In modern battlefield environments, the coordinated penetration of multiple unmanned aerial vehicles (UAVs) has emerged as a critical operational paradigm. The successful execution of such missions hinges on the ability to generate safe, efficient, and cooperative trajectories that allow a formation of UAVs to navigate through complex threat zones while maintaining formation integrity and avoiding collisions. Our research focuses on developing a hierarchical trajectory planning framework that integrates an improved Rapidly-exploring Random Tree Star (RRT*) algorithm for the leader aircraft with an enhanced Artificial Potential Field (APF) model for the follower aircraft. This approach addresses the fundamental challenges of coordinating multiple drones in dynamic and hostile environments, where traditional planning methods often fall short in terms of efficiency, safety, and adaptability. The core contribution of our work lies in the seamless fusion of these two methodologies, leveraging the asymptotic optimality of RRT* for global path generation and the reactive capabilities of APF for local trajectory adjustments. Throughout this paper, we emphasize the pivotal role of drone technology in enabling such sophisticated cooperative behaviors, and we demonstrate through extensive simulations that our proposed method significantly outperforms conventional approaches in terms of path length, threat avoidance, and formation stability.
Introduction to Multi-UAV Cooperative Penetration
The domain of drone technology has witnessed remarkable advancements over the past decade, transforming the way military and civilian operations are conceived and executed. Among the most challenging applications is the coordinated penetration of multiple UAVs into defended airspace, where the aircraft must simultaneously avoid radar detection, anti-aircraft artillery, and other threats while maintaining a predefined formation. The complexity of this task arises from the need to balance multiple, often conflicting, objectives: minimizing exposure to threats, preserving fuel efficiency through optimal path lengths, ensuring collision avoidance among formation members, and adapting to dynamic changes in the environment. Our research addresses this problem by decomposing the planning process into two distinct layers: a global planning layer for the leader UAV, which computes a primary penetration path using an improved version of the RRT* algorithm, and a local planning layer for the follower UAVs, which uses an enhanced APF model to adjust their trajectories in real-time. This hierarchical architecture not only simplifies the computational burden but also provides the flexibility needed to respond to unforeseen disturbances. The improvements we introduce to the RRT* algorithm include a heuristic random probability sampling strategy that biases node generation toward the goal, a greedy algorithm for eliminating redundant path nodes, and a cubic Bezier curve smoothing technique to ensure path continuity. For the APF model, we design a multi-coefficient force field that combines formation maintenance attraction, leader-following attraction and repulsion, and threat avoidance repulsion. This integrated approach represents a significant advancement in drone technology, offering a practical solution for real-world multi-UAV cooperative missions.
Problem Formulation for Multi-UAV Penetration
We consider a scenario where a pair of UAVs, designated as the leader and the follower, must penetrate a defended area containing multiple threat zones. The leader UAV is responsible for determining the primary penetration path, while the follower UAV must maintain a specified formation relative to the leader while avoiding threats and preventing collisions. The operational environment is modeled as a three-dimensional space with dimensions 15 km by 20 km by 5 km. The mission begins at a starting point and ends at a target point, with the UAVs flying at a constant speed of 60 m/s. The threats in the environment are modeled as hemispherical radar zones and cylindrical anti-aircraft artillery zones, each defined by a center coordinate, a radius, and a height. The objective is to generate a trajectory for the leader that minimizes the total path length while avoiding all threat zones, and to generate a corresponding trajectory for the follower that maintains the formation, avoids threats, and prevents collisions with the leader. This problem is inherently challenging due to the coupling between the leader’s path planning and the follower’s local adjustments. Our proposed method addresses this coupling by first planning the leader’s path using the improved RRT* algorithm, and then using this path as a reference for the follower’s APF-based trajectory optimization. The formation geometry is defined using a distance-angle description, where the follower’s ideal position is offset from the leader’s position by a fixed vector in the leader’s body frame. This offset is transformed into the inertial frame using the leader’s pitch and yaw angles, ensuring that the formation is maintained regardless of the leader’s orientation. The following table summarizes the key parameters used in our problem formulation.
