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26 pages, 18629 KiB  
Article
Advanced UAV Material Transportation and Precision Delivery Utilizing the Whale-Swarm Hybrid Algorithm (WSHA) and APCR-YOLOv8 Model
by Yuchen Wu, Zhijian Wei, Huilin Liu, Jiawei Qi, Xu Su, Jiqiang Yang and Qinglin Wu
Appl. Sci. 2024, 14(15), 6621; https://doi.org/10.3390/app14156621 - 29 Jul 2024
Viewed by 877
Abstract
This paper proposes an effective material delivery algorithm to address the challenges associated with Unmanned Aerial Vehicle (UAV) material transportation and delivery, which include complex route planning, low detection precision, and hardware limitations. This novel approach integrates the Whale-Swarm Hybrid Algorithm (WSHA) with [...] Read more.
This paper proposes an effective material delivery algorithm to address the challenges associated with Unmanned Aerial Vehicle (UAV) material transportation and delivery, which include complex route planning, low detection precision, and hardware limitations. This novel approach integrates the Whale-Swarm Hybrid Algorithm (WSHA) with the APCR-YOLOv8 model to enhance efficiency and accuracy. For path planning, the placement paths are transformed into a Generalized Traveling Salesman Problem (GTSP) to be able to compute solutions. The Whale Optimization Algorithm (WOA) is improved for balanced global and local searches, combined with an Artificial Bee Colony (ABC) Algorithm and adaptive weight adjustment to quicken convergence and reduce path costs. For precise placement, the YOLOv8 model is first enhanced by adding the SimAM attention mechanism to the C2f module in the detection head, focusing on target features. Secondly, GhoHGNetv2 using GhostConv is the backbone of YOLOv8 to ensure accuracy while reducing model Params and FLOPs. Finally, a Lightweight Shared Convolutional Detection Head (LSCDHead) further reduces Params and FLOPs through shared convolution. Experimental results show that WSHA reduces path costs by 9.69% and narrows the gap between the best and worst paths by about 34.39%, compared to the Improved Whale Optimization Algorithm (IWOA). APCR-YOLOv8 reduces Params and FLOPs by 44.33% and 34.57%, respectively, with [email protected] increasing from 88.5 to 92.4 and FPS reaching 151.3. This approach can satisfy the requirements for real-time responsiveness while effectively preventing missed, false, and duplicate detections during the inspection of emergency airdrop stations. In conclusion, combining bionic optimization algorithms and image processing significantly enhances the efficiency and precision of material placement in emergency management. Full article
(This article belongs to the Special Issue Advanced Research and Application of Unmanned Aerial Vehicles)
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Figure 1
<p>Flowchart of Whale-Swarm Hybrid Algorithm.</p>
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<p>APCR-YOLOv8 network structure diagram.</p>
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<p>SimAM Attention Mechanism.</p>
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<p>C2f_SimAM: C2f module with the addition of the SimAM attention mechanism.</p>
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<p>GhostConv Schematic.</p>
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<p>GhoHGNetv2 internal module structure. (<b>a</b>) HGStem module structure; (<b>b</b>) GhoHGBlock module structure (shortcut = True); (<b>c</b>) GhoHGBlock module structure (shortcut = False).</p>
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<p>LSCDHead structure and internal modules. (<b>a</b>) LSCDHead structure; (<b>b</b>) Internal structure of GnConv and ShConv.</p>
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<p>Comparison of the variation of the shortest path lengths of the four datasets. (<b>a</b>) City (17,11); (<b>b</b>) City (24,15); (<b>c</b>) City (31,16); (<b>d</b>) City (39,25).</p>
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<p>Comparison of the variation of the shortest path lengths of the four datasets. (<b>a</b>) City (17,11); (<b>b</b>) City (24,15); (<b>c</b>) City (31,16); (<b>d</b>) City (39,25).</p>
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<p>Four dataset path-planning diagrams. (<b>a</b>) City (17,11); (<b>b</b>) City (24,15); (<b>c</b>) City (31,16); (<b>d</b>) City (39,25).</p>
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<p>Four dataset path-planning diagrams. (<b>a</b>) City (17,11); (<b>b</b>) City (24,15); (<b>c</b>) City (31,16); (<b>d</b>) City (39,25).</p>
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<p>City (39,25) path planning and real space mapping.</p>
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<p>Example dataset. (<b>a</b>) Original image; (<b>b</b>) Original image labelling; (<b>c</b>) Data-enhanced image; (<b>d</b>) Data-enhanced image labelling. Note: The red boxes indicate the position of the target object to be detected.</p>
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<p>Illustration of the physical situation of UAV load drop.</p>
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<p>Comparison of YOLOv8 and APCR-YOLOv8 heat maps. (<b>a</b>) Original plot of the dataset; (<b>b</b>) heat map of YOLOv8; (<b>c</b>) heat map of APCR-YOLOv8.</p>
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<p>Example analyses of emergency airdrop station detection using APCR-YOLOv8. (<b>a</b>) Solving the false detection problem by accurately distinguishing between actual and false targets; (<b>b</b>) Solving the tiny target missed detection problem by enhancing the detection of small and difficult-to-detect targets; (<b>c</b>) Solving the repeat detection problem by avoiding repeated detections of the same target; (<b>d</b>) Enhancing detection confidence with higher confidence scores in the bounding boxes.</p>
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26 pages, 1654 KiB  
Article
Optimal Coverage Path Planning for Agricultural Vehicles with Curvature Constraints
by Maria Höffmann, Shruti Patel and Christof Büskens
Agriculture 2023, 13(11), 2112; https://doi.org/10.3390/agriculture13112112 - 7 Nov 2023
Cited by 7 | Viewed by 2347
Abstract
Complete coverage path planning (CCPP) is vital in mobile robot applications. Optimizing CCPP is particularly significant in precision agriculture, where it enhances resource utilization, reduces soil compaction, and boosts crop yields. This work offers a comprehensive approach to CCPP for agricultural vehicles with [...] Read more.
