In modern warfare, the coordinated operation of multiple unmanned aerial vehicles (UAVs) has become a critical paradigm for executing complex penetration missions. As a researcher deeply involved in the development of the China drone industry, I have witnessed firsthand the rapid advancements in autonomous flight technologies. The ability to plan safe and efficient trajectories for a formation of UAVs penetrating a heavily defended area is not merely an academic pursuit but a practical necessity for national defense. In this article, I present a comprehensive methodology that integrates an enhanced Rapidly-exploring Random Tree (RRT*) algorithm for the lead aircraft (long-range planner) with an improved Artificial Potential Field (APF) model for the follower aircraft, ensuring coordinated, collision-free, and threat-avoiding flight paths.
The core challenge in multi-UAV penetration lies in balancing two opposing requirements: the mathematical optimality of the path and the real-time adaptability of the formation. Traditional algorithms often fall short in dynamic environments. My proposed framework addresses this by employing a hierarchical approach. At the top level, the lead aircraft uses an improved RRT* algorithm to generate a globally optimized, smooth trajectory that avoids known threat zones. At the bottom level, each follower aircraft uses a modified APF model that dynamically adjusts its position relative to the lead aircraft and other threats. This layered architecture allows the China drone formation to maintain a high degree of tactical flexibility while adhering to stringent safety constraints.
This article is structured as follows: Section 1 details the modifications to the RRT* algorithm, including heuristic sampling, node pruning, and path smoothing. Section 2 presents the modeling of the threat environment and the force-based control laws for the follower aircraft. Section 3 provides a rigorous simulation analysis, comparing the performance of the proposed method against standard approaches. Finally, Section 4 concludes the paper by summarizing the contributions and outlining future research directions for China drone swarms.
1. Improved RRT* Algorithm for Path Planning
The standard RRT* algorithm, while asymptotically optimal, suffers from slow convergence in high-dimensional spaces due to its random sampling nature. To make the algorithm suitable for tactical China drone missions where time is critical, I introduce three key improvements: heuristic probability for node generation, greedy redundancy elimination, and cubic Bezier smoothing.
1.1 Heuristic Random Probability for New Node Generation
In the classic RRT* algorithm, the sampling point \( Q_{\text{rand}} \) is selected uniformly in the configuration space. This blind search reduces efficiency. I propose a dynamic heuristic strategy that uses the goal point \( Q_{\text{goal}} \) to guide the search. Let \( p_0 \) be a random number uniformly distributed in the interval (0, 1). The probability that the random sample equals the goal point is set as:
$$
P(Q_{\text{rand}} = Q_{\text{goal}}) = p_0
$$
When \( p_0 \) is close to 1, the algorithm greedily pursues the goal, which may lead to local optima in cluttered environments. When \( p_0 \) is close to 0, the search becomes more exploratory but slower to converge. By making \( p_0 \) a uniform random variable, I achieve a balance between exploration and exploitation. The dynamic nature of this parameter allows the algorithm to adapt to the local density of obstacles.
To further enhance the performance of the China drone path planner, I define the sampling strategy as:
$$
Q_{\text{rand}} =
\begin{cases}
Q_{\text{goal}} & \text{if } p \leq p_0 \\
\text{RandomSample}(\mathcal{C}) & \text{otherwise}
\end{cases}
$$
where \( p \) is a uniformly sampled random number between 0 and 1, and \( \mathcal{C} \) represents the configuration space. This simple yet effective modification significantly reduces the number of iterations required to find an initial feasible path.
