In the rapidly evolving field of autonomous systems, China drone technology has become a pivotal element in applications ranging from agricultural monitoring to logistics delivery and disaster relief. The efficiency and safety of China drone operations heavily depend on the ability to generate optimal flight paths that avoid obstacles, minimize energy consumption, and ensure smooth maneuvers. Traditional path planning algorithms, such as A* and Dijkstra, are intuitive but suffer from high computational costs in complex environments, especially when dealing with the intricate constraints of real-world China drone missions. Swarm intelligence optimization algorithms have emerged as promising alternatives, yet many suffer from issues like premature convergence, poor local exploitation, and insufficient path smoothness. To address these challenges, we propose an improved GOOSE optimization algorithm, termed GOOSE_DE, specifically tailored for China drone path planning in two-dimensional grid environments. Our method integrates Sobol sequences, opposition-based learning, bidirectional information exchange, and adaptive parameter control to achieve superior convergence accuracy, stability, and path quality. Extensive experiments on standard benchmark functions and grid maps demonstrate that GOOSE_DE consistently outperforms existing algorithms in terms of path length, smoothness, and robustness, thereby providing a reliable solution for China drone autonomous navigation.
Path planning for China drone involves finding a collision-free trajectory from a start point to a target point while optimizing multiple objectives: total path length, turning smoothness, and motion continuity. In low-altitude cruise scenarios, the motion of a China drone can be reasonably abstracted as a two-dimensional planar path planning problem, where the drone is modeled as a point mass moving at a constant altitude. The path length directly affects flight time and energy consumption, while smoothness influences attitude stability and flight safety. Existing classic algorithms like A* and Dijkstra provide deterministic solutions but become inefficient in large-scale or dynamic environments. Swarm intelligence algorithms, such as the Sparrow Search Algorithm (SSA), Grey Wolf Optimization (GWO), and Particle Swarm Optimization (PSO), have been widely applied. However, they often exhibit limitations: SSA has strong global exploration but weak local refinement; GWO suffers from premature convergence; PSO tends to stagnate early. The recently proposed GOOSE algorithm mimics the vigilance and foraging behavior of geese, offering robust global search capability but still lacking fine-grained local exploitation. To overcome these drawbacks, we integrate the GOOSE algorithm with the Differential Evolution (DE) algorithm, which is known for its simple structure, strong local search, and resistance to premature convergence. The resulting hybrid algorithm, GOOSE_DE, synergistically combines the strengths of both methods.
The core contributions of our work are summarized as follows:
- A hybrid optimization framework that seamlessly integrates GOOSE and DE through bidirectional information interaction and solution-quality-driven selection, enabling complementary cooperative search.
- An initialization strategy combining Sobol sequences, Gaussian perturbation, and opposition-based learning to enhance population coverage and early search efficiency.
- An adaptive stage-wise regulation mechanism based on iteration progress and population diversity, which dynamically adjusts information sharing intensity and key DE parameters (differential weight \(F_k\) and crossover probability \(c_k\)), thereby improving convergence stability and reducing the risk of local optima.
- Comprehensive simulation validation for China drone path planning under different grid complexities, demonstrating superior performance in path length, smoothness, and robustness compared to five state-of-the-art algorithms.
The remainder of this paper is organized as follows. Section 2 describes the proposed GOOSE_DE algorithm and its key strategies. Section 3 presents experimental results on benchmark functions. Section 4 evaluates the algorithm on China drone path planning scenarios. Section 5 concludes the paper.
2. GOOSE_DE Algorithm and Improvement Strategies
2.1 Sobol Sequence, Gaussian Perturbation, and Opposition-Based Learning
The Sobol sequence is a low-discrepancy sequence that generates uniformly distributed points in the hypercube \([0,1]^d\), providing excellent spatial coverage. To further enhance solution diversity and search efficiency, we combine Sobol points with a Gaussian perturbation. The perturbed point is given by:
$$ S_i’ = S_i + \epsilon_i, \quad \epsilon_i \sim \mathcal{N}(0, \sigma^2) $$
where \(S_i\) is the original Sobol point, and \(\epsilon_i\) is a random sample from a normal distribution with mean 0 and variance \(\sigma^2\). Additionally, we apply opposition-based learning to each perturbed point:
$$ S_i^{\text{opp}} = 1 – S_i $$
The fitness of both \(S_i’\) and \(S_i^{\text{opp}}\) is evaluated, and the better one is retained in the initial population. This three-pronged initialization strategy generates a high-quality starting population that accelerates convergence.
