Mobile relay auxiliary target positioning method based on semi-definite programming optimization

Abstract

The invention discloses a mobile relay auxiliary target positioning method based on semi-definite programming optimization, which relates to the technical field of high-precision positioning and mainly comprises the following steps: after a signal transmitted by a sensor in a positioning period is forwarded to a target through a mobile relay, the target reflects the signal and reversely returns the signal to the corresponding sensor through the same signal link; extracting the time delay of signal propagation and the measured value of the angle of arrival; reconstructing the time delay measurement model into a distance conversion model, and obtaining a linearization expression of the angle of arrival; constructing a constraint weighted least square problem including the position and speed of the mobile relay and the distance offset through a joint distance conversion model and a linearization expression; an auxiliary variable is added and finally converted into a semi-definite programming problem, and a preliminary estimation value is solved; and constructing a joint refined weighted least square problem based on the initial estimation value, and solving a final estimation value. According to the method, the target and the relay state are decoupled, and the positioning precision is improved by combining an SDP optimization algorithm with a combined refining process.

Description

Technical Field

[0001] This invention relates to the field of high-precision positioning technology, specifically to a mobile relay-assisted target positioning method based on semidefinite programming optimization. Background Technology

[0002] The widespread application of modern wireless communication, IoT sensing, telemetry, and location services has created an urgent need for high-precision positioning technology. For the past two decades, positioning technology research has primarily focused on signal parameter measurements in line-of-sight (LOS) environments, including key indicators such as time delay (TD), time difference of arrival (TDOA), and azimuth of arrival (AOA), as well as their combined applications. However, in complex propagation scenarios, such as indoor environments with multiple obstacles or urban canyons, non-line-of-sight (NLOS) propagation of signals leads to a sharp deterioration in positioning accuracy. Simultaneously, signal strength attenuation in long-distance transmission scenarios severely restricts the reliability of traditional positioning systems.

[0003] Against this backdrop, Reconfigurable Intelligent Surface (RIS) technology has emerged, offering an innovative approach to solving communication challenges in complex propagation environments. This technology intelligently modulates the reflection characteristics of electromagnetic surfaces, enabling the establishment of efficient reflection links between base stations and terminals. Of particular note is the proposal by some researchers to integrate RIS into UAV platforms to improve signal coverage quality in urban three-dimensional spaces. Compared to conventional relay solutions, while RIS-assisted systems can improve spectral efficiency, the computational complexity arising from their phase modulation algorithms cannot be ignored. Therefore, researchers are exploring hybrid networking schemes combining RIS and relay technologies. Recent research indicates that far-field positioning technology has achieved a breakthrough with the aid of mobile transceivers—traditional far-field positioning relies on DOA measurements under the plane wavefront assumption, while the new scheme introduces mobile nodes and utilizes the characteristics of spherical wavefronts to achieve absolute position calculation. The high-sensitivity amplifiers equipped on these transceiver nodes effectively overcome long-distance transmission losses, ensuring accurate extraction of positioning parameters.

[0004] Notably, this transceiver-assisted positioning technology exhibits unique advantages in NLOS scenarios. Its working principle is similar to RIS, improving signal propagation conditions by establishing multi-hop line-of-sight links. Compared to fixed RIS, transceiver systems mounted on mobile platforms (such as UAVs or autonomous underwater robots) can dynamically optimize the positioning topology. However, the self-localization problem of the mobile platform in GNSS failure environments becomes a key factor restricting system performance and urgently needs to be addressed. Summary of the Invention

[0005] To achieve target localization in environments lacking line-of-sight propagation, this invention proposes a mobile relay-assisted target localization method based on semidefinite programming optimization, comprising the following steps: S1: Construct a three-dimensional coordinate system, deploy a corresponding number of sensors with known positions, mobile relays with unknown positions and speeds, and a target with unknown positions according to the task requirements, and set the mobile relays to maintain uniform linear motion within the positioning cycle. S2: During the positioning period, the signal emitted by the sensor is forwarded to the target via a mobile relay. The target reflects the signal and transmits it back to the corresponding sensor via the same signal link to extract the signal propagation delay and angle of arrival measurement values. S3: Construct time delay measurement model and angle of arrival measurement model based on time delay measurement value and angle of arrival measurement value, respectively; S4: By introducing auxiliary constants to construct independent variables of target position, the time delay measurement model is reconstructed into a distance transformation model, and the linearized expression of the angle of arrival is obtained by separating the noise term from the angle of arrival measurement model; S5: Construct a constrained weighted least squares problem that includes the location, velocity, and distance offset of the mobile relay by using a joint distance transformation model and a linearized expression; S6: The initial transformation of the constrained weighted least squares problem by auxiliary variables, the removal of rank 1 constraints and the addition of equality constraints to tighten the solution space, and the transformation into a semidefinite programming problem are used to solve the problem and obtain the initial estimates of the target position, the mobile relay position and the velocity. S7: Based on the initial estimates, perform a first-order Taylor expansion on the time delay measurement model and the angle of arrival measurement model, construct a joint refined weighted least squares problem, and solve it to obtain the final estimates of the target position, mobile relay position and velocity.

