Method for calibrating attitude deviation of radar on moving platform based on semi-definite relaxation and target positioning method

Abstract

The application provides a semi-definite relaxation-based motion platform radar attitude deviation calibration and target positioning method, adopts a deviation conversion formula to convert spherical coordinate measurement to a radar body coordinate system to obtain a target position vector as measurement input, and calculates a Cartesian domain measurement noise covariance matrix based on the spherical coordinate measurement, takes the inverse matrix thereof as a weight matrix to construct a rotation matrix to solve an optimization problem, adopts a semi-definite relaxation method to convert the optimization problem into a convex semi-definite programming problem and solve to obtain real three-axis attitude angles at each moment, calculates a constant three-axis attitude deviation estimation value according to the difference between attitude measurement data and the real three-axis attitude angles at each moment, compensates the attitude measurement data to obtain a modified real attitude, and converts the target position vector to a global coordinate system by using the modified real attitude to obtain a target positioning result. The application can realize global optimal estimation of attitude deviation, and significantly improve the target positioning precision of the motion platform radar.

Description

Technical Field

[0001] This application belongs to the field of radar detection technology, specifically relating to a method for calibrating the attitude deviation of a moving platform radar and locating the target based on semi-definite relaxation. Background Technology

[0002] With the continuous development of radar technology, moving platform radar has been increasingly widely used in military-civilian integration fields such as environmental perception, target detection, and collaborative surveillance. High-precision target positioning is the core foundation for maximizing the effectiveness of these systems, and the accuracy of platform attitude measurement directly determines the quality of positioning. In engineering practice, attitude deviations caused by factors such as inertial navigation device drift and radar antenna misalignment are the main sources of systematic errors affecting the positioning accuracy of moving platform radar. This deviation usually manifests as a constant angular offset along three axes, causing misalignment in the transformation between the radar body coordinate system and the global inertial coordinate system, thus introducing systematic positioning errors and severely restricting the performance of multi-sensor data fusion and collaborative detection. Therefore, high-precision attitude deviation calibration is a prerequisite and core element for achieving high-precision positioning of moving platform radar.

[0003] For the attitude deviation calibration problem of motion platform radar, existing technologies are mainly divided into two categories: filtering methods and batch processing methods. Among them, filtering methods, represented by extended Kalman filtering, usually rely on fixed reference targets in practical applications, which has poor versatility and is difficult to maintain stability in complex dynamic scenarios. While traditional weighted least squares method and its improved iterative batch processing method are simple to implement and have high computational efficiency, they do not strictly consider the inherent SO(3) orthogonal nonconvex constraint of the rotation matrix when establishing the optimization model. The solution process is essentially a local optimization, which makes the algorithm extremely sensitive to the selection of the initial value of the iteration. Under large angle measurement noise interference, the calibration robustness drops sharply, making it difficult to meet the requirements of high-precision positioning. In recent years, semidefinite relaxation techniques have emerged. With the help of convex optimization theory, they can effectively handle the SO(3) orthogonal constraint and theoretically obtain the global optimal solution, showing great application potential.

[0004] However, most existing attitude solving frameworks based on semidefinite relaxation are designed for static scenarios and fail to adapt to the actual situation of continuous dynamic operation of motion radar platforms. They also lack effective mechanisms to utilize redundant information from multi-time observations to suppress attitude measurement noise. More importantly, current research mostly focuses on bias estimation algorithms and has failed to form a complete engineering processing chain from "bias calibration" to "attitude compensation" and then to "target localization," resulting in insufficient practicality and applicability. In addition, most existing calibration methods require three or more cooperative targets to achieve a fully observable estimate of three-axis attitude bias, which imposes stringent requirements on the deployment conditions of application scenarios and limits their widespread use in complex environments. Summary of the Invention

[0005] To address the aforementioned problems in the existing technology, this application provides a method for radar attitude deviation calibration and target localization of a moving platform based on semi-definite relaxation. The technical problem to be solved by this application is achieved through the following technical solution: A method for radar attitude deviation calibration and target localization of a moving platform based on semi-definite relaxation includes: S1: Acquire the spherical coordinate measurement of the ground cooperative target by the motion platform radar and the attitude measurement data output by the motion platform radar at each moment; S2: The spherical coordinate measurement is transformed to the radar volume coordinate system using the debiasing transformation formula to obtain the target position vector in the radar volume coordinate system; the measurement noise covariance matrix in the Cartesian domain is calculated based on the spherical coordinate measurement. S3: Using the target position vector as the measurement input and the inverse matrix of the measurement noise covariance matrix as the weight matrix, construct a weighted least squares cost function, and use the rotation matrix belonging to the three-dimensional special orthogonal group SO(3) as a constraint condition to establish the rotation matrix optimization problem; S4: The rotation matrix optimization problem is transformed into a convex semidefinite programming problem by using a semidefinite relaxation method, and the true three-axis attitude angles at each time are obtained by solving the convex semidefinite programming problem; S5: Calculate the estimated value of constant three-axis attitude deviation based on the difference between the attitude measurement data at each time point and the actual three-axis attitude angle; S6: The attitude measurement data at each moment is compensated using the estimated value of the constant three-axis attitude deviation to obtain the corrected true attitude at each moment, and the target position vector is transformed to the global coordinate system using the corrected true attitude to obtain the target positioning result.

