The rapid development of unmanned aerial vehicle (UAV) technology has brought significant changes to various industries, especially in China, where UAVs are widely used for reconnaissance, disaster rescue, agricultural monitoring, and logistics. However, the complexity of three-dimensional environments and the need for multi-UAV coordination pose severe challenges to trajectory planning. Traditional methods such as A* and Dijkstra often fail to find feasible paths within acceptable time constraints, while meta-heuristic algorithms often struggle with the trade-off between accuracy and computational efficiency. In this work, we propose a novel multi-UAV path planning algorithm that integrates a cell-like spiking neural P system (CSNPS) with a hybrid grey wolf optimizer (GWO), termed CSP-GWO. Our algorithm leverages the inherent parallelism of membrane computing to simulate the cooperative behavior of multiple China UAVs, ensuring both high-quality solutions and low computational overhead. Extensive experiments on benchmark functions and realistic multi-UAV scenarios demonstrate the superiority of our approach.
1. Problem Formulation and Objective Function
We consider a scenario where multiple China UAVs must navigate from designated start points to target areas in a 3D environment containing obstacles, no-fly zones, and radar threats. Each UAV must satisfy physical constraints (maximum flight distance, altitude limits, turning angles), environmental constraints (avoidance of threats), and cooperative constraints (collision avoidance and temporal synchronization). The path of each UAV is discretized into a sequence of waypoints. The overall cost function combines six weighted sub-costs:
$$
J = w_1 F_1 + w_2 F_2 + w_3 F_3 + w_4 F_4 + w_5 F_5 + w_6 F_6 .
$$
The definitions of each sub-cost are given below:
$$
\begin{aligned}
F_1 &= k_1 \sum_{i=1}^{n} l_i, \\
F_2 &= k_2 \begin{cases} p_1 (H_{\min} – z_i), & z_i \le H_{\min} \\ 0, & H_{\min} < z_i < H_{\max} \\ p_2 (z_i – H_{\max}), & z_i \ge H_{\max} \end{cases}, \\
F_3 &= k_3 \sum_{i=1}^{n} (p_3 f_{\alpha_i} + p_4 f_{\psi_i}), \\
F_4 &= k_4 \begin{cases} p_5 \sum_{j=1}^{m} \frac{1}{R_m}, & d_w \le R_m \\ 0, & d_w > R_m \end{cases}, \\
F_5 &= k_5 \begin{cases} 0, & d_{ab} \ge d_{\min} \\ p_6, & d_{ab} < d_{\min} \end{cases}, \\
F_6 &= k_6 \begin{cases} 0, & [t_{\text{start}},t_{\text{end}}] \cap [t_{i,\text{start}},t_{i,\text{end}}] \ne \emptyset \\ p_7, & \text{otherwise} \end{cases}.
\end{aligned}
$$
Here, $l_i$ denotes the Euclidean distance between consecutive waypoints; $z_i$ is the altitude; $\alpha_i$ and $\psi_i$ are the horizontal and vertical turning angles; $d_w$ is the distance to the nearest threat; $d_{ab}$ is the distance between two UAVs; and the time intervals ensure that all UAVs can reach the target simultaneously. The weights $w_1,\dots,w_6$ satisfy $\sum w_i = 1$. To generate smooth and flyable trajectories, we apply B-spline interpolation after the optimization.
Table 1 summarizes the main parameters used in our simulations.