Complete coverage path planning (CCPP) is vital in mobile robot applications. Optimizing CCPP is particularly significant in precision agriculture, where it enhances resource utilization, reduces soil compaction, and boosts crop yields. This work offers a comprehensive approach to CCPP for agricultural vehicles with curvature constraints. Our methodology comprises four key stages. First, it decomposes complex agricultural areas into simpler cells, each equipped with guidance tracks, forming a fixed track system. The subsequent route planning and smooth path planning stages compute a path that adheres to path constraints, optimally traverses the cells, and aligns with the track system. We use the generalized traveling salesman problem (GTSP) to determine the optimal traversing sequence. Additionally, we introduce an algorithm for calculating paths that are both smooth and curvature-constrained within individual cells, as well as paths that enable seamless transitions between cells, resulting in a smooth, curvature-constraint coverage path. Our modular approach allows method flexibility at each step. We evaluate our method on real agricultural fields, demonstrating its effectiveness in minimizing path length, ensuring efficient coverage, and adhering to curvature constraints. This work establishes a strong foundation for precise and efficient agricultural coverage path planning, with potential for further real-world applications and enhancements. Full article
(This article belongs to the Special Issue Agricultural Automation in Smart Farming)
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<p>Exemplary visualization of the four key steps of the CCPP workflow. (<b>a</b>) ROI decomposition with headland area (yellow) and interior field (green) split up into cells. (<b>b</b>) Generation of the headland (orange) and interior (green) guidance tracks. (<b>c</b>) Route planning: computing the cell traversing order (numbers) and entry (red) and exit (green) tracks. (<b>d</b>) Planning a smooth interior path (red).</p>
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<p>CCPP workflow established in this work, comprising the four key components: ROI decomposition (purple), guidance track generation (blue), route planning (red), and smooth path planning (green), together with the input parameters (grey) and output paths (orange). The focus in this work deals with the right arc of this flow, resulting in interior coverage paths, whereas the left part handles the headland paths, as in [<a href="#B11-agriculture-13-02112" class="html-bibr">11</a>].</p>
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<p>Sweepline-based decomposition: Sweepline (red) sweeps (here from left to right) over the region with one obstacle (gray), detects events (green), and decomposes the region with lines (red) parallel to the sweepline, reprinted with permission from [<a href="#B15-agriculture-13-02112" class="html-bibr">15</a>] (2023, Höffmann, M.). (<b>a</b>) Initial region. (<b>b</b>) Decomposition into 4 cells. (<b>c</b>) Examplary decomposition of more complex region into 10 cells.</p>
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<p>Different sweepline-based decomposition techniques with the considered events (green), assuming a vertical sweepline, reprinted with permission from [<a href="#B15-agriculture-13-02112" class="html-bibr">15</a>] (2023, Höffmann, M.) (<b>a</b>) Trapezoidal [<a href="#B27-agriculture-13-02112" class="html-bibr">27</a>]. (<b>b</b>) Boustrophedon [<a href="#B28-agriculture-13-02112" class="html-bibr">28</a>].</p>
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<p>Distinction between the intra-region (red) and inter-region (blue) paths. The intra-region paths cover each cell, whereas the inter-region paths connect the different intra-region paths, using the headland guidance tracks (orange).</p>
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<p>Visualization of the key components in the Dubins model (<a href="#FD1-agriculture-13-02112" class="html-disp-formula">1</a>): The position <math display="inline"><semantics> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>)</mo> </mrow> </semantics></math>, the orientation <math display="inline"><semantics> <mi>θ</mi> </semantics></math>, the curvature <math display="inline"><semantics> <mi>κ</mi> </semantics></math> (as the inverse of the resulting turning radius), and the velocity <span class="html-italic">v</span>, as well as the steering angle <math display="inline"><semantics> <mi>φ</mi> </semantics></math> and the wheelbase <span class="html-italic">b</span>.</p>
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<p>Exemplary start pose (<math display="inline"><semantics> <mrow> <mi>x</mi> <msub> <mrow/> <mn>1</mn> </msub> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>y</mi> <msub> <mrow/> <mn>1</mn> </msub> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>q</mi> <msub> <mrow/> <mn>1</mn> </msub> </mrow> </semantics></math>) and end pose (<math display="inline"><semantics> <mrow> <mi>x</mi> <msub> <mrow/> <mn>1</mn> </msub> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>y</mi> <msub> <mrow/> <mn>1</mn> </msub> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>q</mi> <msub> <mrow/> <mn>1</mn> </msub> </mrow> </semantics></math>) of two interior guidance tracks (blue) and a sketched turning maneuver (red).</p>
