In recent years, the development of the low-altitude economy in China has received strong policy support, making China drone technology a core enabler for applications such as urban logistics, emergency rescue, and surveillance. Path planning for multiple China drones is a critical challenge that requires finding optimal, collision-free trajectories while satisfying constraints like flight distance, altitude, smoothness, and no-fly zones. Existing grey wolf optimization (GWO) algorithms often suffer from poor search capability and premature convergence when applied to multi-UAV path planning. To address these issues, we propose a multi-strategy improved grey wolf optimization (MIGWO) algorithm tailored for China drone missions. This paper presents the MIGWO algorithm, validates its performance on CEC2017 benchmark functions, and demonstrates its effectiveness in three simulated environments.
The rest of this paper is organized as follows. Section 2 formulates the UAV path planning constraints and objective function. Section 3 details the MIGWO algorithm with multiple enhancements. Section 4 presents performance tests and multi-UAV path planning simulations. Section 5 concludes the work.
1. Problem Formulation for China Drone Path Planning
We consider a scenario where multiple China drones start from different origins and fly to assigned targets while avoiding obstacles and adhering to flight constraints. The path is discretized into waypoints. The total cost function consists of four terms: flight distance, flight height, flight smoothness, and no-fly zone penalty.
1.1 Flight Distance Cost
The Euclidean distance between consecutive waypoints is summed over the path. For a drone with n waypoints (including start and target), the cost F1 is:
$$
F_1(X_i) = \sum_{j=1}^{n-1} \| p_{i,j+1} – p_{i,j} \|
$$
where pi,j = (xij, yij, zij) is the coordinate of the j-th waypoint for the i-th drone.
1.2 Flight Height Cost
Each China drone must fly within a safe altitude range [hmin, hmax]. The penalty is:
$$
F_2(X_i) = \sum_{j=1}^{n} \left| h_{ij} – \frac{h_{\max} + h_{\min}}{2} \right|
$$
where hij is the actual altitude of the waypoint.
1.3 Flight Smoothness Cost
Smoothness is evaluated by turning angles and climb angles. Let βij be the turning angle and θij the climb angle at waypoint j. The cost is:
$$
F_3(X_i) = \alpha_1 \sum_{j=1}^{n-1} \beta_{ij} + \alpha_2 \sum_{j=1}^{n-1} |\theta_{i,j+1} – \theta_{ij}|
$$
where βij = arctan( (pi,j+1 × pi,j) / (pi,j+1 · pi,j) ) and θij = arctan( (zi,j+1 – zij) / ∥ (xi,j+1 – xij, yi,j+1 – yij) ∥ ).
1.4 No-Fly Zone Cost
No-fly zones are modeled as cylinders. The penalty is:
$$
F_4(X_i) = \sum_{j=1}^{n} L_{ij}, \quad L_{ij} = \begin{cases} 1 & \sqrt{(x_{ij}-a)^2 + (y_{ij}-b)^2} < R \\ 0 & \text{otherwise} \end{cases}
$$
where (a,b) is the center of the no-fly zone and R its radius.
1.5 Overall Objective
The total cost for each drone is the weighted sum:
$$
F = \alpha_1 F_1 + \alpha_2 F_2 + \alpha_3 F_3 + \alpha_4 F_4, \quad \alpha_1+\alpha_2+\alpha_3+\alpha_4 = 1
$$
We aim to minimize the summed cost for all China drones simultaneously.
2. Multi-Strategy Improved Grey Wolf Optimization
The standard GWO algorithm relies on the top three wolves (α, β, δ) to guide the pack. However, this often leads to insufficient exploration and premature convergence. We introduce five enhancements to design MIGWO specifically for China drone path planning.
2.1 Piecewise Chaotic Initialization
To improve population diversity, we initialize the wolf positions using the Piecewise chaotic map:
$$
x_{n+1} = \begin{cases}
\frac{x_n}{P}, & 0 \le x_n < P \\
\frac{x_n – P}{0.5 – P}, & P \le x_n < 0.5 \\
\frac{1 – P – x_n}{0.5 – P}, & 0.5 \le x_n < 1-P \\
\frac{1 – x_n}{P}, & 1-P \le x_n \le 1
\end{cases}
$$
with P=0.4. This generates a more uniformly distributed initial population than random initialization.
