In recent years, the rapid development of unmanned aerial vehicles (UAVs) has significantly impacted both military and civilian sectors, with China UAV industry playing a pivotal role in global technological advancement. Among the core challenges in autonomous UAV navigation, trajectory planning in complex threat environments remains a critical issue due to its high-dimensional, multi-constrained, and strongly coupled nature. This paper presents an improved hybrid intelligent optimization algorithm named HGWOSCA, which integrates the strengths of the Grey Wolf Optimizer (GWO) and the Sine Cosine Algorithm (SCA). By incorporating a Circle chaotic initialization strategy, a piecewise sine selection strategy, and a nonlinear oscillation control parameter, the proposed method achieves enhanced global search capability and local exploitation precision. Extensive simulations under multiple threat scenarios validate that HGWOSCA consistently outperforms seven benchmark metaheuristic algorithms, demonstrating superior robustness and applicability for China UAV trajectory planning.
1. Problem Formulation for UAV Trajectory Planning
In a known three-dimensional mission space, a UAV must navigate from a start point to a target point while avoiding various threats and satisfying flight constraints. The environment is modeled with obstacles (e.g., cubic no-fly zones, cylindrical interference regions) and terrain features approximated by Gaussian-shaped mountains. The goal is to minimize a weighted cost function that accounts for obstacle threat, no-fly zone penalty, path length, turning smoothness, and altitude fluctuation. The mathematical model is defined as follows.
The flight space is bounded by:
$$ \begin{cases} x_{\min} \le x \le x_{\max} \\ y_{\min} \le y \le y_{\max} \\ z_{\min} \le z \le z_{\max} \end{cases} $$
Terrain elevation is modeled as a sum of Gaussian peaks:
$$ Z(x,y) = \sum_{i=1}^{m} H_i \cdot \exp\left( -\left(\frac{x – x_i}{x_{s,i}}\right)^2 – \left(\frac{y – y_i}{y_{s,i}}\right)^2 \right) $$
Obstacle threat cost is given by:
$$ F_o = \begin{cases} 0, & d_r \ge R \\ (k / d_r)^4, & d_r < R \end{cases} $$
where \( d_r = \sqrt{(x_i – x_R)^2 + (y_i – y_R)^2 + (z_i – z_R)^2} \), \( R \) is the detection radius, and \( k \) is the threat weight. No-fly zone penalty is defined as:
$$ F_n = \begin{cases} 1, & x_{\text{NFZ}_l} \le x_i \le x_{\text{NFZ}_u} \text{ and } y_{\text{NFZ}_l} \le y_i \le y_{\text{NFZ}_u} \\ 0, & \text{otherwise} \end{cases} $$
Path length cost is normalized by the Euclidean distance between start and goal:
$$ F_L = \frac{\sum_{i=1}^{N+1} l_i}{L_{\text{EC}}}, \quad L_{\text{EC}} = \sqrt{(x_g – x_s)^2 + (y_g – y_s)^2 + (z_g – z_s)^2} $$
Turning smoothness is evaluated by the deviation from a reference turning angle \(\phi = 90^\circ\):
$$ F_{\text{Turn}} = \sum_{i=1}^{M-2} (\cos\phi – \cos\theta_i) $$
where \(\theta_i\) is the turning angle at the \(i\)-th waypoint. Altitude fluctuation cost is expressed as:
$$ F_{\text{Height}} = \sum_{i=1}^{M} (Z_i – \bar{Z})^2, \quad \bar{Z} = \frac{1}{M}\sum_{i=1}^{M} Z_i $$
The overall objective function linearly combines these five terms with weight factors \(m_1, m_2, m_3, m_4\):
$$ F_{\text{Whole}} = m_1 \cdot (F_o + F_n) + m_2 \cdot F_L + m_3 \cdot F_{\text{Height}} + m_4 \cdot F_{\text{Turn}} $$
To ensure flight feasibility, cubic spline interpolation is employed to smooth the piecewise linear path defined by control points, generating a continuous and differentiable trajectory.
