A distributed relative positioning method based on dynamic partitioning and semi-definite programming
By employing dynamic partitioning and semidefinite programming, the problems of high communication overhead, high computational complexity, and limited network connectivity in large-scale aircraft clusters of distributed relative positioning algorithms are solved, achieving high-precision positioning results.
Patent Information
- Authority / Receiving Office
- CN · China
- Patent Type
- Applications(China)
- Current Assignee / Owner
- BEIJING INST OF TECH
- Filing Date
- 2026-03-27
- Publication Date
- 2026-05-29
AI Technical Summary
Existing distributed relative positioning algorithms suffer from problems such as high communication overhead, high computational complexity, susceptibility to getting trapped in local optima, and limited network connectivity in large-scale, highly dynamic aircraft swarms, making it difficult to achieve high-precision positioning.
We employ a connectivity-based dynamic block partitioning strategy and semidefinite programming technique to reduce communication overhead and computational complexity by dynamically dividing the blocks. We also transform the non-convex optimization problem into a convex relaxation problem and use a signal-to-noise ratio weighting mechanism and singular value decomposition to achieve accurate localization fusion.
It achieves high-precision distributed relative positioning in low connectivity and complex noise environments, reduces communication overhead and computational complexity, avoids local optima, and improves the robustness and accuracy of the positioning algorithm.
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Figure CN122108147A_ABST
Abstract
Description
Technical Field
[0001] This invention belongs to the field of navigation and positioning technology, specifically relating to a distributed relative positioning method based on dynamic block division and semidefinite programming. Background Technology
[0002] In recent years, low-altitude aircraft such as drones have been widely deployed in fields such as military reconnaissance, environmental monitoring, disaster relief, and logistics transportation due to their significant advantages such as high flexibility, strong scalability, and low deployment costs. With the continuous expansion of application scenarios and the increasing complexity of tasks, their application model is gradually shifting from "single-aircraft operation" to "aircraft swarms," which have stronger collaborative capabilities and wider coverage. In tasks such as collaborative control, formation maintenance, task allocation, and inter-aircraft networking communication within aircraft swarms, high-precision positional awareness is a key prerequisite for ensuring that the swarm can efficiently and safely complete its designated tasks.
[0003] Currently, low-altitude aircraft primarily rely on Global Navigation Satellite Systems (GNSS) to obtain absolute position information. However, in practical applications, relying solely on GNSS for positioning has certain limitations. First, the positioning error of conventional GNSS often reaches several meters in the horizontal direction, and the error in the vertical direction is even greater. Second, GNSS signals suffer from severe obstruction in complex geographical environments (such as cities and canyons) and are susceptible to man-made electromagnetic interference and spoofing attacks. Therefore, in environments where GNSS signals are limited or unavailable, using relative observations between aircraft for relative positioning has become an important means of maintaining the safe operation of a cluster.
[0004] Traditional relative positioning techniques can be broadly categorized into centralized and distributed methods. Centralized relative positioning algorithms require all nodes to transmit their observation data to a central node for unified computation. While theoretically capable of achieving a globally optimal solution, these algorithms place extremely high demands on the central node's computational power and communication bandwidth. As the cluster size increases, communication overhead and computational complexity grow exponentially, making them unsuitable for large-scale, highly dynamic aircraft swarm scenarios. In contrast, distributed relative positioning algorithms distribute computational tasks across various sub-blocks or nodes, offering advantages such as smaller computational scale, stronger scalability, and higher robustness, and have become a current research hotspot.
[0005] Despite the significant advantages of distributed positioning algorithms, existing classic algorithms still face numerous technical challenges in practical applications, mainly in the following aspects:
[0006] (1) The sub-block partitioning strategy has an efficiency bottleneck: Existing distributed algorithms usually adopt a simple partitioning strategy, which has problems such as excessive information redundancy between sub-blocks and too many sub-blocks. This not only increases the overall communication overhead of the system, but also limits the real-time performance of the positioning algorithm.
[0007] (2) Non-convex optimization is prone to getting trapped in local optima: The cost function of classical positioning algorithms (such as least squares method and multidimensional scaling algorithm) is often non-convex. The convergence of the algorithm is highly dependent on the selection of initial values. If the initial conditions are poor or the measurement noise is large, the algorithm is very likely to get trapped in local optima, resulting in a large positioning error.
[0008] (3) High network connectivity requirements: Classical positioning algorithms typically require dense distance matrices within sub-blocks, meaning high network connectivity is needed to ensure positioning accuracy. However, in actual flight missions, aircraft networks are often sparse due to limitations in communication distance and power. Under low connectivity conditions, the use of classical algorithms is severely limited.
