wind loads and ground vibrations. This swaying obscures bridge vibration information in the radar monitoring signal, leading to distortion of dynamic monitoring signals for urban bridges. Existing research suggests that real-time radar sway tracking can be achieved by deploying contact sensors, such as accelerometers and gyroscopes. However, contact sensors face challenges in deployment and synchronization of monitoring data with radar sampling frequencies, making it difficult to achieve fusion compensation of radar sway tracking and target monitoring information. Non-contact monitoring methods, such as visual measurement, can also be used for non-contact radar sway tracking and compensation. However, visual measurement involves capturing radar sway images with a camera and obtaining radar sway values ​​based on feature point detection algorithms. This method is susceptible to interference from strong light and rain / snow, resulting in poor calculation accuracy, and cannot guarantee all-weather radar sway trajectory tracking, making it difficult to achieve real-time compensation for high-dynamic monitoring. Therefore, a dynamic monitoring and sway compensation method for urban bridges using a real aperture radar tower is needed. Summary of the Invention

[0004] The purpose of this invention is to provide a method for dynamic monitoring and sway compensation of urban bridges using tower-end true aperture radar, so as to solve the problems existing in the prior art.

[0005] To achieve the above objectives, the present invention is implemented according to the following technical solution:

[0006] On one hand, the present invention includes the following steps:

[0007] Acquire the radar coordinates and coordinates of each control point at the initial moment, and collect the line-of-sight distance from the radar to each control point in real time;

[0008] A radar-control point spatial pose coupled trajectory tracking model with extended Kalman filter is constructed to obtain preliminary coordinate estimates of the radar and control points.

[0009] Based on the preliminary coordinate estimates, a Levenberg-Marquardt optimization model is constructed to obtain high-precision sway trajectory coordinates:

[0010] The bridge is divided into multiple distance monitoring units, and the line-of-sight offset of the control point is calculated based on the high-precision sway trajectory coordinates.

[0011] A sway compensation model is constructed based on the line-of-sight sway offset. Segmented compensation is performed on different distance units to obtain the time-series signal of bridge target point displacement that eliminates the sway effect.

[0012] Furthermore, at least three fixed control points are set up within the monitoring area, and these fixed control points are located on stable bedrock or rigid structures around the bridge.

[0013] Furthermore, the preliminary coordinate estimates of the radar and control points specifically include:

[0014] Step 201: Define a state vector containing the three-dimensional coordinates (X_A, Y_A, Z_A) of the radar at time t and the three-dimensional coordinates (X_Cm, Y_Cm, Z_Cm, where m = 1, 2, 3…) of the m-th control point at time t. The state vector is represented as X_S(t) = [X_A, Y_A, Z_A, X_Cm, Y_Cm, Z_Cm]^T;

[0015] Step 202: Construct the linearized state transition equation:

[0016] X S (t)=FX S (t-1)+δ(t)

[0017] Where F is the state transition matrix, δ(t) ~ (0, D) δ ) represents the state model error, D δLet Q be the process noise covariance, representing the random disturbance between the radar and control point positions.

[0018] Step 203: Construct the observation model, which can be represented as:

[0019] S'(t)=HX S (t)+v(t)

[0020] Where S'(t) is the observed time series signal, v(t)~(0,R) represents the observation model error, R is the observation noise covariance matrix, and H is the observation operator, representing the relationship between the monitored time series signal S'(t) and the state vector X. S (t)=[X A ,Y A Z A XC m YC m ZC m ] T Relationship equation;

[0021] The observation operator is:

[0022]

[0023] Among them, H m Let X be the geometric distance from the radar to the m-th control point. A (t), Y A (t), Z A (t) represents the three-dimensional coordinates of the radar antenna at time t, XC m (t), YC m (t), ZC m (t) represents the three-dimensional coordinates of the m-th control point at time t;

[0024] Calculate the Jacobian matrix of the radar position at this moment:

[0025]

[0026] in

[0027] Step 204: Iterative solution of extended Kalman filter

[0028] Covariance prediction at this point:

[0029] P S (t)=F·P S (t-1)F T +Q

[0030] Among them, P S (t) is the state covariance matrix, and Q represents the process noise covariance matrix;

[0031] At this point, the Kalman gain is:

[0032] K(t)=P(t-1)·J H T ·[J H ·P(t-1)·J T H +R] -1

[0033] Where K(t) is the Kalman gain matrix and R is the observation process noise covariance matrix;

[0034] The covariance is now updated as follows:

[0035]

[0036] The state vector of the radar position is obtained after calculation using the Extended Kalman Filter (EKF) method:

[0037]

[0038] X S (t) EKF The state vector estimate of the extended Kalman filter output includes radar position coordinates and control point coordinates.