[0006] This invention utilizes the dynamic motion of mobile relays to overcome the physical aperture limitations of traditional fixed sensor networks, providing spatial geometric constraints through multi-view observation; a hybrid measurement model (TD provides absolute distance information, AOA provides direction vectors) collaboratively decouples the target and mobile relay states; and the SDP optimization algorithm combined with a joint refinement process brings the positioning accuracy close to the Cramer-Rhodes lower bound.

[0007] Furthermore, the mobile relay is mounted on a drone or autonomous mobile robot platform, the target is a concealed target that passively radiates electromagnetic signals, and the time delay and angle of arrival measurements are passively acquired based on the target's own radiation signals. Neither the sensor nor the mobile relay actively transmits calibration signals during the positioning process.

[0008] Furthermore, in step S3, the delay measurement model quantifies the total propagation delay by superimposing the geometric distance of the bidirectional signal link with the mobile relay hardware offset to obtain the target relative distance information. The formula is expressed as: In the formula, Where i is the mobile trunk number and M is the number of mobile trunks. j is the sensor number, and N is the number of sensors. k is the sampling time number, and K is the total number of samples in the positioning cycle. Let c be the measured distance of the signal at the k-th sampling time, transmitted from the j-th sensor to the target via the i-th mobile relay, contaminated by measurement noise. Let be the time delay measurement value of the signal at the k-th sampling time being forwarded from the j-th sensor to the target via the i-th mobile relay. Let i be the location of the i-th mobile relay at the k-th sampling time. The true coordinates of the target in the three-dimensional coordinate system. Let j be the true coordinate position of the j-th sensor in the three-dimensional coordinate system. The distance offset caused by the i-th mobile relay forwarding signal. The distance measurement noise along the propagation path of the signal at the k-th sampling time, from the j-th sensor to the target via the i-th mobile relay, is... This is the symbol for the 2-norm.

[0009] Furthermore, in step S3, the angle of arrival measurement model is based on spherical wavefront characteristics. It utilizes the inverse trigonometric relationship between azimuth and elevation angles to project the signal propagation direction from the mobile relay to the sensor onto a three-dimensional coordinate system to provide spatial directional constraints. The formula is expressed as: In the formula, The azimuth angle measurement value contaminated by measurement noise along the propagation path of the signal from the i-th mobile relay to the j-th sensor at the k-th sampling time. Let be the azimuth angle measurement value at the k-th sampling time, which is unaffected by measurement noise along the propagation path from the i-th mobile relay to the j-th sensor. The azimuth measurement noise is defined as the propagation path of the signal at the k-th sampling time from the i-th mobile relay to the j-th sensor. They are respectively scalar components on the X, Y, and Z axes of a three-dimensional coordinate system They are respectively scalar components on the X, Y, and Z axes of a three-dimensional coordinate system for Vector components in the X and Y coordinate system for Vector components in the X and Y coordinate system The pitch angle measurement value contaminated by measurement noise along the propagation path of the signal from the i-th mobile relay to the j-th sensor at the k-th sampling time is given. The pitch angle measurement value at the k-th sampling time, unaffected by measurement noise, represents the path of the signal from the i-th mobile relay to the j-th sensor. The pitch angle measurement noise is the propagation path of the signal at the k-th sampling time from the i-th mobile relay to the j-th sensor.

[0010] Furthermore, in step S4, the specific process of reconstructing the distance transformation model includes: Will Move to the left side of the equation, square both sides of the equation and ignore the second-order noise term to obtain the distance approximation model; Based on the distance approximation model, a constant translation vector is introduced. The location of the mobile relay is represented as an auxiliary variable. This leads to the distance transformation model, expressed by the following formula: In the formula, , Let be the actual coordinate position of the i-th mobile relay in the three-dimensional coordinate system at the initial moment. The sampling interval duration. Let be the initial moving speed of the i-th mobile relay; Let T be the unit direction vector calculated from the angle of arrival measurement, and let T be the transpose operation. This is a composite error term.

[0011] Further, in step S5, the constrained weighted least squares problem is described as follows: In the formula, x is the optimization variable in the constrained weighted least squares problem. For the target location variable, The coefficient matrix introduced in the constrained weighted least squares problem. The coefficient vector introduced in the constrained weighted least squares problem. The weight matrix is ​​introduced to constrain the weighted least squares problem.

[0012] Furthermore, in step S6, the auxiliary variable is... , The specific transformation operation of the semidefinite programming problem is as follows: Based on auxiliary variables The constrained weighted least squares problem is equivalently transformed into: ; Using the semidefinite relaxation technique, the non-convex rank 1 constraints in the transformed problem are discarded, resulting in the semidefinite programming problem: In the formula, The trace of the matrix; Based on the semidefinite programming problem, the solution space is tightened by adding equality constraints, transforming it into a semidefinite programming problem: .

[0013] Furthermore, in step S7, the formula for the joint refined weighted least squares problem is expressed as: In the formula, To jointly refine the optimization variables in the weighted least squares problem, including the target position, the position of the mobile relay, and the velocity, To constrain the objective function of the weighted least squares problem, The coefficient vector introduced in the joint refinement of the weighted least squares problem. The coefficient matrix introduced in the joint refinement of the weighted least squares problem is... The weight matrix is ​​introduced to refine the weighted least squares problem.

[0014] Furthermore, in step S4, the linearized expression for the angle of arrival is obtained by performing a first-order Taylor expansion on the angle of arrival measurement model to separate the noise term.