| Parameter | Symbol | Value | Description |
|---|---|---|---|
| Environment dimensions | X × Y × Z | 15 km × 20 km × 5 km | Spatial bounds for the battlefield |
| Starting point | Pstart | (1500, 2000, 800) m | Initial position of the leader UAV |
| Target point | Pgoal | (9210, 16200, 1500) m | Destination of the mission |
| Flight speed | v | 60 m/s | Constant speed for all UAVs |
| Formation offset vector | Δ | (150, -150, -150) m | Offset in leader body frame |
| Threat radius (radar) | Rr | 500 m | Radius of hemispherical threat zones |
| Threat radius (artillery) | Rp | 400 m | Radius of cylindrical threat zones |
Improved RRT* Algorithm for Leader Path Planning
The RRT* algorithm is a sampling-based path planning method that guarantees asymptotic optimality by iteratively rewiring the tree to minimize path cost. However, the standard RRT* algorithm suffers from slow convergence due to the randomness of its sampling strategy, which often results in redundant explorations and suboptimal paths. To address this limitation, we propose three key improvements to the RRT* algorithm, specifically tailored for the multi-UAV penetration scenario. These improvements are designed to accelerate convergence, reduce path length, and ensure smooth trajectories, thereby enhancing the overall efficiency of drone technology in complex missions.
Heuristic Random Probability Sampling
In the standard RRT* algorithm, the random sample point Qrand is selected uniformly at random from the entire configuration space. This lack of bias leads to undirected tree expansion and slow convergence toward the goal. Our first improvement introduces a heuristic bias by setting the random sample point equal to the goal point Qgoal with a dynamically adjusted probability. Instead of using a fixed probability, we employ a random probability p0 uniformly distributed in the interval (0, 1). At each iteration, Qrand is set to Qgoal with probability p0, and to a uniformly random point otherwise. This dynamic bias balances exploration and exploitation: when p0 is close to 1, the algorithm aggressively pursues the goal, potentially leading to faster convergence but risking local optima; when p0 is close to 0, the algorithm explores more broadly, avoiding local traps but slowing down convergence. The random nature of p0 allows the algorithm to naturally transition between these two behaviors, improving overall performance. The probability of selecting the goal point is given by:
$$P(Q_{rand} = Q_{goal}) = p_0$$
where p0 is sampled uniformly from (0, 1) at each iteration. This simple yet effective modification significantly reduces the number of iterations required to find a feasible path, while maintaining the algorithm’s ability to escape local minima. In the context of drone technology, this translates to faster mission planning and reduced computational overhead, which are critical for time-sensitive operations.
Greedy Redundancy Elimination
Even with improved sampling, the path generated by RRT* often contains redundant nodes that increase the total path length without contributing to threat avoidance. The second improvement applies a greedy algorithm to eliminate these redundant nodes, further optimizing the path. The process begins at the start node and iteratively checks each subsequent node. For a given node, we find the farthest node in the sequence that can be directly connected without intersecting any threat zone. This farthest node becomes the new parent of the current node, and the intermediate nodes are removed. This process is repeated until no further reductions are possible. The algorithm can be summarized as follows:
- Initialize the path as a sequence of nodes from start to goal.
- Set the current index i to 0 (start node).
- Find the farthest node j > i such that the straight line segment between node i and node j does not intersect any threat zone.
- Remove all nodes between i and j from the path.
- Set i = j and repeat steps 3-4 until j is the goal node.
- Repeat the entire process starting from the second node until no further reductions occur.
This greedy approach effectively shortens the path by eliminating unnecessary detours, resulting in a more direct and efficient trajectory. The reduction in path length is particularly significant in environments with sparse threat distributions, where the initial RRT* path may contain large deviations from the optimal route. By integrating this technique into our drone technology framework, we ensure that the leader UAV follows a near-optimal path that minimizes fuel consumption and exposure time.
Cubic Bezier Curve Smoothing
The path obtained after redundancy elimination is still piecewise linear, with sharp corners at the nodes. Such paths are unsuitable for UAV flight due to the physical constraints on turning angles and acceleration. The third improvement applies cubic Bezier curve smoothing to generate a continuous and smooth trajectory. A cubic Bezier curve is defined by four control points P0, P1, P2, and P3, where P0 and P3 are the endpoints of the path segment, and P1 and P2 determine the shape of the curve. The curve is parametrized by a variable t ranging from 0 to 1:
$$B(t) = (1-t)^3 P_0 + 3t(1-t)^2 P_1 + 3t^2(1-t) P_2 + t^3 P_3$$
For each segment of the path, we select P0 and P3 as the endpoints of the segment. The intermediate control points P1 and P2 are chosen to ensure that the curve is tangent to the incoming and outgoing directions at the endpoints. This is achieved by placing P1 along the direction from P0 to the next node, and P2 along the direction from P3 to the previous node. The distance of P1 and P2 from their respective endpoints is set to a fraction of the segment length to control the curvature. The following table compares the performance of the improved RRT* algorithm with the standard RRT* and basic RRT algorithms across three different battlefield scenarios.