2.2 Nonlinear Dynamic Factor
Instead of the linear decreasing parameter a in GWO, we adopt a nonlinear factor to balance exploration and exploitation adaptively:
$$
a(t) = 1 – \frac{1}{1 + e^{\alpha (1 – t/T_{\max})}}, \quad \alpha=2
$$
where t is the current iteration and Tmax the maximum iterations. This factor decays slowly initially (emphasizing exploration) and rapidly later (emphasizing exploitation).
2.3 Dynamic Position Update Strategy
The standard GWO updates positions by averaging the contributions of the top three wolves equally. To reflect the varying leadership quality, we weight each leader by its fitness:
$$
X(t+1) = \frac{\lambda_1 X_1 + \lambda_2 X_2 + \lambda_3 X_3}{\sum_{i=1}^3 \lambda_i}, \quad \lambda_i = \frac{1}{f_i(X(t+1))}
$$
where X1, X2, X3 are the candidate positions influenced by α, β, δ wolves, and fi is the fitness. This ensures the best current leader has more influence.
2.4 Cross-Fusion ω Wolf Position Update
The standard algorithm ignores the role of lower-ranking ω wolves. We introduce three new wolves generated by crossover between the top three wolves:
$$
X_{\text{new}1} = \text{cross}(X_\alpha, X_\delta), \quad X_{\text{new}2} = \text{cross}(X_\beta, X_\delta), \quad X_{\text{new}3} = \text{cross}(X_\delta, X_\delta)
$$
Then, the ω wolf position is updated using a combination of the original three leaders and the new wolves:
$$
X_\omega(t+1) = \frac{X_1 + X_2 + X_3 + X_4 + X_5 + X_6}{6}
$$
where X4, X5, X6 are calculated similarly to the original GWO but using the new wolves as reference points. This enhances diversity and prevents stagnation.
2.5 Cuckoo Search Secondary Optimization
To further avoid local optima, after each GWO iteration, we apply a cuckoo search (CS) step using Lévy flights and random elimination. The best solution from GWO is used as one of the CS initial nests. The CS pseudocode is given below:
Begin
Generate initial population of n host nests (including the GWO best)
while t < MaxGeneration
Get a cuckoo randomly by Lévy flights
Evaluate its fitness Fi
Choose a nest j randomly
if (Fi > Fj) then replace j
Abandon a fraction (pa) of worst nests and build new ones
Keep the best solutions
Rank and find current best
end while
End
3. Numerical Experiments
3.1 Benchmark Function Validation
We tested MIGWO on four CEC2017 functions: one unimodal (F3) and three multimodal (F4, F5, F6). The dimensions were set to 30, population size 30, and maximum iterations 500. Each run was repeated 30 times. The results are compared with standard GWO and Dung Beetle Optimizer (DBO).
| Function | Metric | GWO | DBO | MIGWO |
|---|---|---|---|---|
| F3 (Unimodal) | Mean | 3.82×10² | 4.22×10² | 3.00×10² |
| Std | 2.89×10¹ | 1.01×10² | 1.01×10⁻¹² | |
| Best | 3.42×10² | 3.88×10² | 3.00×10² | |
| F4 (Multimodal) | Mean | 4.89×10² | 4.92×10² | 4.00×10² |
| Std | 2.81×10¹ | 4.63×10² | 1.03×10⁻¹⁴ | |
| Best | 4.42×10² | 4.51×10² | 4.00×10² | |
| F5 (Multimodal) | Mean | 5.67×10² | 5.77×10² | 5.03×10² |
| Std | 2.20×10¹ | 7.23×10¹ | 1.04×10⁻⁵ | |
| Best | 5.42×10² | 5.69×10² | 5.00×10² | |
| F6 (Multimodal) | Mean | 6.56×10² | 6.68×10² | 6.01×10² |
| Std | 4.58×10¹ | 6.13×10¹ | 1.11×10⁻⁶ | |
| Best | 6.22×10² | 6.31×10² | 6.01×10² |
MIGWO achieves the theoretical optimum for F3 and F4, and significantly lower means and standard deviations for all functions, demonstrating superior accuracy and robustness. This indicates that the proposed enhancements effectively boost both exploration and exploitation, making MIGWO well-suited for complex China drone path planning problems.