2.2 Review of Base Algorithms
2.2.1 GOOSE Algorithm
The GOOSE algorithm simulates the vigilance and foraging behavior of geese. A random variable dynamically switches between exploration and exploitation phases. In the exploration phase, the position update follows:
$$ X_{i}^{k+1} = g^k + \epsilon_i^k m^k a^k $$
where \(g^k\) is the global best position at iteration \(k\), \(\epsilon_i^k \sim \mathcal{N}(0, I_d)\), \(m^k\) is the minimum total response time, and \(a^k = 2 – 2k/T\) is a decay factor. In the exploitation phase, the update is:
$$ X_{i}^{k+1} = \begin{cases}
g^k + f_i^k + d_i^k (\tau_i^k)^2, & p_i^k > 0.2, \omega_i^k \ge 12, \\
g^k + f_i^k \odot d_i^k (\tau_i^k)^2 c_i^k, & \text{otherwise}
\end{cases} $$
where \(f_i^k\) is the disturbance term from falling stones, \(d_i^k\) is the effective wake radius, \(\tau_i^k\) is the average response time, \(\odot\) denotes element-wise multiplication, and \(c_i^k\) is a local perturbation coefficient.
2.2.2 Differential Evolution Algorithm
DE algorithm iteratively improves candidate solutions through mutation, crossover, and selection. The mutation operation is:
$$ V_{i,j}^k = X_{r_1,j}^k + F^k (X_{r_2,j}^k – X_{r_3,j}^k) $$
where \(F^k\) is the scaling factor, and \(r_1, r_2, r_3\) are distinct random indices. The crossover operation produces a trial vector:
$$ U_{i,j}^k = \begin{cases}
V_{i,j}^k, & p_{i,j}^k \le c^k \ \text{or} \ j = j_i^k \\
X_{i,j}^k, & \text{otherwise}
\end{cases} $$
where \(c^k\) is the crossover probability. Selection retains the better between \(U_i^k\) and \(X_i^k\) based on fitness.
2.3 Proposed GOOSE_DE Algorithm
GOOSE_DE constructs a stage-adaptive optimization framework that enables bidirectional information exchange between GOOSE and DE populations. The algorithm uses solution quality to intelligently select which algorithm’s best solution to share, and dynamically adjusts DE parameters \(F_k\) and \(c_k\) based on iteration progress and population diversity.
2.3.1 Stage-Wise Adjustment and Population Diversity Complementarity
The interaction probability between GOOSE and DE is set according to three stages of iteration:
| Stage | \(k\) Range | \(P_{\text{goose}}(k)\) | \(P_{\text{de}}(k)\) |
|---|---|---|---|
| Early exploration | \(k < T/3\) | 0.2 | 0.4 |
| Middle refinement | \(T/3 \le k < 2T/3\) | 0.4 | 0.5 |
| Late convergence | \(k \ge 2T/3\) | 0.6 | 0.7 |
To further adapt to the current search status, we calculate population diversity as the average Euclidean distance among all individuals:
$$ D_{\text{Diversity}} = \frac{2}{n(n-1)} \sum_{i=1}^{n-1} \sum_{j=i+1}^{n} \|x_i – x_j\|_2 $$
A diversity adjustment factor \(\alpha\) is then applied to the base probabilities:
$$ P_{\gamma}^{\text{final}}(k) = P_{\gamma}(k) \times \alpha \times \beta $$
where \(\beta\) is a normalization factor that constrains the final probability within \([0.05, 0.95]\) to avoid extreme values. The factor \(\alpha\) is determined by thresholding the normalized diversity \(D\): if \(D < D_{\text{low}}\), \(\alpha = 1.5\) (increase interaction to escape stagnation); if \(D > D_{\text{high}}\), \(\alpha = 0.5\) (reduce interaction to preserve exploration); otherwise \(\alpha = 1.0\).
2.3.2 Adaptive Parameter Control
The differential weight \(F_k\) and crossover probability \(c_k\) are linearly adjusted over iterations to balance exploration and exploitation:
$$ F_k = 0.5 + 0.3 \times \left(1 – \frac{k}{T}\right) $$
$$ c_k = 0.9 – 0.2 \times \frac{k}{T} $$
Early iterations use larger \(F_k\) to enhance global search, while later iterations use smaller \(F_k\) for refined local search. Similarly, higher crossover probability early on promotes diversity, and lower probability later intensifies selection pressure.