| Algorithm | Scenario 1 | Scenario 2 | Scenario 3 | Average |
|---|---|---|---|---|
| Improved RRT* | 16905 m | 17979 m | 17377 m | 17420 m |
| Standard RRT* | 17453 m | 18070 m | 17464 m | 17662 m |
| Basic RRT | 17533 m | 18469 m | 18106 m | 18036 m |
The results clearly demonstrate the superiority of the improved RRT* algorithm. On average, the improved RRT* reduces the path length by 1.37% compared to the standard RRT* and by 3.42% compared to the basic RRT. The improvement is most pronounced in Scenario 1, where the threat distribution is moderate, with an optimization of over 3% relative to the standard RRT*. This confirms that the combination of heuristic sampling, greedy redundancy elimination, and Bezier smoothing effectively enhances the efficiency of path planning in the context of drone technology.
Artificial Potential Field Model for Follower Trajectory Optimization
While the improved RRT* algorithm provides a high-quality global path for the leader UAV, the follower UAV requires a local trajectory planning mechanism that can react to dynamic disturbances and maintain formation integrity. The Artificial Potential Field (APF) method is well-suited for this purpose due to its computational efficiency and reactive nature. We develop an enhanced APF model that incorporates three distinct force components: a formation maintenance attractive force, a leader-following attractive or repulsive force, and a threat avoidance repulsive force. The resultant force determines the instantaneous acceleration of the follower UAV, guiding it toward its ideal position while avoiding obstacles and maintaining safe distances.
Formation Maintenance Attractive Force
The formation maintenance attractive force is designed to keep the follower UAV at its ideal position relative to the leader. The ideal position of the follower at the i-th time step is computed from the leader’s path point and the formation offset vector. The transformation from the leader’s body frame to the inertial frame is accomplished using the leader’s pitch and yaw angles. The rotation matrices for pitch and yaw are defined as:
$$R_{Pitch,i} = \begin{bmatrix} 1 & 0 & 0 \\ 0 & \cos\theta_{Pitch,i} & -\sin\theta_{Pitch,i} \\ 0 & \sin\theta_{Pitch,i} & \cos\theta_{Pitch,i} \end{bmatrix}$$
$$R_{Yaw,i} = \begin{bmatrix} \cos\theta_{Yaw,i} & -\sin\theta_{Yaw,i} & 0 \\ \sin\theta_{Yaw,i} & \cos\theta_{Yaw,i} & 0 \\ 0 & 0 & 1 \end{bmatrix}$$
The ideal follower position Tari is then given by:
$$Tar_i = Path_i + R_{Pitch,i} R_{Yaw,i} \Delta$$
where Δ = [150, -150, -150] m is the fixed offset vector. The formation maintenance attractive force Fk is proportional to the distance between the follower’s actual position and its ideal position:
$$F_k = \frac{k_0}{d_0} \begin{bmatrix} x_k(t+1) – x_c(t) \\ y_k(t+1) – y_c(t) \\ z_k(t+1) – z_c(t) \end{bmatrix}$$
where k0 is the attractive gain coefficient, (xk, yk, zk) is the ideal follower position, (xc, yc, zc) is the actual follower position, and d0 is the Euclidean distance between them. When d0 exceeds a predefined threshold, the force becomes active and pulls the follower toward its ideal position. This mechanism ensures that the follower maintains the desired formation geometry, which is essential for coordinated penetration in drone technology applications.
