In this study, I address the challenge of deploying fixed-wing drones for rapid and complete coverage search of submarines in large, complex, and irregular sea areas. Traditional approaches often simplify search regions to convex polygons, but real-world maritime boundaries are frequently concave and irregular, leading to inefficient search paths with excessive turning costs. I propose an improved strategy that combines selective concave point removal with Dubins curve-based path planning for fixed-wing drones. The key contributions include: (1) a mathematical model for search detection cost accounting for flight distance, time, and energy; (2) a method to determine the optimal search direction by computing the polygon width; (3) a selective concave point removal algorithm that avoids “violent” removal of all concavities, using edge-length and angle thresholds; and (4) integration of Dubins curves to handle the minimum turning radius constraint of fixed-wing drones. Simulation results demonstrate that the proposed approach significantly reduces invalid navigation cost compared to conventional methods, providing theoretical support for efficient submarine search operations.
1. Introduction
The detection of submarines has become increasingly difficult due to advances in noise reduction and stealth technology. When a submarine is near the surface (e.g., at periscope depth, snorkeling, or shallow-water transit), non-acoustic detection methods such as magnetic anomaly detection, infrared, laser, and radar can be employed. Fixed-wing drones are ideal platforms for such sensors because of their speed, endurance, and wide-area coverage capability. However, the operational constraints of fixed-wing drones—particularly their minimum turning radius—make coverage path planning nontrivial, especially over irregular sea regions.
Existing research on multi-drone cooperative search often focuses on convex or simplified rectangular areas. For concave polygonal regions, a common approach is to remove all concave points to convert the region into a convex hull, then apply standard convex polygon search strategies. This “brute-force” removal, however, introduces large redundant search areas and increases invalid flight paths. My work improves upon this by selectively removing concave points based on geometric criteria, thereby preserving the original search region’s shape while reducing unnecessary turns.
Furthermore, I incorporate Dubins curves to model the turning trajectory of fixed-wing drones when the minimum turning radius exceeds half the sensor swath width, ensuring no coverage gaps occur. The overall search cost is minimized by aligning the search direction with the polygon’s width direction, which yields the fewest turns.
2. Mathematical Model of Search and Detection Cost
2.1 Fixed-Wing Drone Sensor Field of View
When a fixed-wing drone flies at a constant altitude \( h \), its onboard sensor covers a rectangular footprint on the sea surface. The along-track length \( D_l \) and across-track width \( D_s \) of this footprint are given by:
$$
D_l = h \cdot \left[ \cot(\eta – \lambda) – \cot(\eta + \lambda) \right]
$$
$$
D_s = \frac{2h \tan \gamma}{\sin \eta}
$$
where \( \gamma \) is the maximum lateral detection angle, \( \lambda \) is the maximum longitudinal detection angle, and \( \eta \) is the sensor projection angle. These parameters depend on the sensor’s specifications and the drone’s altitude.
2.2 Fixed-Wing Drone Motion Model
I model the fixed-wing drone as a point mass moving in a 2D plane at constant altitude. Its kinematics in the inertial frame are:
$$
\begin{cases}
\dot{x}(t) = v(t) \cos \phi(t) \\
\dot{y}(t) = v(t) \sin \phi(t) \\
\dot{\phi}(t) = \omega(t) \theta_{\max}
\end{cases}
$$
where \( (x, y) \) is the position, \( v \) is the speed, \( \phi \) is the yaw angle, \( \omega \) is the normalized yaw rate (between -1 and 1), and \( \theta_{\max} \) is the maximum yaw rate. For a single drone, the index \( i \) is omitted.
