Multi-aircraft cooperative autonomous navigation method in satellite denial environment

Abstract

The application discloses a kind of satellite rejection environment under multi-aircraft cooperative autonomous navigation method, comprising: constructing side measurement network adjustment model, utilize inherent distance constraint in cluster topological network to reduce distance estimation error, real-time generation high-precision distance measurement network;Based on distance measurement adjustment theory, cooperative relative positioning is carried out, and distance network under the side measurement network adjustment model is fully utilized to realize cluster global optimal positioning, while improving the positioning accuracy and robustness of cluster system, the implementation range of cooperative positioning in local connection topology is expanded;Based on multidimensional scaling technology and coordinate registration technology, construct two-stage cooperative positioning framework based on TOSA, realize multi-aircraft cluster absolute space coordinate system high-precision positioning.The method effectively utilizes the distance network information composed of paired relative distance measurement to improve the positioning accuracy of cluster.

Description

Technical Field

[0001] The present invention relates to the field of collaborative navigation, and in particular to a method for collaborative autonomous navigation of multiple aircraft in a satellite-denied environment. Background Art

[0002] In a strong satellite denial environment, due to the lack of stable and reliable external measurement information, the accuracy of the aircraft's autonomous navigation will decrease as the cumulative error increases.

[0003] In a satellite-denied environment, the navigation and positioning of an aircraft mainly relies on its own inertial navigation system for track calculation or relative positioning through other relative measurement sensors. Based on the utilization of the above two positioning results in different scenarios, the means of achieving navigation and positioning of aircraft in a denied environment can be divided into three categories: first, relying on the track calculation of the pure inertial navigation system to reduce the inertial navigation positioning error from the perspective of the inertial navigation error model; second, relative positioning through external measurement information of the aircraft; and third, combining the track calculation results and relative measurement results to achieve collaborative navigation and positioning of aircraft. In the current progress, many scholars have studied collaborative navigation in denied environments, including relative collaborative navigation and positioning with fixed anchor points, relative collaborative positioning under partial node denial of multi-aircraft clusters, collaborative positioning under inconsistent node positioning accuracy of multi-aircraft clusters, relative collaborative navigation and positioning under fully connected topology of small multi-aircraft clusters, and relative collaborative positioning under non-fully connected topology of large multi-aircraft clusters. A key advantage of collaborative navigation of multi-vehicle swarms is that the geometric topological relationships between formation configurations serve as crucial reference information for collaborative missions. Using observation information between vehicles, relative positioning can be achieved based on the principle of triangulation. Within the constraints of the formation's topological relationships, the system adaptively suppresses and compensates for track-dead reckoning errors that accumulate over time, improving relative positioning accuracy. However, the relative positioning results of swarms still lack a reference correction in absolute space, making it difficult to avoid global navigation divergence when the swarm drifts as a whole. Furthermore, sensor nodes are highly susceptible to loss in complex scenarios, resulting in significant deviations in the track-dead reckoning or observation results of individual nodes. Directly integrating these nodes into global fusion without prior information evaluation can reduce the navigation accuracy of the entire system. By utilizing redundant position information from each node in the collaborative network to evaluate its positioning results and adaptively adjusting the weights of the information interaction fusion, the navigation accuracy of the entire system can be guaranteed. However, this also increases the computational burden of the system. These pressing challenges hinder the development of collaborative navigation and positioning in denied environments. Summary of the Invention

[0004] The purpose of the present invention is to provide a method for cooperative autonomous navigation of multiple aircraft in a satellite-denied environment, which effectively utilizes distance network information composed of paired relative distance measurements to improve cluster positioning accuracy.

[0005] The technical solution to achieve the purpose of the present invention is: a method for collaborative autonomous navigation of multiple aircraft in a satellite-denied environment, comprising:

[0006] Step 1: construct a boundary measurement network adjustment model, use the inherent distance constraints in the cluster topology network to reduce the distance estimation error, and generate a high-precision distance measurement network in real time;

[0007] Step 2: Based on the distance measurement adjustment theory, collaborative relative positioning is carried out, which makes full use of the distance network under the edge measurement network adjustment model to achieve the cluster global optimal positioning, while improving the positioning accuracy and robustness of the cluster system, and expanding the scope of collaborative positioning in the local connection topology;

[0008] Step 3: Based on multi-dimensional scaling technology and coordinate registration technology, a two-stage collaborative positioning framework based on TOSA is constructed to achieve high-precision positioning of the absolute spatial coordinate system of the multi-aircraft cluster.

[0009] A computer device comprises a memory, a processor and a computer program stored in the memory and executable on the processor, wherein the steps of the above method are implemented when the processor executes the program.

[0010] A computer-readable storage medium stores a computer program, which implements the steps of the above method when executed by a processor.

[0011] A computer program product comprises a computer program, which implements the steps of the above method when executed by a processor.

[0012] Compared with the prior art, the present invention has the following beneficial effects:

[0013] (1) Aiming at the uncertainty caused by inaccurate distance estimation in collaborative localization, the relationship between relative measurement results and relative positioning results is explored, and an improved distance measurement error adjustment model is proposed. This model calculates distance measurement and node coordinate errors by utilizing the inherent constraints in the cluster network;

[0014] (2) A new measurement and adjustment theory for clustered aircraft (TOSA) is proposed. While achieving global optimal positioning and generating distance networks, it also expands the scope of collaborative positioning in local connection topologies and improves the positioning accuracy and robustness of cluster systems.