[0015] Compared with the prior art, the present invention has at least the following beneficial effects: (1) The present invention proposes a mobile relay-assisted target localization method based on semidefinite programming optimization. It utilizes the dynamic motion of the mobile relay to overcome the physical aperture limitation of traditional fixed sensor networks and provides spatial geometric constraints through multi-view observation. The hybrid measurement model (TD provides absolute distance information and AOA provides direction vector) decouples the target and relay status in a coordinated manner. The SDP optimization algorithm combined with the joint refinement process makes the positioning accuracy approach the Cramer-Rao lower bound. (2) Passive positioning is achieved by directly utilizing the radiation signal of the target itself, completely eliminating the active signal transmission link between the sensor and the mobile relay, and avoiding network location exposure caused by electromagnetic feature leakage; (3) By linearization preprocessing (separation of distance transformation model and AOA noise), the nonlinear localization problem is transformed into a convex optimization problem. The SDP relaxation technique is used to process the constrained weighted least squares problem. While discarding the rank 1 constraint, equality constraints are added to tighten the solution space, which greatly reduces the computational complexity. Attached Figure Description

[0016] Figure 1 A flowchart illustrating the steps of a mobile relay-assisted target localization method based on semidefinite programming optimization; Figure 2 The present invention provides the method and the Cramer-Rao lower bound (CRLB) in... A comparison of the mean square error (MSE) of the target's coordinate position estimation as a function of TD measurement noise power; Figure 3 The present invention provides the method and the Cramer-Rao lower bound (CRLB) in... A comparison of the mean square error (MSE) of mobile speed estimation for mobile relays as a function of TD measurement noise power; Figure 4 The present invention provides the method and the Cramer-Rao lower bound (CRLB) in... A comparison of the mean square error (MSE) of the coordinate position estimation of the mobile relay as a function of TD measurement noise power; Figure 5 The present invention provides the method and the Cramer-Rao lower bound (CRLB) in... A comparison of the mean square error (MSE) of the target's coordinate position estimation as a function of AOA measurement noise power; Figure 6 The present invention provides the method and the Cramer-Rao lower bound (CRLB) in... A comparison of the mean square error (MSE) of mobile speed estimation for mobile relays as a function of AOA measurement noise power; Figure 7 The present invention provides the method and the Cramer-Rao lower bound (CRLB) in... A comparison of the mean square error (MSE) of coordinate position estimation for mobile relays as a function of AOA measurement noise power; Figure 8 The present invention provides the method and the Cramer-Rao lower bound (CRLB) in... A comparison graph showing the mean square error (MSE) of target coordinate position estimation as a function of the number of sensors; Figure 9 The present invention provides the method and the Cramer-Rao lower bound (CRLB) in... A comparison of the mean square error (MSE) of mobile relay speed estimation as a function of the number of sensors; Figure 10 The present invention provides the method and the Cramer-Rao lower bound (CRLB) in... A comparison of the mean square error (MSE) of mobile relay coordinate position estimation as a function of the number of sensors; Figure 11 The present invention provides the method and the Cramer-Rao lower bound (CRLB) in... A comparison chart showing how the mean square error (MSE) of the target's coordinate position estimation varies with the number of mobile relays; Figure 12 The present invention provides the method and the Cramer-Rao lower bound (CRLB) in... A comparison chart showing the mean square error (MSE) of mobile speed estimation for mobile relays as a function of the number of mobile relays; Figure 13 The present invention provides the method and the Cramer-Rao lower bound (CRLB) in... A comparison chart showing how the mean square error (MSE) of mobile relay coordinate location estimation varies with the number of mobile relays. Detailed Implementation

[0017] Example 1 In existing mobile target localization technologies, while the deployment of static relay nodes can solve some non-line-of-sight propagation problems, their inherent limitations significantly restrict localization performance in complex environments. On one hand, static relays cannot actively adapt to the dynamic movement trajectory of the target. When the target is in a rapid maneuvering state or in a relay coverage blind spot, the signal transmission link is prone to interruption, leading to missing localization data or distorted time delay measurements. On the other hand, traditional algorithms lack the ability to jointly process heterogeneous measurements such as time delay and angle of arrival: the nonlinear terms in the time delay model (such as the square operation of geometric distance) are coupled with the inverse trigonometric function of the angle of arrival model, introducing strong non-convexity. Existing linear approximation methods struggle to effectively balance noise sensitivity (such as the cumulative effect of hardware offset and angle measurement noise) with model fidelity, often resulting in localization results deviating from the global optimum. Furthermore, the mechanism relying on actively transmitting calibration signals not only increases system power consumption and complexity but also easily exposes the localization intent, making it difficult to meet the tactical requirements of concealed target detection. To address these issues, such as... Figure 1 As shown, this invention proposes a mobile relay-assisted target localization method based on semidefinite programming optimization, comprising the following steps:

[0018] S1: Construct a three-dimensional coordinate system, deploy a corresponding number of sensors with known positions, mobile relays with unknown positions and speeds, and a target with unknown positions according to the task requirements, and set the mobile relays to maintain uniform linear motion within the positioning cycle. S2: During the positioning period, the signal emitted by the sensor is forwarded to the target via a mobile relay. The target reflects the signal and transmits it back to the corresponding sensor via the same signal link to extract the signal propagation delay and angle of arrival measurement values. S3: Construct time delay measurement model and angle of arrival measurement model based on time delay measurement value and angle of arrival measurement value, respectively; S4: By introducing auxiliary constants to construct independent variables of target position, the time delay measurement model is reconstructed into a distance transformation model, and the linearized expression of the angle of arrival is obtained by separating the noise term from the angle of arrival measurement model; S5: Construct a constrained weighted least squares problem that includes the location, velocity, and distance offset of the mobile relay by using a joint distance transformation model and a linearized expression; S6: The initial transformation of the constrained weighted least squares problem by auxiliary variables, the removal of rank 1 constraints and the addition of equality constraints to tighten the solution space, and the transformation into a semidefinite programming problem are used to solve the problem and obtain the initial estimates of the target position, the mobile relay position and the velocity. S7: Based on the initial estimates, perform a first-order Taylor expansion on the time delay measurement model and the angle of arrival measurement model, construct a joint refined weighted least squares problem, and solve it to obtain the final estimates of the target position, mobile relay position and velocity.

[0019] Specifically, in one embodiment, a mobile relay assist system contains A known and unknown sensor, Taking a mobile relay with unknown location and speed, and a target with unknown location as examples, a corresponding three-dimensional coordinate system is established as the reference coordinate system. In this embodiment (matrix and vector parameters are represented in bold, and scalar parameters are represented in thin font), j The true coordinates of a sensor at a known location in the reference coordinate system are denoted as and the true coordinates of a target at an unknown location in the reference coordinate system are denoted as . And ensure that the mobile relay maintains uniform linear motion during the current positioning task, and record the true coordinates of the mobile relay at the i-th unknown location at the reference time (time 0) in the reference coordinate system as Movement speed is denoted as Let the sampling interval be... Then, the position of the i-th mobile relay at the k-th sampling time can be represented as ,in (K is the total number of samples in the positioning cycle).

[0020] At the k-th sampling time, the signal emitted by the sensor at the j-th known location is received, amplified, and forwarded to the target by the mobile relay at the i-th unknown location. The target at the unknown location reflects the signal, and the echo signal is forwarded again from the mobile relay at the i-th unknown location to the sensor at the j-th known location. The sensor extracts the TD (time delay) and AOA (angle of arrival) measurements from the received signal, denoted as […]. and .in, Let be the time delay measurement value of the signal at the k-th sampling time being forwarded from the j-th sensor to the target via the i-th mobile relay. The azimuth angle measurement value contaminated by measurement noise along the propagation path of the signal from the i-th mobile relay to the j-th sensor at the k-th sampling time. The pitch angle measurement value contaminated by measurement noise along the propagation path of the signal from the i-th mobile relay to the j-th sensor at the k-th sampling time.

[0021] Based on the data collected in practical applications, a TD measurement model is constructed for the signal at the k-th sampling time, which is relayed from the j-th sensor to the target via the i-th mobile relay. This model is described as follows: Simultaneously, an AOA measurement model is constructed for the signal arriving at the j-th sensor from the i-th mobile relay at the k-th sampling time, described as follows:

[0022] , In the above formula, Let c be the measured distance of the signal at the k-th sampling time, transmitted from the j-th sensor to the target via the i-th mobile relay, contaminated by measurement noise. Let i be the location of the i-th mobile relay at the k-th sampling time. The distance offset caused by the i-th mobile relay forwarding signal. The distance measurement noise along the propagation path of the signal at the k-th sampling time, from the j-th sensor to the target via the i-th mobile relay, is... The symbol for the second norm; Let be the azimuth angle measurement value at the k-th sampling time, which is unaffected by measurement noise along the propagation path from the i-th mobile relay to the j-th sensor. The azimuth measurement noise is defined as the propagation path of the signal at the k-th sampling time from the i-th mobile relay to the j-th sensor. They are respectively scalar components on the X, Y, and Z axes of a three-dimensional coordinate system They are respectively scalar components on the X, Y, and Z axes of a three-dimensional coordinate system for Vector components in the X and Y coordinate system for Vector components in the X and Y coordinate system The pitch angle measurement value at the k-th sampling time, unaffected by measurement noise, represents the path of the signal from the i-th mobile relay to the j-th sensor. The pitch angle measurement noise is the propagation path of the signal at the k-th sampling time from the i-th mobile relay to the j-th sensor.

[0023] Considering that the original TD measurement model includes the target location and mobile relay location The Euclidean norm nonlinear coupling of the target position requires iterative optimization for direct solution, which is computationally complex and prone to getting trapped in local optima. Therefore, the original TD measurement model needs to be preprocessed. In this embodiment, algebraic transformation is used to determine the target position. Separating the terms into independent linear terms allows the target location to be explicitly expressed. Specific operations include: (1) Moving to the left side of the equation, squaring both sides and ignoring the second-order noise term yields an approximate model for distance measurement. ; Among them, norm terms It can be represented as , The true unit direction vector is defined as follows: ; Next according to Performing a first-order Taylor expansion on the norm terms above, we obtain: , in For composite error terms, , Substituting the above equation into the distance approximation model, we get: ; Because this formula lacks the unknown target location The independent terms are not conducive to solving subsequent problems, so this invention introduces a constant translation vector here. As an auxiliary constant, the mobile relay location is represented as ,in This leads to the final transformation model for distance measurement: , in, This is the unit direction vector calculated from the angle of arrival measurement. This separates the independent linear term of the target position. Furthermore, by eliminating the geometric nonlinearity of the distance equation, a mathematical form compatible with convex optimization is constructed.