| Parameter | Value | Description |
|---|---|---|
| $L_{\max}$ | 20 km | Maximum flight distance per UAV |
| $H_{\min}, H_{\max}$ | 20 m, 2 km | Altitude limits |
| $\alpha_{\max}$ | 45° | Maximum horizontal turning angle |
| $\psi_{\max}$ | 60° | Maximum vertical turning angle |
| $d_{\min}$ | 0.5 km | Minimum safe inter-UAV distance |
| $R_m$ | [0.5, 1.5] km | Threat radii |
| $v$ | [20, 40] m/s | UAV speed range |
| Cooperation time | 160 s | Target arrival window |
2. The CSP-GWO Algorithm
2.1 Cell-like Spiking Neural P Systems
A cell-like spiking neural P system (CSNPS) is a distributed parallel computing model inspired by the structure and functioning of biological neurons. A CSNPS of degree $m\ge 1$ is formally defined as:
$$
\Pi = (O, H, \omega_1, \dots, \omega_n, \sigma_1, \dots, \sigma_m, R_1, \dots, R_m, \text{syn}, I_{\text{in}}, I_{\text{out}}),
$$
where $O$ is the alphabet of objects, $H$ is a set of membrane labels, $\sigma_i = (n_i, R_i)$ represents a neuron containing $n_i$ spikes and a set of rules $R_i$. The rules are of the form $E / a^c \to a^m$ (firing) or $a^s \to \lambda$ (forgetting). The synapses $\text{syn}$ connect neurons, and $I_{\text{in}}$, $I_{\text{out}}$ are input and output neurons. In our implementation, we use a nested membrane structure: the outer membrane contains the global information of all China UAVs, while inner membranes handle parallel optimization for each UAV individually. Pulses (spikes) carry waypoint information and are transmitted via synaptic rules.
2.2 Grey Wolf Optimizer and Its Improvements
The standard GWO mimics the social hierarchy of grey wolves: $\alpha$ (leader), $\beta$, $\delta$, and $\omega$ (followers). The hunting process is modeled by:
$$
\begin{aligned}
\mathbf{D} &= | \mathbf{C} \cdot \mathbf{x}_p(t) – \mathbf{x}(t) |, \\
\mathbf{x}(t+1) &= \mathbf{x}_p(t) – \mathbf{A} \cdot \mathbf{D},
\end{aligned}
$$
where $\mathbf{A} = 2a \cdot \mathbf{r}_1 – a$, $\mathbf{C} = 2\mathbf{r}_2$, and $a$ decreases linearly from 2 to 0. The three best wolves update the positions of the $\omega$ wolves:
$$
\begin{aligned}
\mathbf{D}_\alpha &= | \mathbf{C}_1 \cdot \mathbf{x}_\alpha – \mathbf{x} |, \\
\mathbf{D}_\beta &= | \mathbf{C}_2 \cdot \mathbf{x}_\beta – \mathbf{x} |, \\
\mathbf{D}_\delta &= | \mathbf{C}_3 \cdot \mathbf{x}_\delta – \mathbf{x} |,
\end{aligned}
$$
$$
\mathbf{x}_1 = \mathbf{x}_\alpha – \mathbf{A}_1 \cdot \mathbf{D}_\alpha,\quad
\mathbf{x}_2 = \mathbf{x}_\beta – \mathbf{A}_2 \cdot \mathbf{D}_\beta,\quad
\mathbf{x}_3 = \mathbf{x}_\delta – \mathbf{A}_3 \cdot \mathbf{D}_\delta,
$$
$$
\mathbf{x}(t+1) = \frac{\mathbf{x}_1 + \mathbf{x}_2 + \mathbf{x}_3}{3}.
$$
2.3 Proposed CSP-GWO Algorithm
We introduce three key improvements to enhance the performance for China UAV path planning:
- Population initialization via Tent map: To improve diversity, we generate chaotic sequences using $x_{n+1} = r x_n$ for $x_n \le 0.5$ and $x_{n+1} = r (1 – x_n)$ for $x_n > 0.5$, with $r \in [1,2]$.
- Fuzzy weighting for leader wolves: Instead of equal weights ($1/3$), we design a Mamdani fuzzy system that dynamically assigns weights $W_1$, $W_2$, $W_3$ to $\alpha$, $\beta$, $\delta$ based on the iteration number. The update rule becomes:
$$
\mathbf{x}(t+1) = W_1 \cdot \mathbf{x}_1 + W_2 \cdot \mathbf{x}_2 + W_3 \cdot \mathbf{x}_3.
$$
The fuzzy system uses triangular membership functions and centroid defuzzification. Early iterations emphasize exploration (higher $\delta$ weight), while later iterations emphasize exploitation (higher $\alpha$ weight). - Parallel membrane architecture: We embed the GWO into a CSNPS with four inner neurons ($P_1$, $P_2$, $P_3$, $P_4$). In each iteration, each inner neuron independently runs a GWO subpopulation and selects its three best wolves. The outer membrane collects the global best and broadcasts it via spikes to guide the inner membranes. This parallel structure dramatically reduces computation time.
