UAV positioning method and system based on semidefinite programming and reconstruction linearization technology

Abstract

The present invention provides a method and system for positioning a UAV based on semidefinite programming and reconstruction linearization technology. The method comprises: step 1: establishing a three-dimensional relative positioning model of the UAV according to the distance between the target UAV and the beacon UAV and the distance between any two target UAVs; step 2: establishing a semidefinite programming constraint for the three-dimensional relative positioning model of the UAV, and adjusting it to a standard semidefinite programming problem; step 3: tightening the feasible domain of the solution of the standard semidefinite programming problem by using reconstruction linearization technology and solving it, that is, obtaining the preliminary position of the target UAV; step 4: modeling the distance measurement process of the UAV with interference factors, taking the preliminary position of the target UAV as the initialization position of the wolf pack, and using the improved gray wolf optimization algorithm to solve and obtain the final position of each target UAV.

Description

Technical Field

[0001] The present invention relates to the technical field of flying ad-hoc networks, and in particular, to a method and system for positioning unmanned aerial vehicles (UAVs) based on semidefinite programming and reconfiguration linearization technology. Background Art

[0002] Flying ad-hoc networks have broad application prospects in military and civilian fields, such as search and rescue, forest fire detection, emergency communication network deployment, and formation flight. Compared with ordinary sensor networks, UAVs have multiple advantages: they have higher altitude and distance coverage capabilities, can be easily moved and flexibly deployed, and can provide fast on-demand services in dangerous and harsh environments. However, due to the highly mobile and dynamic three-dimensional environment, the topology of UAV networks often changes, and the frequent disconnection of communication links caused by topology changes will affect the communication quality of UAV networks. In most applications, the data captured by UAV nodes is only useful when it is associated with the geographical locations of these nodes. Therefore, it is crucial to know the accurate and low-latency position information of each UAV.

[0003] In the most common positioning methods, the Global Positioning System (GPS) or manual configuration can be used to locate UAV nodes, but they usually require high costs and cannot be deployed in all scenarios. In addition, the positioning performance may be affected by path attenuation and building blockage in harsh environments (such as receivers in dense cities, canyons, or enemy airspaces or GPS signal interference). Summary of the Invention

[0004] Aiming at the problems of high cost in equipping UAVs with GPS to achieve positioning between UAVs in the existing methods and the decrease in the accuracy of distance measurement values when there is noise interference, the present invention provides a method and system for positioning UAVs based on semidefinite programming and reconfiguration linearization technology.

[0005] On the one hand, the present invention provides a method for positioning UAVs based on semidefinite programming and reconfiguration linearization technology, including:

[0006] Step 1: Establish a three-dimensional relative positioning model of UAVs according to the distances between the target UAV and beacon UAVs and the distances between any two target UAVs;

[0007] Step 2: Establish semidefinite programming constraints for the three-dimensional relative positioning model of UAVs and adjust it to a standard semidefinite programming problem;

[0008] Step 3: Use the reconfiguration linearization technology to tighten the feasible region of the solution of the standard semidefinite programming problem and solve it, that is, obtain the preliminary position of the target UAV;

[0009] Step 4: Model the distance measurement process of the UAVs with interference factors. Use the preliminary positions of the target UAVs as the initial positions of the wolf pack, and solve for the final positions of each target UAV using the improved grey wolf optimization algorithm.

[0010] Further, step 1 specifically includes:

[0011] Step 1.1: Set up a three-dimensional network space with m + n UAVs, specifically: m beacon UAVs with positioning functions and n target UAVs without self-positioning functions; among them, any two UAVs estimate the distance between each other through communication.

[0012] Denote the node pair composed of any two target UAVs i and j within the communication range R as node pair (i, j), and denote the node pair composed of beacon UAV k and target UAV j within the communication range R as (k, j).

[0013] Step 1.2: Define the distance between node pair (i, j) as the Euclidean distance d ji , and define the distance between node pair (k, j) as the Euclidean distance Then the three-dimensional relative positioning model of the UAVs is expressed as formula (1):

[0014]

[0015] where, x 1 , x 2 ,..., x n respectively represent the positions of the n target UAVs to be solved, and a k represents the known position of beacon UAV k.