[0024] (2) Then, the AOA measurement model constructed in step S3, which shows the signal at the k-th sampling time from the i-th mobile relay to the j-th sensor, needs to be preprocessed. Specifically, the following operations are included: The azimuth measurement model can be equivalently rewritten as: Using a first-order Taylor expansion to separate the noise term, we obtain: ; The model for measuring pitch angle can be equivalently rewritten as: , Using a first-order Taylor expansion to separate the noise term, we obtain: , Similarly, the non-convexity of the original measurement model was removed, making it suitable for subsequent convex optimization algorithms. At the same time, the model was linearized through a first-order Taylor expansion, which facilitated the solution.

[0025] Then, by combining the distance transformation model, azimuth measurement model, and pitch measurement model obtained after the above preprocessing, a constrained weighted least squares problem is constructed, described as follows: , , , , , For example, in this function express The Middle To the A subvector consisting of elements express The Middle Each element.

[0026] This function is a minimum value function. To constrain the objective function of the weighted least squares problem, using the values ​​of... As an optimization variable. Furthermore, after the joint distance transformation model and linearization expression, this optimization variable is expressed as:

[0027] , in, For the target location variable, To obtain A scalar of one element. This is the coefficient vector introduced in the current least squares problem. , The propagation distance coefficient, This is the azimuth coefficient. Let be the pitch angle coefficient, where: The first in The elements are , ; The first in The elements are , ; The first in The elements are , .

[0028] The coefficient matrix introduced in the current least squares problem, , For the propagation distance coefficient vector, This is the azimuth coefficient vector. Let be the pitch angle coefficient vector, where: The first in Vector composed of row elements , A d kMN+ j−1 M+i,: =[2 κ i T , 0 3 i−1 ,−2 d ij k γ ij k T −2 s j T , 0 3 M−1 -2kT d ij k γ ij k T -2kT s j T , 0 3 M−i +i−1 ,−2 d ij k −2 γ ij k T s j +2 γ ij k T κ i , 0 M−i+3 i−1 ,2 γ ij k T , 0 3 M−1 2kT γ ij k T , 0 3 M−i +i−1 ,1, 0 M−1 2kT 0 M−i ] ,in, It is a row vector with dimension 1 * index; The first in Vector composed of row elements , ; The first in Vector composed of row elements , .

[0029] The weight matrix introduced in the current least squares problem, , The coefficient matrix, ,in: The first in Vector composed of row elements , B d kMN+ j−1 M+i ,; =[ 0 kMN+(j−1)M+i ,2 t i o k − u o , 0 K−1 MN+ N−1 M+M−1 ,−2 d ij k − δ i o ρ ij θ k T t i o k − s j , 0 K−1 MN+ N−1 M+M−1 ,−2 d ij k − δ i o ρ ij ϕ k T t i o k − s j , 0 K−k−1 MN+ N−j M+M−i ] ; The non-zero elements in are , ; The first in Vector composed of row elements , .

[0030] The noise covariance matrix is... ,in , These are the measurement noise vectors for distance, azimuth, and elevation, respectively. To obtain the sign of the desired symbol, It is the inverse of the matrix. Based on measurements provided by K sampling times, M mobile relays, and N sensors, the measurement provided by the j-th sensor and the ith mobile relay at the k-th time is represented as... .

[0031] Since the constrained weighted least squares problem is non-convex and has multiple local optima, it is difficult to solve directly. To address this issue, this invention introduces auxiliary variables. This problem can be equivalently transformed into: , , , , , , , Using the semidefinite relaxation technique, the non-convex rank 1 constraints in the above problem are discarded, resulting in a convex semidefinite programming (SDP) problem: , , , , , , in, Let be the trace of the matrix.

[0032] Furthermore, since semidefinite relaxation causes performance loss, the following equality constraint is added to tighten the problem based on the equation relationship between variables: , , , , in, This leads to the joint estimation of the SDP problem: , , , , , , , , , , The solution to this problem is denoted as The estimates of the target location, mobile relay location, and velocity are as follows: , , .

[0033] To ensure optimal performance of each parameter estimate, a joint refinement is performed based on the estimates obtained from the SDP solution. Through a first-order Taylor expansion, the time delay measurement model constructed in step S3 is approximated as follows: ,in .

[0034] By combining the preprocessed azimuth and elevation measurement models, a joint refined weighted least squares problem is constructed. Its optimal solution is The estimates of the target location, mobile relay location, and velocity are as follows: , , ,in, To jointly refine the optimization variables in the weighted least squares problem, including the target position, the position of the mobile relay, and the velocity, Let be the objective function of the constrained weighted least squares problem, where: The coefficient vector introduced for the current least squares problem. , The first in The elements are , .

[0035] The coefficient matrix introduced for the current least squares problem. , The first in The vector formed by the row elements is , .

[0036] The weight matrix introduced for the current least squares problem. , The coefficient matrix introduced , For dimension is The identity matrix.

[0037] Example 2 To verify the feasibility and effectiveness of the method of the present invention, this embodiment verifies the method of the present invention through a simulation experiment.