[0039] Furthermore, the step of constructing a Levenberg-Marquardt optimization model based on the preliminary coordinate estimates to obtain high-precision sway trajectory coordinates specifically includes: constructing a Levenberg-Marquardt optimization objective function, using the coordinate estimates output by the extended Kalman filter as initial values, constructing an objective function that includes distance residuals and regularization terms, calculating the Jacobian matrix, constructing an incremental equation, iteratively optimizing and solving for the state correction amount, and updating to obtain high-precision sway trajectory coordinates.

[0040] Furthermore, we construct an objective function that includes distance residuals and a regularization term:

[0041]

[0042] Where X S (t) EKF For coordinate estimates, ||P m (t)-P0(t)|| represents the control point micro-motion compared to the initial time, P m (t) represents the estimated position of the m-th control point, which can be calculated from the extended Kalman filter control point coordinate estimates; P0(t) represents the initial coordinates of the control point; r_coord m (t)=D m (t)-H m (X S (t) represents the residual expression of the distance monitoring from the radar system to the control point, Dm (t) represents the measured distance from the radar system to the control point; κ is the regularization coefficient, set to 10. -3 Up to 10 -1 Magnitude.

[0043] Furthermore, the sway monitoring compensation value for each radar range cell can be expressed as:

[0044] Δd virbration (Rbin n ,t)=

[0045] sinθ·Δd vibration (Rbin n ,X S (t) EKF+LM (n = 1, 2, 3, ... N)

[0046] Where Δd vibration (Rbin n ,t) is the sway monitoring compensation value, Rbin n The range cell represents the continuous distribution of radar signals on the bridge, where n is the number of range cells, and sinθ is the line-of-sight angle between the radar and a certain range cell on the bridge.

[0047] The bridge deflection correction value within each distance resolution unit is calculated using a sway monitoring and compensation model and then removed from the actual data to achieve precise compensation.

[0048] The beneficial effects of this invention are:

[0049] This invention provides a method for dynamic monitoring and sway compensation of urban bridges using a tower-end true aperture radar. Compared with existing technologies, this invention has the following technical advantages:

[0050] This invention employs a trajectory tracking method that combines extended Kalman filtering with Levenberg-Marquardt algorithm cyclic optimization, along with a dynamic compensation model based on distance unit segmentation, to solve the accuracy problems of traditional monitoring technologies affected by obstruction and swaying, thereby achieving high-precision, long-term monitoring of bridge dynamic deflection. Attached Figure Description

[0051] Figure 1 This is a schematic diagram of the adjustment system framework for a method for dynamic monitoring and sway compensation of urban bridges using a tower-end true aperture radar according to the present invention.

[0052] Figure 2 This is a schematic diagram of the observation operator for a method of dynamic monitoring and sway compensation for urban bridges using a tower-end true aperture radar according to the present invention (the dashed and solid lines represent the positional changes of the control point before and after monitoring due to radar sway).

[0053] Figure 3This is a schematic diagram of a segmented bridge with range resolution, illustrating a method for dynamic monitoring and sway compensation of urban bridges using a tower-end true aperture radar according to the present invention.

[0054] Figure 4 This is a schematic diagram showing the radar deployment and bridge geometry of a method for dynamic monitoring and sway compensation of urban bridges using a tower-end true aperture radar according to the present invention. Detailed Implementation

[0055] The present invention will be further described below through specific embodiments. The illustrative embodiments and descriptions herein are used to explain the present invention, but are not intended to limit the present invention.

[0056] In this implementation example, "tower end" refers to the radar being deployed at the top of the communication tower, providing wide-field-of-view monitoring capabilities; "nonlinear sway" refers to the complex motion trajectory of the tower under the influence of wind loads, vibrations, etc., which combines low-frequency large-amplitude swaying with high-frequency micro-vibrations; and "range unit segmentation" refers to dividing the bridge monitoring area into multiple independent compensation segments based on the radar range resolution, thereby achieving refined error correction.

[0057] like Figure 1 As shown, a method for dynamic monitoring and sway compensation of urban bridges using a tower-end true aperture radar includes the following steps:

[0058] Acquire the radar coordinates and coordinates of each control point at the initial moment, and collect the line-of-sight distance from the radar to each control point in real time;

[0059] A radar-control point spatial pose coupled trajectory tracking model with extended Kalman filter is constructed to obtain preliminary coordinate estimates of the radar and control points.