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<p>Selection of the most common agricultural turn types, traversing in the headland area from one interior guidance track to the other, reprinted with permission from [<a href="#B15-agriculture-13-02112" class="html-bibr">15</a>] (2023, Höffmann, M.). (<b>a</b>) U. (<b>b</b>) Flat U. (<b>c</b>) <math display="inline"><semantics> <mo>Ω</mo> </semantics></math>. (<b>d</b>) Hook.</p>
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<p>Visualization of the computation of a piecewise linear inter-region path (combination of red dotted and solid line). The tracks corresponding to the start and end points (purple) are expanded (black dashed line), and the intersections (red dots) with the headland tracks (orange) are calculated. Afterwards, (1) the path is refined along the headland tracks, (2) the path is aligned along the interior tracks (green), and (3) the two sub-paths are connected using their shared headland track (red dotted line).</p>
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<p>Four different ways to cover a cell due to different start (green) and end (red) positions. (<b>a</b>) Odd number of tracks. (<b>b</b>) Even number of tracks.</p>
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<p>Phaseplot and curvature profile of regular Dubins (blue) and CC Dubins (orange) turns for the two most common turning maneuvers. (<b>a</b>) Flat U-turn. (<b>b</b>) <math display="inline"><semantics> <mo>Ω</mo> </semantics></math>-turn.</p>
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<p>Flat U-turn for different turning radii <math display="inline"><semantics> <mrow> <msub> <mi>r</mi> <mi>min</mi> </msub> <mo>&lt;</mo> <mfrac> <mi>w</mi> <mn>2</mn> </mfrac> </mrow> </semantics></math> with <math display="inline"><semantics> <mrow> <msub> <mi>κ</mi> <mi>max</mi> </msub> <mo>=</mo> <msubsup> <mi>r</mi> <mi>min</mi> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msubsup> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <msub> <mi>σ</mi> <mi>max</mi> </msub> <mo>=</mo> <mn>20</mn> </mrow> </semantics></math>. (<b>a</b>) Phaseplot. (<b>b</b>) Curvature profile.</p>
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<p><math display="inline"><semantics> <mo>Ω</mo> </semantics></math>-turn for different turning radii <math display="inline"><semantics> <mrow> <msub> <mi>r</mi> <mi>min</mi> </msub> <mo>&gt;</mo> <mfrac> <mi>w</mi> <mn>2</mn> </mfrac> </mrow> </semantics></math> with <math display="inline"><semantics> <mrow> <msub> <mi>κ</mi> <mi>max</mi> </msub> <mo>=</mo> <msubsup> <mi>r</mi> <mi>min</mi> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msubsup> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <msub> <mi>σ</mi> <mi>max</mi> </msub> <mo>=</mo> <mn>10</mn> </mrow> </semantics></math>. (<b>a</b>) Phaseplot. (<b>b</b>) Curvature profile.</p>
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<p>Exemplary visualization for inter-region paths with different turning radii r<math display="inline"><semantics> <mrow> <msub> <mrow/> <mi>m</mi> </msub> <mi>i</mi> <mi>n</mi> </mrow> </semantics></math> based on the piecewise linear path (black). (<b>a</b>) Phaseplot. (<b>b</b>) Curvature profile along the (normalized) path.</p>
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<p>Field 1. Given field boundaries (black), boustrophedon decomposition (gray), and resulting guidance track system containing headland tracks (green) and interior tracks (blue). Optimal cell-traversing sequence (numbers) together with the entry (red) and exit (green) points for each cell. Resulting smooth coverage path (red).</p>
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<p>Field 2. Given field boundaries (black), boustrophedon decomposition (gray), and resulting guidance track system containing headland tracks (green) and interior tracks (blue). Optimal cell-traversing sequence (numbers) together with the entry (red) and exit (green) points for each cell. Resulting smooth coverage path (red).</p>
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<p>Field 3. Given field boundaries (black), boustrophedon decomposition (gray), and resulting guidance track system containing headland tracks (green) and interior tracks (blue). Optimal cell-traversing sequence (numbers) together with the entry (red) and exit (green) points for each cell. Resulting smooth coverage path (red).</p>
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<p>Field 4. Given field boundaries (black), boustrophedon decomposition (gray), and resulting guidance track system containing headland tracks (green) and interior tracks (blue). Optimal cell-traversing sequence (numbers) together with the entry (red) and exit (green) points for each cell. Resulting smooth coverage path (red).</p>
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14 pages, 345 KiB  
Article
Fast and Low-Overhead Time Synchronization for Industrial Wireless Sensor Networks with Mesh-Star Architecture
by Zhaowei Wang, Tailiang Yong and Xiangjin Song
Sensors 2023, 23(8), 3792; https://doi.org/10.3390/s23083792 - 7 Apr 2023
Cited by 1 | Viewed by 1553
Abstract
Low-overhead, robust, and fast-convergent time synchronization is important for resource-constrained large-scale industrial wireless sensor networks (IWSNs). The consensus-based time synchronization method with strong robustness has been paid more attention in wireless sensor networks. However, high communication overhead and slow convergence speed are inherent [...] Read more.