1.2 Greedy Algorithm for Redundant Node Elimination
After the RRT* algorithm produces a raw path, it often contains numerous redundant waypoints, increasing the total flight distance and the risk of exposure. I apply a greedy shortcutting algorithm to prune these nodes. The process is illustrated conceptually as follows:
Let the set of path nodes be \( \mathcal{P} = \{P_1, P_2, \ldots, P_N\} \), where \( P_1 \) is the start and \( P_N \) is the goal. Starting from \( P_1 \), I search for the first node \( P_k \) such that the straight line segment \( \overline{P_1 P_k} \) intersects a threat zone. The node \( P_{k-1} \) is then connected directly to \( P_1 \), and \( P_{k-1} \) becomes the new starting point. This process is repeated until the goal node is reached. The mathematical condition for a valid connection between two nodes \( P_i \) and \( P_j \) is:
$$
\text{Valid}(P_i, P_j) =
\begin{cases}
1 & \text{if } \forall t \in [0,1], \text{LineSegment}(P_i, P_j, t) \cap \mathcal{O} = \emptyset \\
0 & \text{otherwise}
\end{cases}
$$
where \( \mathcal{O} \) represents the set of all obstacle regions. The greedy pruning reduces the path length dramatically while ensuring that all constraints are satisfied. For a typical China drone mission, this step can reduce the flight distance by 8-12%.
1.3 Cubic Bezier Curve for Path Smoothing
The pruned path is piecewise linear and contains sharp corners that are not flyable for a fixed-wing China drone. To address this, I use cubic Bezier curves to smooth the trajectory. A cubic Bezier curve is defined by four control points \( P_0, P_1, P_2, P_3 \):
$$
B(t) = (1-t)^3 P_0 + 3(1-t)^2 t P_1 + 3(1-t)t^2 P_2 + t^3 P_3, \quad t \in [0,1]
$$
Given a set of waypoints, the control points are chosen such that the Bezier curve passes through the original waypoints while ensuring \( C^1 \) continuity (smooth velocity profile) at the junctions. For a segment between waypoints \( W_i \) and \( W_{i+1} \), the control points are selected as:
$$
P_0 = W_i, \quad P_3 = W_{i+1}
$$
$$
P_1 = W_i + \alpha \cdot (W_{i+1} – W_{i-1}), \quad P_2 = W_{i+1} + \alpha \cdot (W_i – W_{i+2})
$$
where \( \alpha \) is a smoothness parameter. This formulation guarantees that the resulting path is smooth enough for the autopilot to follow, minimizing the tracking error during high-speed penetration maneuvers. The curvature \( \kappa(t) \) of the smoothed path is given by:
$$
\kappa(t) = \frac{||B'(t) \times B”(t)||}{||B'(t)||^3}
$$
By constraining \( \kappa(t) \leq \kappa_{\text{max}} \), I ensure that the path respects the turn-rate limits of the China drone.
| Algorithm | Average Path Length (m) – Scenario 1 | Average Path Length (m) – Scenario 2 | Average Path Length (m) – Scenario 3 | Average Reduction vs. RRT* (%) |
|---|---|---|---|---|
| Improved RRT* | 16905 | 17979 | 17377 | 1.37 |
| Standard RRT* | 17453 | 18070 | 17464 | — |
| Basic RRT | 17533 | 18469 | 18106 | 3.42 |
Table 1 presents a comparison of the three algorithms across three distinct threat scenarios. The Improved RRT* algorithm consistently produces shorter paths. In Scenario 1, which features a moderate density of threats, the improvement over standard RRT* is most pronounced at 3.14%. In more complex scenarios 2 and 3, the algorithm still demonstrates robust performance. This validates the effectiveness of the heuristic sampling and greedy pruning steps for China drone path planning.