[0006] Beneficial effects: 1. This application employs a semidefinite relaxation method to transform the non-convex attitude problem, which contains orthogonal constraints of the three-dimensional special orthogonal group SO(3) rotation matrix, into a standard convex semidefinite programming problem. By relaxing the non-convex rank-one constraint into a convex positive semidefinite constraint and transforming the orthogonality of the rotation matrix into six linear trace constraints, the non-convex optimization problem, which originally depended on the selection of initial values ​​and was prone to getting trapped in local optima, now has a globally unique optimal solution. This technique completely eliminates the fundamental defects of traditional batch processing methods such as the Gauss-Newton method and iterative weighted least squares method, which are sensitive to the initial values ​​of iteration and suffer from a sharp decrease in calibration robustness under large angle measurement noise. Simulation results show that the root mean square error of the single-axis estimation of the three-axis attitude deviation is within 0.03 degrees, and the RMSE of the roll axis deviation angle estimation is as low as 0.0166 degrees, approaching the lower bound of Cramer-Rao, indicating extremely high calibration accuracy.

[0007] 2. This application directly constructs a weighted least squares cost function with the rotation matrix as the optimization variable in the radar volume coordinate system, and uses the inverse matrix of the Cartesian domain measurement noise covariance matrix as the weight matrix. This ensures that the measurement noise maintains its original zero-mean Gaussian statistical characteristics in the optimization model, avoiding the noise distortion and statistical characteristic destruction caused by rotation transformation when constructing the cost function in the traditional global coordinate system. Thus, the optimal estimation is achieved in a statistical sense, ensuring the theoretical optimality of the attitude solution.

[0008] 3. This application utilizes the statistical characteristics of zero-mean and time-domain independence of attitude measurement noise to perform an arithmetic average of the instantaneous attitude deviations at all times within the observation period, thereby obtaining an estimate of the constant three-axis attitude deviation. This average estimate is a minimum variance unbiased estimate, which can effectively suppress random noise in the attitude measurement channel, fully utilize redundant time-domain observation information, and significantly improve the accuracy and robustness of attitude deviation calibration without additional hardware investment.

[0009] 4. This application only requires two non-collinear stationary cooperative targets with precisely known positions to complete the fully observable estimation of constant attitude deviations of the three axes: yaw angle, platform pitch angle, and roll angle. It does not require a large number of calibration targets to be densely deployed, which significantly reduces the deployment requirements and costs for application scenarios. It is suitable for complex environments where the deployment of cooperative targets is limited, such as mountainous areas, near-shore areas, and urban canyons.

[0010] 5. This application establishes a complete engineering processing chain, from cooperative target spherical coordinate measurement and acquisition, debiased spherical to Cartesian coordinate transformation, volume coordinate system weighted least squares modeling, semi-definite relaxation convex optimization solution, multi-time average estimation of constant attitude deviation, to attitude compensation and target positioning output. The data flow between each step is closed-loop, and the input-output relationship is clear. Simulation results show that after calibration by this application, the root mean square error of the target steady-state positioning reaches 0.0236 km, which is 99.38% higher than the 3.7937 km positioning accuracy of the uncalibrated scheme. The performance is close to the ideal benchmark scheme. Furthermore, the statistical results of 1000 Monte Carlo simulations show that the algorithm has no systematic estimation bias and excellent statistical stability, which can effectively support the engineering application needs of airborne, vehicle-mounted, shipborne, and other moving platform radars in long-range high-precision detection scenarios.

[0011] The present application will be further described in detail below with reference to the accompanying drawings and embodiments. Attached Figure Description

[0012] Figure 1 This is a flowchart illustrating the radar attitude deviation calibration and target localization method for a motion platform based on semi-definite relaxation provided in this application. Figure 2 This is a schematic diagram of the process for solving the target position vector in the radar volume coordinate system and the weighted least squares optimization modeling provided in this application; Figure 3 This is a bar chart comparing the three-axis attitude deviation estimation RMSE and CRB provided in this application; Figure 4 This is a panoramic comparison image of the moving target's trajectory in the XY plane provided in this application; Figure 5 This is a magnified comparison image of the trajectory of a moving target in the XY plane provided in this application; Figure 6 This is a logarithmic curve of the positioning RMSE of the three positioning schemes provided in this application as a function of tracking time. Detailed Implementation

[0013] The present application will be described in further detail below with reference to specific embodiments, but the implementation of the present application is not limited thereto.