3.2 Multi-UAV Path Planning in Three Environments
We evaluated MIGWO for three China drones in three maps with increasing complexity. The drones’ start and goal coordinates are listed in the table below. Obstacle positions and radii are given accordingly. All drones have speed ≤ 30 m/s, altitude limit 0–20 km, and turning/climb angles within ±45°.
| Drone | Map 1 | Map 2 | Map 3 | ||||||
|---|---|---|---|---|---|---|---|---|---|
| x | y | z | x | y | z | x | y | z | |
| UAV1 start | 50 | 50 | 250 | 50 | 50 | 250 | 50 | 50 | 150 |
| UAV2 start | 20 | 100 | 240 | 20 | 100 | 240 | 20 | 100 | 180 |
| UAV3 start | 100 | 20 | 240 | 100 | 20 | 240 | 100 | 20 | 150 |
| UAV1 goal | 575 | 800 | 230 | 575 | 800 | 230 | 290 | 370 | 300 |
| UAV2 goal | 565 | 700 | 230 | 810 | 400 | 270 | 300 | 400 | 300 |
| UAV3 goal | 610 | 650 | 230 | 560 | 700 | 230 | 320 | 400 | 300 |
The obstacles for each map are listed below. Each obstacle is a cylinder defined by its center (x,y,z) and radius.
| Map | Obstacle | x (m) | y (m) | z (m) | Radius (m) |
|---|---|---|---|---|---|
| Map 1 | 1 | 400 | 500 | 190 | 100 |
| 2 | 600 | 200 | 220 | 100 | |
| 3 | 300 | 250 | 200 | 100 | |
| 4 | 700 | 500 | 200 | 100 | |
| Map 2 | 1 | 400 | 400 | 210 | 90 |
| 2 | 500 | 170 | 210 | 70 | |
| 3 | 200 | 150 | 220 | 60 | |
| 4 | 820 | 300 | 170 | 80 | |
| 5 | 350 | 700 | 190 | 90 | |
| 6 | 170 | 350 | 220 | 100 | |
| 7 | 620 | 500 | 240 | 90 | |
| 8 | 650 | 270 | 200 | 80 | |
| 9 | 800 | 600 | 150 | 90 | |
| Map 3 | 1 | 100 | 100 | 120 | 40 |
| 2 | 200 | 200 | 150 | 50 | |
| 3 | 300 | 80 | 170 | 50 | |
| 4 | 300 | 340 | 160 | 50 | |
| 5 | 200 | 80 | 150 | 40 | |
| 6 | 130 | 280 | 150 | 40 | |
| 7 | 180 | 370 | 300 | 60 |
We compared MIGWO with GWO, DBO, Marine Predators Algorithm (MPA), and Whale Optimization Algorithm (WOA). Each algorithm ran 50 times with population 30 and maximum iterations 200. The best trajectories for each map are compared qualitatively and quantitatively.
In Map 1, MIGWO produced smooth paths with minimal turns. In contrast, GWO and DBO paths exhibited large angle turns and altitude fluctuations. MPA and WOA trajectories sometimes collided with obstacles. The total flight distances for all three drones are summarized in the table below.
| Algorithm | Map 1 | Map 2 | Map 3 |
|---|---|---|---|
| GWO | 4.2 | 6.4 | 3.7 |
| DBO | 5.6 | 7.2 | 4.6 |
| MPA | 4.3 | 6.5 | 3.4 |
| WOA | 4.8 | 6.6 | 3.2 |
| MIGWO | 3.5 | 5.5 | 2.8 |
MIGWO consistently obtained the shortest total flight distance: 3.5 km for Map 1, 5.5 km for Map 2, and 2.8 km for Map 3. Compared to the second-best algorithm, MIGWO reduced the distance by 0.7 km (Map 1), 0.9 km (Map 2), and 0.4 km (Map 3). The convergence curves also show that MIGWO reaches lower fitness values faster and more stably than the competing algorithms. These results confirm that MIGWO effectively plans safe, smooth, and energy-efficient paths for multiple China drones.
4. Conclusion
We propose a multi-strategy improved grey wolf optimization algorithm (MIGWO) for multi-UAV path planning, specifically targeting China drone applications. By incorporating Piecewise chaotic initialization, nonlinear dynamic factor, dynamic position update based on fitness weights, cross-fusion ω wolf update, and cuckoo search secondary optimization, MIGWO significantly enhances population diversity, balances exploration and exploitation, and avoids premature convergence. Benchmark tests on CEC2017 functions demonstrate that MIGWO outperforms GWO and DBO in both unimodal and multimodal optimization. In three simulated environments with varying complexity, MIGWO produces the shortest total flight distances (3.5 km, 5.5 km, 2.8 km) while satisfying all constraints, outperforming GWO, DBO, MPA, and WOA. The proposed algorithm provides a reliable and efficient solution for multi-China drone path planning in real-world scenarios.