3. Experimental Results
3.1 Benchmark Function Tests
We evaluated the proposed CSP-GWO on 21 IEEE CEC benchmark functions (including unimodal, multimodal, and fixed-dimension functions). For each function, we performed 25 independent runs with 50 individuals and 500 iterations. Table 2 compares the mean and standard deviation of the best fitness values obtained by CSP-GWO and six other algorithms (GWO, WOA, ALO, SCA, PSO, SOGWO, MP-GWO).
| Function | GWO | CSP-GWO | WOA | ALO | SCA | PSO | SOGWO | MP-GWO |
|---|---|---|---|---|---|---|---|---|
| F1 (Sphere) | 3.35e-33±6.19e-33 | 6.54e-55±2.33e-54 | 4.57e-83 | 9.73e-5 | 3.864 | 1.08e-5 | 2.12e-33 | 4.58e-43 |
| F2 (Schwefel 2.22) | 6.86e-20±7.41e-20 | 1.27e-32±5.10e-32 | 4.32e-54 | 46.70 | 0.012 | 0.0066 | 6.40e-20 | 1.84e-25 |
| F3 (Schwefel 1.2) | 2.59e-8±5.41e-8 | 2.78e-14±8.08e-12 | 28788.7 | 1989.8 | 799.2 | 46.61 | 3.33e-8 | 1.02e-9 |
| F4 (Schwefel 2.21) | 3.66e-8±6.43e-8 | 2.14e-12±5.57e-12 | 38.12 | 12.37 | 26.70 | 0.869 | 2.19e-8 | 4.91e-11 |
| F5 (Rosenbrock) | 26.71±0.74 | 25.76±0.34 | 27.52 | 205.8 | 6606 | 83.96 | 26.74 | 28.22 |
| F6 (Step) | 0.4905±0.2434 | 0.1137±0.0128 | 0.0822 | 8.81e-5 | 9.575 | 6.14e-6 | 0.609 | 3.788 |
| F7 (Quartic) | 0.00121±0.00074 | 0.00098±0.00046 | 0.00171 | 0.1310 | 0.0763 | 0.1210 | 0.00107 | 0.00336 |
| F8 (Ackley) | −6235.3±877.7 | −7.74e80±2.23e81 | −10730.5 | −5820.8 | −3825.2 | −6053.9 | −6375.0 | −4491.8 |
| F9 (Rastrigin) | 1.552±2.365 | 2.27e-15±1.13e-14 | 6.82e-15 | 77.57 | 28.12 | 46.46 | 1.39 | 151.5 |
| F10 (Griewank) | 4.31e-14±3.55e-15 | 7.41e-15±7.11e-16 | 4.57e-15 | 2.304 | 14.82 | 0.0571 | 4.27e-14 | 1.138 |
| F11 (Penalized1) | 0.00608±0.0118 | 0.00±0.00 | 0.00 | 0.0252 | 0.8747 | 0.00916 | 0.00211 | 0.00422 |
| F12 (Penalized2) | 0.026±0.012 | 0.00±0.00 | 0.007 | 11.48 | 10519 | 1.92e-7 | 0.025 | 9.09 |
| F13 (Schwefel 2.13) | 0.4007±0.2166 | 0.0881±0.0134 | 0.1556 | 4.425 | 676.7 | 0.00351 | 0.4369 | 1.8117 |
| F14 (Shekel 5) | 3.238±3.225 | 1.077±0.396 | 1.943 | 1.790 | 1.157 | 1.474 | 3.235 | 8.096 |
| F15 (Shekel 7) | 0.00199±0.0055 | 0.00031±1.85e-5 | 0.00062 | 0.00543 | 0.00087 | 0.00091 | 0.00121 | 0.00617 |
| F16 (Shekel 10) | −1.03163±1.29e-8 | −1.03163±1.38e-6 | −1.03163 | −1.03163 | −1.03161 | −1.03163 | −1.03163 | −1.01898 |
| F17 (Hartman 3) | 0.39788±6.42e-4 | 0.39773±9.39e-5 | 0.39788 | 0.39788 | 0.39954 | 0.39788 | 0.39788 | 0.39807 |
As shown in Table 2, CSP-GWO achieves significantly better solutions on most functions, especially on multimodal functions where it avoids local optima. The fuzzy weighting mechanism and the Tent map initialization contribute to faster convergence and higher final accuracy. The parallel membrane structure does not degrade solution quality; conversely, it helps by maintaining multiple search agents with diverse exploration behaviors.