[0014] Combination Figure 1 and Figure 2 This application provides a method for radar attitude deviation calibration and target localization of a motion platform based on semi-definite relaxation, including: S1: Acquire the spherical coordinate measurement of the ground cooperative target by the motion platform radar and the attitude measurement data output by the motion platform radar at each moment; This application addresses a scenario involving an airborne radar platform undergoing uniform linear motion. Two non-collinear stationary cooperative targets with precisely known positions are deployed on the ground. The distance between the platform and the targets is approximately 100 km, consistent with the typical operating range of airborne radar. The cooperative targets are at least two non-collinear stationary targets with precisely known positions, and their global positions are pre-calibrated. The spherical coordinate measurements include range, azimuth, and target pitch angle, while the attitude measurement data includes yaw angle, platform pitch angle, and roll angle.

[0015] In this application, the position of the cooperative target is pre-calibrated using high-precision measurement equipment, and its global position is accurately known. Two non-collinear stationary targets are sufficient to achieve a fully observable estimate of the three-axis constant attitude deviation, eliminating the need for a large number of calibration targets densely deployed. The three components of the spherical coordinate measurement—range, azimuth, and target pitch angle—and the three components of the attitude measurement data—yaw angle, platform pitch angle, and roll angle—are used as input data for subsequent steps.

[0016] Assume the observations contain a total of At any given time, the sampling interval is... In the first At that moment, the radar detected the first... The spherical coordinate measurement of each cooperative objective is as follows: These correspond to range, azimuth, and target elevation angle, respectively; all measurements are superimposed with independent zero-mean Gaussian white noise, with noise variances of [missing values]. , , .

[0017] The attitude measurements output by the platform navigation system include constant attitude deviation and random noise. ; In the formula, Corresponding to the true three-dimensional attitude angle, the target moves at a uniform and smooth speed, while the sensor attitude is a slowly changing continuous curve. The constant attitude angle system deviation to be estimated comes from installation error, device zero bias, and initial alignment error, corresponding to the three channels of yaw, pitch, and roll, respectively. Zero-mean Gaussian random noise.

[0018] S2: The spherical coordinate measurement is transformed to the radar volume coordinate system using the debiasing transformation formula to obtain the target position vector in the radar volume coordinate system; the measurement noise covariance matrix in the Cartesian domain is calculated based on the spherical coordinate measurement. Direct spherical-to-Cartesian coordinate transformation introduces nonlinear transformation bias. Therefore, a bias-correction transformation formula is used to transform radar spherical coordinate measurements to the radar volume coordinate system, obtaining the target position vector in the volume coordinate system. Simultaneously, based on a first-order Taylor expansion, the noise covariance matrix of the transformed Cartesian coordinate measurements is calculated, providing a statistical basis for subsequent weighted optimization. Direct spherical-to-Cartesian conversion introduces nonlinear transformation bias; this application uses a bias-correction transformation formula to obtain the target's measured position in the volume coordinate system, expressed as: ; in, Indicates the first Time of the first The target position vectors of the cooperative targets in the radar volume coordinate system; Indicates the first Time of the first Distance measurements of the cooperative objectives; Indicates the first Time of the first Azimuth measurements of the cooperative targets; Indicates the first Time of the first Target pitch angle measurement values ​​for each cooperative objective; This represents the noise variance of the azimuth measurement; Indicates the noise variance of the target pitch angle measurement; superscript Indicates the radar volume coordinate system, superscript This represents the transpose of a vector.

[0019] In this application, a direct spherical-Cartesian coordinate transformation would introduce a nonlinear transformation bias. Therefore, the above-mentioned bias-reduction transformation formula is used to eliminate the nonlinear transformation bias of spherical coordinates to Cartesian coordinates, and to obtain the target position vector in the radar volume coordinate system.

[0020] In one specific embodiment of this application, calculating the measurement noise covariance matrix in the Cartesian domain based on the spherical coordinate measurement includes: S21, based on the measured noise variance of the preset distance. azimuth measurement noise variance and the measurement noise variance of the target pitch angle The noise covariance matrix in spherical coordinates is represented as: ; S22, using the noise covariance matrix in the spherical coordinate domain, calculate the measurement noise covariance matrix in the Cartesian domain, expressed as: ; in, Indicates the first Time of the first The covariance matrix of the noise of the cooperative targets in the Cartesian measurement in the radar volume coordinate system ( ), For the debiasing transformation formula for spherical coordinate vectors The Jacobian matrix obtained by taking the partial derivative ( ), The noise covariance matrix in spherical coordinates (a diagonal matrix, where the diagonal elements are the measurement noise variances of range, azimuth, and target elevation angles, respectively), with superscript... This represents the matrix transpose. This expression, based on a first-order Taylor expansion, transfers spherical coordinate noise to the Cartesian domain, providing a weight matrix (i.e., ...) for subsequent weighted least squares optimization. (the inverse matrix).

[0021] ; This application approximates the measurement noise covariance matrix in the Cartesian domain through a first-order Taylor expansion, providing a noise statistical basis for subsequent weighted optimization.