[0016] Further, step 2 specifically includes:

[0017] Step 2.1: Establish the position matrix X of the target UAVs as X = [x 1 , x 2 ,..., x n , and re-express formula (1) as formula (2):

[0018]

[0019] where, e ij represents a vector with 1 at the i-th position, -1 at the j-th position, and 0 at other positions, e j represents a vector of all 0s except -1 at the j-th position, I 3denotes the 3D identity matrix; T denotes the transpose of a matrix;

[0020] Step 2.2: Introduce semidefinite programming constraints and relax Y = X T X to The constraint condition is equivalent to a linear matrix inequality

[0021] Step 2.3: The standard semidefinite programming problem of formula (1) is to find a symmetric matrix such that formula (3) holds:

[0022]

[0023] where Z 1:3,1:3 denotes the three-dimensional principal submatrix of matrix Z.

[0024] Furthermore, step 3 specifically includes:

[0025] Step 3.1: Determine the upper bound u and lower bound l of each element in the position matrix of the target UAV;

[0026] Step 3.2: Set two variables φ i , φ j ∈ X, and establish four constraints φ i - l ≥ 0, u - φ i ≥ 0, φ j - l ≥ 0, u - φ j ≥ 0. Multiply all the constraints containing φ i by the constraints containing φ j and compound the new constraints obtained by multiplication into the semidefinite programming to obtain formula (4):

[0027]

[0028] where Y ij = φ i φ j , i, j = 1, 2,..., n.

[0029] Furthermore, step 4 specifically includes:

[0030] Step 4.1: Model the interference factor as typical Gaussian noise, then the UAV distance measurement process with interference factors is modeled as formula (5):

[0031]

[0032] where ε ij and γ kj respectively represent the interference factors in the distance measurement processes between target UAVs and between target UAVs and beacon UAVs, N(0, σ2 ) represents a normal random variable with a mean of 0 and a variance of σ 2 ; r ij and respectively represent the true distances between target UAVs and between a target UAV and a beacon UAV;

[0033] Step 4.2: Define the objective function of the improved grey wolf optimization algorithm as formula (6):

[0034]

[0035] wherein, is the optimized position estimate of target UAV node i, is the three-dimensional coordinates of beacon UAV k,

[0036] Step 4.3: Take the preliminary position of the target UAV as the initial position of the wolf pack, iterate the improved grey wolf optimization algorithm, and the optimal solution obtained is the final position of each target UAV.

[0037] On the other hand, the present invention also provides a UAV positioning system based on semi-definite programming and reconstruction linearization technology, including:

[0038] A relative positioning model construction module, configured to establish a three-dimensional relative positioning model of UAVs according to the distances between target UAVs and beacon UAVs and the distances between any two target UAVs;

[0039] A semi-definite programming modeling module, configured to establish semi-definite programming constraints for the three-dimensional relative positioning model of UAVs and adjust it to a standard semi-definite programming problem;

[0040] A reconstruction linearization module, configured to tighten the feasible region of the solution of the standard semi-definite programming problem by using reconstruction linearization technology and solve it, that is, obtain the preliminary position of the target UAV;

[0041] An improved grey wolf optimization module, configured to model the UAV distance measurement process with interference factors, take the preliminary position of the target UAV as the initial position of the wolf pack, and use the improved grey wolf optimization algorithm to solve and obtain the final positions of each target UAV.

[0042] Advantages of the present invention:

[0043] The UAV positioning method and system based on semidefinite programming and reformulation linearization technology provided by the present invention first represent the target UAV node positioning problem as an optimization problem with quadratic constraints quadratic programming by using the information of a limited number of beacon UAV nodes and the pairwise distance measurements between UAV nodes; the solution of this problem is NP-hard and non-convex. Further, to solve this problem, semidefinite programming (SDP) is used to transform the above positioning problem into a convex optimization problem, and then SDP is combined with reformulation linearization technology (RLT) to tighten the feasible region of the result. When the noise level of the distance measurement process is relatively high, in order to improve the positioning accuracy, an improved grey wolf optimization algorithm is subsequently introduced to optimize the positioning result. In the present invention, it is not necessary to equip all UAVs with positioning modules, so the cost can be saved; at the same time, when comparing the method of the present invention with existing positioning schemes, the test results show that the present invention is superior to existing positioning schemes in terms of improving positioning accuracy. BRIEF DESCRIPTION OF THE DRAWINGS

