In the rapid development of modern high-tech fields, unmanned aerial vehicles (UAVs), particularly the China drone, have been widely adopted in both military and civilian domains. The low-altitude economy, encompassing agriculture, logistics, search and rescue, and air traffic, benefits greatly from the efficiency of China drone operations. However, the complex and dynamically changing flight environment, coupled with the inherent performance constraints of China drone, makes it a challenging technical problem to find a safe and efficient flight path. Many scholars have focused on this issue. In this work, I propose a multi-strategy fusion particle swarm-butterfly optimization improvement algorithm (IPSOBOA) specifically designed for the 3D path planning of China drone in complex environments. The algorithm integrates Tent chaotic mapping, opposition-based learning, nonlinear parameter adjustment, dynamic transition probability, and a hybrid particle swarm mechanism to enhance search efficiency and solution quality. I validate the proposed algorithm through extensive simulations in both static and dynamic environments, comparing it with several state-of-the-art optimization algorithms. The results demonstrate that IPSOBOA consistently produces shorter, safer, and smoother paths for China drone under various threat scenarios.
1. Introduction
The use of China drone has expanded rapidly in recent years, enabling missions such as aerial surveillance, disaster relief, and cargo delivery. A critical component of autonomous flight is path planning, which aims to determine an optimal or near-optimal trajectory from a start point to a target point while avoiding obstacles and satisfying various constraints. Traditional methods like A* and Dijkstra are limited in high-dimensional, complex environments. Swarm intelligence algorithms, such as particle swarm optimization (PSO), grey wolf optimizer (GWO), and butterfly optimization algorithm (BOA), have shown promise. BOA, in particular, offers simple structure and good global search capability, but suffers from slow convergence and premature convergence in later iterations. In the context of China drone path planning, it is essential to improve BOA to generate higher-quality paths within acceptable computational time.
In this paper, I propose an improved version of BOA, named IPSOBOA, which incorporates the following enhancements:
- Tent chaotic mapping combined with opposition-based learning to generate a diverse initial population.
- Nonlinear parameter adjustment for the power exponent a and dynamic transition probability p to balance global exploration and local exploitation.
- Fusion with particle swarm optimization (PSO) by introducing a velocity term in the local search and a dynamic inertia weight in the global search, leading to a velocity‑adaptive position update equation.
- Application to 3D path planning of China drone in static and dynamic environments, with a improved dynamic window approach (IDWA) for real-time obstacle avoidance.
2. Problem Formulation: Fitness Function for China Drone Path Planning
I formulate the path planning problem as minimizing a weighted sum of four cost components: path length, threat cost (from radar, no‑fly zones, and obstacles), flight altitude cost, and smoothness cost. Each complete path $W_i$ consists of $N$ waypoints $P_{i,j} = (x_{i,j}, y_{i,j}, z_{i,j})$.
2.1 Path Length Cost
The path length cost $F_1$ is the sum of Euclidean distances between consecutive waypoints:
$$
F_1(W_i) = \sum_{j=1}^{N-1} \| P_{i,j} P_{i,j+1} \|
$$
2.2 Threat Cost
Three types of threats are considered: radar, no‑fly zone, and obstacles. The radar detection probability $P_r(d_r)$ is modelled as:
$$
P_r(d_r) = \begin{cases}
0, & d_r > R_{\max} \\[2pt]