[0038] Assume there are 5 (1≤m≤5) sensors with known positions and 4 (1≤i≤4) mobile relays with unknown positions in a 3D space (3D coordinate system), as well as 1 target with an unknown position. Determine the true Cartesian coordinates of the sensors with known positions in the 3D space. s j =[ s j,x , s j,y , s j,z ] T Randomly generated, where rice, rice, Meters; the true Cartesian coordinates of a mobile relay at an unknown location in three-dimensional space. t i o =[ t i,x o , t i,y o , t i,z o ] T Randomly generated, where rice, rice, Meters, the actual Cartesian coordinate velocity of a mobile relay in three-dimensional space. v i o =[ v i,x o , v i,y o , v i,z o ] T Randomly generated, where meters per second meters per second meters per second; the target's actual Cartesian coordinate position in three-dimensional space is assumed to be... u 0 =[ u x o , u y o , u z o ] T The actual Cartesian coordinates are also randomly generated, where, rice, rice, meters; unknown distance offset set to Randomly generated, where Meters. The azimuth measurement noise from the mobile relay to the sensor is assumed to have a mean of 0 and a covariance matrix of... The Gaussian noise, the elevation angle measurement noise on the reflection path from the target at an unknown location to the receiver, is assumed to have a mean of 0 and a covariance matrix of... Gaussian noise, where, This represents the variance of the AOA measurement noise. Indicates dimension as The identity matrix; the TD measurement noise of the target relayed to the sensor is assumed to have a mean of 0 and a covariance matrix of... Gaussian noise, where, This represents the variance of the noise measured by TD.

[0039] Based on the above parameter settings, simulation experiments were conducted to test the performance of the method of this invention under four conditions: different TD measurement noise power, different AOA measurement noise power, different numbers of mobile relays, and different numbers of sensors. The mean square error (MSE) describing the positioning performance under the four conditions was obtained from 10 random scenarios, with each scenario running 1000 Monte Carlo experiments. Simultaneously, the performance benchmark Cramer-Rhodes lower bound (CRLB) was introduced for comparison.

[0040] Figure 2 The present invention provides the method and the Cramer-Rao lower bound (CRLB) in... The mean square error (MSE) of the target's coordinate position estimation in the case of... ) with the change in noise power measured by TD ( A comparison chart of ) Figure 3 The present invention provides the method and the Cramer-Rao lower bound (CRLB) in... The mean square error (MSE) of mobile relay speed estimation in the case of mobile relay A comparison graph showing the change in noise power as measured by TD. Figure 4 The present invention provides the method and the Cramer-Rao lower bound (CRLB) in... In the case of a mobile relay, the mean square error (MSE) of the coordinate position estimation is... A comparison graph showing the change in noise power as measured by TD.

[0041] Figure 5 The present invention provides the method and the Cramer-Rao lower bound (CRLB) in... The mean square error (MSE) of the target's coordinate position estimation in the case of... ) with the change in noise power measured by AOA () A comparison chart of ) Figure 6 The present invention provides the method and the Cramer-Rao lower bound (CRLB) in... The mean square error (MSE) of mobile relay speed estimation in the case of mobile relay A comparison graph showing the variation of noise power with AOA measurement. Figure 7 The present invention provides the method and the Cramer-Rao lower bound (CRLB) in... In the case of a mobile relay, the mean square error (MSE) of the coordinate position estimation is... A comparison graph showing the change in noise power as measured by AOA.

[0042] Figure 8 The present invention provides the method and the Cramer-Rao lower bound (CRLB) in... The mean square error (MSE) of the target's coordinate position estimation in the given case. A comparison chart showing how the number of sensors varies. Figure 9 The present invention provides the method and the Cramer-Rao lower bound (CRLB) in... The mean square error (MSE) of mobile relay speed estimation in the case of mobile relay A comparison chart showing how the number of sensors varies. Figure 10 The present invention provides the method and the Cramer-Rao lower bound (CRLB) in... The mean square error (MSE) of mobile relay coordinate position estimation in the case of A comparison chart showing how the number of sensors varies.

[0043] Figure 11 The present invention provides the method and the Cramer-Rao lower bound (CRLB) in... The mean square error (MSE) of the target's coordinate position estimation in the given case. A comparison chart showing how the number of mobile relays changes. Figure 12 The present invention provides the method and the Cramer-Rao lower bound (CRLB) in... The mean square error (MSE) of mobile relay speed estimation in the case of mobile relay A comparison chart showing how the number of mobile relays changes. Figure 13 The present invention provides the method and the Cramer-Rao lower bound (CRLB) in... The mean square error (MSE) of mobile relay coordinate position estimation in the case of A comparison chart showing how the number of mobile relays changes.