[0044] Figure 1 is a flowchart of the UAV positioning method based on semidefinite programming and reformulation linearization technology provided by an embodiment of the present invention;

[0045] Figure 2 is a scene diagram of relative positioning of UAVs in a three-dimensional space provided by an embodiment of the present invention;

[0046] Figure 3 is a positioning schematic diagram of implementing the composite semidefinite programming and reformulation linearization technology algorithm provided by an embodiment of the present invention;

[0047] Figure 4 is a schematic diagram of the wolf pack position update rule in the improved grey wolf optimization algorithm provided by an embodiment of the present invention;

[0048] Figure 5 is a comparison chart of the positioning accuracy between the method of the present invention and other existing optimization algorithms provided by an embodiment of the present invention. DETAILED DESCRIPTION OF THE EMBODIMENTS

[0049] To make the objectives, technical solutions and advantages of the present invention clearer, the technical solutions in the embodiments of the present invention will be clearly described below with reference to the accompanying drawings in the embodiments of the present invention. Obviously, the described embodiments are some, but not all, of the embodiments of the present invention. All other embodiments obtained by those of ordinary skill in the art based on the embodiments of the present invention without creative efforts shall fall within the protection scope of the present invention.

[0050] Embodiment 1

[0051] As Figure 1 shown, an embodiment of the present invention provides a UAV positioning method based on semidefinite programming and reformulation linearization technology, including:

[0052] S101: Establish a three-dimensional relative positioning model of unmanned aerial vehicles (UAVs) based on the distance between the target UAV and the beacon UAV and the distance between any two target UAVs.

[0053] Specifically, this step specifically includes:

[0054] S1011: As shown in the scenario of three-dimensional relative positioning of UAVs, assume that there are m + n UAVs in a three-dimensional network space, specifically: m beacon UAVs with positioning functions and n target UAVs without self-positioning functions; among them, the distance between any two UAVs is estimated through communication. Figure 2 As shown in the three-dimensional network space, there are m + n UAVs, specifically: m beacon UAVs with positioning functions and n target UAVs without self-positioning functions; among them, the distance between any two UAVs is estimated through communication.

[0055] Denote the node pair composed of any two target UAVs i and j within the communication range R as node pair (i, j). Denote the node pair composed of the beacon UAV k and the target UAV j within the communication range R as (k, j).

[0056] S1012: Define the distance between node pairs (i, j) as the Euclidean distance d ji (or d ij , in this invention, d ji is used to represent), and define the distance between node pairs (k, j) as the Euclidean distance Then the three-dimensional relative positioning model of UAVs is expressed as formula (1):

[0057]

[0058] where x 1 , x 2 ,..., x n respectively represent the positions of the n target UAVs to be solved, and a k represents the known position of the beacon UAV k.

[0059] S102: Establish a semidefinite programming constraint for the three-dimensional relative positioning model of UAVs and adjust it to a standard semidefinite programming problem.

[0060] Specifically, this step specifically includes:

[0061] S1021: Establish the position matrix X = [x 1 , x 2 ,..., x n of the target UAVs, and re-express formula (1) as formula (2):

[0062]

[0063] Among them, e ij represents a vector where the \(i\)-th position is 1, the \(j\)-th position is -1, and other positions are 0. e j represents a vector of all 0s except that the \(j\)-th position is -1. I 3 represents a 3D identity matrix; \(T\) represents the transpose of a matrix.

[0064] S1022: Introduce a semidefinite programming constraint and relax \(Y = X\) T to The constraint condition is equivalent to a linear matrix inequality

[0065] S1023: The standard semidefinite programming problem of formula (1) is to find a symmetric matrix such that formula (3) holds:

[0066]

[0067] 1:3,1:3 represents the three-dimensional principal submatrix of matrix \(Z\).

[0068] S103: Use the reconstruction linearization technique to tighten the feasible region of the solution of the standard semidefinite programming problem and solve it, then the preliminary position of the target UAV can be obtained; on the basis of semidefinite programming, by compounding the use of the reconstruction linearization technique to tighten the feasible region of the solution of the positioning problem, the positioning efficiency and the accuracy of the solution can be further improved.

[0069] Specifically, this step specifically includes:

[0070] S1031: Determine the upper bound \(u\) and lower bound \(l\) of each element in the position matrix of the target UAV.

