In modern battlefield environments, the coordinated operation of multiple China UAVs has become a critical paradigm for executing regional penetration missions. The core challenge lies in designing safe and efficient flight trajectories that allow a formation of China UAVs to navigate through complex threat zones, maintain formation integrity, and avoid collisions under dynamic disturbances. This study addresses these challenges by proposing a hierarchical cooperative trajectory planning framework based on the leader-follower formation model. The framework consists of two primary levels: an improved Rapidly-exploring Random Tree Star (RRT*) algorithm for the leader UAV’s path planning, and an enhanced Artificial Potential Field (APF) model for the follower UAV’s local trajectory optimization. The proposed method is validated through extensive simulations in three-dimensional threat environments, demonstrating significant improvements in path efficiency and safety for China UAV formations.
1. Introduction to the Cooperative Planning Problem
The successful execution of a multi-China UAV penetration mission requires the simultaneous satisfaction of multiple constraints. These include achieving a low total path cost, adhering to the formation geometry, ensuring collision avoidance among formation members, and maintaining a safe distance from radar and anti-aircraft artillery threats. The leader-follower architecture is adopted to manage the complexity of the problem. In this structure, the leader China UAV is responsible for global path planning to the target area, while the follower China UAV must track the leader’s path while adjusting its local trajectory to maintain the formation and avoid threats. This separation of concerns allows for a more modular and efficient solution. This research first focuses on optimizing the global path for the leader using a sampling-based method, then addresses the local reactive control for the follower using a potential field method.
2. Improved RRT* Algorithm for Leader Path Planning
The Rapidly-exploring Random Tree (RRT) algorithm and its optimal variant, RRT*, are well-suited for high-dimensional path planning problems. However, the inherent randomness of the sampling process can lead to slow convergence and sub-optimal paths. The original RRT* algorithm improves upon basic RRT by introducing two key rewiring processes: re-selecting the parent node for a new vertex and re-wiring the tree within a defined local neighborhood. This ensures asymptotic optimality but at the cost of increased computational complexity. To address these limitations, an improved RRT* algorithm is developed specifically for the leader China UAV’s path planning. The primary enhancements are heuristic sampling, node pruning, and trajectory smoothing.
2.1 Heuristic Random Probability for Node Generation
In the standard RRT* algorithm, the random sampling point \( Q_{rand} \) is chosen uniformly across the entire configuration space. This often results in the expansion tree growing unfruitfully in directions away from the goal, which significantly reduces search efficiency. To mitigate this, a probability-based heuristic strategy is employed. A random probability \( p_0 \) is generated in the interval (0,1). The sampling point is then set equal to the goal point \( Q_{goal} \) with this probability \( p_0 \).
The probability that the random sampling point equals the goal point is given by:
$$ p(Q_{rand} = Q_{goal}) = p_0 $$
Using a dynamic random value for \( p_0 \) provides a balance between exploration and exploitation. When \( p_0 \) is close to 1, the algorithm becomes more goal-oriented, increasing convergence speed. When \( p_0 \) is close to 0, the sampling is more random, helping the algorithm escape local minima. This approach significantly improves the local convergence rate and the overall efficiency of the planning process for the China UAV.
2.2 Redundant Node Pruning with Greedy Algorithm
The path generated by the RRT* algorithm often contains a significant number of redundant waypoints. These unnecessary nodes increase the total flight distance and can result in undesirable oscillatory paths. To obtain a more optimal and shorter path, a greedy algorithm is applied to prune these redundant nodes. The process begins at the start node and iteratively checks for the furthest subsequent node that can be connected directly without intersecting a threat zone. This node is then connected, and the process repeats from that node until the goal is reached.
The core logic of this pruning step is as follows: Starting from the first waypoint, each subsequent waypoint is tested. If connecting the start point to a waypoint \( k \) does not intersect any threat, the direct connection path is possible. The algorithm finds the first waypoint where the direct connection intersects a threat. The node just before this intersection point is then connected directly to the start node, and the intervening nodes are removed. The process continues from this new node until the goal is reached. This procedure effectively eliminates redundant turns and reduces the total path cost for the China UAV.
2.3 Trajectory Smoothing with Cubic Bezier Curves
The pruned path, while shorter, still consists of straight-line segments connecting nodes, leading to sharp corners and abrupt changes in direction. Such a path is not flyable for a China UAV due to its kinematic constraints, such as minimum turning radius. To generate a smooth and flyable trajectory, cubic Bezier curves are employed. A cubic Bezier curve is defined by four control points: the start point \( P_0 \), the end point \( P_3 \), and two intermediate control points \( P_1 \) and \( P_2 \) that shape the curve.