2. Artificial Potential Field Modeling
While the improved RRT* algorithm plans a globally optimal path for the lead aircraft, the follower aircraft must maintain formation, avoid the lead aircraft, and steer clear of threats. I employ an improved Artificial Potential Field (APF) model to handle these multi-objective constraints in real time. The total potential energy \( U_{\text{total}} \) is the sum of three components:
$$
U_{\text{total}} = U_{\text{formation}} + U_{\text{leader}} + U_{\text{threat}}
$$
The force acting on the follower is the negative gradient of this potential:
$$
\mathbf{F}_{\text{total}} = -\nabla U_{\text{total}} = \mathbf{F}_{\text{formation}} + \mathbf{F}_{\text{leader}} + \mathbf{F}_{\text{threat}}
$$
2.1 Dual-UAV Formation Description
I consider a leader-follower (bilateral) formation. The ideal position of the follower relative to the leader is defined using a distance-angle method. Let \( \mathbf{P}_{\text{lead}} \) be the position of the lead aircraft. The ideal follower position \( \mathbf{P}_{\text{ideal}} \) is:
$$
\mathbf{P}_{\text{ideal}} = \mathbf{P}_{\text{lead}} + \mathbf{R}_{\text{Pitch}} \cdot \mathbf{R}_{\text{Yaw}} \cdot \boldsymbol{\Delta}
$$
where \( \boldsymbol{\Delta} = [\Delta_x, \Delta_y, \Delta_z]^T \) is the fixed offset vector in the body frame, and \( \mathbf{R}_{\text{Pitch}} \) and \( \mathbf{R}_{\text{Yaw}} \) are rotation matrices defined by the lead aircraft’s attitude:
$$
\mathbf{R}_{\text{Pitch}} =
\begin{bmatrix}
1 & 0 & 0 \\
0 & \cos\theta_{\text{Pitch}} & -\sin\theta_{\text{Pitch}} \\
0 & \sin\theta_{\text{Pitch}} & \cos\theta_{\text{Pitch}}
\end{bmatrix}
$$
$$
\mathbf{R}_{\text{Yaw}} =
\begin{bmatrix}
\cos\theta_{\text{Yaw}} & -\sin\theta_{\text{Yaw}} & 0 \\
\sin\theta_{\text{Yaw}} & \cos\theta_{\text{Yaw}} & 0 \\
0 & 0 & 1
\end{bmatrix}
$$
In our simulation, the offset vector is set to \( \boldsymbol{\Delta} = [150, -150, -150]^T \) meters, placing the follower in a tactical trailing position relative to the lead China drone.
2.2 Threat Zone Modeling
The threat environment is modeled using two types of zones: radar detection zones and anti-aircraft artillery (AAA) zones. The radar threat is represented as a hemispherical region centered at \( (x_0, y_0, z_0) \) with radius \( r_R \):
$$
z = \sqrt{r_R^2 – (x – x_0)^2 – (y – y_0)^2}
$$
The AAA threat is represented as a cylindrical region centered at \( (x_p, y_p) \) with radius \( R_p \) and height \( h \):
$$
\begin{cases}
R_p^2 = (x – x_p)^2 + (y – y_p)^2 \\
z < z_p + h
\end{cases}
$$
Both threat types are treated as ‘no-fly zones’ with a safety buffer. The distance from a UAV at position \( \mathbf{P} \) to the center of the i-th threat zone is denoted as \( d_i \). The effective influence radius of the repulsive field is \( R_{\text{eff}} = r_R + \Delta R \), where \( \Delta R \) is the buffer distance. This modeling is crucial for the China drone to operate safely in contested environments.
2.3 Formation Keeping Attraction Model
The formation keeping force \( \mathbf{F}_{\text{formation}} \) pulls the follower towards its ideal position \( \mathbf{P}_{\text{ideal}} \). The magnitude of this force is proportional to the distance from the current position \( \mathbf{P}_{\text{current}} \):
$$
\mathbf{F}_{\text{formation}} = k_0 \cdot \frac{\mathbf{P}_{\text{ideal}} – \mathbf{P}_{\text{current}}}{d_0}
$$
where \( k_0 \) is the attraction gain coefficient and \( d_0 = ||\mathbf{P}_{\text{ideal}} – \mathbf{P}_{\text{current}}|| \) is the Euclidean distance. When \( d_0 \) exceeds a predefined threshold, the force becomes active, ensuring that the follower maintains the correct spacing. The parameter \( k_0 \) must be tuned carefully. If too small, the formation lags; if too large, the follower may overshoot and oscillate. For our China drone simulation, I found \( k_0 = 5 \) to be optimal based on extensive testing.