2. Background of GWO and SCA
The Grey Wolf Optimizer (GWO) mimics the social hierarchy and hunting behavior of grey wolves. The algorithm maintains four roles: alpha, beta, delta, and omega wolves. Positions are updated based on the distance to the prey and the encircling mechanism:
$$ \mathbf{D} = |\mathbf{C} \cdot \mathbf{X}_p(t) – \mathbf{X}(t)|, \quad \mathbf{X}(t+1) = \mathbf{X}_p(t) – \mathbf{A} \cdot \mathbf{D} $$
where \(\mathbf{A} = 2a \cdot \mathbf{r}_1 – a\), \(\mathbf{C} = 2 \cdot \mathbf{r}_2\), and \(a\) decreases linearly from 2 to 0. Three best wolves (alpha, beta, delta) guide the search:
$$ \begin{aligned} \mathbf{D}_\alpha &= |\mathbf{C}_1 \cdot \mathbf{X}_\alpha – \mathbf{X}| \\ \mathbf{D}_\beta &= |\mathbf{C}_2 \cdot \mathbf{X}_\beta – \mathbf{X}| \\ \mathbf{D}_\delta &= |\mathbf{C}_3 \cdot \mathbf{X}_\delta – \mathbf{X}| \\ \mathbf{X}_1 &= \mathbf{X}_\alpha – \mathbf{A}_1 \cdot \mathbf{D}_\alpha \\ \mathbf{X}_2 &= \mathbf{X}_\beta – \mathbf{A}_2 \cdot \mathbf{D}_\beta \\ \mathbf{X}_3 &= \mathbf{X}_\delta – \mathbf{A}_3 \cdot \mathbf{D}_\delta \\ \mathbf{X}(t+1) &= \frac{\mathbf{X}_1 + \mathbf{X}_2 + \mathbf{X}_3}{3} \end{aligned} $$
The Sine Cosine Algorithm (SCA) uses trigonometric functions for position updates:
$$ \mathbf{X}_i^{t+1} = \begin{cases} \mathbf{X}_i^t + r_1 \cdot \sin(r_2) \cdot |r_3 \mathbf{P}_i^t – \mathbf{X}_i^t|, & r_4 < 0.5 \\ \mathbf{X}_i^t + r_1 \cdot \cos(r_2) \cdot |r_3 \mathbf{P}_i^t – \mathbf{X}_i^t|, & r_4 \ge 0.5 \end{cases} $$
where \(r_1, r_2, r_3, r_4\) are random numbers in \([0,1]\). While SCA offers good exploration through oscillation, it can suffer from instability and premature convergence in complex landscapes. This motivates the integration with GWO to balance exploration and exploitation.
3. Proposed HGWOSCA Algorithm
3.1 Circle Chaotic Initialization
To enhance population diversity and coverage of the search space, we replace random initialization with Circle chaotic mapping. The Circle map is defined as:
$$ x_{n+1} = \left( x_n + \Omega – \frac{K}{2\pi} \sin(2\pi x_n) \right) \bmod 1 $$
With \(K \ge 1\) and appropriate \(\Omega\), the map exhibits chaotic behavior with excellent ergodicity. This initialization ensures that the initial solutions are uniformly distributed across the feasible domain, which is particularly beneficial for high-dimensional China UAV trajectory planning problems. The generated chaotic sequence is scaled to the variable bounds to form the initial wolf positions.
3.2 Piecewise Sine Selection Strategy
We introduce a dynamic probability \(C_r\) that controls whether the position update follows the improved GWO mechanism or the SCA-inspired perturbation. The probability is modulated by a sine function:
$$ C_r = 0.5 \times \left( \sin\left(2\pi \times 0.25 t + \pi\right) \times \frac{t}{t_{\max}} + 1 \right) $$
This oscillatory behavior balances exploration (when GWO is dominant) and exploitation (when SCA perturbation is applied) throughout the iterative process. Specifically, when a random number \(rand < C_r\), the standard GWO update (using alpha, beta, delta wolves) is executed; otherwise, a differential perturbation inspired by SCA is applied:
$$ \mathbf{X}(t+1) = \mathbf{X}(t) + r_1 \cdot \sin(r_2) \cdot |r_3 \mathbf{X}_{\text{best}} – \mathbf{X}(t)| $$
where \(\mathbf{X}_{\text{best}}\) is the current best solution. This strategy reduces computational complexity while maintaining high diversity in early iterations and precision in later ones.
3.3 Nonlinear Convergence Factor
The linear decrease of parameter \(a\) in standard GWO may cause imbalance between global and local search. We adopt a nonlinear oscillation update for \(a\):
$$ a = 2 – 2 \times \left(\frac{t}{t_{\max}}\right)^\eta $$
where \(\eta = 0.5\) in our implementation, enabling a slower decay in early iterations to promote exploration and a faster decay later to enhance exploitation. The complete pseudo-code of HGWOSCA is summarized below.