[0022] S3: Using the target position vector as the measurement input and the inverse matrix of the measurement noise covariance matrix as the weight matrix, construct a weighted least squares cost function, and assume that the rotation matrix belongs to a special three-dimensional orthogonal group. As a constraint, a rotation matrix is ​​established to solve the optimization problem. Unlike traditional optimization modeling in a global coordinate system, this application directly constructs a weighted least squares optimization problem in the radar volume coordinate system to ensure that the measurement noise retains its original zero-mean Gaussian statistical characteristics and avoids noise distortion caused by rotation transformation. The rotation matrix is ​​used as the optimization variable, and the weight matrix is ​​taken as the inverse matrix of the Cartesian measurement noise covariance to achieve the optimal estimation in a statistical sense. The optimization constraint is that the rotation matrix belongs to the three-dimensional special orthogonal group SO(3).

[0023] Optimization in the global coordinate system involves rotational transformation noise, which corrupts its statistical properties. Therefore, this application directly constructs the cost function in the volume coordinate system. The Mahalanobis norm is defined. Take the weight matrix Then the first The optimization problem for finding the rotation matrix at time step 1 is: ; in, To realize the radar volume coordinate system using a true rotation matrix. Mapping to the global coordinate system; This is the relative position vector between the platform and the target in the global coordinate system, calculated from the known target and platform positions; For the weight matrix, take the inverse of the Cartesian domain measurement noise covariance matrix; Let represent the square of the Mahalanobis norm; I is defined as the total number of cooperative objectives. The specific expression for the rotation matrix is: ; In the formula, the parameter These correspond to yaw angle, platform pitch angle, and roll angle, respectively. It is used to transform the target position vector from the radar volume coordinate system to the global coordinate system. It is the analytical form of the rotation matrix of the optimization variable in S3. Its nine elements are all composed of the sine and cosine combination of the three-axis attitude angles and satisfy the SO(3) orthogonal constraint.

[0024] This application directly constructs a weighted least squares cost function in the radar volume coordinate system, ensuring that the measurement noise retains its original zero-mean Gaussian statistical characteristics and avoiding noise distortion caused by rotation transformation. The weight matrix is ​​taken as the inverse matrix of the measurement noise covariance in the Cartesian domain, achieving the optimal estimation in a statistical sense. The optimization constraint is that the rotation matrix belongs to a special three-dimensional orthogonal group. .

[0025] S4: The rotation matrix optimization problem is transformed into a convex semidefinite programming problem by using a semidefinite relaxation method, and the true three-axis attitude angles at each time are obtained by solving the convex semidefinite programming problem; The semidefinite relaxation method described in this application transforms the optimization problem of solving the rotation matrix into a convex semidefinite programming problem, including: S41, using a vectorized rotation matrix expansion to solve the rotation matrix optimization problem and eliminating constant terms irrelevant to the optimization variables, the rotation matrix optimization problem is transformed into a standard quadratic form, expressed as: ; In the formula, the coefficient matrix With coefficient vector Obtained by operating on the target's relative position and the weight matrix; This is a 9-dimensional column vector after the rotation matrix has been vectorized; superscript Represents the transpose of a matrix or vector.

[0026] S42, Construct a 10-dimensional augmented vector and the corresponding rank-1 augmented matrix ; Define the cost coefficient matrix: ; The optimization problem for solving the rotation matrix of the quadratic form is then: .

[0027] S43, Solving the non-convex rank-1 constraint in the optimization problem of solving the rotation matrix of the standard quadratic form. Relaxation is a convex semi-definite constraint and rotation matrix Orthogonality is transformed into six linear trace constraints, resulting in a standard convex semidefinite programming problem, expressed as: ; In the formula, For augmented matrices; for The corresponding rotation matrix vector Submatrix; This is a preset orthogonal constraint matrix; The Kronecker function; Represents the trace operation of a matrix; Representation matrix It is a positive semi-definite matrix.

[0028] Through the above semi-definite relaxation operation, the non-convex... The problem of constrained attitude solving is transformed into a standard convex semidefinite programming problem, which can be solved globally optimally, completely eliminating the defects of traditional iterative algorithms that rely on initial values ​​and are prone to getting trapped in local optima.

[0029] This application solves the standard convex semidefinite programming problem to obtain the optimal augmented matrix. From the optimal augmented matrix Extract the principal eigenvectors to reconstruct the initial rotation matrix. ; for the initial rotation matrix Perform singular value decomposition projection correction to obtain the desired result. Optimal rotation matrix under constraints , is represented as: ; ; in, The initial rotation matrix obtained from the reconstruction; This represents the singular value decomposition operation; and The orthogonal matrix obtained from singular value decomposition; It is a singular value diagonal matrix; Represents the determinant operation of a matrix; To satisfy the projection correction Optimal rotation matrix under constraints; superscript This represents the transpose of a matrix.

[0030] Finally, based on the ZYX Euler angle definition, from the optimal rotation matrix The true three-axis attitude angles at each moment are obtained by solving the problem.