[0009] In summary, overcoming the local optima problem caused by non-convex optimization while ensuring low communication overhead and computational complexity, and achieving high-precision distributed relative positioning under conditions of limited network connectivity, are key issues that urgently need to be addressed in the field of aircraft swarm positioning technology. Therefore, developing a positioning method that can dynamically optimize the sub-block structure and is robust to noise and sparse observations has significant application value. Summary of the Invention
[0010] In view of this, the present invention provides a distributed relative positioning method based on dynamic block division and semidefinite programming, which can achieve high-precision positioning in large-scale aircraft clusters in low connectivity and complex noise environments.
[0011] To achieve the objectives of this invention, the following technical solutions are provided.
[0012] A distributed relative positioning method based on dynamic block partitioning and semidefinite programming includes the following steps: Step 1: Dynamic sub-block partitioning based on connectivity. Real-time perception of aircraft cluster connectivity information. Recursive block partitioning is performed using the node with the highest connectivity, thereby minimizing the number of sub-blocks and reducing communication overhead, and dividing the large-scale cluster into several overlapping sub-blocks. Step 2: Based on the local relative position estimation of semi-positive definite programming, within each sub-block, the non-convex distance measurement constraint is convexly relaxed, and after being transformed into a semi-positive definite programming problem, the accurate distance matrix is solved. Then, the local relative coordinates of the nodes within the sub-block are obtained by projection through the multi-dimensional scaling transformation algorithm. Step 3: Local relative position merging. The relative positions of each sub-block are globally merged. By utilizing the common node information between adjacent sub-blocks, the local coordinate system is unified to the global coordinate system by solving the rotation matrix and translation vector, thus achieving global relative position merging.
[0013] Specifically, the dynamic sub-block partitioning based on connectivity in step 1 includes: Step 11, Neighborhood Information Interaction and Initialization: Each node broadcasts its own ID and connectivity to its single-hop neighbors within its communication radius, where connectivity is defined as the number of single-hop neighbor nodes of that node; Each node receives and records the ID of the node with the highest connectivity in its neighborhood. Step 12, Construction of the first sub-block: Select the node with the highest connectivity in the cluster as the head node of the first sub-block. The head node and all its single-hop neighbors together constitute the first sub-block, and mark the nodes in the sub-block as the set of blocks. Step 13, Dynamically expand the head nodes of subsequent sub-blocks: To ensure the geometric uniqueness of sub-block fusion in three-dimensional space, subsequent sub-blocks and the already constructed sub-block group must have at least 4 non-coplanar common nodes. Based on this constraint, the node with the most undivided nodes among the candidate nodes is selected as the head node of the next sub-block. Step 14, Iterative partitioning: Update the set of partitioned nodes, and repeat step 13 until all nodes are partitioned into sub-blocks.
[0014] Specifically, step 13, which involves dynamically expanding the head node of subsequent sub-blocks, includes: calculating the number of nodes in the neighborhood of a candidate node that belong to the set of already segmented nodes as the number of common nodes, and selecting nodes that satisfy the condition that the number of common nodes is greater than or equal to 4; among the selected nodes, counting the number of nodes in their neighborhood that do not belong to the set of already segmented nodes as the number of unsegmented nodes; and selecting the node with the most unsegmented nodes as the head node of the next sub-block.
[0015] Specifically, step 2, the local relative position estimation based on semidefinite programming, includes: Step 21, construct a weighted ranging model: obtain the ranging values and estimated signal-to-noise ratios between each connected node in the sub-block, construct a weight matrix, where the weights are set to be positively correlated with the signal-to-noise ratio; Step 22: Construct a weighted least squares cost function based on squared distance; Step 23, establish a positive semidefinite programming model: based on the weighted least squares principle, an auxiliary matrix is introduced to transform the position distance into a linear expression of the matrix trace, thereby transforming the non-convex rank constraint problem into a convex positive semidefinite programming problem; Step 24, Solve for local coordinates: After solving the semidefinite programming problem using the convex optimization method, the updated distance matrix is obtained, and the local relative coordinate matrix is obtained by projection through the multidimensional scaling transformation algorithm.
[0016] Specifically, step 3, merging local relative positions, includes: Step 31, Determine transformation parameters: Assume the common node set in the current global coordinate system and the corresponding common node set in the sub-block to be merged. Solve the orthogonal Protodyakonov problem through singular value decomposition to find the optimal rotation matrix and translation vector that minimizes the transformation error. Step 32, Coordinate Fusion and Weighted Update: Transform the coordinates of all nodes in the sub-block to be fused to the global coordinate system using transformation parameters. For common nodes, use a precision-based weighted average method to update and fuse their positions.