[0071] S1032: Set two variables \(\varphi\) i , \(\varphi\) j \(\in X\), and establish four constraints \(\varphi\) i - \(l\geq0\), \(u - \varphi\) i \(\geq0\), \(\varphi\) j - \(l\geq0\), \(u - \varphi\) j \(\geq0\). Multiply all the constraints containing \(\varphi\) i by the constraints containing \(\varphi\) j and compound the new constraints obtained by multiplication (i.e., ) into the semidefinite programming to obtain formula (4):

[0072]

[0073] Among them, \(Y\) ij = \(\varphi\) i \(\varphi\) j , \(i, j = 1, 2, \ldots, n\).

[0074] After performing semidefinite programming and reconstruction linearization techniques in a three-dimensional experimental scenario, the positioning effect as shown in Figure 3 can be achieved, and the average positioning error can be controlled within 1.082 m.

[0075] S104: Model the distance measurement process of the UAV with interference factors, take the preliminary position of the target UAV as the initial position of the wolf pack, and use the improved grey wolf optimization algorithm to solve for the final positions of each target UAV.

[0076] Specifically, this step specifically includes:

[0077] S1041: Model the interference factors as typical Gaussian noise, then the distance measurement process of the UAV with interference factors is modeled as formula (5):

[0078]

[0079] where ε ij and γ kj respectively represent the interference factors in the distance measurement processes between target UAVs and between target UAVs and beacon UAVs, N(0,σ 2 ) represents a normal random variable with a mean of 0 and a variance of σ 2 ; r ij and respectively represent the true distances between target UAVs and between target UAVs and beacon UAVs.

[0080] S1042: Define the objective function of the improved grey wolf optimization algorithm as formula (6):

[0081]

[0082] where, is the optimized position estimate of target UAV node i, is the three-dimensional coordinate of beacon UAV k,

[0083] S1043: Iterate the improved grey wolf optimization algorithm (I-GWO), and the optimal solution obtained is the final position of each target UAV.

[0084] For any one target UAV, the process of solving for this target UAV based on the improved grey wolf optimization algorithm specifically includes:

[0085] (1) Initialization

[0086] Step A1: Set the relevant convergence factor a, coefficient vectors A, C of the grey wolf pack, and random vectors r 1 ,r2 ; the maximum number of iterations t max , the dynamic random weight q 1 , q 2 , q 3 ;

[0087] Step A2: Initialize the position of the wolf pack U = [U 1 , U 2 , …, U w according to the preliminary position of the target UAV, where w represents the number of all grey wolves in the wolf pack U;

[0088] Specifically, the position of each wolf in the wolf pack represents a possible position of the target UAV to be located, also known as a candidate solution; regarding the initialization rule, for example, a regular polyhedron can be generated with the preliminary position of the target UAV to be located as the center and a set value R as the radius, and the position of a vertex of the regular polyhedron represents the position of a grey wolf.

[0089] (2) Loop

[0090] Step B1: Update the position of each wolf during the hunting process according to the fitness value (calculated by formula (6)), and the update rule is where where U(t) and U(t + 1) represent the positions of the wolf pack in the t-th and (t + 1)-th iteration processes respectively, U p (t) represents the position of the prey (i.e., the target UAV) in the t-th iteration process, and D represents the distance vector between the current wolf pack and the prey (i.e., the target UAV);

[0091] During the entire optimization process, the optimal position of the prey (i.e., the target UAV) is unknown. To further enhance the exploration ability of the improved grey wolf optimization algorithm, it is assumed that wolves α, β, and δ have a better understanding of the position of the prey. Therefore, the first three best solutions are saved, and the position of wolf ω and other wolves is updated according to the mean of the random weights of wolves α, β, and δ. The update rule is as Figure 4 shown, and the formula is defined as

[0092]

[0093] where the first three sub-formulas in formula (7) define the step size and direction for the individuals in the wolf pack to move towards wolves α, β, and δ, and the fourth formula defines the final position of wolf ω;

[0094] Step B2: Update the fitness value using formula (6) and iteratively update the convergence factor a;

[0095] (3) Output

[0096] When the maximum number of iterations is reached, the iterative loop ends, and the position U of the alpha wolf is output α , which is the optimal positioning result of the UAVs.