[0044] from Figure 2 , Figure 3 , Figure 4 As can be seen from this, when the TD measures the noise power When variations occur, the method of this invention can achieve positioning accuracy at the lower bound of the Cramer-Rao boundary, provided that the noise power of the DTD measurement is relatively low. From Figure 5 , Figure 6 , Figure 7 As can be seen from this, when the AOA measurement noise power... When variations occur, the method of this invention can achieve positioning accuracy at the lower bound of the Cramer-Rao boundary, provided that the AOA measurement noise power is relatively low. From Figure 8 , Figure 9 , Figure 10 As can be seen, when the number of sensors changes, the method of the present invention can achieve the lower bound of the Cramer-Rhodes positioning accuracy when the number of targets exceeds two. Furthermore, as the number of sensors increases, the MSE of the target position estimation by the method of the present invention decreases significantly, which means that when the number of targets increases, the positioning accuracy of the method of the present invention is significantly improved. Figure 11 , Figure 12 , Figure 13 As can be seen, when the number of mobile relays changes, the method of the present invention can achieve the lower bound of the Cramer-Rao positioning accuracy when the number of mobile relays exceeds 3. In addition, as the number of mobile relays increases, the MSE of the target location estimation by the method of the present invention decreases significantly, which means that the positioning accuracy of the target is improved when the number of mobile relays increases.

[0045] 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.

[0046] In summary, this invention proposes a mobile relay-assisted target localization method based on semidefinite programming optimization. This method utilizes the dynamic motion of the mobile relay to overcome the physical aperture limitations of traditional fixed sensor networks and provides spatial geometric constraints through multi-view observation. A hybrid measurement model (TD provides absolute distance information, and AOA provides direction vectors) collaboratively decouples the target and relay states. The SDP optimization algorithm, combined with a joint refinement process, improves the localization accuracy.

[0047] By directly utilizing the target's own radiated signals to achieve passive positioning, the active signal transmission links of sensors and mobile relays are completely eliminated, avoiding network location exposure caused by electromagnetic feature leakage.

[0048] By linearizing the preprocessing (separating the distance transformation model from AOA noise), the nonlinear localization problem is transformed into a convex optimization problem. The SDP relaxation technique is used to handle the constrained weighted least squares problem. While discarding the rank-1 constraint, equality constraints are added to tighten the solution space, which greatly reduces the computational complexity.

[0049] It should be noted that all directional indications (such as up, down, left, right, front, back, etc.) in the embodiments of the present invention are only used to explain the relative positional relationship and movement of each component in a certain specific posture (as shown in the figure). If the specific posture changes, the directional indication will also change accordingly.

[0050] Furthermore, in this invention, descriptions involving terms such as "first," "second," and "a" are for descriptive purposes only and should not be construed as indicating or implying their relative importance or implicitly specifying the number of technical features indicated. Thus, a feature defined as "first" or "second" may explicitly or implicitly include at least one of that feature. In the description of this invention, "a plurality of" means at least two, such as two, three, etc., unless otherwise explicitly specified.

[0051] In this invention, unless otherwise explicitly specified and limited, the terms "connection," "fixed," etc., should be interpreted broadly. For example, "fixed" can mean a fixed connection, a detachable connection, or an integral part; it can mean a mechanical connection or an electrical connection; it can mean a direct connection or an indirect connection through an intermediate medium; it can mean the internal communication of two components or the interaction between two components, unless otherwise explicitly limited. Those skilled in the art can understand the specific meaning of the above terms in this invention according to the specific circumstances.

[0052] Furthermore, the technical solutions of the various embodiments of the present invention can be combined with each other, but only if they are feasible for those skilled in the art. If the combination of technical solutions is contradictory or cannot be implemented, it should be considered that such combination of technical solutions does not exist and is not within the scope of protection claimed by the present invention.

Claims

1. A mobile relay-assisted target localization method based on semidefinite programming optimization, characterized in that, Including the following steps: S1: Construct a three-dimensional coordinate system, deploy a corresponding number of sensors with known positions, mobile relays with unknown positions and speeds, and a target with unknown positions according to the task requirements, and set the mobile relays to maintain uniform linear motion within the positioning cycle. S2: During the positioning period, the signal emitted by the sensor is forwarded to the target via a mobile relay. The target reflects the signal and transmits it back to the corresponding sensor via the same signal link to extract the signal propagation delay and angle of arrival measurement values. S3: Construct time delay measurement model and angle of arrival measurement model based on time delay measurement value and angle of arrival measurement value, respectively; S4: By introducing auxiliary constants to construct independent variables of target position, the time delay measurement model is reconstructed into a distance transformation model, and the linearized expression of the angle of arrival is obtained by separating the noise term from the angle of arrival measurement model; S5: Construct a constrained weighted least squares problem that includes the location, velocity, and distance offset of the mobile relay by using a joint distance transformation model and a linearized expression; S6: The initial transformation of the constrained weighted least squares problem by auxiliary variables, the removal of rank 1 constraints and the addition of equality constraints to tighten the solution space, and the transformation into a semidefinite programming problem are used to solve the problem and obtain the initial estimates of the target position, the mobile relay position and the velocity. S7: Based on the initial estimates, perform a first-order Taylor expansion on the time delay measurement model and the angle of arrival measurement model, construct a joint refined weighted least squares problem, and solve it to obtain the final estimates of the target position, mobile relay position and velocity.

2. The mobile relay-assisted target localization method based on semidefinite programming optimization as described in claim 1, characterized in that, The mobile relay is mounted on a drone or autonomous mobile robot platform. The target is a concealed target that passively radiates electromagnetic signals. The time delay and angle of arrival measurements are passively acquired based on the target's own radiation signals. Neither the sensor nor the mobile relay actively transmits calibration signals during the positioning process.