| Step | Description |
|---|---|
| 1 | Initialize population using Circle chaotic mapping; set parameters \(t_{\max}\), population size \(N\), \(\eta\) |
| 2 | Evaluate fitness for all wolves; identify \(\mathbf{X}_\alpha\), \(\mathbf{X}_\beta\), \(\mathbf{X}_\delta\) |
| 3 | For \(t = 1\) to \(t_{\max}\): |
| Update \(a\) using nonlinear formula: \(a = 2 – 2(t/t_{\max})^{0.5}\) | |
| Compute \(C_r = 0.5 \times (\sin(2\pi \cdot 0.25 t + \pi) \cdot t/t_{\max} + 1)\) | |
| For each wolf \(\mathbf{X}_i\): | |
| Generate random \(r \in [0,1]\) | |
| If \(r < C_r\): update using GWO equations (with \(\mathbf{A}=2a\mathbf{r}_1-a\), \(\mathbf{C}=2\mathbf{r}_2\)) | |
| Else: update using SCA-inspired perturbation: \(\mathbf{X}_i = \mathbf{X}_i + r_1 \sin(r_2) |r_3 \mathbf{X}_\alpha – \mathbf{X}_i|\) | |
| End for | |
| Re-evaluate fitness; update \(\mathbf{X}_\alpha\), \(\mathbf{X}_\beta\), \(\mathbf{X}_\delta\) | |
| 4 | Return \(\mathbf{X}_\alpha\) as the best trajectory waypoints |
4. Experimental Setup
We design two simulation environments to evaluate the proposed HGWOSCA algorithm for China UAV trajectory planning. Both scenarios use a three-dimensional space of size \(200 \times 200 \times 100\) units, with start point \((10,10,20)\) and target point \((190,190,40)\). The trajectory is represented by five control points, and cubic spline interpolation generates a smooth path. Parameters are set as follows: population size = 30, maximum iterations = 300, weights \(m_1 = 5\), \(m_2 = 2\), \(m_3 = 1\), \(m_4 = 1\). Seven competitor algorithms are considered: GWO, SCA, Particle Swarm Optimization (PSO), Differential Evolution (DE), Grey Wolf with Sine Cosine (GWOSCA), Whale Optimization Algorithm (WOA), and Salp Swarm Algorithm (SSA). Each algorithm is run 30 independent times to gather statistical results.
| Scenario | Obstacles | No-Fly Zones | Terrain Complexity |
|---|---|---|---|
| Simple | 3 cylindrical obstacles | 1 cubic zone | Low (3 mountains) |
| Complex | 8 cylindrical obstacles | 3 cubic zones | High (7 mountains) |
5. Results and Discussion
5.1 Simple Scenario
In the simple environment, HGWOSCA achieves the best trajectory with minimal length and smooth turning. The proposed algorithm converges rapidly within the first 100 iterations, outperforming all baseline methods in both best and average fitness. The best fitness for HGWOSCA is 273.15, compared to 288.07 for GWO and 313.08 for SCA, representing improvements of 5.18% and 12.72%, respectively. The average fitness of HGWOSCA (284.75) is also superior, demonstrating robustness. Figure illustrates a typical optimized trajectory in this scenario.
5.2 Complex Scenario
In the complex scenario with dense threats and rugged terrain, HGWOSCA maintains its advantage. The algorithm produces a path that safely navigates around all obstacles and no-fly zones while keeping the turning angles moderate. The best fitness achieved by HGWOSCA is 273.22, which is 5.13% better than GWO (288.00) and 17.07% better than SCA (329.72). The average fitness (295.22) also surpasses that of all competitors, indicating consistent performance. Convergence curves show that HGWOSCA avoids premature convergence and continues to improve even in later iterations, thanks to the dynamic balance mechanism.
The following table summarizes the statistical results across 30 runs for both scenarios.
| Algorithm | Simple Best | Simple Avg | Complex Best | Complex Avg |
|---|---|---|---|---|
| HGWOSCA | 273.15 | 284.75 | 273.22 | 295.22 |
| GWO | 288.07 | 312.34 | 288.00 | 302.86 |
| SCA | 313.08 | 322.38 | 329.72 | 393.07 |
| PSO | 301.25 | 318.90 | 315.44 | 340.12 |
| DE | 295.60 | 307.45 | 302.10 | 325.83 |
| GWOSCA | 281.40 | 298.20 | 280.88 | 310.45 |
| WOA | 309.77 | 330.01 | 338.45 | 375.60 |
| SSA | 320.45 | 348.22 | 345.30 | 401.10 |
The superior performance of HGWOSCA can be attributed to three key factors. First, Circle chaotic initialization ensures that the initial wolf population covers the search space uniformly, which is crucial for China UAV operations where threat distribution may be irregular. Second, the piecewise sine selection strategy effectively balances exploration and exploitation, preventing the algorithm from being trapped in local optima, especially in the complex scenario with many local minima. Third, the nonlinear convergence factor \(a\) accelerates convergence speed while preserving diversity. Overall, HGWOSCA proves to be a reliable and efficient tool for autonomous trajectory planning in challenging environments.
6. Conclusion
This paper presents HGWOSCA, a hybrid metaheuristic algorithm that synergistically combines the Grey Wolf Optimizer and Sine Cosine Algorithm for China UAV trajectory planning. The integration of Circle chaotic initialization, piecewise sine selection, and nonlinear convergence control significantly enhances the algorithm’s global search ability and local exploitation accuracy. Extensive simulations under both simple and complex threat scenarios demonstrate that HGWOSCA consistently outperforms seven state-of-the-art algorithms in terms of solution quality, convergence speed, and robustness. The resulting trajectories are smooth, collision-free, and energy-efficient, meeting the multi-constraint requirements of real-world UAV missions. Future work will extend HGWOSCA to dynamic environments with moving obstacles and multi-UAV cooperative path planning, further advancing the capabilities of China UAV systems.