2.3.3 Bidirectional Information Interaction and Intelligent Selection
At each iteration, with probability \(P_{\gamma}^{\text{final}}(k)\), the GOOSE population receives the current best solution from DE, and vice versa. The quality of each algorithm’s best solution is measured by:
$$ Q_{\gamma}^k = \frac{1}{1 + \|b_{\gamma}^k – g^k\|_2} $$
where \(b_{\gamma}^k\) is the best solution of algorithm \(\gamma\) at iteration \(k\), and \(g^k\) is the global best. If \(Q_{DE} > Q_{GOOSE}\), the DE best solution is selected as the guiding center; otherwise the GOOSE best is used. This mechanism ensures that the algorithm always follows the most promising search direction.
2.4 Algorithm Flow
The GOOSE_DE algorithm proceeds as follows:
- Initialization: Generate initial population using Sobol sequence, apply Gaussian perturbation, and retain the best between original and opposition-based solutions. Set parameters: population size \(n\), maximum iterations \(T\), \(F_0\), \(c_0\).
- Main Loop: For each iteration \(k = 0, 1, \ldots, T-1\):
- Update \(F_k\) and \(c_k\) according to equations.
- Compute population diversity \(D_{\text{Diversity}}\) and determine adjustment factor \(\alpha\).
- Calculate final interaction probabilities \(P_{\text{goose}}^{\text{final}}(k)\) and \(P_{\text{de}}^{\text{final}}(k)\).
- Perform GOOSE update for all individuals; perform DE mutation, crossover, and selection for all individuals.
- With probability \(P_{\text{goose}}^{\text{final}}(k)\), replace a randomly selected GOOSE individual with the DE best solution (if quality is higher). Similarly, with probability \(P_{\text{de}}^{\text{final}}(k)\), replace a DE individual with the GOOSE best solution.
- Evaluate fitness of all individuals and update global best \(g^k\).
- Termination: Stop when \(k = T\) and output the global best solution.
3. Performance Evaluation on Benchmark Functions
We evaluate the proposed GOOSE_DE algorithm on five standard benchmark functions, covering unimodal and multimodal types. The functions are listed in the table below. We compare GOOSE_DE with SSA, KMGWO, HIDMSPSO, ADE, and the original GOOSE. Parameter settings: population size = 120, maximum iterations = 500, each algorithm runs 10 independent trials.
| Function | Expression | Search Range | Optimum |
|---|---|---|---|
| F1 | \(F_1(x) = \sum_{i=1}^d i x_i^2\) | [-10,10] | 0 |
| F2 | \(F_2(x) = \sum_{i=1}^d |x_i| + \prod_{i=1}^d |x_i|\) | [-10,10] | 0 |
| F3 | \(F_3(x) = \sum_{i=1}^d \left( \sum_{k=0}^{k_{\max}} a^k \cos(2\pi b^k (x_i+0.5)) \right) – d \sum_{k=0}^{k_{\max}} a^k \cos(2\pi b^k \cdot 0.5)\) | [-0.5,0.5] | 0 |
| F4 | \(F_4(x) = 10d + \sum_{i=1}^d (x_i^2 – 10\cos(2\pi x_i))\) | [-5.12,5.12] | 0 |
| F5 | \(F_5(x) = \sum_{i=1}^d x_i^{i+1}\) | [-1,1] | 0 |
Table below summarizes the best, mean, and standard deviation of fitness values over 10 runs. GOOSE_DE achieves the theoretical optimum (0) for all five functions with zero standard deviation, demonstrating perfect convergence and stability. In contrast, SSA also reaches 0 on F1, F3, and F4 but shows non-zero variance on F2 and F5. Other algorithms like HIDMSPSO, GOOSE, and ADE exhibit significantly larger errors and variances.