[0031] S5: Calculate the estimated value of constant three-axis attitude deviation based on the difference between the attitude measurement data at each time point and the actual three-axis attitude angle; According to the attitude measurement model, the instantaneous attitude deviation at time k is: ; in, These are attitude measurement values. To obtain the true posture.

[0032] In this application, based on the attitude measurement model, the first The instantaneous attitude deviation at time step S4 is the difference between the attitude measurement data and the true three-axis attitude angles calculated in step S4. Since the attitude measurement noise has zero mean and is independent in the time domain, the arithmetic mean of the instantaneous attitude deviations at all times within the observation period can effectively suppress random noise and obtain a minimum variance unbiased estimate of the constant three-axis attitude deviation, making full use of redundant time-domain observations to improve calibration accuracy.

[0033] Since the attitude measurement noise has zero mean and is independent in the time domain, The instantaneous deviations at each time step are arithmetically averaged to suppress random noise, yielding the final constant attitude deviation estimate: ; In the formula, This is an estimate of the constant three-axis attitude deviation; The total number of moments within the observation period; For the first Attitude measurement data at any given time; For the first The true three-axis attitude angles at any given moment.

[0034] In this application, the estimated constant attitude deviation is used to compensate for the attitude measurement data of the platform at each moment, thereby eliminating the attitude system deviation from the root.

[0035] S6: The attitude measurement data at each moment is compensated using the estimated value of the constant three-axis attitude deviation to obtain the corrected true attitude at each moment, and the target position vector is transformed to the global coordinate system using the corrected true attitude to obtain the target positioning result.

[0036] The estimated constant attitude deviation is used to compensate for the platform's real-time attitude measurements: ; In the formula, For attitude measurement data; This is an estimate of the constant three-axis attitude deviation obtained; The corrected true posture is obtained after compensation.

[0037] Based on the corrected true attitude, a coordinate transformation matrix is ​​constructed, and the target position vector in the radar volume coordinate system is transformed to the global coordinate system using the coordinate transformation matrix to obtain the target positioning result after attitude deviation calibration, thus completing the entire link from deviation calibration to positioning output. As a preferred embodiment, the compensated attitude sequence is smoothed by moving average to further reduce residual high-frequency random attitude noise before being used for coordinate transformation and target positioning, which can effectively reduce the fluctuation of the positioning result.

[0038] For moving target tracking scenarios, the calibrated attitude and coordinate transformation results can be integrated into conventional target tracking algorithms such as extended Kalman filtering, combined with the target motion model to achieve continuous high-precision target tracking and output the target motion trajectory.

[0039] refer to Figure 2 The method proposed in this application consists of two stages: The first stage is a three-axis attitude deviation calibration based on semi-definite relaxation. Using the spherical coordinate measurements of the radar body towards the cooperative target and its precisely known global position as input, the calibration process sequentially involves constructing a weighted least squares cost function, vectorization and homogenization, transformation into a convex semi-definite programming problem through semi-definite relaxation, SDP solution, and SVD projection correction to obtain the attitude deviation at a single moment. Finally, a multi-moment statistical average is used to output the estimated three-axis deviation. The second stage is target tracking after attitude correction. The spherical coordinate measurements of the moving target to be tracked are input into the attitude error compensation and EKF tracking modules, and the calibrated global position of the target is finally output. The entire process fully presents a two-level data processing chain from deviation calibration to positioning output.

[0040] Simulation verification and effect analysis The performance of this application was quantitatively verified through 1,000 Monte Carlo simulation experiments, fully reproducing the engineering scenario of long-range target detection by airborne radar.

[0041] I. Simulation Condition Settings A three-dimensional airborne far-field radar simulation scenario was constructed, with the radar platform maintaining uniform linear flight. Two spatially non-collinear stationary cooperative targets were deployed within the scenario to calibrate the three-axis attitude system deviations. An additional uniformly moving target was set up to verify the continuous tracking and positioning performance after attitude compensation. The distance between the target and the radar reference point was on the order of hundreds of kilometers. Under far-field conditions, the influence of attitude angle deviation on target positioning error was approximately linear, which facilitated a direct demonstration of the calibration algorithm's correction effect.

[0042] The simulation process is divided into two independent stages: 1. Attitude calibration phase: Total duration 10s, 50 uniform sampling points, using global accurate prior position of cooperative target, radar body spherical coordinate measurement, attitude observation data with Gaussian noise and constant system bias, and solve the three-axis attitude installation deviation based on semidefinite relaxation convex optimization; 2. Target tracking stage: The total duration is 300s, with a total of 1500 uniform sampling points. The three-axis deviations obtained from SDR estimation are compensated to the original attitude observations, and zero-phase moving average is used to smooth the attitude random noise. Finally, extended Kalman filtering is combined to complete the three-dimensional continuous positioning of the target.