[0017] In step 32, the location of the common nodes is updated and merged. The fusion weight is determined based on the semidefinite programming estimation residual of the sub-blocks. The smaller the residual, the greater the weight of the sub-block, thereby suppressing error accumulation.
[0018] In step 2, the original non-convex problem is transformed into a convex semi-positive definite programming problem through convex relaxation, so that the positioning result is not sensitive to the initial value and maintains stable positioning performance in scenarios with low network connectivity.
[0019] In step 1, the dynamic sub-block partitioning based on connectivity maximizes the node coverage of a single sub-block and reduces the total number of sub-block partitions and information redundancy between sub-blocks, thereby reducing the overall running time, communication overhead, and computational complexity of the algorithm.
[0020] In step 2, a weighted ranging model is constructed. By introducing a signal-to-noise ratio-based weighting mechanism, the influence of low-quality observation data on the positioning results is suppressed, and positioning accuracy close to the lower bound of the Cramer-Rao scale is achieved.
[0021] In step 3, the local relative positions are merged by using the common node information between adjacent sub-blocks and solving the rotation matrix and translation vector that minimizes the Euclidean distance error between the common nodes through singular value decomposition, thereby merging the local coordinate system into the global coordinate system.
[0022] Beneficial effects 1. This invention employs a connectivity-based dynamic block partitioning strategy, which maximizes the node coverage of individual sub-blocks and significantly reduces the total number of sub-blocks and information redundancy between them. Simulation results show that, at the same scale, this strategy can reduce the overall algorithm runtime by approximately 90%, significantly reducing communication overhead and computational complexity. Furthermore, by introducing a signal-to-noise ratio-based weighting mechanism, this invention effectively suppresses the impact of low-quality observation data on positioning results, achieving positioning accuracy close to the Cramero lower bound.
[0023] 2. This invention introduces SDP convex relaxation technology in the local localization stage, avoiding the drawback of classic relative localization algorithms that are prone to getting trapped in local optima. This method is insensitive to initial values and maintains stable localization performance even in scenarios with low network connectivity, demonstrating strong algorithm robustness. Attached Figure Description
[0024] Figure 1 This is a schematic diagram of the method flow of an embodiment of the present invention. Detailed Implementation
[0025] The present invention will be further described below with reference to the accompanying drawings and specific embodiments.
[0026] This invention proposes a distributed relative positioning method based on dynamic block partitioning and semidefinite programming, the process of which is as follows: Figure 1 As shown, it includes the following steps: Step 1: Dynamic sub-block partitioning based on connectivity. Real-time connectivity information of the aircraft cluster is sensed, and recursively sub-blocks are performed using the node with the highest connectivity, thereby minimizing the number of sub-blocks and reducing communication overhead. A distributed strategy is used to divide the large-scale cluster into several overlapping sub-blocks to reduce computational complexity and ensure global connectivity, achieving efficient coverage of the cluster topology, as detailed below: Step 11: Neighborhood information exchange and initialization. Each node communicates with its communication radius. The single-hop neighbor within the neighborhood broadcasts its own ID and connectivity. (Connectivity is defined as the number of single-hop neighbor nodes of a node). Each node receives and records the ID of the node with the highest connectivity in its neighborhood.
[0027] Step 12: Constructing the First Sub-Block. Select the node with the highest connectivity in the cluster as the head node of the first sub-block (if multiple maximum values exist, select the one with the smallest ID). This head node and all its single-hop neighbors together constitute the first sub-block, and the nodes within this sub-block are marked as the set of partitioned nodes. .
[0028] Step 13: Dynamically expand the head nodes of subsequent sub-blocks. To ensure the geometric uniqueness of sub-block merging in 3D space, subsequent sub-blocks must share at least 4 non-coplanar common nodes with the already constructed sub-block group. Based on this constraint, the head node of the next sub-block is selected according to the following strategy: Calculate the set of nodes that belong to the block within the neighborhood of the candidate node. The number of nodes (i.e., the number of common nodes) ), filter out those that meet the requirements The nodes are selected. Among these selected nodes, the number of nodes whose neighborhood does not belong to the set of already segmented nodes is counted. The number of nodes (i.e., the number of undivided nodes) is determined. The node with the largest number of undivided nodes is selected as the head node of the next sub-block. This strategy maximizes the coverage efficiency of a single sub-block for unknown regions, thereby significantly reducing the total number of sub-blocks.
[0029] Step 14: Iterative partitioning. Update the set. Repeat step 13 until all nodes are assigned to sub-blocks.