[0097] For each target UAV to be positioned, the above steps (1) to (3) are executed, and the final positions of all target UAVs can be obtained.

[0098] Figure 5 It is a schematic diagram for comparing the positioning results of different existing optimization algorithms and the method of the present invention under the same standard, which can prove that the method of the present invention has certain advantages in improving the three-dimensional relative positioning accuracy of UAVs. Among them, "I-GWO" represents the improved grey wolf optimization algorithm proposed by the present invention, "GWO" represents the traditional grey wolf optimization algorithm, "PSO" represents the particle swarm optimization algorithm, and "DE" represents the differential evolution algorithm.

[0099] Embodiment 2

[0100] Corresponding to the above method, an embodiment of the present invention further provides a UAV positioning system based on semi-definite programming and reconstruction linearization technology, including a relative positioning model construction module, a semi-definite programming modeling module, a reconstruction linearization module, and an improved grey wolf optimization module;

[0101] Among them, the relative positioning model construction module is used to establish a three-dimensional relative positioning model of UAVs according to the distances between target UAVs and beacon UAVs and the distances between any two target UAVs; the semi-definite programming modeling module is used to establish semi-definite programming constraints for the three-dimensional relative positioning model of UAVs and adjust it to a standard semi-definite programming problem; the reconstruction linearization module is used to tighten the feasible region of the solution of the standard semi-definite programming problem by using the reconstruction linearization technology and solve it, that is, to obtain the preliminary position of the target UAV; the improved grey wolf optimization module is used to model the UAV distance measurement process with interference factors, use the preliminary position of the target UAV as the initial position of the wolf pack, and solve to obtain the final positions of each target UAV by using the improved grey wolf optimization algorithm.

[0102] It should be noted that the UAV positioning system based on semi-definite programming and reconstruction linearization technology provided by the embodiment of the present invention is to implement the above method, and its functions can be specifically referred to the method embodiment above, which will not be elaborated here.

[0103] The UAV positioning method and system based on semidefinite programming and reconstruction linearization technology provided by the present invention first establish a relative positioning model by using the three-dimensional coordinates of finite beacon UAVs and the distance measurement information between the target UAV; secondly, establish a three-dimensional relative positioning algorithm for UAVs with constraints of composite semidefinite programming and reconstruction linearization technology according to the known information to obtain a preliminary solution of the target UAV position; then, aiming at the problem that the distance measurement between UAVs in the actual positioning scenario is contaminated by noise, the preliminary solution of the target UAV position obtained in the previous step is optimized by combining a bionic algorithm improved by grey wolf optimization, so as to realize the three-dimensional relative positioning of UAVs.

[0104] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention, rather than to limit them; although the present invention has been described in detail with reference to the foregoing embodiments, those of ordinary skill in the art should understand that they can still modify the technical solutions recorded in the foregoing embodiments, or perform equivalent replacements on some of the technical features; and these modifications or replacements do not make the essence of the corresponding technical solutions deviate from the spirit and scope of the technical solutions of the embodiments of the present invention.

Claims

1. The UAV positioning method based on semidefinite programming and reconstruction linearization technology is characterized by: include: Step 1: Establish a 3D relative positioning model of the UAV based on the distance between the target UAV and the beacon UAV and the distance between any two target UAVs; Step 2: Establishing semidefinite programming constraints for the UAV 3D relative positioning model and adjusting it to a standard semidefinite programming problem; Step 3: Using the reconstruction linearization technique, the feasible domain of the solution of the standard semidefinite programming problem is tightened and solved, that is, the preliminary position of the target UAV is obtained; Step 4: Model the distance measurement process of the UAV with interference factors, use the initial position of the target UAV as the initial position of the wolf pack, and use the improved gray wolf optimization algorithm to solve the final position of each target UAV; specifically including: Step 4.1: Model the interference factor as typical Gaussian noise. Then the UAV distance measurement process with interference factors is modeled as formula (5): Among them, ε ij and γ kj They represent the interference factors in the distance measurement process between target UAVs and between target UAV and beacon UAV, N(0,σ 2 ) means the mean is 0 and the variance is σ 2 A normal random variable; r ij and They represent the real distances between target UAVs and between target UAV and beacon UAV respectively; Step 4.2: Define the objective function of the improved gray wolf optimization algorithm as formula (6): in, is the optimized position estimate of the target UAV node i, is the three-dimensional coordinate of the beacon drone k, Step 4.3: Use the initial position of the target UAV as the initial position of the wolf pack, iteratively improve the gray wolf optimization algorithm, and the optimal solution obtained is the final position of each target UAV.