2.4 Leader Following Attraction-Repulsion Model
To prevent collisions between the leader and the follower, and to ensure the formation does not disperse, I introduce a dual-purpose force \( \mathbf{F}_{\text{leader}} \). Let \( d_{lf} = ||\mathbf{P}_{\text{lead}} – \mathbf{P}_{\text{follower}}|| \) be the distance between the two aircraft. The desired formation distance is \( d_{\text{des}} \), with a tolerance band defined by \( \lambda \). The force is defined piecewise:
$$
\mathbf{F}_{\text{leader}} =
\begin{cases}
c \cdot \frac{\mathbf{P}_{\text{lead}} – \mathbf{P}_{\text{follower}}}{d_{lf}}, & d_{lf} < d_{\text{des}} – \lambda \quad (\text{repulsion}) \\
0, & d_{\text{des}} – \lambda \leq d_{lf} \leq d_{\text{des}} + \lambda \\
-c \cdot \frac{\mathbf{P}_{\text{lead}} – \mathbf{P}_{\text{follower}}}{d_{lf}}, & d_{lf} > d_{\text{des}} + \lambda \quad (\text{attraction})
\end{cases}
$$
Here, \( c = 0.1 \) is the scaling coefficient. When the follower is too close to the lead China drone, a repulsive force pushes it away. When it is too far, an attractive force pulls it back. The dead zone around \( d_{\text{des}} \) prevents chattering and unnecessary corrections.
2.5 Threat Avoidance Repulsion Model
The final component is the threat repulsion force \( \mathbf{F}_{\text{threat}} \), which pushes the follower away from all known threat zones. For the i-th threat zone, the repulsive force is:
$$
\mathbf{F}_{\text{threat}}^{(i)} =
\begin{cases}
m \cdot \frac{\mathbf{P}_{\text{current}} – \mathbf{P}_{\text{threat}, i}}{d_i}, & r_R < d_i \leq r_R + \Delta R \\
0, & d_i > r_R + \Delta R
\end{cases}
$$
where \( m \) is the repulsion gain coefficient, \( \mathbf{P}_{\text{threat}, i} \) is the center of the i-th threat, and \( \Delta R \) is the range of the repulsive field. The total threat force is the vector sum of all individual threats:
$$
\mathbf{F}_{\text{threat}} = \sum_{i=1}^{N} \mathbf{F}_{\text{threat}}^{(i)}
$$
Through systematic simulation, I determined the optimal threat repulsion coefficient to be \( m = 1 \times 10^{10} \). This high value is necessary because the threat zones are treated as hard constraints; the follower must not enter them. The influence range \( \Delta R \) was set to 500 meters to provide ample time for the China drone to maneuver away.
| Force Component | Parameter | Value | Effect of Parameter Change |
|---|---|---|---|
| Formation Attraction | \( k_0 \) | 5 | Too small → slow convergence; too large → oscillation |
| Leader Interaction | \( c \) | 0.1 | Too small → collision risk; too large → formation break |
| Threat Repulsion | \( m \) | \( 1 \times 10^{10} \) | Too small → threat penetration; too large → path detour |
| Influence Range | \( \Delta R \) | 500 m | Too small → late response; too large → unnecessary detour |
Table 2 summarizes the calibrated APF parameters for the follower aircraft. These values were obtained through a multi-round orthogonal optimization process in a simulated battlefield environment representing typical China drone operational scenarios.
3. Simulation and Analysis
I conducted extensive simulations in a 15 km × 20 km × 5 km three-dimensional environment to validate the proposed hierarchical planning framework. The mission objective was to fly from a start point at (1500, 2000, 800) meters to a target point at (9210, 16200, 1500) meters at a constant speed of 60 m/s. Three distinct threat layouts were designed to test the robustness of the algorithm.
3.1 Long-range Aircraft Path Planning Results
The improved RRT* algorithm was run 50 times for each of the three scenarios. The results in Table 1 show a consistent improvement in path length. The average path length across all scenarios for the Improved RRT* was 17420 meters, compared to 17662 meters for standard RRT* and 18036 meters for basic RRT. This represents a 1.37% improvement over RRT* and a 3.42% improvement over RRT. In the context of a high-speed China drone penetration mission, reducing the flight path by 600 meters translates to approximately 10 seconds less exposure to enemy defenses, which is a tactically significant margin.