[0060] Based on the preliminary coordinate estimates, a Levenberg-Marquardt optimization model is constructed to obtain high-precision sway trajectory coordinates:

[0061] The bridge is divided into multiple distance monitoring units, and the line-of-sight offset of the control point is calculated based on the high-precision sway trajectory coordinates.

[0062] A sway compensation model is constructed based on the line-of-sight sway offset. Segmented compensation is performed on different distance units to obtain the time-series signal of bridge target point displacement that eliminates the sway effect.

[0063] I. Iterative Optimization Model of Extended Kalman Filter (EKF) and Levenberg-Marquardt (LM) Algorithms

[0064] Step 1: Construct an EKF radar-control point spatial pose coupled trajectory tracking model

[0065] 1. Construction of Spatial Pose Coupling Model

[0066] At least three fixed control points are deployed within the monitoring area (preferably on stable bedrock or rigid structures around the bridge). A spatial pose coupling model of the radar and control points is established, and the radar sway trajectory is retrieved by real-time calculation of the deformation field of the control points. At this time, the monitoring data not only includes radar antenna sway information but also control point micro-motion information. Therefore, this invention uses the radar spatial coordinates (X, Y, X) at time t. A ,Y A Z A ) and control point spatial coordinates (XC m YC m ZC m Define the state vector as m = 1, 2, 3, ...

[0067] X S (t)=[X A ,Y A Z A XC m YC m ZC m ] T

[0068] Among them, X S (t) is the state vector (containing the coordinates of the radar antenna and control points), X A ,Y A Z A Let XC be the three-dimensional coordinates of the radar antenna at time t. m YC m ZC m The three-dimensional coordinates of the m-th control point at time t (m = 1, 2, 3, ...).

[0069] 2. Construct the state transition equation

[0070] Considering the radar sampling characteristics (high sampling frequency ≥200Hz, short sampling time interval 0.05 seconds), the changes are small over a short period of time, and a linearized state transition equation is constructed:

[0071] X S (t)=FX S (t-1)+δ(t)

[0072] Where F is the state transition matrix, which is the identity matrix here (ignoring short-term nonlinear changes), δ(t)~(0,Q) δ Q represents the state model error. δ Let Q be the process noise covariance, representing the random disturbance of the radar position;

[0073] 3. Construction of the observation model

[0074] The observation model can then be expressed as:

[0075] S'(t)=HX S (t)+v(t)

[0076] Where S'(t) is the observed time series signal, v(t)~(0,R) represents the observation model error, and R is the observation noise covariance matrix. H is the observation operator, representing the monitoring time series signal S'(t) and the state vector X. S (t)=[X A ,Y A Z A XC m YC m ZC m ] T The relational equation (geometric distance calculation model between radar and each control point).

[0077] The observation operator is:

[0078]

[0079] Among them, H m Let X be the geometric distance from the radar to the m-th control point. A (t), Y A (t), Z A (t) represents the three-dimensional coordinates of the radar antenna at time t. m (t), YC m (t), ZC m (t) represents the three-dimensional coordinates of the m-th control point at time t.

[0080] Calculate the Jacobian matrix of the radar position at this time (assuming there are 3 control points (m = 1, 2, 3), and only take the partial derivative with respect to the radar position):

[0081]

[0082] in

[0083] 4. Iterative solution of Extended Kalman Filter (EKF)

[0084] Covariance prediction at this point:

[0085] P S (t)=F·P S (t-1)F T +Q

[0086] Among them, P S (t) is the state covariance matrix, and Q represents the process noise covariance matrix.

[0087] At this point, the Kalman gain is:

[0088] K(t)=P(t-1)·J H T ·[J H ·P(t-1)·J H T +R] -1

[0089] Where K(t) is the Kalman gain matrix and R is the observation process noise covariance matrix.

[0090] The covariance is now updated as follows:

[0091]

[0092] The state vector of the radar position is obtained after calculation using the Extended Kalman Filter (EKF) method:

[0093]

[0094] X S (t) EKF The state vector estimate output by the Extended Kalman Filter (EKF) includes radar position coordinates and control point coordinates.

[0095] Step 2: Levenberg-Marquardt (LM) linearization error correction radar sway trajectory tracking optimization 1. Construct the Levenberg-Marquardt (LM) optimization objective function

[0096] The state vector X output by the Extended Kalman Filter (EKF) S (t) EKF Using the initial values, construct an objective function that includes distance residuals and a regularization term:

[0097]

[0098] Where X S (t) EKF For coordinate estimates, ||P m (t)-P0(t)|| represents the control point micro-motion compared to the initial time, P m (t) represents the estimated position of the m-th control point, which can be calculated from the coordinate estimates of the control point using the Extended Kalman Filter (EKF) method; P0(t) represents the initial coordinates of the control point; r_coord m (t)=D m (t)-H m (X S (t) represents the residual expression of the distance monitoring from the radar system to the control point, D m(t) represents the measured distance from the radar system to the control point; κ is the regularization coefficient, which prevents excessive shift of the control point position due to noise or model errors and avoids ill-conditioned singularities in the Jacobian matrix. Its value is determined through experimental calibration or empirical formulas, and is usually set to 10. -3 Up to 10 -1 The magnitude and specific value need to be adjusted according to the radar observation noise level to ensure that the Jacobian matrix condition number is within a stable range.

