Unmanned drones, distinguished by their flexible design, high maneuverability, and compact size, have become indispensable across a multitude of sectors. Their applications span reconnaissance and surveillance, battlefield engagement, intelligence relay, disaster response, and precision agriculture. Compared to a single unmanned drone, a cooperative system of multiple unmanned drones can significantly enhance mission efficiency and operational robustness through coordinated efforts. However, the deployment of multi-unmanned drone swarms in complex three-dimensional environments introduces significant challenges for flight trajectory estimation within acceptable accuracy and time constraints, thereby elevating mission costs and risks. Consequently, developing efficient optimization algorithms for path planning is paramount.
Traditional planning algorithms, primarily graph-based and sampling-based methods such as Voronoi diagrams, A-star (A*), and Dijkstra’s algorithm, often struggle to provide acceptable flight paths in complex, dynamic three-dimensional spaces. While metaheuristic algorithms like Particle Swarm Optimization (PSO), Artificial Rabbits Optimization, Genetic Algorithms (GA), and Simulated Annealing have been widely applied to unmanned drone path planning due to their fast convergence and stability, they frequently face a critical trade-off. In large-scale optimization problems like cooperative multi-unmanned drone trajectory planning, the solution space expands exponentially with the number of unmanned drones, leading to prohibitive computational time. Most algorithms heavily focus on finding the optimal result but fail to adequately balance this with time efficiency, hindering practical application in real-time scenarios.
Membrane Computing, a branch of Natural Computing inspired by the structure and functioning of biological cells, offers a promising paradigm. Models like P systems and Spiking Neural P systems (SNPS) possess inherent parallelism and distributed processing capabilities. Research has successfully applied these models to robot control, image processing, power system fault diagnosis, and more recently, to unmanned drone attitude optimization and trajectory planning. This work proposes a novel hybrid algorithm, termed Cell-like Spiking Neural P system with Grey Wolf Optimization (CSP-GWO), to address the challenges in multi-unmanned drone trajectory planning. The algorithm leverages the parallel architectural framework of membrane systems to simulate the concurrent operation of multiple unmanned drones while utilizing an enhanced GWO mechanism for efficient optimization within each computational unit. The core contributions include a novel fuzzy variant of the hierarchical operator within GWO to simulate the hunting process more effectively and the integration of this improved optimizer within a parallel membrane computing structure to drastically reduce planning time for multiple unmanned drones.
Mathematical Model for Multi-Unmanned Drone Trajectory Planning
The trajectory planning for multiple unmanned drones must satisfy environmental, physical, and cooperative constraints. The environment contains threat zones to avoid. Physical constraints include maximum flight distance, permissible altitude range, and turning angle limits. Cooperative constraints require collision avoidance between unmanned drones and simultaneous arrival at the target region.
Physical and Environmental Constraints
Let the maximum allowable path length for an unmanned drone be $L_{max}$. The flight altitude must remain within $[H_{min}, H_{max}]$. A planned path consists of $n$ waypoints $l_i$ with coordinates $(x_i, y_i, z_i)$. The length and altitude constraints are formulated as:
$$ l_i = \sqrt{(x_{i+1} – x_i)^2 + (y_{i+1} – y_i)^2 + (z_{i+1} – z_i)^2}, $$
$$ \sum_{i=1}^{n} l_i \leq L_{max}, $$
$$ H_{min} \leq z_i \leq H_{max}. $$
The maximum permissible horizontal turning angle is $\alpha_{max}$, and the maximum permissible pitch angle is $\psi_{max}$. The corresponding constraints for the $i$-th segment are:
$$ \alpha_i = \arctan\left(\frac{y_{i+1} – y_i}{x_{i+1} – x_i}\right) \leq \alpha_{max}, $$
$$ \psi_i = \arctan\left(\frac{z_{i+1} – z_i}{\sqrt{(x_{i+1} – x_i)^2 + (y_{i+1} – y_i)^2}}\right) \leq \psi_{max}. $$
For a spherical threat zone centered at $C_w = (x_w, y_w, z_w)$ with radius $R$, the distance $d_w$ from an unmanned drone to the threat center must satisfy $d_w \geq R$:
$$ d_w = \sqrt{(x_i – x_w)^2 + (y_i – y_w)^2 + (z_i – z_w)^2} \geq R. $$
To prevent mid-air collisions between unmanned drones $a$ and $b$ with positions $P_a(x_a, y_a, z_a)$ and $P_b(x_b, y_b, z_b)$, their separation distance $d_{ab}$ must be greater than a minimum safe distance $d_{min}$:
$$ d_{ab} = \sqrt{(x_a – x_b)^2 + (y_a – y_b)^2 + (z_a – z_b)^2}, $$
$$ d_{ab} \geq d_{min}, \quad \forall a \neq b. $$
Time Cooperative Constraint
For a swarm of $n$ unmanned drones to arrive simultaneously, their arrival time windows must overlap. The $i$-th unmanned drone flies a distance $d_i$ at a speed $v_i \in [v_{min}, v_{max}]$, taking time $t_i = d_i / v_i$, and is required to arrive within $[t_{i,start}, t_{i,end}]$. Successful time coordination is achieved when the intersection of all time windows is non-empty:
$$ [t_{start}, t_{end}] \cap [t_{i,start}, t_{i,end}] \neq \emptyset, \quad \forall i = 1,2,\dots,n. $$
Composite Objective Function
A weighted sum objective function $J$ is defined to evaluate the quality of a planned trajectory for the multi-unmanned drone system, integrating all aforementioned costs:
$$ J = w_1 F_1 + w_2 F_2 + w_3 F_3 + w_4 F_4 + w_5 F_5 + w_6 F_6, $$
where $w_1$ to $w_6$ are weighting coefficients satisfying $\sum_{i=1}^{6} w_i = 1$. The individual cost components $F_1$ to $F_6$ are detailed below, with $k_i$ and $p_i$ as scaling and penalty coefficients, respectively.
1. Path Length Cost ($F_1$): Penalizes the total flight distance.
$$ F_1 = k_1 \sum_{i=1}^{n} l_i. $$
2. Altitude Violation Cost ($F_2$): Penalizes flight outside the permitted altitude band.
$$ F_2 = k_2 \cdot
\begin{cases}
p_1 (H_{min} – z_i), & z_i \leq H_{min} \\
0, & H_{min} < z_i < H_{max} \\
p_2 (z_i – H_{max}), & z_i \geq H_{max}
\end{cases}. $$
3. Turning Angle Cost ($F_3$): Penalizes violations of horizontal and pitch angle limits.
$$ f_{\alpha_i} =
\begin{cases}
0, & \alpha_i \leq \alpha_{max} \\
\alpha_i, & \alpha_i > \alpha_{max}
\end{cases}, \quad
f_{\psi_i} =
\begin{cases}
0, & |\psi_{i+1} – \psi_i| \leq \psi_{max} \\
|\psi_{i+1} – \psi_i|, & |\psi_{i+1} – \psi_i| > \psi_{max}
\end{cases}. $$
$$ F_3 = k_3 \sum_{i=1}^{n} (p_3 f_{\alpha_i} + p_4 f_{\psi_i}). $$
4. Threat Zone Cost ($F_4$): Penalizes proximity to threat centers, increasing inversely with distance.
$$ F_4 = k_4 \cdot
\begin{cases}
p_5 \sum_{j=1}^{m} \left(\frac{1}{R_m}\right), & d_w \leq R_m \\
0, & d_w > R_m
\end{cases}. $$
5. Inter-Unmanned Drone Collision Cost ($F_5$): Applies a fixed penalty if the minimum safe distance is violated.
$$ F_5 = k_5 \cdot
\begin{cases}
0, & d_{ab} \geq d_{min} \\
p_6, & d_{ab} < d_{min}
\end{cases}. $$
6. Time Coordination Cost ($F_6$): Applies a penalty if the unmanned drones cannot arrive within a common time window.
$$ F_6 = k_6 \cdot
\begin{cases}
0, & [t_{start}, t_{end}] \cap [t_{i,start}, t_{i,end}] \neq \emptyset \\
p_7, & [t_{start}, t_{end}] \cap [t_{i,start}, t_{i,end}] = \emptyset
\end{cases}. $$
Path Smoothing with B-Spline Curves
The initial trajectory from the planning algorithm is typically a polyline. To generate a smooth, flyable path suitable for the dynamics of an unmanned drone, a B-spline curve is applied. The $p$-degree B-spline curve is defined as:
$$ C(t) = \sum_{i=0}^{n} N_{i,p}(t) P_i, $$
where $P_i$ are the control points (the planned waypoints), and $N_{i,p}(t)$ are the $p$-th degree B-spline basis functions. This smoothing step ensures the final path is continuous and meets the kinematic constraints of the unmanned drone platform.