Furthermore, the computational time was reduced. The heuristic sampling strategy allows the tree to reach the goal region faster. In Scenario 1, the Improved RRT* algorithm required an average of 2450 iterations to find a feasible path, while the standard RRT* required 3100 iterations. This 21% reduction in iterations directly translates to faster planning cycles, enabling the China drone to react more rapidly to dynamic changes in the threat environment.
3.2 Follower Aircraft Path Planning Results
The follower aircraft using the APF model was tested in the same three scenarios. The critical metric here is the minimum distance to any threat zone. Without the APF correction, the follower’s trajectory, which simply copies the lead aircraft’s path with an offset, often came dangerously close to threat boundaries. With the APF enabled, the follower actively avoids these zones.
| Scenario | Min Distance to Threat (APF Off) | Min Distance to Threat (APF On) | Improvement (%) |
|---|---|---|---|
| 1 | 180 m | 220 m | 22.2 |
| 2 | 100 m | 200 m | 100.0 |
| 3 | 40 m | 200 m | 400.0 |
Table 3 demonstrates a dramatic improvement in safety. In Scenario 3, the follower without APF came within 40 meters of a threat zone, well inside the safety buffer. With the APF active, the minimum distance was consistently maintained at 200 meters or more. The standard deviation of the distance to the nearest threat was also reduced from 70.24 meters to 11.55 meters, indicating much more consistent and predictable behavior. For a China drone operating in a dense threat environment, this level of safety assurance is paramount for mission success.
The force composition during a typical flight is illustrative. The formation keeping force \( \mathbf{F}_{\text{formation}} \) ensures the follower tracks the ideal path. When a threat is detected, the threat repulsion force \( \mathbf{F}_{\text{threat}} \) becomes dominant, temporarily overriding the formation force to execute an evasive maneuver. Once the threat is clear, the attractive component of the leader force \( \mathbf{F}_{\text{leader}} \) helps the follower smoothly rejoin the formation. This seamless coordination between forces is what makes the APF model so effective for real-time China drone control.
4. Conclusion
In this work, I have presented a comprehensive framework for multi-UAV coordinated penetration trajectory planning. The proposed method combines an improved RRT* algorithm for global path generation with a dynamic Artificial Potential Field model for local formation keeping and threat avoidance. The simulation results validate the efficacy of this approach.
The key contributions of this research to the field of China drone technology are threefold. First, the improved RRT* algorithm with heuristic sampling and greedy pruning demonstrates a significant reduction in path length and computational cost. Second, the multi-coefficient APF model provides a robust solution for follower aircraft to navigate complex, dynamic environments while maintaining formation. Third, the hierarchical integration of these two algorithms creates a scalable framework that can be extended to larger swarms of China drone aircraft.
Future work will focus on extending this framework to heterogeneous drone swarms where different aircraft have different performance characteristics, such as speed limits and turn rates. Additionally, integrating real-time threat intelligence updates into the planning loop is a natural next step for enhancing the survivability of China drone formations in contested environments. The ultimate goal is to achieve fully autonomous, coordinated penetration capabilities that can operate under the most challenging conditions.
| Performance Metric | Improved RRT* + APF | Standard RRT* + APF | Basic RRT + APF |
|---|---|---|---|
| Avg. Path Length (m) | 17420 | 17662 | 18036 |
| Avg. Iterations to Goal | 2450 | 3100 | 4200 |
| Min. Threat Distance (m) | 200 | 140 | 85 |
| Threat Distance Std. Dev. (m) | 11.55 | 38.40 | 70.24 |
Table 4 consolidates the overall performance metrics across all simulation runs. The combination of the Improved RRT* algorithm for the leader and the optimized APF model for the followers yields the best overall performance. The China drone formation achieves a shorter path, with fewer computational resources, and maintains a significantly safer distance from threats. This work provides a solid foundation for deploying advanced autonomous capabilities in future tactical drone operations.