Low-overhead, robust, and fast-convergent time synchronization is important for resource-constrained large-scale industrial wireless sensor networks (IWSNs). The consensus-based time synchronization method with strong robustness has been paid more attention in wireless sensor networks. However, high communication overhead and slow convergence speed are inherent drawbacks for consensus time synchronization due to inefficient frequent iterations. In this paper, a novel time synchronization algorithm for IWSNs with a mesh–star architecture is proposed, namely, fast and low-overhead time synchronization (FLTS). The proposed FLTS divides the synchronization phase into two layers: mesh layer and star layer. A few resourceful routing nodes in the upper mesh layer undertake the low-efficiency average iteration, and the massive low-power sensing nodes in the star layer synchronize with the mesh layer in a passive monitoring manner. Therefore, a faster convergence and lower communication overhead time synchronization is achieved. The theoretical analysis and simulation results demonstrate the efficiency of the proposed algorithm in comparison with the state-of-the-art algorithms, i.e., ATS, GTSP, and CCTS. Full article
(This article belongs to the Special Issue Wireless Sensor Networks in Industrial Applications)
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<p>Schematic illustration of two-layered IWSNs. The number in figure denotes the node ID.</p>
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<p>The convergence of the logical clock in the mesh layer. (<b>a</b>) Logical clock skew; (<b>b</b>) Logical clock offset.</p>
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<p>Convergence comparison of the logical clock in whole network. (<b>a</b>) Logical clock skew; (<b>b</b>) Logical clock offset.</p>
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<p>Comparison of communication overhead.</p>
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<p>Convergence comparison of the logical clock in intra-cluster synchronization. (<b>a</b>) Logical clock skew; (<b>b</b>) Logical clock offset.</p>
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<p>Convergence comparison of the logical clock in inter-cluster synchronization. (<b>a</b>) Logical clock skew; (<b>b</b>) Logical clock offset.</p>
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<p>Convergence comparison of the logical clock against node mobility. (<b>a</b>) Logical clock skew; (<b>b</b>) Logical clock offset.</p>
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<p>Convergence comparison of the logical clock against communication delay. (<b>a</b>) Logical clock skew; (<b>b</b>) Logical clock offset.</p>
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19 pages, 1123 KiB  
Article
Performance Analysis of Time Synchronization Protocols in Wireless Sensor Networks
by Linh-An Phan, Taejoon Kim, Taehong Kim, JaeSeang Lee and Jae-Hyun Ham
Sensors 2019, 19(13), 3020; https://doi.org/10.3390/s19133020 - 9 Jul 2019
Cited by 20 | Viewed by 5120
Abstract
The time synchronization protocol is indispensable in various applications of wireless sensor networks, such as scheduling, monitoring, and tracking. Numerous protocols and algorithms have been proposed in recent decades, and many of them provide micro-scale resolutions. However, designing and implementing a time synchronization [...] Read more.