[0043] To eliminate the randomness of results caused by random noise in a single simulation, 1000 independent Monte Carlo cyclic statistical calculations were conducted to obtain the RMSE performance curve. The simulation did not use a fixed constant attitude; instead, it generated continuous and smooth true attitude values ​​with periodic vibrations and slow heading drift, closely reflecting the real engineering characteristics of attitude dynamic changes during airborne platform flight. The system sampling interval $T = 0.2 s$, and the motion parameters of the radar platform, cooperative target, and the target to be tracked are shown in Table 1. The parameters of three-axis attitude constant deviation, radar measurement noise, and attitude observation noise are shown in Table 2.

[0044] Table 1. Platform and target motion parameter settings

[0045] Table 2 Attitude Deviation and Sensor Noise Parameters

[0046] The root mean square error of attitude angle deviation estimation and the root mean square error of target 3D global position (RMSE) are used as quantitative evaluation indicators for estimation accuracy and positioning accuracy. ; ; In the formula, Let be the target position estimate at time k in the i-th Monte Carlo simulation. To represent the true 3D position of the target at the corresponding time, M=1000 represents the total number of Monte Carlo simulations. To verify the gain effect of the calibration scheme in this application in a stratified manner, three sets of control positioning schemes were set up: To verify the gain effect of the calibration scheme in this application, three sets of control positioning schemes were set up: 1. Scheme a (ideal baseline scheme): directly use the real physical attitude without any systematic bias to complete coordinate transformation and EKF tracking, representing the theoretical upper limit of radar positioning accuracy in the current scenario; 2. Scheme b (uncalibrated comparison scheme): The target position is calculated directly using the original attitude observations containing three-axis constant system bias and superimposed random Gaussian noise, simulating the traditional radar positioning process without attitude calibration module in engineering. 3. Scheme c (SDR calibration scheme of this application): The semi-definite relaxation convex optimization algorithm is used to estimate the deviation of the three-axis attitude system. After the deviation compensation is completed for the original attitude observation, the attitude random noise is suppressed by zero phase moving average, and then the extended Kalman filter is substituted to realize the three-dimensional tracking and positioning of the target.

[0047] II. Simulation Results and Performance Analysis Accuracy verification of three-axis attitude deviation estimation First, the ability of the SDR algorithm in this application to estimate constant deviations in three-axis attitude is verified by a single deterministic simulation. The deviation estimation results of the single calibration experiment are shown in Table 3.

[0048] Table 3. Attitude deviation estimation results for a single SDR.

[0049] This application employs a semidefinite relaxation strategy to transform the non-convex orthogonal constraint optimization problem corresponding to the rotation matrix into a convex SDP problem that can be solved globally optimally. This eliminates the need for initial iteration values ​​and avoids local optimum convergence defects. As shown in Table 3, the absolute errors of the three-axis attitude deviation estimation are all controlled within 0.06°, and the absolute error of the yaw axis estimation is as low as 0.0062°. This allows for high-precision reproduction of the platform's three-axis attitude installation deviations, providing reliable correction values ​​for subsequent attitude compensation.

[0050] The mean of the three-axis bias estimates and the comparison results of RMSE and CRB lower bounds obtained from 1000 Monte Carlo simulations are shown in Table 4. Figure 3 The bar chart comparing the attitude deviation RMSE with the lower limit of CRB shows that the mean of the three-axis deviation estimation is very close to the true value, with no systematic estimation bias; the estimated RMSE of pitch and yaw axes is slightly higher than the lower limit of CRB, which is in line with the theoretical law; the estimated RMSE of roll axis is slightly higher than the lower limit of CRB. The estimated RMSE of the three axes is maintained within 0.03° overall, and the calibration accuracy meets the attitude compensation requirements of long-range radar.

[0051] Table 4. Comparison of Monte Carlo Attitude Deviation Statistics with CRB Lower Bound

[0052] Target trajectory visualization and comparative analysis like Figure 4 As shown in the XY-plane panoramic track comparison diagram, the black solid line represents the target's actual motion track, the dashed circle represents the ideal reference scheme a track, the triangular dotted line represents the uncalibrated scheme b track, and the solid square line represents the calibrated track of scheme c in this application. Uncalibrated scheme b is affected by constant three-axis attitude deviations, and the target's global coordinate transformation continuously introduces systematic offsets, resulting in an overall misalignment of several kilometers between the track and the true value. The track of this application, after SDR deviation compensation, closely matches the true trajectory, and the systematic positioning offset is significantly eliminated. Figure 5 As shown, the magnified view of the XY plane clearly distinguishes the subtle differences between scheme c and the ideal reference scheme a. The two schemes only exhibit small random fluctuations caused by radar measurement noise, without any long-term fixed offset. The comparison results of the three-dimensional tracking trajectories further verify the algorithm's full-dimensional correction capability: the X, Y, and Z-axis tracks effectively suppress the overall offset caused by attitude system errors, with no significant layered deviation in the height dimension, and good consistency in three-dimensional positioning.

