Comparison of Innovative Strategies for the Coverage Problem: Path Planning, Search Optimization, and Applications in Underwater Robotics
Abstract
1. Introduction
Problem Definition
- TSP path on a point lattice—gold standard for minimum-length tours, but constrained to a fixed selection of way-points, only varying the order.
- MST–Hex and MST–Square—fast, scalable heuristics. The literature lacks a quantitative evaluation of their coverage loss.
- The OCP [3]—originally formulated for energy-aware path planning, and we adapt it here to address the coverage problem, supplying guidelines for the coverage terms and for warm-starting with TSP.
- Benchmark canonical planners. We deliver a head-to-head quantitative comparison of TSP, MST and an OCP baseline, using the bi-objective metric “uncovered area vs. path length”.
- Lightweight MST variant for on-board use. There are two key novelties:
- (a)
- Tree-to-contour. Dilate the MST and keep its outer boundary, obtaining a single closed sweep line.
- (b)
- Real-time runtime. Coarse grid, Kruskal algorithm, and two image operations complete in under 0.5 s on an Intel i7-12700H.
- Coverage-tuned OCP. We replace the original energy cost with a “missed-area + path-length” objective, warm-start the solver with the TSP solution, and enforce obstacle constraints directly in the collocation mesh.
2. Literature Review
2.1. Search Theory Problem
2.2. Coverage Problem
2.3. Minimum Spanning Tree
- The surface can be divided into N areas of equal surface a priori, but this does not guarantee optimality, since some areas can be much more difficult to explore;
- The problem can be posed and solved for a single robot; then, the trajectory will be divided into N trajectories of equal length, which does not allow for recovering the robots at the place from where they started;
- Finally, the calculated minimum spanning tree can be divided into a forest of trees (by removing some branches) of the same size, which then allows them to recover N closed trajectories. If N = 2, both robots will be released and recovered at the same point.
2.4. Traveling Salesman Problem
2.5. Optimal Control Problem
- A mathematical model of the system to be controlled,
- A cost function,
- A specification of all boundary conditions on states and constraints to be satisfied by states and controls, and
- A statement of what variables are free.
3. Methodology
3.1. Optimal Control Problem (OCP)
3.1.1. Detection Model Implementation
3.1.2. Optimal Control Problem Formulation
3.2. Minimum Spanning Tree
- The presence of a third dimension;
- Winds and sea currents fluctuating over time;
- The difficulty of navigating certain robots (sailboats, gliders, etc.);
- The risk of losing a robot (risks of collision, breakdowns, etc.);
- The specificity of sensors in a marine environment;
- The difficulty of communicating with fixed stations or between robots when they are underwater.
- The extraction of a minimum spanning tree is performed using Kruskal’s algorithm, which is a greedy algorithm that can be proven to be optimal. It proceeds by sorting the edges in ascending order and iteratively adding them to the tree, ensuring no cycles are formed. Since the number of vertices is proportional to the area to be explored, the number of edges is proportional to the square of this area.
- The dilation operation is equivalent to a convolution with a disk of radius r. Assuming r is small relative to the exploration area, the cost of this operation is linear with respect to the surface area.
- Contour extraction can also be formulated as a convolution, for instance, using the Sobel edge detection method, which remains a linear-time operation in terms of surface area. However, to obtain the final trajectory, it is more practical to perform contour following, which can also be executed in linear time.
- The use of a regular grid ensures minimal-length trajectories in a rectangular domain. However, it may be beneficial to introduce additional points in the discretization to refine exploration in specific areas or to navigate through narrow passages.
3.3. Traveling Salesman Problem
3.3.1. Problem Statement
3.3.2. Objective Function
3.3.3. Constraints
3.3.4. Implementation Considerations
4. Numerical Results
4.1. Single-Sample Illustration
4.2. Aggregate Performance over 10 Samples
4.3. Discussion
4.4. Marine Application
- Zero stream current.
- Vehicle traveling at a constant speed.
- The problem is considered to be a simple planar motion at the sea surface (i.e., pitch, roll, and heave motions are zero).
- Yaw rate can be directly controlled. (The Nomoto time constant is considered to be very large)
- (a)
- Segment selection. Starting from the last active index k, advance to the first segment whose end-point is still at least ahead of .
- (b)
- Orthogonal projection. With and , compute
- (c)
- Carrot point and desired heading. The look-ahead (“carrot”) point is . Bearing to that point defines the desired heading
4.5. Outer Polygon Extraction from Map Images
5. Conclusions
Future Work
Author Contributions
Funding
Data Availability Statement
Acknowledgments
Conflicts of Interest
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Algorithm | Avg. UAR (%) | Avg. ALOP | Avg. Time (s) |
---|---|---|---|
TSP | 4.76 | 15.80 | |
MST-hex | 14.06 | 15.14 | |
MST-square | 16.13 | 14.54 | |
OCP | 3.67 | 14.92 | 2306 |
Boustrophedon | 3.52 | 18.35 |
Design Parameter | Value |
---|---|
Nomoto Gain Constant, K | 0.5 1/s |
Nomoto Time Constant, T | 5.0 s |
Velocity, V | 2.5 m/s |
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Ibrahim, A.; Rego, F.F.C.; Busvelle, É. Comparison of Innovative Strategies for the Coverage Problem: Path Planning, Search Optimization, and Applications in Underwater Robotics. J. Mar. Sci. Eng. 2025, 13, 1369. https://doi.org/10.3390/jmse13071369
Ibrahim A, Rego FFC, Busvelle É. Comparison of Innovative Strategies for the Coverage Problem: Path Planning, Search Optimization, and Applications in Underwater Robotics. Journal of Marine Science and Engineering. 2025; 13(7):1369. https://doi.org/10.3390/jmse13071369
Chicago/Turabian StyleIbrahim, Ahmed, Francisco F. C. Rego, and Éric Busvelle. 2025. "Comparison of Innovative Strategies for the Coverage Problem: Path Planning, Search Optimization, and Applications in Underwater Robotics" Journal of Marine Science and Engineering 13, no. 7: 1369. https://doi.org/10.3390/jmse13071369
APA StyleIbrahim, A., Rego, F. F. C., & Busvelle, É. (2025). Comparison of Innovative Strategies for the Coverage Problem: Path Planning, Search Optimization, and Applications in Underwater Robotics. Journal of Marine Science and Engineering, 13(7), 1369. https://doi.org/10.3390/jmse13071369