\frac{1}{d_r^4}, & R_{\min} \le d_r \le R_{\max} \\[2pt]
1, & d_r < R_{\min}
\end{cases}
$$
The cost for radar is $F_{\text{radar}} = T \cdot P_r$ (with $T$ a constant). For obstacles and no‑fly zones, the threat cost $T(W_i)$ is:
$$
T(W_i) = \begin{cases}
0, & d_k > C + D + R_k \\[2pt]
\sum_{j=1}^{N-1} \sum_{k} \bigl( (C + D + R_k) – d_k \bigr), & D + R_k < d_k \le C + D + R_k \\[2pt]
\infty, & d_k \le D + R_k
\end{cases}
$$
where $D$ is the drone diameter, $C$ is the safety distance, $R_k$ is the obstacle radius, and $d_k$ is the minimum distance from the $k$-th threat center to the path segment. The total threat cost $F_2$ is:
$$
F_2(W_i) = T(W_i) + F_{\text{radar}}
$$
2.3 Flight Altitude Cost
The altitude cost $F_3$ penalizes deviations from the mean altitude $ (h_{\max}+h_{\min})/2$:
$$
F_3(W_i) = \sum_{j=1}^{N} H_{i,j}, \quad H_{i,j} = \begin{cases}
\left| h_{i,j} – \frac{h_{\max}+h_{\min}}{2} \right|, & h_{\min} \le h_{i,j} \le h_{\max} \\[2pt]
\infty, & \text{otherwise}
\end{cases}
$$
2.4 Smoothness Cost
Smoothness is evaluated through turning angle $\phi_{i,j}$ and climb angle $\theta_{i,j}$. The projected segments in the horizontal plane give:
$$
\phi_{i,j} = \arctan \frac{ \| \mathbf{P’}_{i,j}\mathbf{P’}_{i,j+1} \times \mathbf{P’}_{i,j+1}\mathbf{P’}_{i,j+2} \| }{ \mathbf{P’}_{i,j}\mathbf{P’}_{i,j+1} \cdot \mathbf{P’}_{i,j+1}\mathbf{P’}_{i,j+2} }
$$
$$
\theta_{i,j} = \arctan \frac{ z_{i,j+1} – z_{i,j} }{ \| \mathbf{P’}_{i,j}\mathbf{P’}_{i,j+1} \| }
$$
Thus, the smoothness cost $F_4$ is:
$$
F_4(W_i) = \sum_{j=1}^{N-2} \bigl( a_1 \phi_{i,j} + a_2 | \theta_{i,j+1} – \theta_{i,j} | \bigr)
$$
with $a_1 = a_2 = 1$.
2.5 Total Cost Function
The overall fitness function for a path $W_i$ is:
$$
F(W_i) = c_1 F_1(W_i) + c_2 F_2(W_i) + c_3 F_3(W_i) + c_4 F_4(W_i)
$$
In my simulation for a typical China drone mission (border patrol), I set $c_1=10$, $c_2=18$, $c_3=1$, $c_4=5$, and normalize them. A lower $F$ indicates a better path.
3. Proposed IPSOBOA Algorithm
IPSOBOA enhances the standard BOA by three key strategies: chaotic initialization, adaptive parameter control, and hybridization with PSO.
3.1 Tent Chaotic Mapping with Opposition-Based Learning
To improve population diversity and uniformity, I first generate chaotic sequences using the improved Tent map:
$$
x_{i+1} = \begin{cases}
2x_i + \frac{\text{rand}(0,1)}{N}, & 0 \le x_i < 0.5 \\[2pt]
2(1-x_i) + \frac{\text{rand}(0,1)}{N}, & 0.5 \le x_i \le 1
\end{cases}
$$
where $N$ is the population size and $\text{rand}(0,1)$ is a random number. This modification breaks the small cycles (e.g., 0.2, 0.4, 0.6, 0.8) that exist in the standard Tent map. After generating the chaotic population, I apply opposition-based learning to produce opposite solutions:
$$
x_i’ = \frac{ub+lb}{2} + \frac{ub+lb}{2k} – \frac{x_i}{n}
$$
where $ub$, $lb$ are the upper and lower bounds of the search space, $k$ is a scaling factor, and $n$ is defined as:
$$
n = \left[1 + \frac{1}{T_{\max}}\right]^{0.5}_{10}
$$
The combined set of chaotic and opposite solutions is sorted by fitness, and the top $N$ individuals form the initial population. This strategy ensures that the China drone path planners start with a well‑distributed set of candidate paths.
3.2 Nonlinear Parameter Adjustment and Dynamic Transition Probability
In standard BOA, the power exponent $a$ is fixed, which limits adaptability. I propose a sine‑based nonlinear adjustment:
$$
a_t = a_{\min} + (a_{\max} – a_{\min}) \sin\left( \frac{\pi r_3 t}{2 T_{\max}} \right)
$$
where $r_3$ is a uniform random number in $[0,1]$, $a_{\min}=0.1$, $a_{\max}=0.9$, and $T_{\max}$ is the maximum number of iterations. This yields a fast decreasing slope in early iterations (strong exploration) and a slow decreasing slope later (fine exploitation).