[0030] Step 2: Local relative position estimation based on semidefinite programming (SDP). Within each sub-block, the non-convex distance measurement constraints are relaxed to a convex state, transforming the problem into a semidefinite programming problem. The exact distance matrix is then solved, and the local relative coordinates of the nodes within the sub-block are obtained through projection using a multi-dimensional scaling algorithm. The original non-convex problem is transformed into a convex semidefinite programming problem through convex relaxation, and the local relative coordinates of the nodes within the sub-block are solved as follows: Step 21: Construct a weighted ranging model. Obtain the connected nodes within each sub-block. , Distance values between and estimated signal-to-noise ratio Construct the weight matrix The weights It is set to be positively correlated with the signal-to-noise ratio, so that high-precision observations dominate.
[0031] Step 22: Construct a weighted least squares cost function based on squared distance.
[0032] (1) Let be the coordinate matrix of the nodes to be solved; It is a weighting coefficient set based on the link signal-to-noise ratio, used to adjust the impact of different quality ranging values on the positioning results; This is the incomplete set of distance measurements between nodes at the current time. and Representing nodes respectively and 3D position vector; Step 23: Establish a semidefinite programming model. Based on the weighted least squares principle, define the optimization objective as minimizing the weighted error. Introduce an auxiliary matrix. (in , (where the coordinates are local relative positions), the positional distance is transformed into a linear expression of the matrix trace. This transforms the non-convex rank-constant problem into the following convex positive semi-definite programming problem: (2) Step 23: Solve for the local coordinates. After solving the SDP problem using convex optimization methods, the updated distance matrix can be obtained. The local relative coordinate matrix can then be obtained by projecting the matrix using a multidimensional scaling transformation algorithm. .
[0033] Step 3: Local relative position merging. The relative positions of each sub-block are globally fused. This can be achieved using global position fusion based on Protodyakonov analysis. Utilizing the common node information between adjacent sub-blocks, singular value decomposition (SVD) is used to solve for the rotation matrix and translation vector that minimizes the Euclidean distance error between common nodes. The transformation parameters obtained from the solution are then used to unify the local coordinate system to the global coordinate system, achieving global relative position fusion. The details are as follows: Step 31: Determine the transformation parameters. Let the set of common nodes in the current global coordinate system be... The set of corresponding common nodes in the sub-blocks to be merged is Solve the following orthogonal Protodyakonov problem using singular value decomposition to find the optimal rotation matrix. Translation vector This minimizes the transformation error: (3) Step 32: Coordinate Fusion and Weighted Update. The coordinates of all nodes within the sub-block to be fused are transformed to the global coordinate system using transformation parameters. For non-common nodes, their coordinates are directly retained. For common nodes (i.e., overlapping areas), a precision-based weighted average method is used for position updating and fusion. (4) in, To update the merged position, For the current position, weight The weight of a sub-block is determined based on the SDP estimation residual of the sub-block. The smaller the residual, the greater the weight of the sub-block, thereby suppressing error accumulation.
[0034] This invention includes, but is not limited to, the above embodiments. Any equivalent substitutions or partial improvements made under the spirit and principles of this invention shall be considered within the scope of protection of this invention.
Claims
1. A distributed relative positioning method based on dynamic block partitioning and semidefinite programming, characterized in that, Includes the following steps: Step 1: Dynamic sub-block partitioning based on connectivity. Real-time perception of aircraft cluster connectivity information. Recursive block partitioning is performed using the node with the highest connectivity, thereby minimizing the number of sub-blocks and reducing communication overhead, and dividing the large-scale cluster into several overlapping sub-blocks. Step 2: Based on the local relative position estimation of semi-positive definite programming, within each sub-block, the non-convex distance measurement constraint is convexly relaxed, and after being transformed into a semi-positive definite programming problem, the accurate distance matrix is solved. Then, the local relative coordinates of the nodes within the sub-block are obtained by projection through the multi-dimensional scaling transformation algorithm. Step 3: Local relative position merging. The relative positions of each sub-block are globally merged. By utilizing the common node information between adjacent sub-blocks, the local coordinate system is unified to the global coordinate system by solving the rotation matrix and translation vector, thus achieving global relative position merging.