2. The UAV positioning method based on semidefinite programming and reconstruction linearization technology according to claim 1 is characterized in that: Step 1 specifically includes: Step 1.1: Set up the 3D network space There are m+n drones, specifically: m beacon drones with positioning function and n target drones without self-positioning function; any two drones estimate the distance between them through communication; The node pair consisting of any two target drones i and j within the communication range R is recorded as node pair (i, j). The node pair consisting of the beacon drone k and the target drone j located in the communication range R is recorded as (k, j). Step 1.2: Define the distance between node pairs (i, j) as the Euclidean distance d ji , define the distance between node pairs (k, j) as the Euclidean distance The three-dimensional relative positioning model of the UAV is expressed as formula (1): where x1,x2,...,x n They represent the positions of the n target drones to be found, a k represents the known location of beacon drone k.

3. The UAV positioning method based on semidefinite programming and reconstruction linearization technology according to claim 2 is characterized in that: Step 2 specifically includes: Step 2.1: Create the target drone’s position matrix X = [x1, x2, ..., x n ], and reformulate formula (1) as formula (2): Among them, e ij represents a vector with 1 at the i-th position, -1 at the j-th position, and 0 at other positions. e j represents a vector of all zeros except the jth position which is -1. I3 represents the 3D identity matrix; T represents the transpose of the matrix; Step 2.2: Introduce semidefinite programming constraints and set Y = X T X relaxes to Y ≥ X T X, the constraint condition is equivalent to the linear matrix inequality Step 2.3: The standard semidefinite programming problem of formula (1) is to find the symmetric matrix So that formula (3) holds true: Among them, Z 1:3,1:3 Represents the three-dimensional principal submatrix of the matrix Z.

4. The UAV positioning method based on semidefinite programming and reconstruction linearization technology according to claim 3 is characterized in that: Step 3 specifically includes: Step 3.1: Establish the upper bound u and lower bound l of each element in the position matrix of the target UAV; Step 3.2: Set two variables φ i ,φ j ∈X, and establish four constraints φ i -l≥0,u-φ i ≥0,φ j -l≥0,u-φ j ≥0, will include φ i Multiply all constraints including φ j The constraints obtained by multiplication are combined into the semidefinite programming to obtain formula (4): Among them, Y ij =φ i f j ,i,j=1,2,...,n.

5. The UAV positioning system based on semidefinite programming and reconstruction linearization technology is characterized by: include: A relative positioning model building module is used to build a three-dimensional relative positioning model of the UAV according to the distance between the target UAV and the beacon UAV and the distance between any two target UAVs; A semidefinite programming modeling module is used to establish semidefinite programming constraints for the three-dimensional relative positioning model of the UAV and adjust it to a standard semidefinite programming problem; A reconstruction linearization module is used to tighten the feasible domain of the solution of the standard semidefinite programming problem by using the reconstruction linearization technology and solve it, that is, to obtain the preliminary position of the target UAV; An improved gray wolf optimization module is used to model the distance measurement process of UAVs with interference factors, taking the initial position of the target UAV as the initial position of the wolf pack, and using the improved gray wolf optimization algorithm to solve the final position of each target UAV; Specifically, the interference factor is modeled as a typical Gaussian noise, and the UAV distance measurement process with interference factors is modeled as formula (5): Among them, ε ij and γ kj They represent the interference factors in the distance measurement process between target UAVs and between target UAV and beacon UAV, N(0,σ 2 ) means the mean is 0 and the variance is σ 2 A normal random variable; r ij and They represent the real distances between target UAVs and between target UAV and beacon UAV respectively; The objective function of the improved gray wolf optimization algorithm is defined as formula (6): in, is the optimized position estimate of the target UAV node i, is the three-dimensional coordinate of the beacon drone k, And the initial position of the target UAV is used as the initial position of the wolf pack, and the gray wolf optimization algorithm is iteratively improved, and the optimal solution obtained is the final position of each target UAV.

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