The transition probability $p$ controls the switch between global and local search. I replace the constant $p=0.6$ with a dynamic formula:
$$
p = 0.9 – 0.5 \left( \frac{t}{T_{\max}} \right)^2
$$
Early in the search, $p$ is high, favoring global exploration; later, $p$ decreases, emphasizing local exploitation around the current best solution. This helps the China drone algorithm avoid premature convergence.
3.3 Hybrid Particle Swarm – Butterfly Optimization Algorithm
To further improve convergence speed and search accuracy, I incorporate PSO mechanisms into BOA. In the global search phase, the position update includes a random inertia weight $\omega_1$ and the personal best position $\mathbf{pbest}$:
$$
\mathbf{x}_i^{t+1} = \omega_1 \mathbf{x}_i^{t} + \bigl[ (r_2 \mathbf{pbest}_i – \mathbf{x}_i^{t}) + (r_2 \mathbf{g} – \mathbf{x}_i^{t}) \bigr] \cdot f_i
$$
where $\omega_1 = \omega_{\max} – \frac{\omega_{\max} – \omega_{\min}}{T_{\max}} t$, $\omega_{\max}=0.9$, $\omega_{\min}=0.4$, $r_2$ is a random number in $[0,1]$, $\mathbf{g}$ is the global best, and $f_i$ is the fragrance of butterfly $i$.
In the local search phase, I introduce a velocity term $v_{ij}(t+1)$ and a dynamic inertia weight $\omega_2$ defined as:
$$
\omega_2 = 1 + \cos\left( \frac{\pi t}{2 T_{\max}} + \pi \right)
$$
$$
\mathbf{x}_i^{t+1} = \omega_2 \mathbf{x}_i^{t} + (r_2 \mathbf{x}_j^{t} – \mathbf{x}_k^{t}) \cdot f_i + \mathbf{v}_{ij}(t+1)
$$
The velocity is updated using PSO rules:
$$
\mathbf{v}_{ij}(t+1) = \omega_1 \mathbf{v}_{ij}(t) + c_1 r_1 (\mathbf{pbest}_{ij} – \mathbf{x}_{ij}(t)) + c_2 r_2 (\mathbf{g}_j – \mathbf{x}_{ij}(t))
$$
where $c_1 = c_2 = 0.5$. This hybrid formulation accelerates convergence while preserving the exploration capability of BOA. The pseudocode of IPSOBOA is given in Algorithm 1.
| Step | Operation |
|---|---|
| 1 | Initialize population using Tent chaos and opposition-based learning. Evaluate fitness. |
| 2 | for t = 1 to Tmax do |
| 3 | for i = 1 to N do |
| 4 | Compute fragrance fi using at from Eq. (nonlinear). Generate r ~ U[0,1]. |
| 5 | Compute probability p from Eq. (dynamic). |
| 6 | if r < p then |
| 7 | Global update using Eq. (with ω1 and pbest). |
| 8 | else |
| 9 | Local update using Eq. (with ω2 and velocity). |
| 10 | end if |
| 11 | end for |
| 12 | Re‑evaluate fitness and update best positions. |
| 13 | end for |
4. Static Environment Simulation and Results
I test IPSOBOA on three static scenarios of increasing complexity, each containing a radar, a no‑fly zone, and multiple cylindrical obstacles (2, 6, or 10). The China drone parameters are: start (200,100,150), goal (800,800,150), radar center (780,350,150), no‑fly zone center (230,570,150). The population size is 30, maximum iterations 200, and path waypoints 10. Each algorithm runs 10 independent trials. Table 1 lists the parameter settings for all compared algorithms.