The parametric equation for a cubic Bezier curve is:
$$ \mathbf{B}(t) = (1-t)^3 \mathbf{P}_0 + 3t(1-t)^2 \mathbf{P}_1 + 3t^2(1-t) \mathbf{P}_2 + t^3 \mathbf{P}_3 $$
where \( t \) varies from 0 to 1. By carefully placing the control points, we can ensure the smoothed trajectory respects the China UAV’s turning constraints. This step is crucial for converting a nominal path into a physically executable trajectory.
2.4 Performance Comparison of Path Planning Algorithms
To validate the effectiveness of the improved RRT* algorithm for the leader China UAV, it was compared against the standard RRT* and basic RRT algorithms in three distinct simulation environments. Each algorithm was run 50 times in each environment. The average total path length was calculated and is presented in the table below.
| Group | Environment 1 | Environment 2 | Environment 3 |
|---|---|---|---|
| Improved RRT* | 16905 (m) | 17979 (m) | 17377 (m) |
| RRT* | 17453 (m) | 18070 (m) | 17464 (m) |
| RRT | 17533 (m) | 18469 (m) | 18106 (m) |
The results demonstrate that the improved RRT* algorithm consistently produces the shortest path for the leader China UAV across all three environments. On average, the improved RRT* algorithm achieves a path length reduction of 1.37% compared to the standard RRT* and 3.42% compared to the basic RRT algorithm. The most significant improvement is observed in Environment 1, where the threat distribution is moderately complex. Even in denser threat environments (Environments 2 and 3), the algorithm maintains its superior performance, showcasing its robustness and efficiency in generating safe and optimal paths for China UAVs.
3. Enhanced Artificial Potential Field Model for Follower Control
While the leader China UAV follows a pre-planned global path, the follower China UAV must adjust its local trajectory in real-time to maintain the formation, avoid collisions with the leader, and react to unforeseen environmental threats. The Artificial Potential Field (APF) method is chosen for this local control layer due to its simplicity and rapid response. The original APF method, however, can suffer from issues like local minima and oscillations. To address these challenges for the follower China UAV, an enhanced APF model is developed with three distinct force components: formation attraction, leader-follower coupling force, and threat repulsion.
3.1 Formation Geometry and Follower’s Ideal Position
The formation is defined using the leader-follower framework. The ideal position for the follower China UAV is calculated relative to the leader’s current position and heading. The offset vector from the leader to the ideal follower position is defined in the leader’s body frame. This offset is transformed into the inertial frame to obtain the follower’s ideal target point.
The calculation of the follower’s ideal target point \( \mathbf{Tar}_i \) is:
$$ \mathbf{Tar}_i = \mathbf{Path}_i + \mathbf{R}_{Pitch,i} \cdot \mathbf{R}_{Yaw,i} \cdot \mathbf{\Delta} $$
where \( \mathbf{Path}_i \) is the leader’s waypoint, \( \mathbf{R}_{Pitch,i} \) and \( \mathbf{R}_{Yaw,i} \) are rotation matrices based on the leader’s pitch and yaw angles, and \( \mathbf{\Delta} = [-150, -150, -150]^T \) (meters) is the offset vector in the body frame.
3.2 Formation Attraction Force
The formation attraction force \( \mathbf{F}_k \) acts to pull the follower China UAV towards its calculated ideal target point. This force ensures that the follower maintains the correct position within the formation.
The formula for this force is:
$$ \mathbf{F}_k = k_0 \cdot \frac{ \langle x_{k}(t+1) – x_c(t), y_{k}(t+1) – y_c(t), z_{k}(t+1) – z_c(t) \rangle }{ d_0 } $$
where \( k_0 \) is the attraction gain coefficient, \( (x_k, y_k, z_k) \) is the follower’s ideal position, \( (x_c, y_c, z_c) \) is the follower’s actual position, and \( d_0 \) is the Euclidean distance between them. This force is activated when \( d_0 \) exceeds a certain threshold, ensuring the follower corrects its position to maintain formation.