[0015] (3) A two-stage collaborative positioning framework based on TOSA was constructed, a classic relative positioning algorithm was derived, and the relationship between classic multidimensional scaling (MDS) and coordinate alignment was explored to achieve accurate relative positioning based on multidimensional scaling (MDS) and procrustes analysis (PA) method. BRIEF DESCRIPTION OF THE DRAWINGS

[0016] Figure 1 is the principle diagram of multi-aircraft target cooperative positioning based on distance measurement in the application.

[0017] Figure 2 is the error adjustment model of distance measurement between aircraft nodes in the application.

[0018] Figure 3 is the aircraft formation configuration and trajectory diagram under simulation experiment in the application.

[0019] Figure 4 is the x-axis divergence diagram of each node under pure inertial navigation in the simulation experiment in the application.

[0020] Figure 5 is the y-axis divergence diagram of each node under pure inertial navigation in the simulation experiment in the application.

[0021] Figure 6 is the z-axis divergence diagram of each node under pure inertial navigation in the simulation experiment in the application.

[0022] Figure 7 is the xoy plane divergence curve of each node under pure inertial navigation in the simulation experiment in the application.

[0023] Figure 8 is the anchor point and node trajectory curve diagram in the simulation experiment in the application.

[0024] Figure 9 is the formation topology comparison diagram in the simulation experiment in the application.

[0025] Figure 10 is the positioning RMSE comparison diagram under different algorithms in the simulation experiment in the application.

[0026] Figure 11 is the topology area error comparison diagram under different algorithms in the simulation experiment in the application.

[0027] Figure 12 is the positioning error component comparison diagram under different algorithms in the simulation experiment in the application.

[0028] Figure 13 is the PIR and RMSE comparison diagram under different algorithms in the simulation experiment in the application.

[0029] Figure 14 is the performance analysis under different fixed error parameters α in the simulation experiment in the application.

[0030] Figure 15 is the performance analysis under different proportional error parameters β in the simulation experiment in the application.

[0031] Figure 16 is the positioning performance improvement diagram of different algorithms compared with CG algorithm under the change of node number in the simulation experiment in the application.

[0032] Figure 17 is the simulation experiment in the application under the node number change under the different algorithm topological area maintenance compared with CG algorithm promotion schematic diagram.

[0033] Figure 18 is the simulation experiment in the application under the node number change under the algorithm performance comparison schematic diagram. DETAILED DESCRIPTION

[0034] The application provides a multi-aircraft cooperative autonomous navigation method in a satellite denial environment, which utilizes relative position relationships between aircrafts to correct aircraft cluster navigation information.Aiming at uncertainty caused by inaccurate distance estimation in cooperative positioning, a relationship between relative measurement results and relative positioning results is explored, and an improved edge network adjustment model is proposed.Meanwhile, a new aircraft cluster measurement adjustment (TOSA) theory is proposed, aiming to simultaneously realize global optimal positioning and generation of a range network, expand the application range of cooperative positioning in a local connection topology, and improve positioning accuracy and robustness of the cluster system.In addition, a two-stage cooperative positioning framework based on TOSA is constructed, a relationship between classic multi-dimensional scaling (MDS) and coordinate registration is explored, and accurate relative positioning based on MDS and Procrustes analysis (PA) methods is realized.

[0035] The method first designs an error adjustment function and a measurement adjustment model from the perspective of cluster configuration to reduce distance measurement errors.Secondly, a TOSA (Theory of Surveying Adjustment) algorithm is proposed based on the robustness of the edge network to realize global optimal positioning of the cluster and feedback correction of distance measurement by using the distance robustness of the edge network.Then, relative distance measurements based on adjustment correction are used to form a distance matrix, and MDS (Multi-Dimensional Scaling) technology is used to further improve the positioning accuracy of the cluster.Meanwhile, a coordinate registration strategy based on PA (Procrustes Analysis) is combined with relative positioning to realize mapping of the cluster positioning in the relative space to the absolute space coordinate system.

[0036] The technical solutions of the application will be described in detail below with reference to the drawings.

[0037] From the perspective of overall configuration, this paper establishes a measurement adjustment function and measurement adjustment model to reduce distance errors. Based on this, a multidimensional scaling (MDS) algorithm that exploits pairwise distance redundancy is proposed to further improve positioning accuracy. Furthermore, a two-stage collaborative positioning framework based on the TOSA method is constructed, thereby expanding the applicability of collaborative positioning in locally connected topologies. The implementation process includes the following three specific steps.

[0038] Step 1: Build a boundary measurement network adjustment model, use the inherent distance constraints in the cluster topology network to reduce distance estimation errors, and generate a high-precision distance measurement network in real time.