Leader-Following Attraction and Repulsion
In addition to formation maintenance, the follower must also regulate its distance to the leader to prevent collisions and avoid separation. This is achieved through a distance-dependent force Fc that switches between attraction and repulsion based on the relative distance dx between the leader and the follower. When dx is too large (the follower is straying from the formation), a attractive force pulls the follower toward the leader. When dx is within the desired range, the force is zero. When dx is too small (the risk of collision exists), a repulsive force pushes the follower away. The force is defined as:
$$F_c = \begin{cases}
c \frac{1}{d_x} \begin{bmatrix} x_t(t) – x_c(t) \\ y_t(t) – y_c(t) \\ z_t(t) – z_c(t) \end{bmatrix}, & d_x < d – \lambda \\
0, & d – \lambda \leq d_x < d + \lambda \\
-c \frac{1}{d_x} \begin{bmatrix} x_t(t) – x_c(t) \\ y_t(t) – y_c(t) \\ z_t(t) – z_c(t) \end{bmatrix}, & d_x \geq d + \lambda
\end{cases}$$
where c is the leader repulsion coefficient, d is the desired formation distance, and λ is a threshold that defines the dead zone. The distance dx is computed as the Euclidean distance between the leader’s position (xt, yt, zt) and the follower’s position (xc, yc, zc). This bi-directional force ensures that the follower maintains a safe and stable distance from the leader, which is critical for avoiding mid-air collisions in multi-UAV operations. The incorporation of this force into our APF model enhances the safety and reliability of drone technology in cooperative missions.
Threat Avoidance Repulsive Force
The third component of the APF model is the threat avoidance repulsive force Fp, which pushes the follower away from threat zones when it comes too close. Each threat zone i is defined by its center (xoi, yoi, zoi), its radius Rr, and an influence range ΔR. The repulsive force is activated when the follower’s distance to the threat center di falls within the range Rr < di ≤ Rr + ΔR. The force magnitude is inversely proportional to the distance, ensuring that the repulsion increases as the follower approaches the threat boundary:
$$F_p = \begin{cases}
m \frac{1}{d_i} \begin{bmatrix} x_c(t) – x_{oi} \\ y_c(t) – y_{oi} \\ z_c(t) – z_{oi} \end{bmatrix}, & R_r < d_i \leq R_r + \Delta R \\
0, & d_i > R_r + \Delta R
\end{cases}$$
where m is the threat avoidance repulsion coefficient. The direction of the force is away from the threat center, effectively pushing the follower to a safer location. The influence range ΔR determines how early the follower begins to react to the threat, providing a trade-off between safety and path efficiency. In our implementation, we set ΔR = 500 m to give the follower sufficient time to adjust its trajectory without causing excessive deviations. The following table summarizes the force coefficients used in our APF model, which were optimized through iterative simulation experiments.
| Force Component | Symbol | Value | Description |
|---|---|---|---|
| Formation attraction gain | k0 | 5 | Strength of attraction to ideal position |
| Leader repulsion coefficient | c | 0.1 | Strength of leader-following force |
| Threat repulsion coefficient | m | 1 × 1010 | Strength of threat avoidance force |
| Threat influence range | ΔR | 500 m | Range within which threat force acts |
The resultant force acting on the follower UAV is the vector sum of the three components: Fa = Fk + Fc + Fp. This resultant force determines the acceleration of the follower, which is then integrated to update its velocity and position. The follower continuously computes this force at each time step, allowing it to react dynamically to changes in the leader’s position, the presence of threats, and any external disturbances. This reactive capability is a key advantage of the APF method, making it highly suitable for real-time applications in drone technology.
Simulation Results and Analysis
We conducted extensive simulations to validate the effectiveness of our proposed hierarchical planning framework. The simulations were performed in three distinct battlefield scenarios, each characterized by different configurations of threat zones. Scenario 1 features a moderate number of threats with relatively sparse distribution. Scenario 2 presents a more complex environment with densely packed threats. Scenario 3 offers an intermediate complexity with threats arranged in a mixed pattern. For each scenario, we compared the performance of our improved RRT* algorithm for the leader with the standard RRT* and basic RRT algorithms. We also evaluated the impact of the APF-based trajectory optimization for the follower by comparing the minimum distance to threat zones achieved with and without the APF correction.
The path length results for the leader UAV are presented in Table 2. The improved RRT* algorithm consistently achieves shorter path lengths across all three scenarios. In Scenario 1, the improvement over the standard RRT* is 3.14%, while in Scenarios 2 and 3, the improvements are 0.50% and 0.49%, respectively. The larger improvement in Scenario 1 can be attributed to the fact that the greedy redundancy elimination is more effective when the threat distribution allows for more direct connections between nodes. In more cluttered environments, the potential for redundancy is lower, but the improved algorithm still maintains a competitive advantage. Across all scenarios, the improved RRT* demonstrates superior convergence and path quality, underscoring its value in drone technology applications where efficiency is paramount.