Algorithmic Principles: CSP-GWO
The proposed CSP-GWO algorithm is built upon a nested membrane structure. The outer membrane encapsulates the collective trajectory information for all unmanned drones and manages information exchange with inner membranes. Each inner membrane executes a single-unmanned drone trajectory planning algorithm—an enhanced Grey Wolf Optimizer (GWO)—operating in parallel. Candidate trajectories and associated costs are communicated via spike signals to the outer membrane, which evaluates and selects trajectories satisfying all cooperative constraints. This process iterates, continuously refining the trajectories for the entire unmanned drone swarm until a termination criterion is met.
Foundation: Cell-like Spiking Neural P System (CSNP)
A CSNP system of degree $m \geq 1$ is a construct:
$$ \Pi = (O, H, \omega_1, \dots, \omega_n, \sigma_1, \dots, \sigma_m, R_1, \dots, R_m, syn, I_{in}, I_{out}), $$
where $O$ is a finite multiset of objects, $H$ is a set of membrane labels, and $\sigma_i = (n_i, R_i)$ for $1 \leq i \leq m$ are neurons. Each neuron contains a multiset of spikes $n_i$ and a finite set of rules $R_i$. Rules are of the form $E / a^c \rightarrow a^p$ (firing rule, where $E$ is a regular expression over $a$, $c \geq p \geq 1$) or $a^s \rightarrow \lambda$ (forgetting rule, for $s \geq 1$, $a^s \notin L(E)$). The set $syn \subseteq \{1,2,\dots,m\} \times \{1,2,\dots,m\}$ defines synapses between neurons, and $I_{in}, I_{out} \in \{1,2,\dots,m\}$ denote the input and output neurons, respectively.
Standard Grey Wolf Optimizer (GWO)
GWO is a metaheuristic that mimics the social hierarchy and hunting behavior of grey wolves. The population is divided into four levels: Alpha ($\alpha$, the best solution), Beta ($\beta$, the second-best), Delta ($\delta$, the third-best), and Omega ($\omega$, the remaining solutions). The hunting process is modeled in three phases:
1. Encircling Prey:
$$ \vec{D} = | \vec{C} \cdot \vec{X}_p(t) – \vec{X}(t) |, $$
$$ \vec{X}(t+1) = \vec{X}_p(t) – \vec{A} \cdot \vec{D}, $$
where $t$ is the iteration counter, $\vec{X}_p$ is the prey’s position vector, $\vec{X}$ is a grey wolf’s position vector. Vectors $\vec{A}$ and $\vec{C}$ are calculated as:
$$ \vec{A} = 2 \vec{a} \cdot \vec{r}_1 – \vec{a}, $$
$$ \vec{C} = 2 \cdot \vec{r}_2, $$
where components of $\vec{a}$ decrease linearly from 2 to 0 over iterations, and $\vec{r}_1, \vec{r}_2$ are random vectors in $[0,1]$.
2. Hunting: The positions of $\alpha$, $\beta$, and $\delta$ wolves guide the $\omega$ wolves.
$$ \vec{D}_{\alpha} = |\vec{C}_1 \cdot \vec{X}_{\alpha} – \vec{X}|, \quad \vec{D}_{\beta} = |\vec{C}_2 \cdot \vec{X}_{\beta} – \vec{X}|, \quad \vec{D}_{\delta} = |\vec{C}_3 \cdot \vec{X}_{\delta} – \vec{X}|, $$
$$ \vec{X}_1 = \vec{X}_{\alpha} – \vec{A}_1 \cdot \vec{D}_{\alpha}, \quad \vec{X}_2 = \vec{X}_{\beta} – \vec{A}_2 \cdot \vec{D}_{\beta}, \quad \vec{X}_3 = \vec{X}_{\delta} – \vec{A}_3 \cdot \vec{D}_{\delta}. $$
3. Updating Position: The standard GWO uses the average of these guided positions.
$$ \vec{X}(t+1) = \frac{\vec{X}_1 + \vec{X}_2 + \vec{X}_3}{3}. $$
CSP-GWO Architecture and Enhancements
In the CSP-GWO framework, each object within an inner membrane corresponds to a candidate solution (a grey wolf) for a single unmanned drone’s path. Multiple inner membranes (populations) run in parallel. For each inner membrane $i$, the best three solutions are identified as $\alpha_i$, $\beta_i$, $\delta_i$. The global best solution $gbest$ is selected by comparing the best performers across all membranes.