3.2 Multi-UAV Path Planning Results
We simulated a 20 km × 20 km × 20 km 3D environment with multiple mountain peaks and radar threats. Three scenario sizes were tested: 4, 8, and 16 China UAVs. Each algorithm was run 25 times independently. Table 3 reports the best cost, average cost, variance, number of failed coordinations, number of collisions, and average runtime.
| #UAVs | Metric | GWO | SOGWO | MP-GWO | PSO | WOA | CSP-GWO |
|---|---|---|---|---|---|---|---|
| 4 | Best cost | 2.402 | 2.365 | 2.636 | 2.601 | 2.760 | 2.098 |
| Average cost | 2.626 | 2.473 | 2.743 | 2.723 | 2.877 | 2.156 | |
| Cost variance | 0.106 | 0.075 | 0.066 | 0.061 | 0.062 | 0.041 | |
| Coordination failures | 0 | 0 | 0 | 0 | 0 | 0 | |
| Collisions | 0 | 0 | 0 | 0 | 0 | 0 | |
| Runtime (s) | 13.6 | 12.4 | 20.5 | 13.1 | 16.1 | 11.1 | |
| 8 | Best cost | 4.793 | 4.624 | 4.838 | 4.823 | 4.822 | 4.159 |
| Average cost | 5.080 | 4.897 | 5.110 | 5.150 | 5.220 | 4.311 | |
| Cost variance | 0.098 | 0.113 | 0.161 | 0.161 | 0.160 | 0.093 | |
| Coordination failures | 1 | 0 | 0 | 1 | 0 | 0 | |
| Collisions | 0 | 0 | 0 | 0 | 0 | 0 | |
| Runtime (s) | 26.4 | 24.6 | 38.8 | 25.2 | 31.4 | 11.6 | |
| 16 | Best cost | 9.002 | 8.995 | 9.421 | 9.780 | 9.786 | 8.100 |
| Average cost | 9.428 | 9.380 | 9.849 | 10.170 | 10.213 | 8.323 | |
| Cost variance | 0.255 | 0.239 | 0.213 | 0.219 | 0.266 | 0.104 | |
| Coordination failures | 2 | 2 | 1 | 2 | 1 | 0 | |
| Collisions | 2 | 0 | 2 | 1 | 2 | 0 | |
| Runtime (s) | 49.3 | 48.2 | 73.2 | 49.0 | 60.7 | 12.8 |
The results clearly demonstrate the advantages of CSP-GWO. For the 16-UAV scenario, our algorithm achieves a 11.7% reduction in average cost compared to GWO, with zero collisions and zero coordination failures. Most importantly, the runtime remains nearly constant (approximately 12 seconds) regardless of the number of UAVs, thanks to the parallel membrane architecture. In contrast, other algorithms show an almost linear increase in runtime as the number of UAVs grows. This makes CSP-GWO highly suitable for real-time deployment in China UAV cooperative missions, where both solution quality and computational efficiency are critical.
Figure 1 (see the image above) illustrates a sample 3D path generated by CSP-GWO for 8 China UAVs in a simulated threat environment. The B-spline smoothed trajectories are smooth, safe, and well-coordinated. The algorithm successfully avoids all obstacles and ensures that all UAVs reach the target within the required time window.
4. Conclusion
In this paper, we have presented a novel multi-UAV path planning algorithm that synergistically combines cell-like spiking neural P systems with a hybrid grey wolf optimizer. The key innovations include a fuzzy-weighted hierarchy that adapts the influence of leader wolves, a Tent-map initialization for better diversity, and a parallel membrane structure that dramatically reduces computational time. Extensive experiments on standard benchmarks show that CSP-GWO outperforms state-of-the-art meta-heuristics in terms of both accuracy and robustness. When applied to multi-UAV path planning for China UAVs, our method consistently produces lower-cost, collision-free, and temporally synchronized trajectories with minimal runtime. The near-constant runtime across different fleet sizes underscores the scalability of the membrane-inspired parallelism. Future work will extend the model to dynamic threats and more complex cooperative constraints, further improving the applicability of CSP-GWO in real-world China UAV operations.