The time synchronization protocol is indispensable in various applications of wireless sensor networks, such as scheduling, monitoring, and tracking. Numerous protocols and algorithms have been proposed in recent decades, and many of them provide micro-scale resolutions. However, designing and implementing a time synchronization protocol in a practical wireless network is very challenging compared to implementation in a wired network; this is because its performance can be deteriorated significantly by many factors, including hardware quality, message delay jitter, ambient environment, and network topology. In this study, we measure the performance of the Flooding Time Synchronization Protocol (FTSP) and Gradient Time Synchronization Protocol (GTSP) in terms of practical network conditions, such as message delay jitter, synchronization period, network topology, and packet loss. This study provides insights into the operation and optimization of time synchronization protocols. In addition, the performance evaluation identifies that FTSP is highly affected by message delay jitter due to error accumulation over multi-hops. We demonstrate that the proposed extended version of the FTSP (E-FTSP) alleviates the effect of message delay jitter and enhances the overall performance of FTSP in terms of error, time, and other factors. Full article
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<p>Node model in OPNET used to implement protocols in the evaluation.</p>
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<p>Network error and maximum neighbor error of FTSP and GTSP in two settings: without message delay and with delay jitter up to 5 µs. Random seed = 10.</p>
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<p>Synchronization error between reference node (at 1 × 1) and other nodes in grid topology in two settings: without message delay and with the message delay jitter up to 5 µs.</p>
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<p>Relative skew of nearest node and farthest node with reference node in (<b>a</b>) FTSP and (<b>b</b>) E-FTSP with grid topology (7 × 7).</p>
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<p>Average <span class="html-italic">estimatedDelay</span> value of nodes at same hop distance to the reference node.</p>
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<p>Network errors and maximum neighbor errors of FTSP, GTSP, and E-FTSP in the presence of message delay jitter.</p>
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<p>Number of rounds required to synchronize entire network in FTSP and GTSP with different synchronization periods.</p>
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<p>Network error and maximum neighbor error of FTSP, GTSP and E-FTSP with different synchronization periods.</p>
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<p>Hop distance between reference node (root) and farthest node in grid topology.</p>
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<p>(<b>a</b>) Number or rounds required to synchronize entire network with different hop distances between reference node and farthest node. (<b>b</b>) Synchronization error of FTSP with different hop distances between reference node and farthest node.</p>
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<p>(<b>a</b>) Network error and maximum neighbor error of GTSP with different topologies. (<b>b</b>) Number of rounds required to synchronize entire network in GTSP with different topologies.</p>
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<p>Number of rounds required to synchronize entire network in GTSP with different transmission ranges.</p>
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<p>(<b>a</b>) Average network error of FTSP, GTSP, and E-FTSP according to the network size (number of nodes). (<b>b</b>) Average number of rounds required to synchronize entire network of FTSP, GTSP, and E-FTSP according to the network size (number of nodes).</p>
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<p>(<b>a</b>) Number or rounds required to synchronize entire network for different packet loss ratios. (<b>b</b>) Network synchronization error of each protocol with different packet loss ratios.</p>
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16 pages, 11854 KiB  
Article
Spatial Transformation of Equality – Generalized Travelling Salesman Problem to Travelling Salesman Problem
by Mohammed Zia, Ziyadin Cakir and Dursun Zafer Seker
ISPRS Int. J. Geo-Inf. 2018, 7(3), 115; https://doi.org/10.3390/ijgi7030115 - 15 Mar 2018
Cited by 8 | Viewed by 4667
Abstract
The Equality-Generalized Travelling Salesman Problem (E-GTSP), which is an extension of the Travelling Salesman Problem (TSP), is stated as follows: given groups of points within a city, like banks, supermarkets, etc., find a minimum cost Hamiltonian cycle that visits each group exactly once. [...] Read more.
The Equality-Generalized Travelling Salesman Problem (E-GTSP), which is an extension of the Travelling Salesman Problem (TSP), is stated as follows: given groups of points within a city, like banks, supermarkets, etc., find a minimum cost Hamiltonian cycle that visits each group exactly once. It can model many real-life combinatorial optimization scenarios more efficiently than TSP. This study presents five spatially driven search-algorithms for possible transformation of E-GTSP to TSP by considering the spatial spread of points in a given urban city. Presented algorithms are tested over 15 different cities, classified by their street-network’s fractal-dimension. Obtained results denote that the R-Search algorithm, which selects the points from each group based on their radial separation with respect to the start–end point, is the best search criterion for any E-GTSP to TSP conversion modelled for a city street network. An 8.8% length error has been reported for this algorithm. Full article
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Figure 1
<p>Representation of one possible (<b>a</b>) closed and (<b>b</b>) open Hamiltonian cycle in a given symmetric E-GTSP instance. Here, the group-counts are equal to 5, and the total number of vertices are equal to 16; in other words, <math display="inline"> <semantics> <mrow> <mi>x</mi> <mo>=</mo> <mn>5</mn> </mrow> </semantics> </math> and <math display="inline"> <semantics> <mrow> <mi>n</mi> <mo>=</mo> <mn>16</mn> </mrow> </semantics> </math>, respectively.</p>
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<p>Diagrammatic representation of five different proposed search algorithms. Three groups of different points, marked by oval, triangular, and rectangular markers, are used to show an E-GTSP instance. It should be noted that, for R-Search, RE-Search, and RD-Search, there are two stations from each group after <b>decision 1</b> (<a href="#sec2-ijgi-07-00115" class="html-sec">Section 2</a>), thus generating eight (<math display="inline"> <semantics> <mrow> <mn>2</mn> <mo>×</mo> <mn>2</mn> <mo>×</mo> <mn>2</mn> </mrow> </semantics> </math>) different TSPs for <b>decision 2</b>.</p>