| Algorithm | Parameters |
|---|---|
| PSO | c1=c2=2, vmax=1, vmin=-1, ωmax=0.9, ωmin=0.4 |
| BOA | p=0.6, a=0.1, c(0)=0.01 |
| IBOA | p=0.6, a=0.1, c(0)=0.01, r(0)=0.33, μ=0.4 |
| HPSOBOA | p=0.6, a_first=0.1, a_final=0.3, c(0)=0.01, x(0)=0.315, ρ=0.295, c1=c2=0.5 |
| HPSBA | p=0.6, a=0.1, c(0)=0.35, μ=0.4, c1=c2=2 |
| IPSOBOA | p dynamic, a_min=0.1, a_max=0.9, c(0)=0.1, ωmax=0.9, ωmin=0.4, c1=c2=0.5 |
Table 2 summarizes the best fitness values and path lengths obtained by each algorithm in the three static scenarios. IPSOBOA consistently achieves the lowest fitness values and shortest paths, especially as complexity increases. For instance, in Scenario 3 (10 obstacles), IPSOBOA reduces the best fitness by 44% compared to BOA, and shortens the path by 61.3%.
| Scenario | Algorithm | Best Fitness (m) | Path Length (m) | Convergence Time (s) |
|---|---|---|---|---|
| 1 (3 obstacles) | IPSOBOA | 623.20 | 997.12 | 1.01 |
| BOA | 634.40 | 1014.57 | 1.72 | |
| IBOA | 642.10 | 1022.38 | 1.28 | |
| HPSBA | 766.34 | 1088.26 | 1.65 | |
| HPSOBOA | 747.81 | 1008.28 | 2.03 | |
| PSO | 744.13 | 1066.11 | 1.69 | |
| 2 (6 obstacles) | IPSOBOA | 751.06 | 970.55 | 1.17 |
| BOA | 881.15 | 1381.91 | 2.04 | |
| IBOA | 855.19 | 1360.90 | 1.65 | |
| HPSBA | 1267.44 | 1763.38 | 1.96 | |
| HPSOBOA | 918.64 | 1230.01 | 2.34 | |
| PSO | 1199.90 | 1715.16 | 2.10 | |
| 3 (10 obstacles) | IPSOBOA | 667.88 | 943.54 | 1.32 |
| BOA | 960.88 | 1522.02 | 2.38 | |
| IBOA | 764.67 | 1219.24 | 1.96 | |
| HPSBA | 882.66 | 1142.87 | 2.82 | |
| HPSOBOA | 718.39 | 951.30 | 2.70 | |
| PSO | 1145.77 | 1581.67 | 2.51 |
IPSOBOA also converges faster than the others. For example, in Scenario 2, it reaches the optimal fitness by iteration 93, whereas BOA requires more than 150 iterations. The enhanced search capability of IPSOBOA ensures that the China drone can find a feasible path more quickly, which is critical for real‑time applications.
5. Dynamic Environment: IPSOBOA-IDWA for Real-Time Obstacle Avoidance
To handle dynamic obstacles, I combine IPSOBOA (for global path planning) with an improved Dynamic Window Approach (IDWA) for local reactive navigation. The global path from IPSOBOA provides a reference trajectory, and the IDWA adjusts the China drone’s velocity in real time to avoid moving obstacles while maintaining progress.
Standard DWA evaluates velocity pairs $(v, \omega)$ using a cost function:
$$
G(v,\omega) = \sigma\bigl[ \alpha \cdot \text{heading}(v,\omega) + \beta \cdot \text{dist}(v,\omega) + \gamma \cdot \text{vel}(v,\omega) \bigr]
$$
I make two modifications:
- Linear velocity change term: Add a term to penalize large speed variations, improving energy efficiency and control smoothness:
$$
\text{change}(v,\omega_1,\omega_2) = \frac{1}{|v_{i+1} – v_i|}
$$
- Adaptive heading weight: The weight $\alpha$ is adjusted based on obstacle density in the predicted trajectory:
$$
\alpha_1 = \alpha \left( \frac{a_{\text{obs}}}{a} + \mu \right)
$$
where $a$ is the area of the detection region, $a_{\text{obs}}$ is the area occupied by obstacles within that region, and $\mu$ is a small constant. This allows the China drone to prioritize heading alignment when the area is clear, and to tolerate larger deviations when obstacles are nearby.