2.3 Search Cost Formulation
The total search cost \( C \) is composed of three parts: travel distance cost \( L \), time cost \( T \), and energy cost \( P \). The weighted sum is:
$$
C = \omega_1 L + \omega_2 T + \omega_3 P
$$
The total path length \( L \) includes the straight-line distance to the start point \( L_s \), the straight search legs \( L_l \), and the turning arcs \( L_c \). Similarly, time \( T \) is calculated using straight-flight speed \( v_l \) and turning speed \( v_t \):
$$
T = \frac{L_s + L_l}{v_l} + \frac{L_c}{v_t}
$$
The energy consumption \( P \) integrates power during straight flight \( P_l(t) \) and turning flight \( P_t(t) \):
$$
P = \int_{0}^{(L_s + L_l)/v_l} P_l(t)\, dt + \int_{0}^{L_c/v_t} P_t(t)\, dt
$$
In my simulations, I set \( v_l = 600 \) km/h, \( v_t = 350 \) km/h, \( P_l = 2 \) kW, \( P_t = 6.5 \) kW, and equal weights \( \omega_1 = \omega_2 = \omega_3 = 1 \) for simplicity.
3. Coverage Search Strategy for Irregular Sea Areas
3.1 Convex Polygon Search with Span Direction
For a convex polygonal search region, the optimal flight direction is perpendicular to the polygon’s width (the minimum distance between two parallel supporting lines). The width can be found using the “point-edge” span algorithm. Let the convex polygon have vertices \( V = \{v_1, v_2, \dots, v_n\} \) in counterclockwise order. For each edge \( v_i v_{i+1} \), the distance to every other vertex is computed, and the maximum distance is the span for that edge. The width is the minimum span among all edges:
$$
W = \min_{i=1,\dots,n} \left( \max_{j \neq i,i+1} d(v_i v_{i+1}, v_j) \right)
$$
Once the width direction is identified, the fixed-wing drone flies along that direction in a back-and-forth (S-shaped) pattern. The number of turns required is:
$$
N_t = \left\lceil \frac{W}{D_s} \right\rceil – 1
$$
where \( \lceil \cdot \rceil \) is the ceiling function. When the minimum turning radius \( R_t \le D_s/2 \), the turn can be a simple semicircle; otherwise, a Dubins curve must be used to avoid coverage gaps.
3.2 Dubins Curve for Turning
For fixed-wing drones with \( R_t > D_s/2 \), I employ Dubins curves to connect consecutive search swaths. A Dubins path is the shortest smooth curve between two directed points (configurations) under curvature constraints. The path is composed of arcs of radius \( R_t \) and straight segments. The six possible Dubins path types are: LSL, RSR, RSL, LSR, RLR, LRL, where L/R denotes left/right turn and S denotes straight. The optimal type minimizes the total path length.
Given start configuration \( (x_s, y_s, \alpha) \) and end configuration \( (x_e, y_e, \beta) \), the Dubins path length \( \Omega \) is computed analytically. I use the standard Dubins method to generate the turning trajectory for each leg transition. This ensures full coverage of the search area without blind spots.
3.3 Selective Concave Point Removal for Irregular Polygons
When the search region is a concave polygon, directly applying convex polygon methods leads to many unnecessary turns inside concavities. I propose a selective concave point removal algorithm that only removes concave points when doing so significantly reduces the turn count without excessively increasing the search area.
First, I identify concave points using vector cross products. For a polygon with vertices in counterclockwise order, a vertex \( v_k \) is concave if the cross product of vectors \( \overrightarrow{v_k v_{k-1}} \) and \( \overrightarrow{v_k v_{k+1}} \) is negative.
Define the edge set \( G = \{(c_1, c_2), \dots, (c_i, c_j)\} \) and the decision distance \( D_d(p, q) = \min_{q_k \in q} \| p – q_k \| \). For each concave point \( p \), I compute the ratio:
$$
\rho = \frac{\text{length of the edge adjacent to } p}{D_d(p, \text{other vertices})}
$$
If \( \rho < N \) (where \( N \) is a threshold, e.g., 2), the concave point is removed by connecting its two neighbors. Additionally, if \( \rho \ge N \), I check the interior angle \( \theta \) at the concave point; if \( \theta < \Theta \) (another threshold, e.g., 30°), the point is also removed because the indentation is very sharp and the added redundant area is small. This two-step criterion avoids both “violent” removal and overly conservative retention.
The algorithm iterates until no more points satisfy the removal conditions. The resulting polygon is a simplified convex hull that closely approximates the original region while smoothing out narrow concavities.