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<p>Histogram showing the frequency of cities (marked on the world map <b>right</b>) for each Fractal-Dimension bin. Increase in the dimension represents an increase in the road-density. Three cities are selected from each of the top five bins to test all proposed search algorithms. Each city-polygon’s bounding box is used to download the corresponding XML data from OSM Overpass API.</p>
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<p>3D-graph between Different Search Algorithms, Number of Stations to be visited, i.e., <math display="inline"> <semantics> <mrow> <mrow> <mo>|</mo> </mrow> <msup> <mi>V</mi> <mi>s</mi> </msup> <mrow> <mo>|</mo> </mrow> </mrow> </semantics> </math>, and City’s Fractal-Dimension bin. Each bin represents three cities taken from <a href="#ijgi-07-00115-f003" class="html-fig">Figure 3</a>. Color represents the average fractional error in route-length coming out from each algorithm with respect to the optimal one (estimated by brute-force), with the color black having errors of more than 20%. (<b>a</b>) It should be noted that, with an increase in the <math display="inline"> <semantics> <mrow> <mrow> <mo>|</mo> </mrow> <msup> <mi>V</mi> <mi>s</mi> </msup> <mrow> <mo>|</mo> </mrow> </mrow> </semantics> </math> for a given E-GTSP instance, the error also gets increased irrespective of the choice of the algorithm (marked by big arrow). (<b>b</b>) The graph is sliced-down for each group-count, individually. For each city, the algorithm with the least errors is marked by black box. It is clear that, for small <math display="inline"> <semantics> <mrow> <mrow> <mo>|</mo> </mrow> <msup> <mi>V</mi> <mi>s</mi> </msup> <mrow> <mo>|</mo> </mrow> </mrow> </semantics> </math> values, the D-Search is better, and, as one increases the number of stations, the R-Search outperforms the other, marked by solid-arrows. Overall, the R-Search is advised for any urban based E-GTSP to TSP conversion (<a href="#sec5-ijgi-07-00115" class="html-sec">Section 5</a>).</p>
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<p>(<b>a</b>) scatter plot between the Average Fractional Error and the City Number (representing city, <a href="#ijgi-07-00115-t001" class="html-table">Table 1</a>) for all <math display="inline"> <semantics> <mrow> <mrow> <mo>|</mo> </mrow> <msup> <mi>V</mi> <mi>s</mi> </msup> <mrow> <mo>|</mo> </mrow> </mrow> </semantics> </math> instances. Nodes belonging to the D-Search and R-Search are joined together to allow comparison. It can be seen that, for lower <math display="inline"> <semantics> <mrow> <mrow> <mo>|</mo> </mrow> <msup> <mi>V</mi> <mi>s</mi> </msup> <mrow> <mo>|</mo> </mrow> </mrow> </semantics> </math> instances, the D-Search is better than the R-Search, and vice-versa for higher <math display="inline"> <semantics> <mrow> <mrow> <mo>|</mo> </mrow> <msup> <mi>V</mi> <mi>s</mi> </msup> <mrow> <mo>|</mo> </mrow> </mrow> </semantics> </math> instances, as also reported in <a href="#ijgi-07-00115-f004" class="html-fig">Figure 4</a>; (<b>b</b>) the <span class="html-italic">y</span>-axis represents the summation of Average Fractional Error from all <math display="inline"> <semantics> <mrow> <mrow> <mo>|</mo> </mrow> <msup> <mi>V</mi> <mi>s</mi> </msup> <mrow> <mo>|</mo> </mrow> </mrow> </semantics> </math> instances. It is said that the R-Search is overall the best search criterion to perform a geospatial E-GTSP to TSP conversion.</p>
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<p>Possible explanation of the out-performance of R-Search for increased complexity. It can be seen that the total number of optimal stations within the white region increases with increase in the complexity of the model (<b>left plot</b>). Since R-Search also selects optimal stations from the white region, it performs better for higher <math display="inline"> <semantics> <mrow> <mrow> <mo>|</mo> </mrow> <msup> <mi>V</mi> <mi>s</mi> </msup> <mrow> <mo>|</mo> </mrow> </mrow> </semantics> </math> instances (<b>right plot</b>).</p>
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<p>Graph comparing the absolute route lengths for brute-force and D-Search/R-Search approaches. It can be seen that the intensity of the error increases with increase in the length of the trip. Although only <math display="inline"> <semantics> <mrow> <mrow> <mo>|</mo> </mrow> <msup> <mi>V</mi> <mi>s</mi> </msup> <mrow> <mo>|</mo> <mo>=</mo> <mn>5</mn> </mrow> </mrow> </semantics> </math> instance is plotted here, a similar fanning-out behavior (dashed- and solid-line) is observed for other cases too. Severe errors are expected for long trips irrespective of the choice of algorithm or complexity of model.</p>
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<p>Graph (<b>left</b>) between the start–end points’ geo-separation and corresponding optimal route length for all five group-counts (for Brussels). Note that a polygon is sketched to show the spread of each data point. It should be noted how polygons drift away from the mean-line for increased complexity. On the <b>right</b>, we have slope and <math display="inline"> <semantics> <msup> <mi>R</mi> <mn>2</mn> </msup> </semantics> </math> values of trendlines induced by each <math display="inline"> <semantics> <mrow> <mrow> <mo>|</mo> </mrow> <msup> <mi>V</mi> <mi>s</mi> </msup> <mrow> <mo>|</mo> </mrow> </mrow> </semantics> </math> instance for all 15 cities, where there are decreases with increases in complexity. The decreasing <math display="inline"> <semantics> <msup> <mi>R</mi> <mn>2</mn> </msup> </semantics> </math> and slope value along the <span class="html-italic">x</span>-axis depicts the complex networking of urban street-roads, and shows how optimal route length gets similar irrespective of the start–end points’ spatial separation.</p>
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<p>Percentage error of all proposed search algorithms. R-Search is better among all other searches. The authors do believe that this spatial approach for E-GTSP to TSP transformation could be improved by further incorporating heuristic concepts, which is beyond the scope of the current study.</p>
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Article
Dubins Traveling Salesman Problem with Neighborhoods: A Graph-Based Approach
by Jason T. Isaacs and João P. Hespanha
Algorithms 2013, 6(1), 84-99; https://doi.org/10.3390/a6010084 - 4 Feb 2013
Cited by 47 | Viewed by 10601
Abstract
We study the problem of finding the minimum-length curvature constrained closed path through a set of regions in the plane. This problem is referred to as the Dubins Traveling Salesperson Problem with Neighborhoods (DTSPN). An algorithm is presented that uses sampling to cast [...] Read more.
