A Relative Coordinate-Based Topology Shaping Method for UAV Swarm with Low Computational Complexity
Abstract
:1. Introduction
- From a global system perspective, a system optimization model for topology shaping of UAV swarm in 3D space is developed without relying on any external localization information, in which the topology shaping and global energy consumption minimization are considered.
- Under the framework of the topology shaping optimization model, the topology shaping problem is transformed into a problem of relative coordinate mapping. The topology shaping with minimum global energy consumption is achieved by obtaining the optimal mapping relationship of the relative coordinates. The LAPJV algorithm is employed to solve this optimization model. The simulation results have demonstrated the effectiveness of this algorithm. The LAPJV algorithm significantly reduces the computational complexity by an average of in the case of 1000 UAV nodes while achieving the same minimum global energy consumption as other algorithms.
2. System Model
3. The Construction of Initial and Target Topology Coordinate Systems
Algorithm 1: Obtaining the coordinate matrix. |
Input: Distance matrix , coordinate dimension ; |
Output: Relative coordinate matrix ; |
1 Derive by Equation (18) from ; |
2 Obtain Matrix from ; |
3 Eigen-decomposition of matrix and obtain the and of ; |
4 Form the diagonal matrix by employ 3 largest ; |
5 Construct matrix by ; |
6 Get by Equation (20); |
7 return; |
4. The Optimal Coordinate Mapping from Initial Topology to Target Topology
- Select an unassigned node in the initial topology .
- Construct the residual (auxiliary, incremental) graph, with costs .
- Find the shortest augmenting path via a modified Dijkstra algorithm (recall ).
- Augment the solution to improve the match, construct the auxiliary network and determine from unassigned row i to unassigned column j an alternating path of minimal total reduced cost.
- Update the dual variables so that CS conditions, i.e., , hold.
- The initialization phase including three sub-phases: column reduction, reduction transfer, augmenting row reduction.
- (a)
- Column reduction. Each element of a column subtracts a positive value. For the input cost matrix , from right to left, subtract the smallest value in the current column from each element in each column of the input matrix, as shown in Algorithm 2 (from line 3 to line 11). In this process, each column is assigned to a minimal row element, and some rows may not be assigned.
- (b)
- Reduction transfer, further reducing the unassigned rows. First, suppose the minimum value in the row i is , then perform the inverse column reduction, i.e., add to all elements in column j. After that, subtract from all elements in row i, as shown in Algorithm 2 (from line 13 to line 16).
- (c)
- Augmenting row reduction, trying to find a set of alternate paths. Starting from an unassigned row i, i.e., a node in initial topology, attempt to find the alternate path by first finding the current minimum value of in row i, and then finding the second minimum value , where . Next, reduce all elements of row i by . If , the new is negative. Assign i to column j with the reverse column reduction for column j. If column j has previously been assigned to row m, repeat this step from row m. This repeats until either row m is matched to an unassigned column, or it becomes impossible to transfer reduction to the selected row m. This process is shown in Algorithm 2 (from line 18 to line 32).
- The augmentation phase, which is the core of the algorithm to construct a bijective graph. For each unassigned row, the augmentation phase will find a shortest alternate path to the unassigned column through a modified Dijkstra’s algorithm. Starting with an unassigned row i, the search returns a shortest path to column j. If column j is assigned to row k, then add row k to the path. If the distances via row k to any given column are shorter, update these distances. After augmentation, all assignments correspond to the minimum value of each row in the cost matrix, which finally leads to the assignment with the lowest weight, i.e., the minimum global energy consumption. The assignment is the mapping relationship from the initial topology coordinates to the target topology coordinates. This process is shown in Algorithm 3.
Algorithm 2: Optimal Mapping Through Linear Assignment |
|
Algorithm 3: Augmentation Process of LAPJV |
|
- Obtain the initial and target coordinates matrices and by Algorithm 1.
- Obtain the optimal coordinate mapping relationship by Algorithms 2 and 3.
- Achieve the topology shaping based on the optimal coordinate mapping relationship .
5. Numerical Results
5.1. Validation of Proposed Topology Shaping Method
5.2. Global Energy Consumption
5.3. Computational Complexity
6. Conclusions and Future Work
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
References
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Number of nodes | 32 |
Spatial area | 1000 m × 1000 m × 1000 m |
Aircraft length | 350 m |
Aircraft wingspan | 350 m |
Aircraft height | 70 m |
Accuracy requirement | 10 |
Number of nodes | 32 |
Spatial area | 1000 m × 1000 m × 1000 m |
Cube side length | 240 m |
Accuracy requirement | 10 |
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Yang, Y.; Zhang, X.; Zhou, J.; Li, B.; Qin, K. A Relative Coordinate-Based Topology Shaping Method for UAV Swarm with Low Computational Complexity. Appl. Sci. 2022, 12, 2631. https://doi.org/10.3390/app12052631
Yang Y, Zhang X, Zhou J, Li B, Qin K. A Relative Coordinate-Based Topology Shaping Method for UAV Swarm with Low Computational Complexity. Applied Sciences. 2022; 12(5):2631. https://doi.org/10.3390/app12052631
Chicago/Turabian StyleYang, Yanxiang, Xiangyin Zhang, Jiayi Zhou, Bo Li, and Kaiyu Qin. 2022. "A Relative Coordinate-Based Topology Shaping Method for UAV Swarm with Low Computational Complexity" Applied Sciences 12, no. 5: 2631. https://doi.org/10.3390/app12052631
APA StyleYang, Y., Zhang, X., Zhou, J., Li, B., & Qin, K. (2022). A Relative Coordinate-Based Topology Shaping Method for UAV Swarm with Low Computational Complexity. Applied Sciences, 12(5), 2631. https://doi.org/10.3390/app12052631