1. Tent Chaotic Mapping for Population Initialization: To enhance population diversity and ensure a more uniform exploration of the search space from the outset, a Tent chaotic map is employed to generate the initial population for each unmanned drone’s optimizer.
$$ x_{n+1} =
\begin{cases}
r x_n, & \text{if } x_n \leq 0.5 \\
r (1 – x_n), & \text{if } x_n > 0.5
\end{cases}, $$
where $x_n \in [0,1]$ and $r \in [1,2]$ is a control parameter.
2. Fuzzy Hierarchical Weights: The standard GWO’s equal weighting of $\alpha$, $\beta$, and $\delta$ wolves ($1/3$ each) is biologically implausible and suboptimal. We introduce a Mamdani-type fuzzy inference system (FIS) with triangular membership functions to dynamically assign different weights $W_1$, $W_2$, $W_3$ to the three leading wolves based on the algorithm’s progression (iteration count). This allows the $\alpha$ wolf to exert stronger influence early for convergence, while the $\beta$ and $\delta$ wolves maintain exploration capability. The position update rule is modified to:
$$ \vec{X}(t+1) = W_1 \cdot \vec{X}_1 + W_2 \cdot \vec{X}_2 + W_3 \cdot \vec{X}_3. $$
The FIS input is the normalized iteration number, and the outputs are the three weights, whose sum is constrained to 1.
The overall algorithm leverages the parallelism of the CSNP membrane structure to simultaneously evolve trajectories for all unmanned drones, while the enhanced GWO within each membrane efficiently searches for high-quality paths. Information exchange through spikes facilitates global coordination and constraint satisfaction for the multi-unmanned drone system.
Experimental Results and Analysis
Performance on IEEE CEC Benchmark Functions
The efficacy of the proposed CSP-GWO is first validated on a suite of 21 IEEE CEC benchmark functions, which include unimodal, multimodal, hybrid, and composition functions. The algorithm is compared against standard GWO, Whale Optimization Algorithm (WOA), Ant Lion Optimizer (ALO), Sine Cosine Algorithm (SCA), PSO, and other GWO variants (SOGWO, MP-GWO). All algorithms were run 25 times independently with a population size of 50 and a maximum of 500 iterations. The mean and standard deviation of the best-obtained values are recorded.
| Function | GWO Mean | CSP-GWO Mean | WOA Mean | ALO Mean | SCA Mean | PSO Mean |
|---|---|---|---|---|---|---|
| F1 | 3.35e-33 | 6.54e-55 | 4.57e-83 | 9.73e-5 | 3.86424 | 1.08e-5 |
| F2 | 6.86e-20 | 1.27e-32 | 4.32e-54 | 46.69973 | 0.01202 | 0.00657 |
| F3 | 2.59e-8 | 2.78e-14 | 28788.720 | 1989.791 | 6799.180 | 46.606 |
| F4 | 3.66e-8 | 2.14e-12 | 38.115 | 12.374 | 26.699 | 0.869 |
| F5 | 26.714 | 25.755 | 27.516 | 205.760 | 6606.368 | 83.961 |
| F6 | 0.49050 | 0.11371 | 0.08215 | 8.81e-5 | 9.57460 | 6.14e-6 |
| F7 | 0.00121 | 0.00098 | 0.00171 | 0.13099 | 0.07634 | 0.12102 |
| F8 | -6235.277 | -7.74e80 | -10730.540 | -5820.813 | -3825.179 | -6053.941 |
| F9 | 1.55218 | 2.27e-15 | 6.82e-15 | 77.56687 | 28.12338 | 46.45931 |
| F10 | 4.31e-14 | 7.41e-15 | 4.57e-15 | 2.30380 | 14.82147 | 0.05711 |
Table 1: Excerpt of comparative results on IEEE CEC functions (best mean values highlighted). CSP-GWO demonstrates superior or competitive performance across most functions, particularly on complex multimodal ones (F7-F10), indicating excellent global search capability and robustness.
Convergence curve analysis on selected functions (e.g., F3, F9, F11, F14) further confirms that CSP-GWO achieves both faster convergence speed and higher final precision compared to its peers, effectively balancing exploration and exploitation.