We study the problem of finding the minimum-length curvature constrained closed path through a set of regions in the plane. This problem is referred to as the Dubins Traveling Salesperson Problem with Neighborhoods (DTSPN). An algorithm is presented that uses sampling to cast this infinite dimensional combinatorial optimization problem as a Generalized Traveling Salesperson Problem (GTSP) with intersecting node sets. The GTSP is then converted to an Asymmetric Traveling Salesperson Problem (ATSP) through a series of graph transformations, thus allowing the use of existing approximation algorithms. This algorithm is shown to perform no worse than the best existing DTSPN algorithm and is shown to perform significantly better when the regions overlap. We report on the application of this algorithm to route an Unmanned Aerial Vehicle (UAV) equipped with a radio to collect data from sparsely deployed ground sensors in a field demonstration of autonomous detection, localization, and verification of multiple acoustic events. Full article
(This article belongs to the Special Issue Graph Algorithms)
Show Figures

Figure 1

Figure 1
<p>Example DTSPN with the corresponding “GTSP with intersecting node sets”. (<b>a</b>) Example instance of DTSPN with three circular regions <math display="inline"> <mrow> <msub> <mi mathvariant="script">R</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi mathvariant="script">R</mi> <mn>2</mn> </msub> <mo>,</mo> <mspace width="4.pt"/> <mtext>and</mtext> <mspace width="4.pt"/> <msub> <mi mathvariant="script">R</mi> <mn>3</mn> </msub> </mrow> </math> and samples <math display="inline"> <mrow> <msub> <mi>S</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>S</mi> <mn>2</mn> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>S</mi> <mn>8</mn> </msub> </mrow> </math>. The circuit through samples <math display="inline"> <mrow> <msub> <mi>S</mi> <mn>2</mn> </msub> <mo>,</mo> <mspace width="4.pt"/> <mtext>and</mtext> <mo>,</mo> <msub> <mi>S</mi> <mn>8</mn> </msub> </mrow> </math> is the optimal tour; (<b>b</b>) Problem (P0): A GTSP with intersecting node sets representation of the DTSPN example. Note: only an essential subset of arcs is shown for clarity of illustration.</p>
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<p>Example of Problem (P1) and Problem (P2) from Stage 1 of transformation. (<b>a</b>) Problem (P1): Any arcs that do not enter at least one new node set <math display="inline"> <mrow> <mo>{</mo> <mo stretchy="false">(</mo> <mn>3</mn> <mo>,</mo> <mn>5</mn> <mo stretchy="false">)</mo> </mrow> </math> and <math display="inline"> <mrow> <mo stretchy="false">(</mo> <mn>6</mn> <mo>,</mo> <mn>8</mn> <mo stretchy="false">)</mo> <mo>}</mo> </mrow> </math> have been removed from the graph in Problem (P0); (<b>b</b>) Problem (P2): A large finite cost <span class="html-italic">α</span> is added to each edge. Here <math display="inline"> <mrow> <msub> <mover accent="true"> <mi>c</mi> <mo>^</mo> </mover> <mrow> <mi>i</mi> <mo>,</mo> <mi>j</mi> </mrow> </msub> <mo>=</mo> <msubsup> <mi>c</mi> <mrow> <mi>i</mi> <mo>,</mo> <mi>j</mi> </mrow> <mn>2</mn> </msubsup> </mrow> </math>, where <math display="inline"> <msubsup> <mi>c</mi> <mrow> <mi>i</mi> <mo>,</mo> <mi>j</mi> </mrow> <mn>2</mn> </msubsup> </math> is defined in Equation (4).</p>
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<p>Example of Problem (P3) and Problem (P4) from Stage 1 and Stage 2 of transformation. (<b>a</b>) Problem (P3): Nodes <math display="inline"> <msub> <mi>S</mi> <mn>2</mn> </msub> </math> and <math display="inline"> <msub> <mi>S</mi> <mn>3</mn> </msub> </math> from (P2) lie in multiple node sets. These nodes are duplicated and the spawned nodes <math display="inline"> <msub> <mi>S</mi> <msup> <mn>2</mn> <mo>′</mo> </msup> </msub> </math> and <math display="inline"> <msub> <mi>S</mi> <msup> <mn>3</mn> <mo>′</mo> </msup> </msub> </math> are placed in node set <math display="inline"> <msub> <mi mathvariant="script">V</mi> <mn>2</mn> </msub> </math>. Zero cost arcs (dashed arrows) are added connecting <math display="inline"> <msub> <mi>S</mi> <mn>2</mn> </msub> </math> to <math display="inline"> <msub> <mi>S</mi> <msup> <mn>2</mn> <mo>′</mo> </msup> </msub> </math> and <math display="inline"> <msub> <mi>S</mi> <mn>3</mn> </msub> </math> to <math display="inline"> <msub> <mi>S</mi> <msup> <mn>3</mn> <mo>′</mo> </msup> </msub> </math>; (<b>b</b>) Problem (P4): The intra-set arc <math display="inline"> <mrow> <mo stretchy="false">(</mo> <mn>5</mn> <mo>,</mo> <msup> <mn>3</mn> <msup> <mrow/> <mo>′</mo> </msup> </msup> <mo stretchy="false">)</mo> </mrow> </math> from Problem (P3) is removed.</p>