[0039] Considering that aircraft can provide stable and reliable altitude information based on barometric altimeters and can achieve relative positioning based on mutual distance measurement information, the navigation problem of aircraft clusters can be transformed into a two-dimensional plane to study the problem of collaborative ranging relative positioning and navigation. Consider a collaborative navigation system composed of n nodes on a two-dimensional plane, and define the adjustment coordinates as i=1,2,...,n,where (·) T represents the transpose operation. The system coordinate configuration can be approximately expressed by the definition of the node position coordinates as All node members are equipped with inertial measurement units, which can measure relative positions and communicate with each other. The aircraft in the cluster are divided into anchor aircraft and node aircraft. The anchor aircraft are equipped with high-precision inertial navigation to ensure their own high-precision positioning. Other aircraft equipped with low-cost inertial navigation that lack accurate positioning results are considered nodes. The schematic diagram of aircraft cluster collaborative positioning is shown below. Figure 1 shown. represents the relative adjustment distance from k to j, which can be expressed by the coordinate vectors of the two nodes:

[0040]

[0041] By using relative distance measurement sensors and cluster communication modules installed between clusters, distance measurement and information sharing can be achieved for all nodes in the network. TOA distance estimation is a practical method for estimating the distance between wireless nodes, which involves measuring the round-trip time of flight. When the distance observation error caused by time synchronization problems is subtracted from the mean, its measured value can be approximated as a zero-mean Gaussian random variable. Here, the noisy distance measurement between two nodes can be expressed in the general form as

[0042]

[0043] Where, Indicates the relative adjustment distance from k to j in the side adjustment network, L jk Indicates the theoretical distance from k to j in the side adjustment network. jkwhere σ denotes the corresponding distance measurement noise, the error function of distance noise can be expressed as

[0044] σ jk = α + βL jk (3)

[0045] where α is the relevant fixed error scalar parameter; β represents the relevant proportional error parameter. It should be noted that the distance noise σ jk and the value of parameter α are in units of meters, the value of distance L jk is in units of kilometers, and β is a dimensionless parameter.

[0046] For a collaborative system consisting of n nodes, the distance measurement network is composed of N = n 2 edges (including the measurement value of the node itself). For ease of problem description and modeling, the relative distance set L jk between nodes is redefined as L i , where the subscript index i = 1, 2, …, N represents the serial number of the distance measurement network, as shown in Figure 2 The black filled points in the figure are anchor points and approximate points for distance measurement calculation, and the red filled points are adjustment points under distance constraints.

[0047] Here, we take an edge of the edge network as an example to describe the error function between the measurement value of the adjustment point and the approximate point. Let the coordinate adjustment value of the unknown point be the parameter, and let

[0048]

[0049] where, denotes the coordinates of the approximate point; denotes the coordinates of the adjusted point; denotes the distance error adjustment value.

[0050] According to the approximate calculation and measurement results, the measurement function of the ith distance can be expressed as

[0051]

[0052] where, is the adjustment distance calculated according to the adjustment coordinates, L i is the distance measurement value, v i is the distance adjustment error;

[0053] Using the first-order Taylor expansion, the distance adjustment equation can be obtained as:

[0054]

[0055] where l i represents the distance measurement error, represents the approximate distance of nodes j and k, Represents the coordinate error between nodes j and k in the two-dimensional direction.

[0056] Generally speaking, when the coordinates of both nodes are unknown, the general form of the distance adjustment error equation is shown in Equation (6). Similarly, this equation can be flexibly applied according to different situations. If the coordinates of point j are given, then Then formula (6) can be rewritten as:

[0057]

[0058] In traditional distance-based relative positioning research, at least three anchor points with known positions and relative distance measurements are usually required to locate unknown nodes. This is reasonable and can avoid the singular value problem due to insufficient constraints. However, for the collaborative system introduced in this section, under the prior information of the rigid formation configuration, the relative spatial relationship between the aircraft nodes is known, so the relative positioning between the nodes can be achieved through the spatial constraints under the configuration topology. In other words, reliable relative positioning can be achieved with only two anchor points under the above prior information. Specifically, Figure 1 As shown, nodes i and j are regarded as anchor points with known positions, and their positions are defined as X i =[X i,x ,X i,y ] T , X j =[X j,x ,X j,y ] T , the position X of node k k =[X k,x ,X k,y ] T It can be calculated based on the relative geometric relationship between the anchor point and the node

[0059]

[0060] Where, represents the horizontal angle between nodes i and j, θ ijk Indicates distance L ij and L jk The vector angles between them are process parameters calculated based on the relative position coordinates, not measured data.

[0061] Step 2: Perform collaborative relative positioning based on distance measurement adjustment theory, making full use of the distance network under the side measurement network adjustment model to achieve the global optimal positioning of the cluster. While improving the positioning accuracy and robustness of the cluster system, it expands the scope of collaborative positioning in the local connection topology.

[0062] First, the definitions of approximate points and anchor points are given. Anchor points refer to nodes with known coordinates and high positioning accuracy, as shown in the figure. Approximate points refer to normal nodes calculated based on geometric relationships based on the relative distance measurement between the anchor point and the given anchor point coordinates, such as Figure 2 As shown in Figure 2. Since the coordinates of the approximate points only consider distance measurements, their positioning accuracy is constrained by measurement uncertainty and noise. In collaborative formations, considering the advantages of cluster networks, the formation topology provides a certain coupling relationship for distance measurement, which can provide additional distance measurement accuracy guarantees based on the topological constraints between cluster members. To this end, this paper proposes a collaborative relative positioning method based on the distance measurement adjustment theory (TOSA) to find the global optimal positioning of the cluster system. Here, the positioning of the cluster nodes obtained based on the TOSA method is recorded as the adjustment point.

[0063] In a two-dimensional cluster distance measurement network, assuming that there are m nodes as normal nodes, the coordinate vector to be determined consists of M = 2m adjustment coordinate parameters. Therefore, in general form, the error function of each distance measurement in the network is derived as follows

[0064]

[0065] Where, represents the adjustment parameter element, b represents the calculation parameter, and is interpreted and represented according to equations (4) to (9).