4. Simulation Results and Validation
I conducted comparative simulations to evaluate the effectiveness of the proposed strategy. The parameters for the fixed-wing drone and sensor are summarized in Table 1.
| Parameter | Symbol | Value |
|---|---|---|
| Sensor swath width | \( D_s \) | 2 km |
| Sensor along-track length | \( D_l \) | 5 km |
| Minimum turning radius | \( R_t \) | 1.5 km |
| Straight-flight speed | \( v_l \) | 600 km/h |
| Turning speed | \( v_t \) | 350 km/h |
| Straight-flight power | \( P_l \) | 2 kW |
| Turning power | \( P_t \) | 6.5 kW |
| Edge-length threshold | \( N \) | 2.0 |
| Angle threshold | \( \Theta \) | 30° |
The test search region is a concave polygon as shown in the following figure (representative of a complex irregular sea area).
4.1 Without Concave Point Removal
First, I applied the convex polygon search method directly to the concave region without any concave point removal. The search direction was varied from 0° to 180°, and the total path length, number of turns, and invalid navigation costs were recorded. The results are shown in Table 2 (for the optimal direction).
| Metric | Value (optimal direction = 90°) |
|---|---|
| Total path length \( L \) | 2,443 km |
| Invalid time cost \( T_{tc} \) | 1.06 h |
| Invalid energy cost \( P_{tc} \) | 9.72×10⁴ (arb. units) |
| Number of turns (left+right) | 14 |
The optimal search direction was found to be 90° (parallel to the width direction), confirming the width-based approach. However, the concave region caused many small turning loops inside the concavities, increasing the number of turns and path length.
4.2 With Selective Concave Point Removal
After applying the selective concave point removal algorithm, the polygon was simplified to a near-convex shape. The simulation was repeated. Table 3 presents the results.
| Metric | Value (optimal direction = 90°) |
|---|---|
| Total path length \( L \) | 3,085 km |
| Invalid time cost \( T_{tc} \) | 0.88 h |
| Invalid energy cost \( P_{tc} \) | 9.16×10⁴ (arb. units) |
| Number of turns (left+right) | 8 |
Although the total path length increased slightly (due to the expanded convex hull), the number of turns dropped significantly from 14 to 8, leading to lower invalid time and energy costs. This demonstrates the benefit of selective concave point removal for fixed-wing drones.
4.3 Comparison of Strategies
Table 4 summarizes the key performance indicators for four representative cases.
| Case | Total Path (km) | Invalid Time Cost (h) | Invalid Energy Cost (×10⁴) |
|---|---|---|---|
| Original concave, non-span direction (0° direction, Fig. 14 equivalent) | 2,409 | 1.11 | 1.03 |
| Original concave, span direction (90°) | 2,443 | 1.06 | 0.972 |
| After concave removal, non-span direction | 3,247 | 0.94 | 0.984 |
| After concave removal, span direction (proposed) | 3,085 | 0.88 | 0.916 |
The proposed method (last row) achieves the lowest invalid time and energy costs, despite a longer total path. This trade-off is favorable for fixed-wing drones where turning is costly in terms of time and fuel.
4.4 Effect of Search Direction
I also analyzed how the search direction rotation angle affects the performance for the simplified polygon. The total path length and number of turns as functions of rotation angle are plotted (although images are not shown here, the trend is clear). The minimal invalidity cost occurs at 90° rotation, aligning with the width direction. The left and right turn counts are nearly symmetric, indicating a well-balanced Dubins search pattern.
5. Conclusion
In this work, I have developed a comprehensive coverage search path planning strategy for fixed-wing drones operating over complex irregular sea areas. The key innovations are: (1) a mathematical cost model that accounts for distance, time, and energy; (2) determination of the optimal search direction via polygon width computation; (3) a selective concave point removal algorithm that avoids excessive redundant area; and (4) integration of Dubins curves to respect the minimum turning radius of fixed-wing drones. Simulation results show that the proposed method reduces invalid navigation cost by up to 20% compared to conventional convex-only approaches, while maintaining full coverage. This provides a practical theoretical foundation for deploying fixed-wing drones in real-world submarine search missions. Future work may extend the strategy to multi-drone cooperative scenarios and dynamic environments.