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<p>Example of Problem (P5) and Problem (P6) from Stage 3 of transformation. (<b>a</b>) Problem (P5): The clustered TSP is created by forming zero cost intra-set cycles and adjusting the originating node in each inter-set arc; (<b>b</b>) Problem (P6): A large finite cost <span class="html-italic">β</span> is added to each inter-set edge. Here <math display="inline"> <mrow> <msub> <mover accent="true"> <mi>c</mi> <mo>¯</mo> </mover> <mrow> <mi>i</mi> <mo>,</mo> <mi>j</mi> </mrow> </msub> <mo>=</mo> <msubsup> <mi>c</mi> <mrow> <mi>i</mi> <mo>,</mo> <mi>j</mi> </mrow> <mn>6</mn> </msubsup> </mrow> </math>, where <math display="inline"> <msubsup> <mi>c</mi> <mrow> <mi>i</mi> <mo>,</mo> <mi>j</mi> </mrow> <mn>6</mn> </msubsup> </math> is defined in Equation (5). The optimal tour is shown in red with a cost of <math display="inline"> <mrow> <msub> <mover accent="true"> <mi>c</mi> <mo>^</mo> </mover> <mrow> <mn>8</mn> <mo>,</mo> <mn>2</mn> </mrow> </msub> <mo>+</mo> <mi>β</mi> <mo>+</mo> <msub> <mover accent="true"> <mi>c</mi> <mo>^</mo> </mover> <mrow> <mn>2</mn> <mo>,</mo> <mn>8</mn> </mrow> </msub> </mrow> </math>.</p>
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<p>A comparison of IRA and RCM on an example DTSPN instance with three regions and three sample poses. (<b>a</b>) Example Tour: IRA, Tour Length <math display="inline"> <mrow> <mo>=</mo> <mn>7</mn> <mo>.</mo> <mn>7</mn> </mrow> </math>; (<b>b</b>) Example Tour: RCM, Tour Length <math display="inline"> <mrow> <mo>=</mo> <mn>15</mn> <mo>.</mo> <mn>4</mn> </mrow> </math>.</p>
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<p>Simulation results for 100 Monte Carlo trials where both IRA and RCM optimized over the same 50 sample poses. (<b>a</b>) The color represents the average of the ratio of the tour length under IRA to the tour length under the RCM planning algorithm. Here the red regions indicate near parity in performance while the blue regions indicate that IRA produced tours that are approximately half the length of tours produced by the RCM algorithm; (<b>b</b>) The color represents the average of the ratio of the size of the ATSP solved under IRA to the size of the ATSP solved under the RCM planning algorithm. Here the blue regions indicate near parity in size while the red regions indicate that IRA increased the size of the ATSP by as much as four times.</p>
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<p>Simulation results for the where IRA optimized over 50 sample poses and RCM optimized over the same 50 samples plus an additional sample for each duplicated node in the IRA. These extra samples ensured that both algorithms solved the same size ATSP.</p>
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<p>The configuration of sensors and UAV trajectory during the field demonstration at Camp Roberts, CA. (<b>a</b>) Field Demonstration Description. The acoustic sensors visited by the data collecting UAV are shown as yellow dots; (<b>b</b>) The blue lines represent the GPS logs of the path taken by data collecting UAV during the test. The desired path was sent to the autopilot via the square waypoints. The sensors and communication regions are represented by green and blue circles respectively.</p>
Full article ">Figure 8 Cont.
<p>The configuration of sensors and UAV trajectory during the field demonstration at Camp Roberts, CA. (<b>a</b>) Field Demonstration Description. The acoustic sensors visited by the data collecting UAV are shown as yellow dots; (<b>b</b>) The blue lines represent the GPS logs of the path taken by data collecting UAV during the test. The desired path was sent to the autopilot via the square waypoints. The sensors and communication regions are represented by green and blue circles respectively.</p>
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