[0066] In order to simplify the expression of all distance measurement errors of multi-aircraft clusters, the error function can be expressed in a general form, that is, the state space function can be expressed as

[0067]

[0068] Where, represents the state vector containing all adjustment parameters, represents the error matrix after all distance adjustments, represents the distance measurement error matrix, Represents the calculation matrix of the error function. The specific expression is as follows:

[0069]

[0070] l=[l1 l2 …l N ] T (17)

[0071] V=[v1 v2 …v N ] T (18)

[0072] According to the adjustment principle, formula (15) should satisfy

[0073]

[0074] Where P is the measurement weight, defined as

[0075] P=[p i …p N ] T (20)

[0076]

[0077] σ jk =α+βL jk (twenty two)

[0078] in, is the unit weight variance defined, is the distance variance calculated according to the following formula. In formula (22), α is the fixed error scalar related to the instrument accuracy (arrival time error), in meters; β represents the scale error related to various measured distances. It should be noted that, in general, the distance noise σ jk The unit of the parameter α is meter, and the distance L jk The value unit of is kilometers, and β is a dimensionless parameter.

[0079] Mathematically speaking, indirect adjustment is the process of finding the unknown parameters in the error equation under the requirement of the least squares criterion. In mathematics, it is a problem of finding the extreme value of a multivariate function. Therefore, the coordinate adjustment value can be solved as

[0080]

[0081] Among them, N BB and W represent two auxiliary indicators for simplified expression. The specific expression is

[0082] N BB =B T PB W=B T Pl (24)

[0083] Then, based on the coordinate adjustment values, the adjustment distance and adjustment coordinates can be recalculated according to equations (4) and (5) respectively. Under the condition that the cluster formation adjustment topology is determined, the centroid, area and other parameters of the polygon topology can also be obtained by calculation. Specifically, the topological area calculation formula is as follows:

[0084]

[0085] Where S is the area of ​​the polygon, n is the number of nodes of the polygon, (x i ,y i) are the ordered vertices of the polygon. The vertices of the polygon should be arranged in reverse order to complete the calculation of the center of mass. For closed-loop topology, let j = i + 1, and the definition related to the node order in the calculation should be specifically given as

[0086]

[0087] Furthermore, the centroid coordinates (c x ,c y ) is calculated as

[0088]

[0089] The presence of the centroid provides an additional reference point for the relative positioning of the entire collaborative system, making it particularly suitable for anchor-less cluster configurations. In anchor-less cluster systems, the topological centroid achieves higher global positioning accuracy and system robustness than node-based reference points. However, in collaborative systems with anchors, anchor-based benchmarking is usually preferred.

[0090] Step 3: Based on multi-dimensional scaling technology and coordinate registration technology, a two-stage collaborative positioning framework based on TOSA is constructed to achieve high-precision positioning of the absolute spatial coordinate system of the multi-aircraft cluster.

[0091] For a set of two-vector distance measurements, the Euclidean range matrix D is constructed as the application basis of the multidimensional scaling theory MDS. Then, the positive semidefinite symmetric matrix is ​​formed

[0092]

[0093] Among them, D ⊙2 represents the element-wise (Hadamard) power, J is a double-centered matrix, and is written as:

[0094]

[0095] Where I represents the n×n unit matrix and 1 represents the n×1 unit column vector

[0096] At this point, the MDS problem is transformed into a mathematical solution F for Ψ, where F represents the coordinate topology of the relative configuration. The singular value decomposition can be used to effectively calculate the characteristic decomposition of Ψ, so that

[0097] Ψ=QΛQ T (31)

[0098] Where Λ is a diagonal matrix containing positive eigenvalues, and Q is an orthogonal eigenvector matrix. After a simple transformation, the equation can be rewritten as

[0099] Ψ=QΛQ T =(QΛ ⊙12)(QΛ ⊙12 ) T =FF T (32)

[0100] In this way, the coordinate topology F of the desired configuration can be calculated according to the following formula:

[0101] F=QΛ ⊙1 / 2 (33)

[0102] However, the coordinate topology F lacks an absolute spatial positioning standard. To address deformations such as rotation and translation caused solely by the MDS, coordinate registration must be considered. Using Procrustes analysis based on the least squares method, the affine transformation from the relative topology F to the absolute topology G is calculated. Under this premise, the coordinate registration problem can be recast as the following optimization problem:

[0103] [R,T]=arg min||FR+TG|| (34)

[0104] Where ||·|| is the Euclidean norm, s is the scale factor, R is the orthogonal rotation matrix, and T is the translation vector. For the optimization problem, we considered and specifically implemented the following three steps:

[0105] (1) Data preprocessing

[0106] Its essence is to find the best match from F to G in the sense of least squares. In order to simplify the calculation process of R, topological transformation and normalization are the two main steps of data preprocessing. The transformed topology is denoted as G c and F c , the conversion process is as follows:

[0107]

[0108] in, represents the coordinate components of the topology G, Represents the topology G c The coordinate components of The definition of is similar.

[0109] The translation transformation converts the coordinates to the coordinate center in the relative coordinate space. In order to eliminate the influence of topological scaling, normalization is required. The normalized topology is recorded as G0 and F0. The normalization process is as follows:

[0110]

[0111] (2) Rotation

[0112] So far, the topological matrices G0and F0satisfy the conditions ||G0|| = 1 and ||F0|| = 1, which are helpful for the rotation transformation. Considering only the rotation operation, the optimization problem (34) can be simplified as

[0113] R = arg min ||F0R - G0|| (39)

[0114] According to the definition of the matrix trace and the properties of the calculation, we have

[0115]

[0116] where tr(·) denotes the trace of a matrix.