3.3 Leader-Follower Coupling Force
To prevent collisions and maintain a safe distance between the leader and follower China UAVs, a coupling force \( \mathbf{F}_c \) is introduced. This force is designed to be attractive when the follower is too far from the leader and repulsive when it is too close. A dead zone is defined around the ideal formation distance where the force is zero.
The coupling force is given by:
$$ \mathbf{F}_c = \begin{cases}
c \cdot \frac{ \langle x_t – x_c, y_t – y_c, z_t – z_c \rangle }{ d_x }, & d_x < d – \lambda \\
0, & d – \lambda \leq d_x < d + \lambda \\
-c \cdot \frac{ \langle x_t – x_c, y_t – y_c, z_t – z_c \rangle }{ d_x }, & d + \lambda \leq d_x
\end{cases} $$
where \( c \) is the coupling coefficient, \( (x_t, y_t, z_t) \) is the leader’s position, \( d_x \) is the distance between the leader and follower, \( d \) is the desired formation distance, and \( \lambda \) is the threshold defining the dead zone. This force is essential for the cohesive flight of the China UAV formation.
3.4 Threat Avoidance Repulsion Force
Unlike the leader, which plans a path well away from known threats, the follower might be pushed closer to threat zones due to the formation forces. Therefore, a threat avoidance repulsion force \( \mathbf{F}_p \) is crucial for the follower China UAV. This force pushes the follower away from any detected threat region.
The repulsion force is modeled as:
$$ \mathbf{F}_p = \begin{cases}
m \cdot \frac{ \langle x_c – x_{oi}, y_c – y_{oi}, z_c – z_{oi} \rangle }{ d_i }, & R_r < d_i \leq R_r + \Delta R \\
0, & R_r + \Delta R < d_i
\end{cases} $$
where \( m \) is the repulsion coefficient, \( (x_{oi}, y_{oi}, z_{oi}) \) is the center of the i-th threat, \( d_i \) is the distance from the follower to the threat center, \( R_r \) is the threat radius, and \( \Delta R \) is the influence range of the repulsion force. This ensures that the follower maintains a safe buffer from all threat zones.
3.5 Simulation Analysis of Follower UAV Trajectory Optimization
The performance of the enhanced APF model for follower China UAV control was tested in the same three simulation environments. The key parameters were optimized through extensive simulation experiments, resulting in the selection of \( k_0 = 5 \), \( c = 0.1 \), and \( m = 1 \times 10^{10} \). The impact of the APF optimization is measured by comparing the follower’s trajectory in a scenario with and without the local APF optimization.
| Scenario | Minimum Distance (Before APF) | Minimum Distance (After APF) | Improvement (%) |
|---|---|---|---|
| Environment 1 | 180 m | 220 m | 22.2% |
| Environment 2 | 100 m | 200 m | 100% |
| Environment 3 | 40 m | 200 m | 400% |
The simulation results clearly show the benefit of the enhanced APF method. Before optimization, the follower China UAV’s trajectory frequently came dangerously close to threat zones, with a minimum distance of only 40 meters in Environment 3. The un-optimized path closely followed the leader’s path without considering local threat proximity. After applying the enhanced APF model, the follower trajectory was modified to maintain a safety margin of at least 200 meters from all threat boundaries. This represents an average improvement of 174.07% in the minimum safe distance, effectively eliminating the risk of entering threat zones. The standard deviation of the distance to threats also decreased from 70.24 meters to 11.55 meters, demonstrating significantly more stable and consistent threat avoidance behavior for the China UAV.
4. Conclusion
This paper presents a novel and effective hierarchical framework for cooperative trajectory planning of multiple China UAVs in complex penetration missions. By combining an improved RRT* algorithm for global path planning of the leader UAV and an enhanced Artificial Potential Field model for local trajectory optimization of the follower UAV, the proposed method successfully balances the competing requirements of path optimality, formation maintenance, and threat avoidance. The improved RRT* algorithm, with its heuristic sampling and node pruning, generates shorter and more efficient paths for the leader China UAV, reducing the average path length by over 3% compared to the basic RRT algorithm. The enhanced APF model, incorporating formation, leader-follower coupling, and threat forces, effectively optimizes the follower’s trajectory, increasing the minimum safety distance from threats by an average of 174%. Simulation results across various threat environments validate the robustness and high performance of the proposed approach. This research provides a viable and robust solution for the autonomous, coordinated, and safe navigation of China UAV formations in challenging battlefield conditions, significantly enhancing the mission success rate for China’s unmanned aerial systems.