2. The distributed relative positioning method based on dynamic block division and semidefinite programming according to claim 1, characterized in that, The dynamic sub-block partitioning based on connectivity in step 1 specifically includes: Step 11, Neighborhood Information Interaction and Initialization: Each node broadcasts its own ID and connectivity to its single-hop neighbors within its communication radius, where connectivity is defined as the number of single-hop neighbor nodes of that node; Each node receives and records the ID of the node with the highest connectivity in its neighborhood. Step 12, Construction of the first sub-block: Select the node with the highest connectivity in the cluster as the head node of the first sub-block. The head node and all its single-hop neighbors together constitute the first sub-block, and mark the nodes in the sub-block as the set of blocks. Step 13, Dynamically expand the head nodes of subsequent sub-blocks: To ensure the geometric uniqueness of sub-block fusion in three-dimensional space, subsequent sub-blocks and the already constructed sub-block group must have at least 4 non-coplanar common nodes. Based on this constraint, the node with the most undivided nodes among the candidate nodes is selected as the head node of the next sub-block. Step 14, Iterative partitioning: Update the set of partitioned nodes, and repeat step 13 until all nodes are partitioned into sub-blocks.
3. The distributed relative positioning method based on dynamic block division and semidefinite programming according to claim 2, characterized in that, The step 13 of dynamically expanding the head node of the subsequent sub-block specifically includes: calculating the number of nodes in the neighborhood of the candidate node that belong to the set of already divided nodes as the number of common nodes, and filtering out nodes that satisfy the condition that the number of common nodes is greater than or equal to 4; among the filtered nodes, counting the number of nodes in their neighborhood that do not belong to the set of already divided nodes as the number of undivided nodes; and selecting the node with the most undivided nodes as the head node of the next sub-block.
4. The distributed relative positioning method based on dynamic block division and semidefinite programming according to claim 1, characterized in that, Step 2, the local relative position estimation based on semidefinite programming, specifically includes: Step 21, construct a weighted ranging model: obtain the ranging values and estimated signal-to-noise ratios between each connected node in the sub-block, construct a weight matrix, where the weights are set to be positively correlated with the signal-to-noise ratio; Step 22: Construct a weighted least squares cost function based on squared distance; Step 23, establish a positive semidefinite programming model: based on the weighted least squares principle, an auxiliary matrix is introduced to transform the position distance into a linear expression of the matrix trace, thereby transforming the non-convex rank constraint problem into a convex positive semidefinite programming problem; Step 24, Solve for local coordinates: After solving the semidefinite programming problem using the convex optimization method, the updated distance matrix is obtained, and the local relative coordinate matrix is obtained by projection through the multidimensional scaling transformation algorithm.
5. The distributed relative positioning method based on dynamic block division and semidefinite programming according to claim 1, characterized in that, The merging of local relative positions in step 3 specifically includes: Step 31, Determine transformation parameters: Assume the common node set in the current global coordinate system and the corresponding common node set in the sub-block to be merged. Solve the orthogonal Protodyakonov problem through singular value decomposition to find the optimal rotation matrix and translation vector that minimizes the transformation error. Step 32, Coordinate Fusion and Weighted Update: Transform the coordinates of all nodes in the sub-block to be fused to the global coordinate system using transformation parameters. For common nodes, use a precision-based weighted average method to update and fuse their positions.
6. The distributed relative positioning method based on dynamic block division and semidefinite programming according to claim 5, characterized in that, In step 32, the common nodes are updated and merged. The fusion weight is determined based on the semidefinite programming estimation residual of the sub-blocks. The smaller the residual, the greater the weight of the sub-block, thereby suppressing error accumulation.
7. The distributed relative positioning method based on dynamic block partitioning and semidefinite programming according to any one of claims 1-6, characterized in that, In step 2, the original non-convex problem is transformed into a convex semi-positive definite programming problem through convex relaxation, so that the positioning result is not sensitive to the initial value and maintains stable positioning performance in scenarios with low network connectivity.
8. The distributed relative positioning method based on dynamic block partitioning and semidefinite programming according to any one of claims 1-6, characterized in that, The dynamic sub-block partitioning based on connectivity in step 1 reduces the overall running time, communication overhead, and computational complexity of the algorithm by maximizing the node coverage of a single sub-block and reducing the total number of sub-block partitions and information redundancy between sub-blocks.
9. The distributed relative positioning method based on dynamic block partitioning and semidefinite programming according to any one of claims 1-6, characterized in that, In step 2, a weighted ranging model is constructed. By introducing a signal-to-noise ratio-based weighting mechanism, the influence of low-quality observation data on the positioning results is suppressed, and positioning accuracy close to the lower bound of the Cramer-Rao scale is achieved.
10. The distributed relative positioning method based on dynamic block partitioning and semidefinite programming according to any one of claims 1-6, characterized in that, In step 3, the local relative positions are merged by using the common node information between adjacent sub-blocks and solving the rotation matrix and translation vector that minimizes the Euclidean distance error between the common nodes through singular value decomposition, thereby merging the local coordinate system into the global coordinate system.