[0117] The expectation of the optimization problem (39) is the maximum of tr(G0 T F0R). According to the singular value decomposition theory, the optimization problem can also be transformed as

[0118] R = arg max (tr(WH T RU)) (41)

[0119] where W denotes the diagonal matrix containing the positive eigenvalues of the matrix G0 T F0, and U and H denote two unit eigenvector matrices.

[0120] So far, the rotation matrix can be calculated by

[0121] R = H T U (42)

[0122] (3) Translation

[0123] According to equations (35)-(38) and equation (39), the approximate solution can be written as

[0124]

[0125] Considering the initial optimization problem (34), the translation vector can be written as

[0126]

[0127] Based on the above research, MDS can make full use of relative ranging information to achieve high-precision spatial relative positioning of formation nodes. However, due to the inevitable affine transformation and the lack of absolute spatial reference, it is not suitable for all configuration tasks. In addition, MDS requires high-precision relative distance measurement and is very sensitive to distance uncertainty. In the present invention, the distance adjustment based on the TOSA algorithm can ensure the accuracy of paired distances. In addition, PA only needs a reference topology G to implement the configuration affine transformation based on MDS. However, although the reference topology is not suitable for non-rigid configuration, TOSA can also provide a topology based on the distance adjustment network with good accuracy. In addition, the selection of translation vectors in PA also has additional support, such as anchor coordinates and topological centroid based on TOSA. Therefore, on this basis, the present invention combines the advantages of these two algorithms, proposes an improved algorithm, and organizes the algorithm in Algorithm 1 as follows.

[0128]

[0129] Co-localization can essentially be viewed as an unbiased distance estimation problem, where the distance measurement follows a Gaussian distribution with a mean of L i , the covariance is Cramer-Rao lower bound (CRLB) is usually used as a performance indicator to explore the expected performance boundary of sensor networks. In the scenario assumed in this article, the distance measurement error between nodes will vary due to different distances, so the variance of different edge measurements in the collaborative topology is different. Therefore, it is necessary to modify the Cramer-Rao bound for the case where the sensor network is used for target positioning. In two-dimensional space, assuming that the collaborative system contains n nodes and the member nodes can observe each other's distances, then the distance measurement network has a total of N = n including its own measurement. 2 Observation. Take a measurement of the edge distance as an example, and let the reference node coordinate be (x i ,y i ), unknown node S(x,y). For reference node i, the independent distance between the reference node and the unknown node is Subject to the mean r i , the variance is The normal distribution of r i is a real position of an unknown node θ=(x,y) T In terms of estimation, it is assumed that is a set of independent distance measurements, is the set of node positions to be estimated. Therefore, the conditional probability density function of node i can be expressed as

[0130]

[0131] Assume that the independent distance estimators obtained by measurement are For distance measurements Obey the mean The variance is The joint conditional probability density equation is as follows:

[0132]

[0133] For ease of calculation, the above formula can be rewritten in logarithmic form:

[0134]

[0135] According to the definition of Fisher information, the Fisher information matrix can be expressed as

[0136]

[0137] Taking into account the relative independence of distance measurement, we have

[0138]

[0139] Specifically, the element expression of the Fibonacci matrix is ​​as follows:

[0140]

[0141]

[0142] Therefore, according to the measurement model, the distance estimation The CRLB expression can be updated as follows:

[0143]

[0144] Example

[0145] Considering the needs of fully connected communication topology in actual application scenarios, and in order to verify the effectiveness of the proposed algorithm, this embodiment constructs a rigid collaborative cluster network with different communication connection topologies, and verifies the contents of the claims of the present invention based on numerical simulation experiments. In combination with practical applications, a simulation experiment of a cluster collaborative system consisting of 8 aircraft was constructed to verify the effectiveness of the proposed algorithm. Among them, 2 nodes with higher positioning accuracy serve as anchor points, and the remaining nodes serve as normal nodes with lower positioning accuracy. Assuming that the aircraft maintains tracking in a relatively rigid configuration, such as Figure 3The figure below illustrates the time-varying trajectory topology of a collaborative system using a typical configuration of five aircraft to complete a target tracking task in two-dimensional space. The dashed lines of different colors represent the trajectory of the aircraft trackers, while the solid black line represents the trajectory of the target. The red and black filled points represent the two anchor points, the unfilled blue points represent the normal nodes, and the unfilled red points are the centroid of the topology throughout the simulation.

[0146] In this simulation, the positioning differences of the aircraft are reflected in the inertial units of the equipment. The parameters of these instruments are listed in Table 1. The sampling interval is 8ms and the total simulation time is 200s. The configuration starts with the initial position of the anchor point (34°, 108°, 100m). There are a total of 8 nodes, of which the first two nodes are used as high-precision anchor points and the last two nodes are used as low-precision nodes. The specific performance of each node is as follows Figure 4-Figure 7 shown.

[0147] Table 1 Component parameter settings

[0148]

[0149] In addition, three other algorithms (referred to as CG, TOSA, and MDSPA in this section) were simulated and compared with the algorithm proposed in this paper. It should be emphasized that the positioning results of the CG algorithm are generally used as performance constraints based on geometric operations. The TOSA algorithm listed here is to prove the effectiveness of the adjustment model and lay the foundation for the algorithm proposed in this paper. According to the actual instrument, the range measurement error is established as (3), the fixed error parameter is set to α = 5, and the scale error parameter is set to β = 1×10 -3 .