3. The mobile relay-assisted target localization method based on semidefinite programming optimization as described in claim 1, characterized in that, In step S3, the delay measurement model quantifies the total propagation delay by superimposing the geometric distance of the bidirectional signal link with the mobile relay hardware offset to obtain the target relative distance information. The formula is expressed as follows: In the formula, Where i is the mobile trunk number and M is the number of mobile trunks. j is the sensor number, and N is the number of sensors. k is the sampling time number, and K is the total number of samples in the positioning cycle. Let c be the measured distance of the signal at the k-th sampling time, transmitted from the j-th sensor to the target via the i-th mobile relay, contaminated by measurement noise. Let be the time delay measurement value of the signal at the k-th sampling time being forwarded from the j-th sensor to the target via the i-th mobile relay. Let i be the location of the i-th mobile relay at the k-th sampling time. The true coordinates of the target in the three-dimensional coordinate system. Let j be the true coordinate position of the j-th sensor in the three-dimensional coordinate system. The distance offset caused by the i-th mobile relay forwarding signal. The distance measurement noise along the propagation path of the signal at the k-th sampling time, from the j-th sensor to the target via the i-th mobile relay, is... This is the symbol for the 2-norm.

4. The mobile relay-assisted target localization method based on semidefinite programming optimization as described in claim 3, characterized in that, In step S3, the angle of arrival measurement model is based on spherical wavefront characteristics. It uses the inverse trigonometric function relationship between azimuth and elevation angles to project the signal propagation direction from the mobile relay to the sensor onto a three-dimensional coordinate system to provide spatial direction constraints. The formula is expressed as follows: In the formula, The azimuth angle measurement value contaminated by measurement noise along the propagation path of the signal from the i-th mobile relay to the j-th sensor at the k-th sampling time. Let be the azimuth angle measurement value at the k-th sampling time, which is unaffected by measurement noise along the propagation path from the i-th mobile relay to the j-th sensor. The azimuth measurement noise is the propagation path of the signal at the k-th sampling time from the i-th mobile relay to the j-th sensor. They are respectively Scalar components on the X, Y, and Z axes of a three-dimensional coordinate system They are respectively Scalar components on the X, Y, and Z axes of a three-dimensional coordinate system for Vector components in the X and Y coordinate system for Vector components in the X and Y coordinate system The pitch angle measurement value contaminated by measurement noise along the propagation path of the signal from the i-th mobile relay to the j-th sensor at the k-th sampling time is given. The pitch angle measurement value at the k-th sampling time, unaffected by measurement noise, represents the path of the signal from the i-th mobile relay to the j-th sensor. The pitch angle measurement noise is the propagation path of the signal at the k-th sampling time from the i-th mobile relay to the j-th sensor.

5. The mobile relay-assisted target localization method based on semidefinite programming optimization as described in claim 3, characterized in that, In step S4, the specific process of reconstructing the distance transformation model includes: Will Move to the left side of the equation, square both sides of the equation and ignore the second-order noise term to obtain the distance approximation model; Based on the distance approximation model, a constant translation vector is introduced. The location of the mobile relay is represented as an auxiliary variable. This leads to the distance transformation model, expressed by the following formula: In the formula, , Let be the actual coordinate position of the i-th mobile relay in the three-dimensional coordinate system at the initial moment. The sampling interval duration. Let be the initial moving speed of the i-th mobile relay; Let T be the unit direction vector calculated from the angle of arrival measurement, and let T be the transpose operation. This is a composite error term.

6. The mobile relay-assisted target localization method based on semidefinite programming optimization as described in claim 5, characterized in that, In step S5, the constrained weighted least squares problem is described as follows: In the formula, To constrain the optimization variables in the weighted least squares problem, including the target position, the position of the mobile relay, and the velocity, For the target location variable, The coefficient matrix introduced in the constrained weighted least squares problem. The coefficient vector introduced in the constrained weighted least squares problem. The weight matrix is ​​introduced to constrain the weighted least squares problem.

7. The mobile relay-assisted target localization method based on semidefinite programming optimization as described in claim 6, characterized in that, In step S6, the auxiliary variable is , The specific transformation operation of the semidefinite programming problem is as follows: Based on auxiliary variables The constrained weighted least squares problem is equivalently transformed into: ; Using the semidefinite relaxation technique, the non-convex rank 1 constraints in the transformed problem are discarded, resulting in the semidefinite programming problem: In the formula, The trace of the matrix; Based on the semidefinite programming problem, the solution space is tightened by adding equality constraints, transforming it into a semidefinite programming problem: .

8. The mobile relay-assisted target localization method based on semidefinite programming optimization as described in claim 6, characterized in that, In step S7, the formula for the joint refined weighted least squares problem is expressed as follows: In the formula, To jointly refine the optimization variables in the weighted least squares problem, including the target position, the position of the mobile relay, and the velocity, To constrain the objective function of the weighted least squares problem, The coefficient vector introduced in the joint refinement of the weighted least squares problem. The coefficient matrix introduced in the joint refinement of the weighted least squares problem is... The weight matrix is ​​introduced to refine the weighted least squares problem.

9. The mobile relay-assisted target localization method based on semidefinite programming optimization as described in claim 1, characterized in that, In step S4, the linearized expression for the angle of arrival is obtained by performing a first-order Taylor expansion on the angle of arrival measurement model to separate the noise term.

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