[0053] Monte Carlo Statistical RMSE Quantitative Performance Analysis The logarithmic curves of RMSE variation with tracking time for the three positioning schemes were obtained from 1000 Monte Carlo simulations, as shown below. Figure 6 As shown, the average RMSE of the tracking steady-state interval (last 50 sampling points) is extracted as the steady-state accuracy index. The quantitative performance of the three schemes is as follows: 1. Scheme b (uncalibrated): The steady-state positioning RMSE is as high as 3.7937km. The systematic offset introduced by the constant attitude deviation cannot be eliminated by EKF filtering iteration. The positioning error is too large in long-distance scenarios and does not meet the requirements of high-precision detection. 2. Scheme a (ideal benchmark): The steady-state RMSE is 0.0154km, which is the theoretical upper limit of positioning accuracy in this simulation scenario; 3. Scheme c (SDR calibration of this application): The steady-state RMSE is 0.0236km, which improves the positioning accuracy by 99.38% compared with the traditional scheme without calibration. The steady-state error is only slightly higher than the ideal reference. After calibration, the tracking performance is much closer to the theoretical optimal level.

[0054] The statistical results of 1000 Monte Carlo average tracks also corroborate the stability of the algorithm: after multiple simulations and averaging, the uncalibrated scheme always has a fixed amplitude global offset, while the statistical tracks of the SDR calibration scheme of this application have no systematic offset, small random fluctuation amplitude, and excellent statistical stability of calibration and tracking links.

[0055] Simulation results show that the semi-definite relaxation-based three-axis attitude deviation convex optimization calibration algorithm of this application can accurately solve the constant system deviation of radar platform attitude, and eliminate the systematic offset of long-range target positioning from the source of coordinate transformation. After being combined with zero-phase smoothing processing to suppress attitude random noise, the random jitter of the tracking trajectory is further reduced. The entire integrated "SDR attitude calibration-EKF target tracking" link logic is complete and the statistical performance is stable, which can be adapted to the engineering application scenario of high-precision detection of airborne long-range moving targets.

[0056] It is worth noting that the terms "first" and "second" in this application are used for descriptive purposes only and should not be construed as indicating or implying relative importance or implicitly specifying the number of technical features indicated. Therefore, a feature defined as "first" or "second" may explicitly or implicitly include one or more of that feature. In the description of this application, "multiple" means two or more, unless otherwise explicitly specified.

[0057] The above description, in conjunction with specific preferred embodiments, provides a further detailed explanation of this application and should not be construed as limiting the specific implementation of this application to these descriptions. For those skilled in the art, various simple deductions or substitutions can be made without departing from the concept of this application, and all such modifications or substitutions should be considered within the scope of protection of this application.

Claims

1. A method for calibrating radar attitude deviation and locating targets for a moving platform based on semi-definite relaxation, characterized in that, include: S1: Acquire the spherical coordinate measurement of the ground cooperative target by the motion platform radar and the attitude measurement data output by the motion platform radar at each moment; S2: The spherical coordinate measurement is transformed to the radar volume coordinate system using the debiasing transformation formula to obtain the target position vector in the radar volume coordinate system; the measurement noise covariance matrix in the Cartesian domain is calculated based on the spherical coordinate measurement. S3: Using the target position vector as the measurement input and the inverse matrix of the measurement noise covariance matrix as the weight matrix, construct a weighted least squares cost function, and use the rotation matrix belonging to the three-dimensional special orthogonal group SO(3) as a constraint condition to establish the rotation matrix optimization problem; S4: The rotation matrix optimization problem is transformed into a convex semidefinite programming problem by using a semidefinite relaxation method, and the true three-axis attitude angles at each time are obtained by solving the convex semidefinite programming problem; S5: Calculate the estimated value of constant three-axis attitude deviation based on the difference between the attitude measurement data at each time point and the actual three-axis attitude angle; S6: The attitude measurement data at each moment is compensated using the estimated value of the constant three-axis attitude deviation to obtain the corrected true attitude at each moment, and the target position vector is transformed to the global coordinate system using the corrected true attitude to obtain the target positioning result.

2. The method for radar attitude deviation calibration and target localization of a motion platform based on semi-definite relaxation according to claim 1, characterized in that, The cooperative targets are at least two non-collinear stationary targets whose positions are precisely known, and the global positions of the cooperative targets are pre-calibrated; the spherical coordinate measurements include distance, azimuth, and target pitch angle, and the attitude measurement data includes yaw angle, platform pitch angle, and roll angle.

3. The method for calibrating and locating the attitude deviation of a motion platform radar based on semi-definite relaxation according to claim 2, characterized in that, The bias removal transformation formula described in S2 is: ； in, Indicates the first Time of the first The target position vectors of the cooperative targets in the radar volume coordinate system; Indicates the first Time of the first Distance measurements of the cooperative objectives; Indicates the first Time of the first Azimuth measurements of the cooperative targets; Indicates the first Time of the first The target pitch angle measurement value of each cooperative objective; This represents the noise variance of the azimuth measurement; Indicates the noise variance of the target pitch angle measurement; superscript Indicates the radar volume coordinate system, superscript This represents the transpose of a vector.