[0150] Without compromising generality, the positioning accuracy of the solution is calculated based on the average Euclidean distance between the estimated position and the actual position. The root mean square error (RMSE) and the precision improvement ratio (PIR) are used to evaluate the tracking performance of the algorithm. In particular, the PTR indicator is defined as a standard line based on positioning measurements. The purpose of the PIR parameter is to compare the accuracy improvement rate of different algorithm results based on the positioning results under the original device. RMSE requires absolute position information, and the calculation definition of RMSE is as follows:

[0151]

[0152] Where (x k ,y k )and are the real position coordinates and estimated coordinates of the target at time k, and M is the number of simulation samplings.

[0153] By analyzing the navigation performance of the cluster system in the scenario, we carried out the verification of the invention points of multi-aircraft cooperative navigation in a denied environment. The specific test results are as follows:

[0154] Compared with the true trajectory, Figure 8 The middle blue dashed line, i.e., the normal node trajectory of the different colored solid lines, deviates significantly due to long-term dead reckoning errors. Meanwhile, the trajectory of the anchor points, i.e., the red and green dashed lines, can maintain a small deviation with high-precision instruments. Specifically, the partially enlarged box shows the accuracy of the anchor points at the end of the trajectory. Due to the presence of low-precision nodes, the configuration of the entire collaborative system will deviate significantly unless high-precision anchor points are used for calibration, as shown in Figure 2. Figure 9 As shown in Figure 3, the initial topology is consistent at the beginning, as shown by the blue solid line and the red dashed line. The blue dashed line between two nodes represents the communication that can be satisfied in the fully connected topology. It can be clearly seen that the topology based on the pure INS algorithm (black dashed line) diverges compared to the reference. At the same time, the cyan solid line topology based on this algorithm remains consistent with the reference graph.

[0155] The root mean square error of positioning and area analysis with the configuration topology time is as follows Figure 10-11 shown. Figure 12 The error analysis of positioning components based on different algorithms is given, and the performance indicators are recorded in detail in Table 2.

[0156] Table 2 Cluster overall topology node error components under different algorithms

[0157]

[0158]

[0159] Figure 10 The root mean square error of the positioning of the entire configuration under different algorithms is plotted. It can be seen that when the distance measurement has serious uncertainty, the positioning RMSE based on classical geometric information, that is, the blue line with CG label, shows a significant deviation. It is reasonable that the irregular divergence of node positioning over time will reduce the positioning accuracy of the entire system. Given sufficient relative distance measurements, the distance network can provide additional guarantees for positioning accuracy. Therefore, the results obtained using the TOSA and MDSPA algorithms both have lower root mean square errors. In addition, for rigid collaborative configurations, the MDSPA algorithm will perform better in the presence of reliable topological criteria. When the collaborative system lacks the ability to maintain a rigid topology, its performance deserves reconsideration and evaluation. As expected from the proposed TOSA+MDSPA algorithm, the TOSA method provides more accurate distance measurements for MDSPA, ultimately leading to a higher level of relative positioning accuracy. Specifically, the positioning errors of the proposed algorithm in components (such as Figure 10The purple dashed line in the middle) always keeps the error close to zero, confirming its excellent performance, while the errors of other compared algorithms show different degrees of oscillation. In addition, from the perspective of topology maintenance, the proposed algorithm ( Figure 11 The purple dashed line in the middle has the smallest topological area error while maintaining topological rigidity.

[0160] In order to prove the effectiveness of the algorithm of the present invention, the RMSE mean and positioning precision improvement ratio (PIR) of the collaborative system during the entire simulation process are calculated and plotted. Figure 13 As shown. It can be clearly seen that the proposed algorithm (as shown in the figure) has the lowest RMSE and the highest PIR value. Performance indicators, including positioning components in the x and y directions, Euclidean distance, and the area of ​​the entire configuration, are calculated and recorded in the error components of different algorithms in Table 2. The positioning errors of the algorithm in the x and y directions in this scenario are 1.2299m and 0.9170m, respectively. Compared with the indicators based on classical geometric techniques, the positioning accuracy in the x direction is improved by 77.91%, and the positioning accuracy in the y direction is improved by 73.94%. The root mean square error of positioning is reduced by 70.3%, and the area error of the overall configuration is reduced by 81.62%.

[0161] Convergence analysis under different error parameters is as follows Figure 14 and Figure 15 As shown in the figure, CRLB increases with the increase of the distance error factor, which also affects the variance of the algorithms with poor convergence. The CG method is essentially a geometric positioning technique that does not take advantage of the distance measurement network, resulting in the largest variance and the lowest positioning accuracy. In contrast, TOSA and MDSPA take advantage of the cluster network and demonstrate excellent positioning accuracy by ensuring the global consistency of the ranging network and significantly reducing the positioning variance. It is worth noting that TOSA makes full use of the distance measurement network based on global adjustment correction, resulting in enhanced convergence performance. In addition, MDSPA can improve the positioning accuracy in the TOSA topology, which helps to improve the overall performance of the proposed TOSAMDSPA algorithm, especially in the case of large measurement errors.

[0162] In addition, another simulation was performed to investigate the relationship between performance exploration and the number of nodes. The results are recorded in Figure 16 and Figure 18 In particular Figure 18 The RMSE analysis of different algorithms under different numbers of nodes is shown. Table 3 is the cluster RMSE analysis under the change of the number of nodes.