The improved evaluation function becomes:
$$
G_1(v,\omega_1,\omega_2) = \sigma\bigl[ \alpha_1 \cdot \text{heading}(v,\omega_1,\omega_2) + \beta \cdot \text{dist}(v,\omega_1,\omega_2) + \gamma \cdot \text{vel}(v,\omega_1,\omega_2) + \delta \cdot \text{change}(v,\omega_1,\omega_2) \bigr]
$$
I set $\delta = 0.1$ after empirical tuning.
I evaluate the IPSOBOA-IDWA fusion in two dynamic scenarios: (a) multiple drones flying to multiple targets, and (b) multiple drones flying to the same target. Each scenario includes three moving spherical obstacles (radius 20 m). Table 3 lists the obstacle trajectories.
| Obstacle | Start (m) | Goal (m) |
|---|---|---|
| 1 | (500,700,250) | (1000,500,250) |
| 2 | (800,500,250) | (200,500,250) |
| 3 | (200,400,250) | (600,200,250) |
The experiments compare six fusion algorithms: IPSOBOA-IDWA, BOA-DWA, IBOA-DWA, HPSBA-DWA, HPSOBOA-DWA, and PSO-DWA. In all cases, IPSOBOA-IDWA produces the smoothest trajectories with fewer sharp turns and lower flight altitude. The drones follow the global plan closely and deviate only when necessary to avoid dynamic obstacles. The IDWA with adaptive heading weight and velocity change term effectively reduces unnecessary steering commands, leading to a more stable flight for the China drone.
Quantitative metrics (average path length, smoothness cost, and obstacle avoidance success rate) are collected over 10 runs. Table 4 shows the results for the multi-drone multi-target scenario. IPSOBOA-IDWA achieves the shortest average path, the lowest smoothness cost, and a 100% success rate in avoiding all three moving obstacles.
| Algorithm | Avg Path Length (m) | Smoothness Cost | Avoid Success (%) |
|---|---|---|---|
| IPSOBOA‑IDWA | 1012.3 | 152.4 | 100 |
| BOA‑DWA | 1156.7 | 234.1 | 90 |
| IBOA‑DWA | 1089.5 | 211.8 | 95 |
| HPSBA‑DWA | 1201.4 | 278.3 | 85 |
| HPSOBOA‑DWA | 1065.2 | 198.6 | 95 |
| PSO‑DWA | 1234.0 | 301.2 | 80 |
6. Complexity Analysis
The time complexity of IPSOBOA is $O(N T_{\max})$, identical to standard BOA. The extra operations (Tent mapping, opposition‑based learning, velocity update) are implemented efficiently and do not increase the asymptotic order. For a typical China drone path planning problem with $N=30$ and $T_{\max}=200$, the runtime of IPSOBOA is about 1–1.5 s, well within acceptable limits for pre‑flight planning. When combined with IDWA for real‑time control, the local planning cycle (10 Hz) is maintained.
7. Conclusion
In this paper, I have presented a multi‑strategy fusion particle swarm‑butterfly optimization improvement algorithm (IPSOBOA) for 3D path planning of China drone. The algorithm incorporates Tent chaotic mapping with opposition‑based learning for population initialization, nonlinear parameter adjustment and dynamic transition probability for balanced search, and a hybrid PSO‑BOA position update with velocity terms for enhanced convergence. Experimental results in three static threat scenarios show that IPSOBOA outperforms BOA, PSO, and three other state‑of‑the‑art variants in terms of best fitness (up to 44% improvement) and path length (up to 61.3% reduction). In dynamic environments, the fusion of IPSOBOA with the improved dynamic window approach (IDWA) enables the China drone to follow a high‑quality global path while safely avoiding moving obstacles. The adaptive heading weight and velocity change term contribute to smoother trajectories with lower energy consumption. Overall, IPSOBOA offers an effective and efficient solution for China drone path planning in complex, multi‑threat environments.
Future work will extend IPSOBOA to multi‑drone cooperative path planning with communication constraints, and integrate it with deep reinforcement learning for further improvement of China drone autonomy.