| Function | Algorithm | Best | Mean | Std |
|---|---|---|---|---|
| F1 | GOOSE_DE | 0 | 0 | 0 |
| F1 | HIDMSPSO | 7.899e-9 | 2.036e-8 | 9.620e-9 |
| F1 | GOOSE | 2.937e-1 | 5.596e-1 | 2.842e-1 |
| F1 | ADE | 4.817e-4 | 1.416e-3 | 6.308e-4 |
| F1 | KMGWO | 1.916e-45 | 4.092e-37 | 9.124e-37 |
| F1 | SSA | 0 | 0 | 0 |
| F2 | GOOSE_DE | 0 | 0 | 0 |
| F2 | HIDMSPSO | 5.252e-5 | 9.486e-5 | 2.939e-5 |
| F2 | GOOSE | 4.667e-1 | 2.180 | 4.699 |
| F2 | ADE | 7.746e-3 | 1.216e-2 | 2.740e-4 |
| F2 | KMGWO | 9.505e-26 | 5.542e-21 | 1.704e-20 |
| F2 | SSA | 0 | 1.579e-227 | 4.994e-227 |
| F3 | GOOSE_DE | 0 | 0 | 0 |
| F3 | HIDMSPSO | 2.999e-2 | 1.712e-1 | 3.070e-1 |
| F3 | GOOSE | 1.751e1 | 2.382e1 | 4.957 |
| F3 | ADE | 1.399e-1 | 1.848e-1 | 3.288e-2 |
| F3 | KMGWO | 0 | 7.105e-16 | 2.247e-15 |
| F3 | SSA | 0 | 0 | 0 |
| F4 | GOOSE_DE | 0 | 0 | 0 |
| F4 | HIDMSPSO | 2.786e1 | 3.343e1 | 6.028 |
| F4 | GOOSE | 6.748e1 | 1.119e2 | 3.693e1 |
| F4 | ADE | 3.424e1 | 5.038e1 | 1.109e1 |
| F4 | KMGWO | 0 | 1.364e-13 | 3.734e-13 |
| F4 | SSA | 0 | 0 | 0 |
| F5 | GOOSE_DE | 0 | 0 | 0 |
| F5 | HIDMSPSO | 2.887e-21 | 6.553e-20 | 1.326e-19 |
| F5 | GOOSE | 3.309e-6 | 9.669e-6 | 3.986e-6 |
| F5 | ADE | 1.469e-23 | 9.606e-22 | 1.984e-21 |
| F5 | KMGWO | 3.918e-157 | 2.076e-110 | 6.564e-110 |
| F5 | SSA | 0 | 3.414e-154 | 1.080e-153 |
The convergence curves further illustrate the superior convergence speed and accuracy of GOOSE_DE. On functions F1–F5, the proposed algorithm reaches the global optimum within fewer iterations than competing methods, while maintaining stable and monotonic descent.
4. China Drone Path Planning Experiments
To validate the effectiveness of GOOSE_DE for China drone path planning, we conducted simulations on two grid maps of sizes 20×20 and 30×30 with obstacle densities of approximately 26% and 27%, respectively. The start point is at the lower-left corner and the target at the upper-right corner. We compared GOOSE_DE with HIDMSPSO, KMGWO, SSA, ADE, and GOOSE. Each algorithm ran 10 independent trials. The metrics include path length, number of iterations to reach the best solution, number of turning points, path curvature, turning angle, and acceleration continuity (a measure of smoothing). Results are summarized in the table below.
| Algorithm | Map | Path Length (m) | Min Iterations | Turning Points | Curvature | Turning Angle (°) | Acceleration Continuity |
|---|---|---|---|---|---|---|---|
| HIDMSPSO | 20×20 | 34.820 ± 1.941 | 150 ± 27.105 | 13 ± 2.494 | 0.402 ± 0.096 | 34.296 ± 5.263 | 0.399 ± 0.049 |
| KMGWO | 20×20 | 32.720 ± 1.494 | 116 ± 28.418 | 13 ± 4.061 | 0.482 ± 0.119 | 33.663 ± 7.845 | 0.382 ± 0.093 |
| SSA | 20×20 | 36.053 ± 2.213 | 33 ± 60.353 | 5 ± 4.191 | 1.249 ± 0.576 | 38.409 ± 10.438 | 0.161 ± 0.137 |
| ADE | 20×20 | 29.651 ± 0.639 | 160 ± 33.190 | 13 ± 1.780 | 0.386 ± 0.078 | 26.145 ± 4.271 | 0.524 ± 0.067 |
| GOOSE | 20×20 | 35.616 ± 1.783 | 49 ± 38.541 | 17 ± 2.055 | 0.501 ± 0.056 | 41.171 ± 6.643 | 0.399 ± 0.051 |
| GOOSE_DE | 20×20 | 28.409 ± 0.301 | 89 ± 29.069 | 8 ± 1.932 | 0.243 ± 0.063 | 16.742 ± 3.917 | 0.625 ± 0.042 |
| HIDMSPSO | 30×30 | 58.859 ± 3.308 | 237 ± 40.530 | 18 ± 3.091 | 0.556 ± 0.152 | 37.419 ± 5.826 | 0.296 ± 0.030 |
| KMGWO | 30×30 | 78.127 ± 14.395 | 139 ± 93.863 | 24 ± 4.077 | 0.686 ± 0.101 | 50.531 ± 8.610 | 0.240 ± 0.039 |
| SSA | 30×30 | 56.443 ± 1.987 | 8 ± 15.629 | 7 ± 6.877 | 1.035 ± 0.336 | 41.920 ± 5.859 | 0.136 ± 0.131 |
| ADE | 30×30 | 48.045 ± 1.524 | 279 ± 17.289 | 18 ± 2.003 | 0.433 ± 0.090 | 31.574 ± 4.948 | 0.417 ± 0.066 |
| GOOSE | 30×30 | 56.010 ± 2.112 | 131 ± 43.652 | 18 ± 2.211 | 0.537 ± 0.151 | 38.936 ± 5.545 | 0.303 ± 0.038 |