[0163] Table 3 Cluster RMSE analysis under changes in the number of nodes (unit: meter)

[0164]

[0165] It can be concluded that as the number of nodes increases, the positioning accuracy of the entire configuration improves. Figure 17 The topology maintenance capabilities of different algorithms under rigid topology conditions are demonstrated. The results show that the proposed algorithm is able to maintain rigid topology structures with minimal area error in collaborative systems of varying scales. Furthermore, the TOSA algorithm exhibits slightly superior performance compared to the MDSPA algorithm, particularly when the number of nodes increases. This further validates the TOSA algorithm's effectiveness in improving distance measurement accuracy and achieving high-precision positioning. Table 4 presents the RMSE and PIR analysis of the topology area of ​​the proposed algorithm.

[0166] Table 4 Topological area analysis under the change of node number (unit: m 2 )

[0167]

[0168] It is clear that although the improvement trend slows down with the increase of the number of nodes, the algorithm of the present invention is still effective in improving the positioning accuracy. This simulation exploration can provide a basis for future research involving large-scale configurations.

Claims

1. A method for cooperative autonomous navigation of multiple aircraft in a satellite-denied environment, characterized in that: include: Step 1: Build a boundary measurement network adjustment model, use the inherent distance constraints in the cluster topology network to reduce distance estimation errors, and generate a high-precision distance measurement network in real time; specifically: For a collaborative navigation system consisting of n nodes on a two-dimensional plane, the adjustment coordinates are defined as in(·) T represents the transpose operation; the system coordinate configuration is approximately expressed by the definition of the node position coordinates as All node members are equipped with inertial measurement units, which can measure relative positions and communicate with each other; each aircraft in the cluster is divided into anchor aircraft and node aircraft. represents the relative adjustment distance from node j to k, which can be expressed by the coordinate vectors of the two nodes: The distance measurement and information sharing of all nodes in the network are achieved through the relative distance measurement sensors and cluster communication modules loaded between clusters. The noise distance measurement between two nodes is expressed as Where, Indicates the relative adjustment distance from k to j in the side adjustment network, L jk represents the theoretical distance from k to j in the side adjustment network; σ jk Represents the corresponding distance measurement noise, and the error function of the distance noise is expressed as s jk =α+βL jk (3) Where α is the relevant fixed error scalar parameter; β represents the relevant proportional error parameter; For a collaborative system consisting of n nodes, the distance measurement network is composed of N = n 2 The relative distance set L between nodes is composed of jk Redefining L i , where the subscript index i = 1, 2, ..., N represents the serial number of the distance measurement network; an edge of the measurement network is used to describe the error function between the measurement value of the adjustment point and the approximate point; let the coordinate adjustment value of the undetermined point be a parameter, and let Where, and Represents the coordinates of the approximate point; and Indicates the coordinates of the point after adjustment; Indicates the distance error adjustment value; According to the approximate calculation and measurement results, the measurement function of the i-th distance is expressed as Where, is the adjustment distance calculated based on the adjustment coordinates, L i is the distance measurement value, v i is the distance adjustment error; Using the first-order Taylor expansion, the distance adjustment equation can be obtained as follows: Where, l i represents the distance measurement error, represents the approximate distance between nodes j and k, and represents the coordinate error between nodes j and k in the two-dimensional direction; When the coordinates of both nodes are unknown, the distance adjustment error equation is as shown in formula (6); If the coordinates of point j are given, then Then formula (6) can be rewritten as: Nodes i and j are considered as anchor points with known positions, and their positions are defined as X i =[X i,x ,X i,y ] T , X j =]X j,x ,X j,y ] T , the position X of node k k =[X k,x ,X k,y ] T Calculated based on the relative geometric relationship between the anchor point and the node Where, represents the horizontal angle between nodes i and j, θ ijk Indicates distance L ij and L jk The vector angles between them are process parameters calculated based on the relative position coordinates, not measured data; Step 2: Based on the distance measurement adjustment theory, collaborative relative positioning is performed, and the distance network under the side measurement network adjustment model is fully utilized to achieve the global optimal positioning of the cluster; Step 3: Based on multi-dimensional scaling technology and coordinate registration technology, a two-stage collaborative positioning framework based on TOSA is constructed to achieve high-precision positioning of the absolute spatial coordinate system of the multi-aircraft cluster.