[0099] Calculate the Jacobian matrix J between the radar and control points (based on r_coord(t)). 2 (Partial derivative with respect to radar coordinates):

[0100]

[0101] in, The same applies to the other partial derivatives.

[0102] Based on the Jacobian matrix, the incremental equation is constructed as follows:

[0103] (J T J+kI)·X S (t) EKF+LM =-J T r_coord m (t)

[0104] I is the identity matrix, and X is solved through multiple iterations. S (t) EKF+LM This includes the radar sway coordinates and the sway coordinates of each control point. In summary, the X output from step 1... S (t) EKF As the initial value for step 2, step 2 (LM) applies to X. S (t) EKF Perform nonlinear optimization and output the corrected state vector X. S (t) EKF+LM , will X S (t) EKF+LM The feedback from step 1 to the next moment is used as input to perform iterative optimization, thereby achieving millimeter-level precision correction of the radar sway trajectory.

[0105] II. Dynamic Fine Compensation Model for Urban Bridges with Radar Range Unit Segmentation

[0106] Step 3: High-dynamic fine-grained compensation for urban bridges segmented by radar system range unit

[0107] 1. Bridge geometric parameter modeling

[0108] The correction for bridge dynamic deflection can be expressed as:

[0109] S'1(t)=Starget (t)-Δd vibration (Rbin j ,t)

[0110] Among them, S target (t) represents the uncompensated bridge target point displacement time-series signal, Δd vibration S'1(t) is the sway correction value of the j-th range unit Rbin, and S'1(t) is the target point displacement monitoring timing signal to eliminate the influence of radar system sway.

[0111] Cell resolution length of radar range element Rbin:

[0112]

[0113] Where, r los Let Δβ represent the line-of-sight distance of the radar to the bridge, and let Δβ represent the radar angular resolution. Based on the bridge length L, the bridge is divided along its length into... There are 1 distance unit, and the position of each distance unit is represented as l. n =n×D R (j=1,2,3,…N), where j is the number of distance resolution units that the radar waves cover the bridge surface during monitoring, and the number can correspond to the actual data monitoring units obtained.

[0114] 2. Calculation of sway correction amount

[0115] Based on X S (t) EKF+LM The sway monitoring correction value Δd corresponding to each control point is calculated. vibration A line-of-sight sway correction model is constructed to calculate the sway correction within each distance cell. This model can be obtained through experimental simulation by setting up multiple control points on the monitored bridge.

[0116] Δd vibration (t)=Δd vibration (Rbin n ,X S (t) EKF+LM ), n = 1, 2, 3, ..., N

[0117] 3. Segmented fine-grained compensation

[0118] like Figure 2 As shown, the system includes monitoring the main bridge structure, radar observation directions, multiple distance monitoring units, and at least three fixed control points (e.g., control point 1, control point 2, and control point 3). The segmentation rule is based on the total bridge length L and the radar range resolution length D. R The bridge is divided into multiple independent distance units along its length, and the location of each unit is marked as l. n =n×D R(n=1,2,3,…N) The bridge is divided into multiple monitoring sections that can be compensated independently.

[0119] Figure 3 and 4 As shown, a sway compensation model is constructed to quantify the spatial attitude relationship between the radar observation direction and the bridge deck, providing geometric parameter support for subsequent calculation of the vertical deflection component (sway correction value) of each range unit. The radar installation height H (vertical distance from the top of the communication tower to the ground), bridge deck height h (vertical distance from the bridge deck to the ground), and line-of-sight distance r from the radar to the bridge are labeled. los And the angle θ between the radar line of sight and the bridge surface.

[0120] The calculation basis of θ is illustrated using a right-angled triangle model. The geometric relationship between the radar and the bridge layout is as follows:

[0121]

[0122] Here, sinθ is the angle between the radar line of sight and the bridge deck, used for subsequent calculations of the vertical deflection component; h is the bridge deck height; and H is the radar installation height. los This represents the line-of-sight distance of the radar to each range cell. Therefore, the sway monitoring compensation value for each radar range cell can be expressed as:

[0123] Δd vibration (Rbin n ,t)=

[0124] sinθ·Δd vibration (Rbin n ,X S (t) EKF+LM (n = 1, 2, 3, ... N)

[0125] Finally, the correction value of bridge deflection within each distance resolution unit is calculated using the sway monitoring compensation model and then removed from the actual data to achieve fine compensation.