For the follower UAV, the APF-based optimization yields dramatic improvements in threat avoidance. Before applying the APF correction, the follower’s trajectory closely follows the leader’s path, often coming dangerously close to threat zones. In Scenario 1, the minimum distance between the follower and any threat zone was 180 m, which is below the safe threshold of 200 m. After applying the APF correction, this distance increased to 220 m, representing a 22.2% improvement. In Scenario 2, the minimum distance improved from 100 m to 200 m, a 100% increase. In Scenario 3, the improvement was from 40 m to 200 m, a 400% increase. The average improvement across all scenarios is 174.07%, demonstrating the effectiveness of the APF model in enhancing safety. The following table summarizes these results.
| Scenario | Without APF | With APF | Improvement |
|---|---|---|---|
| Scenario 1 | 180 m | 220 m | 22.2% |
| Scenario 2 | 100 m | 200 m | 100.0% |
| Scenario 3 | 40 m | 200 m | 400.0% |
| Average | 106.7 m | 206.7 m | 174.1% |
The standard deviation of the follower’s distance to threat zones also decreased significantly after applying the APF correction, from 70.24 m to 11.55 m. This reduction indicates that the APF model not only increases the minimum distance but also stabilizes the follower’s trajectory, preventing it from oscillating between safe and unsafe regions. The consistent performance across all three scenarios demonstrates the robustness of our approach, which is essential for reliable drone technology in unpredictable environments.
An additional observation from the simulations is the interaction between the three force components of the APF model. The formation maintenance force ensures that the follower tracks the leader’s path with high fidelity. The leader-following force prevents the follower from drifting too far or too close to the leader, maintaining a safe separation. The threat avoidance force provides an additional layer of safety by pushing the follower away from danger zones. The combination of these forces results in a trajectory that is both safe and efficient, achieving the dual objectives of formation keeping and threat avoidance without compromising either. This synergy is a testament to the careful design of the force coefficients and the underlying drone technology framework.
Discussion on Parameter Sensitivity
The performance of the APF model is sensitive to the choice of the three force coefficients: k0, c, and m. Through systematic experimentation, we identified the optimal values that balance the competing objectives of formation accuracy, collision avoidance, and threat evasion. If k0 is set too low, the follower’s trajectory converges too slowly to the ideal formation, resulting in large deviations from the leader’s path. If k0 is set too high, the follower responds aggressively to small deviations, potentially causing oscillations and overshoots. The leader repulsion coefficient c must be carefully tuned to maintain a safe distance without causing excessive repulsion that could disrupt the formation. When c is too small, the follower may approach the leader too closely, increasing the risk of collision. When c is too large, the follower may be pushed too far away, breaking the formation. The threat repulsion coefficient m must be sufficiently high to ensure effective threat avoidance, but not so high that it causes the follower to deviate unnecessarily from the desired path. The values presented in Table 3 represent the optimal trade-off obtained through our iterative optimization process.
The robustness of our parameter selection was verified by testing the algorithm under a range of different conditions, including variations in threat density, formation size, and flight speed. The results confirm that the selected coefficients provide consistent performance across a wide range of scenarios, demonstrating the practical applicability of our approach in real-world drone technology deployments. The inclusion of a dead zone and a gradual force transition in the leader-following force further enhances the stability of the system by preventing abrupt changes in the force field.
Conclusion
In this paper, we have presented a comprehensive hierarchical framework for multi-UAV cooperative penetration trajectory planning, integrating an improved RRT* algorithm for global path planning with an enhanced Artificial Potential Field model for local trajectory optimization. The improvements to the RRT* algorithm, including heuristic random probability sampling, greedy redundancy elimination, and cubic Bezier curve smoothing, significantly enhance the efficiency and quality of the leader’s path. The APF model, with its carefully designed multi-coefficient force field, effectively guides the follower UAV to maintain formation, avoid collisions, and evade threats. Extensive simulations in three different battlefield scenarios validate the effectiveness of our approach, demonstrating consistent improvements in path length and threat avoidance. The proposed method represents a significant advancement in drone technology, providing a practical and robust solution for coordinating multiple UAVs in complex and dynamic environments. Future work will extend this framework to larger formations, incorporate dynamic threat models, and explore adaptive parameter tuning mechanisms to further enhance the flexibility and autonomy of multi-UAV systems in real-world operations.