4. The method for calibrating and locating the attitude deviation of a moving platform radar based on semi-definite relaxation according to claim 3, characterized in that, S2 includes calculating the measurement noise covariance matrix in the Cartesian domain based on the spherical coordinate measurement, including: S21, based on the measured noise variance of the preset distance. azimuth measurement noise variance and the measurement noise variance of the target pitch angle The noise covariance matrix in spherical coordinates is represented as: ； S22, using the noise covariance matrix in the spherical coordinate domain, calculate the measurement noise covariance matrix in the Cartesian domain, expressed as: ； in, Indicates the first Time of the first The covariance matrix of the noise of the cooperative targets in Cartesian measurement in radar volume coordinates. For the partial transformation formula of spherical coordinate vector The Jacobian matrix obtained by taking the partial derivative. The noise covariance matrix in spherical coordinates, with superscript... This indicates the matrix transpose.

5. The method for calibrating and locating the attitude deviation of a moving platform radar based on semi-definite relaxation according to claim 4, characterized in that, The optimization problem for solving the rotation matrix described in S3 is: ； In the formula, To realize the radar volume coordinate system using a true rotation matrix. Mapping to the global coordinate system; This is the relative position vector between the platform and the target in the global coordinate system, calculated from the known target and platform positions; For the weight matrix, take the inverse of the Cartesian domain measurement noise covariance matrix; Represents the square of the Markov norm. Represents the residual vector, with superscript To represent the transpose of a matrix or vector, the superscript... This represents the inverse of a matrix.

6. The method for calibrating and locating the attitude deviation of a moving platform radar based on semi-definite relaxation according to claim 5, characterized in that, The semidefinite relaxation method described in S4 transforms the optimization problem of solving the rotation matrix into a convex semidefinite programming problem, including: S41, using a vectorized rotation matrix expansion to solve the rotation matrix optimization problem and eliminating constant terms irrelevant to the optimization variables, the rotation matrix optimization problem is transformed into a standard quadratic form, expressed as: ； In the formula, the coefficient matrix With coefficient vector Obtained by operations on the target's relative position and the weight matrix; This is a 9-dimensional column vector after the rotation matrix has been vectorized; superscript Represents the transpose of a matrix or vector; S42, Construct a 10-dimensional augmented vector and the corresponding rank-1 augmented matrix ; S43, Solving the non-convex rank-1 constraint in the optimization problem of solving the rotation matrix of the standard quadratic form. Relaxation is a convex semidefinite constraint and rotation matrix Orthogonality is transformed into six linear trace constraints, resulting in a standard convex semidefinite programming problem, expressed as: S44, rank 1 constraint on non-convex surfaces. Relaxation is a convex semidefinite constraint , rotate the matrix Orthogonality is transformed into six linear trace constraints, resulting in a standard convex semidefinite programming problem, expressed as: ； In the formula, For augmented matrices; for The corresponding rotation matrix vector Submatrix; This is a preset orthogonal constraint matrix; The Kronecker function; Represents the trace operation of a matrix; Representation matrix It is a positive semi-definite matrix.

7. The method for calibrating and locating the attitude deviation of a moving platform radar based on semi-definite relaxation according to claim 6, characterized in that, Solving the convex semidefinite programming problem in S4 yields the true three-axis attitude angles at each time step, including: Solving the standard convex semidefinite programming problem yields the optimal augmented matrix. ; From the optimal augmented matrix Extract the principal eigenvectors to reconstruct the initial rotation matrix. ; For the initial rotation matrix Perform singular value decomposition projection correction to obtain the desired result. Optimal rotation matrix under constraints ; Based on the ZYX Euler angle definition, from the optimal rotation matrix The true three-axis attitude angles at each moment are obtained by solving the problem.

8. The method for calibrating radar attitude deviation and locating targets for a motion platform based on semi-definite relaxation according to claim 7, characterized in that, The estimated value of the constant three-axis attitude deviation mentioned in S5 is: ； In the formula, This is an estimate of the constant three-axis attitude deviation; The total number of moments within the observation period; For the first Attitude measurement data at any given time; For the first The true three-axis attitude angles at any given moment.

9. The method for calibrating and locating the attitude deviation of a motion platform radar based on semi-definite relaxation according to claim 8, characterized in that, In S6, the estimated value of the constant three-axis attitude deviation is used to compensate for the attitude measurement data at each moment: ； In the formula, For attitude measurement data; This is an estimate of the constant three-axis attitude deviation obtained; The corrected true posture is obtained after compensation.

10. The method for calibrating radar attitude deviation and locating targets for a moving platform based on semi-definite relaxation according to claim 1, characterized in that, The step S6, which involves transforming the target position vector to the global coordinate system using the corrected true pose to obtain the target localization result, includes: Based on the corrected true attitude, a coordinate transformation matrix is ​​constructed, and the target position vector in the radar volume coordinate system is transformed to the global coordinate system using the coordinate transformation matrix.

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