| GOOSE_DE | 30×30 | 43.433 ± 0.092 | 193 ± 57.970 | 13 ± 1.350 | 0.234 ± 0.027 | 17.204 ± 2.008 | 0.570 ± 0.006 |
On the 20×20 grid, GOOSE_DE achieves the shortest average path length (28.409 m) with a very small standard deviation (0.301 m), indicating excellent stability. Its curvature (0.243) and turning angle (16.742°) are the smallest among all algorithms, while acceleration continuity is the highest (0.625), demonstrating a remarkably smooth path. Although the number of turning points is slightly higher than SSA (8 vs. 5), the overall path quality is far superior. On the larger 30×30 grid, GOOSE_DE again obtains the shortest path (43.433 m) with near-zero variance (0.092 m), the lowest curvature and turning angle, and the highest acceleration continuity. In contrast, ADE yields a path length of 48.045 m but with significantly poorer smoothness (curvature 0.433, turning angle 31.574°). SSA exhibits very poor curvature and acceleration continuity, indicating abrupt direction changes that are unsuitable for China drone flight.
The convergence curves for the path planning experiments show that GOOSE_DE converges faster than other algorithms, often reaching a near-optimal solution within the first 50–100 iterations, while maintaining consistent progress without stagnation. The path visualizations reveal that GOOSE_DE generates routes that hug obstacles more efficiently with fewer unnecessary detours, producing smoother turns that reduce the mechanical stress on the China drone airframe.
4.1 Complexity Analysis
We also compare the computational complexity of different algorithms. The table below lists the time and space complexities, where \(n\) is population size, \(d\) is problem dimension, and \(T\) is maximum iterations.
| Algorithm | Time Complexity | Space Complexity |
|---|---|---|
| HIDMSPSO | \(O(T n d)\) | \(O(n d)\) |
| KMGWO | \(O(T n d)\) | \(O(n d)\) |
| SSA | \(O(T n (\log n + d))\) | \(O(n d)\) |
| ADE | \(O(T n (d + \log n))\) | \(O(n d)\) |
| GOOSE | \(O(T n d)\) | \(O(n d)\) |
| GOOSE_DE | \(O(T n^2 d)\) | \(O(n d)\) |
GOOSE_DE introduces an additional diversity computation that increases the per-iteration cost to \(O(n^2 d)\), but in practice the total runtime remains acceptable for offline path planning scenarios. The significant improvements in path quality and robustness justify the moderate increase in computational burden.
5. Conclusion
In this work, we proposed GOOSE_DE, an improved GOOSE optimization algorithm for China drone path planning. By integrating Sobol sequence initialization, opposition-based learning, bidirectional information interaction, solution-quality-driven selection, and adaptive parameter control, the algorithm effectively balances global exploration and local exploitation. Extensive experiments on standard benchmark functions demonstrate that GOOSE_DE achieves exact convergence to the global optimum with zero variance, outperforming five state-of-the-art algorithms. In China drone path planning simulations on 20×20 and 30×30 grid maps, GOOSE_DE consistently yields the shortest average path length, the smallest curvature and turning angle, and the highest acceleration continuity, indicating superior smoothness and robustness. These results confirm that GOOSE_DE is a highly effective and reliable tool for autonomous navigation of China drone in complex environments. Future work will extend the algorithm to three-dimensional space and dynamic obstacle scenarios, and investigate its real-time implementation on embedded flight controllers for practical China drone deployment.