2. The method for cooperative autonomous navigation of multiple aircraft in a satellite-denied environment according to claim 1, characterized in that: Step 2, collaborative relative positioning based on distance measurement adjustment theory, is as follows: The collaborative relative positioning method based on distance measurement adjustment theory is used to find the global optimal positioning of the cluster system; the positioning of the cluster nodes obtained based on the TOSA method is recorded as the adjustment point; In a two-dimensional cluster ranging network, assuming that there are m nodes as normal nodes, the undetermined coordinate vector consists of M = 2m adjustment coordinate parameters; the error function of each distance measurement in the network is derived as follows Where, represents the adjustment parameter element, and b represents the calculation parameter; The state space function is expressed as Where, represents the state vector containing all adjustment parameters, represents the error matrix after all distance adjustments, represents the distance measurement error matrix, Represents the calculation matrix of the error function; the specific expression is as follows: l＝[l1 l2 … l N ] T (17) V=[v1 v2 … v N ] T (18) According to the adjustment principle, formula (15) should satisfy Where P is the measurement weight, defined as P=[p i …p N ] T (20) s jk =α+βL jk (22) in, is the unit weight variance defined, is the distance variance calculated according to the following formula; where α is the fixed error scalar related to the instrument accuracy; β represents the scale error related to various measured distances; The coordinate adjustment value is solved as Among them, N BB and W represent two auxiliary indicators for simplified expression. The specific expression is N BB =B T PB W=B T Pl (24) Then, based on the coordinate adjustment values, the adjustment distance and adjustment coordinates are recalculated according to formula (4) and formula (5) respectively; under the condition that the cluster formation adjustment topology is determined, the centroid and area parameters of the polygon topology are obtained by calculation; the topological area calculation formula is as follows: Where S is the area of ​​the polygon, n is the number of nodes of the polygon, (x i ,y i ) are the ordered vertices of the polygon; the vertices of the polygon should be arranged in reverse order to complete the calculation of the centroid; for closed-loop topology, let j = i + 1, and the node order is defined as The centroid coordinates (c x ,c y ) is calculated as 3. The method for cooperative autonomous navigation of multiple aircraft in a satellite-denied environment according to claim 2, characterized in that: Step 3: Based on multi-dimensional scaling technology and coordinate registration technology, a two-stage collaborative positioning framework based on TOSA is constructed to achieve high-precision positioning of the absolute spatial coordinate system of the multi-aircraft cluster. Specifically: Measure the distance between a set of two vectors and construct the Euclidean range matrix D as the application basis of multidimensional scaling theory MDS; form a positive semidefinite symmetric matrix Among them, D ⊙2 represents the element-by-element power, J is a double-centered matrix, and is written as: Where I represents the n×n unit matrix and 1 represents the n×1 unit column vector; The MDS problem is transformed into a mathematical solution F for Ψ, where F represents the coordinate topology of the relative configuration; the characteristic decomposition of Ψ is calculated using singular value decomposition, so that Ψ=QΛQ T (31) Where Λ is a diagonal matrix containing positive eigenvalues, and Q is an orthogonal eigenvector matrix; after a simple transformation, the equation is rewritten as Ψ=QΛQ T =(QΛ ⊙1 / 2 )(QL ⊙1 / 2 ) T =FF T (32) The coordinate topology F of the desired configuration can be calculated according to the following formula: F=QΛ ⊙1 / 2 (33) The affine transformation from the relative topology F to the absolute topology G is obtained using Procrustes analysis based on the least squares method; the coordinate registration problem is restructured into the following optimization problem: [R,T]=arg min||FR+TG|| (34) Where ||·|| is the Euclidean norm, s is the scale factor, R is the orthogonal rotation matrix, and T is the translation vector. For the optimization problem, the following three steps are included: (1) Data preprocessing Find the best match from F to G in the least squares sense; in order to simplify the calculation process of R, topological transformation and normalization are the two main steps of data preprocessing; the transformed topology is denoted as G c and F c , the conversion process is as follows: in, represents the coordinate components of the topology G, Represents the topology G c The coordinate components of represents the coordinate components of the topology F, Represents the topology F c The coordinate components of The translation transformation converts the coordinates to the coordinate center in the relative coordinate space; the normalized topology is recorded as G0 and F0; the normalization process is as follows: (2) Rotation The topological matrices G0 and F0 satisfy the conditions ||G0||=1and||F0||=1; considering only the rotation operation, the optimization problem (34) is simplified to R=arg min||F0R-G0|| (39) According to the definition and computational properties of matrix trace, we have Where tr(·) represents the trace of the matrix; The expected value of the optimization problem (39) is tr(G0 T According to the singular value decomposition theory, the optimization problem is transformed into R=arg max(tr(WH T RU)) (41) Among them, W represents the matrix G0 T The diagonal matrix of the positive eigenvalues ​​of F0, U and H represent two one-element eigenvector matrices; At this point, the rotation matrix is ​​calculated by the following formula R=H T U (42) (3) Translation According to equations (35)-(38) and (39), the approximate solution is Considering the initial optimization problem (34), the translation vector is written as In two-dimensional space, assuming that the collaborative system contains n nodes and the member nodes can observe each other's distance, then the distance measurement network has a total of N = n including its own measurement. 2 For a measurement of the edge distance, let the reference node coordinate be (x i ,y i ), unknown node S(x,y); for reference node i, the independent distance between the reference node and the unknown node Subject to the mean r i , with variance σ i 2 The normal distribution of r i is a real position of an unknown node θ=(x,y) T Related quantities; assumptions is a set of independent distance measurements, is the set of node positions to be estimated; therefore, the conditional probability density function of node i is expressed as Assume that the independent distance estimators obtained by measurement are For distance measurements The mean is The variance is The normal distribution of , i = 1, ..., N, N is the number of distance measurements obtained by measuring the topology; the joint conditional probability density equation is as follows: The above formula is rewritten in logarithmic form: According to the definition of Fisher information, the Fisher information matrix is ​​expressed as Considering the relative independence of distance measurement, we have The element expression of the Fibonacci matrix is ​​as follows: Therefore, according to the measurement model, the distance estimation The CRLB expression is updated to:

4. A computer device comprising a memory, a processor, and a computer program stored in the memory and executable on the processor, wherein: When the processor executes the program, the steps of the method according to any one of claims 1 to 3 are implemented.

5. A computer-readable storage medium having a computer program stored thereon, characterized in that: When the program is executed by a processor, the steps of the method according to any one of claims 1 to 3 are implemented.

6. A computer program product comprising a computer program, characterized in that When the computer program is executed by a processor, the steps of the method according to any one of claims 1 to 3 are implemented.

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