[0126] The above description is only a preferred embodiment of the present invention and is not intended to limit the present invention. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of the present invention should be included within the protection scope of the present invention.

Claims

1. A method for dynamic monitoring and sway compensation of urban bridges using a tower-end true aperture radar, characterized in that, Includes the following steps: Acquire the radar coordinates and coordinates of each control point at the initial moment, and collect the line-of-sight distance from the radar to each control point in real time; An extended Kalman filter radar-control point spatial pose coupled trajectory tracking model is constructed to obtain preliminary coordinate estimates of the radar and control points: Based on the preliminary coordinate estimates, a Levenberg-Marquardt optimization model is constructed to obtain high-precision sway trajectory coordinates: The bridge is divided into multiple distance monitoring units, and the line-of-sight offset of the control point is calculated based on the high-precision sway trajectory coordinates. A sway compensation model is constructed based on the line-of-sight sway offset. Segmented compensation is performed on different distance units to obtain the time-series displacement signal of the bridge target point that eliminates the sway effect. The specific steps for constructing a Levenberg-Marquardt optimization model based on the preliminary coordinate estimates to obtain high-precision sway trajectory coordinates include: constructing a Levenberg-Marquardt optimization objective function, using the coordinate estimates output by the extended Kalman filter as initial values, constructing an objective function that includes distance residuals and regularization terms, calculating the Jacobian matrix, constructing an incremental equation, iteratively optimizing and solving for the state correction amount, and updating to obtain high-precision sway trajectory coordinates; Construct an objective function that includes distance residuals and a regularization term: in These are coordinate estimates. This indicates a slight adjustment of the control point compared to the initial time. The estimated position of the m-th control point is calculated from the extended Kalman filter control point coordinate estimates. Initial coordinates of control points This represents the residual expression for the distance monitoring from the radar system to the control point. It is the initial measured distance from the radar system to the control point; The regularization coefficient is set to... Magnitude; The sway compensation value for each range cell of the radar is expressed as: in This is the sway monitoring compensation value. The range unit represents the continuous distribution of radar signals on the bridge, where n is its quantity. The angle between the line of sight between the radar and a certain range cell on the bridge; The bridge deflection correction value within each distance resolution unit is calculated using a sway monitoring and compensation model and then removed from the actual data to achieve precise compensation.

2. The method for dynamic monitoring and sway compensation of urban bridges using a tower-end true aperture radar according to claim 1, characterized in that, At least three fixed control points shall be set up within the monitoring area, and the fixed control points shall be set up in stable bedrock or rigid structures around the bridge.

3. The method for dynamic monitoring and sway compensation of urban bridges using a tower-end true aperture radar according to claim 1, characterized in that, The preliminary coordinate estimates of the radar and control points include: Step 201: Define a state vector containing the radar's three-dimensional coordinates (X_A, Y_A, Z_A) at time t and the three-dimensional coordinates (X_Cm, Y_Cm, Z_Cm, where m = 1, 2, 3…) of the m-th control point at time t. The state vector is represented as follows: ; Step 202: Construct the linearized state transition equation: in, Here is the state transition matrix. For state model error, Let Q be the process noise covariance, representing the random disturbance between the radar and control point positions. Step 203: Construct the observation model, represented as: in, Observe time series signals, R represents the observation model error, and R is the observation noise covariance matrix. For observation operators, it represents the monitoring time series signal. With the state vector Relationship equation; The observation operator is: ； in, Let be the geometric distance from the radar to the m-th control point. Let be the three-dimensional coordinates of the radar antenna at time t. , , Let be the three-dimensional coordinates of the m-th control point at time t; Calculate the Jacobian matrix of the radar position at this moment: in , ; Step 204: Iterative solution of extended Kalman filter Covariance prediction at this point: in, Let Q be the state covariance matrix, and let Q be the process noise covariance matrix. At this point, the Kalman gain is: in, R is the Kalman gain matrix, and R is the observation noise covariance matrix. The covariance is now updated as follows: The state vector of the radar position is obtained after calculation using the Extended Kalman Filter (EKF) method: The state vector estimate of the extended Kalman filter output includes radar position coordinates and control point coordinates; H(t) represents the observation operator at time t; K(t) represents the Kalman gain matrix at time t.

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