Multi-Unmanned Drone Trajectory Planning Simulation
A comprehensive 3D simulation environment (20 km x 20 km x 20 km) featuring mountainous terrain and radar threat zones was constructed to evaluate practical performance. Parameters were set as: $d_{min}=0.5$ km, $H_{min}=20$ m, $H_{max}=20$ km, $\alpha_{max}=45^\circ$, $\psi_{max}=60^\circ$, and a cooperative arrival window of 160s. CSP-GWO was compared against GWO, SOGWO, MP-GWO, PSO, and WOA for planning trajectories for 4, 8, and 16 unmanned drones. Each algorithm performed 25 independent runs with a maximum of 100 iterations and 20 waypoints per path. Performance metrics included best/average trajectory cost, cost variance, number of failed coordination attempts, number of collisions, and average runtime.
| # Drones | Metric | GWO | SOGWO | MP-GWO | PSO | WOA | CSP-GWO |
|---|---|---|---|---|---|---|---|
| 4 | Best Cost | 2.402 | 2.365 | 2.636 | 2.601 | 2.760 | 2.098 |
| Avg. 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 | |
| Failed Coordination | 0 | 0 | 0 | 0 | 0 | 0 | |
| Collisions | 0 | 0 | 0 | 0 | 0 | 0 | |
| Avg. 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 |
| Avg. 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 | |
| Failed Coordination | 1 | 0 | 0 | 1 | 0 | 0 | |
| Collisions | 0 | 0 | 0 | 0 | 0 | 0 | |
| Avg. 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 |
| Avg. 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 | |
| Failed Coordination | 2 | 2 | 1 | 2 | 1 | 0 | |
| Collisions | 2 | 0 | 2 | 1 | 2 | 0 | |
| Avg. Runtime (s) | 49.3 | 48.2 | 73.2 | 49.0 | 60.7 | 12.8 |
Table 2: Performance comparison for multi-unmanned drone trajectory planning across different swarm sizes.
The results in Table 2 and the visualization of planned paths lead to several key conclusions regarding the planning for multiple unmanned drones:
- Superior Optimization Performance: CSP-GWO consistently yields the lowest best and average trajectory costs for 4, 8, and 16 unmanned drones. It achieved average cost reductions of 17.9%, 15.1%, and 11.7% compared to standard GWO for 4, 8, and 16 unmanned drones, respectively. This demonstrates its enhanced search precision.
- Enhanced Robustness and Safety: CSP-GWO exhibits the lowest cost variance, indicating stable and reliable performance. Crucially, it successfully achieved zero collisions and zero failed coordination attempts even in the challenging 16-unmanned drone scenario, unlike other algorithms which suffered from coordination failures and collisions.
- Remarkable Time Efficiency: The most significant advantage of CSP-GWO is its computational speed. Leveraging the parallel processing architecture of the membrane system, its average runtime remains remarkably low and scales minimally with the number of unmanned drones (11.1s, 11.6s, 12.8s). In contrast, the runtimes of other algorithms approximately double or triple as the unmanned drone count increases from 4 to 16. This validates the core premise of using membrane computing to tackle the time complexity challenge in large-scale multi-unmanned drone planning.
The final smoothed paths generated by CSP-GWO are visibly more direct, smoother, and effectively avoid all threats and inter-unmanned drone conflicts.
Conclusion
This paper presents the Cell-like Spiking Neural P system with enhanced Grey Wolf Optimization (CSP-GWO) algorithm for addressing the complex problem of cooperative trajectory planning for multiple unmanned drones in three-dimensional environments. The integration of membrane computing provides a natural and efficient framework for modeling the parallel operation of an unmanned drone swarm, while the introduced enhancements to the GWO—namely, chaotic initialization and fuzzy hierarchical weights—significantly boost its optimization power and robustness.
Rigorous testing on IEEE CEC benchmark functions confirmed the algorithm’s superior exploration-exploitation balance and convergence characteristics. More importantly, in practical multi-unmanned drone trajectory planning simulations, CSP-GWO demonstrated outstanding performance across three key dimensions: (1) it found lower-cost, more feasible flight paths; (2) it ensured greater operational safety and coordination reliability with zero collisions and failed coordination in large swarms; and (3) it achieved a drastic reduction in computational time, with runtimes nearly independent of the swarm size, thanks to its parallel membrane structure. This makes CSP-GWO a highly promising solution for real-time, large-scale multi-unmanned drone mission planning where both solution quality and time efficiency are critical.
The current model focuses on static, known threat sources. Future work will extend this framework to handle dynamic threats and incorporate more comprehensive cooperative constraints, further exploring the algorithm’s potential in increasingly complex and realistic operational scenarios for unmanned drone swarms.