Multi-Unmanned Aerial Vehicle (UAV) collaborative path planning represents a complex optimization challenge involving multiple constraints and conflicting objectives, where planning efficacy directly impacts mission safety and robustness. Existing algorithms often struggle with local optima and uneven solution distribution when handling high-dimensional objectives. To address these limitations, we propose a Multi-Objective Jellyfish Algorithm (MOJS) that significantly enhances solution quality for drone technology applications.
The mathematical model for UAV path planning incorporates four critical cost functions. Path length cost ensures minimum segment distance compliance:
$$C_{\text{path},k(P_k) = \sum_{i=1}^{N_k-1} \|p_{k,i+1} – p_{k,i}\|$$
Height constraints maintain operational altitude ranges with penalty terms:
$$H_{k,i} = \begin{cases}
\infty & z_{k,i} < H_{\min,k} \lor z_{k,i} > H_{\max,k} \\
\frac{H_{\max,k} – H_{\min,k}}{2} & \text{otherwise}
\end{cases}$$
$$C_{\text{height},k}(P_k) = \sum_{i=1}^{N_k} H_{k,i}$$
Smoothness cost minimizes abrupt maneuvers using turn ($\phi$) and pitch ($\psi$) angles:
$$\phi_{k,i} = \arctan\left(\frac{(p’_{k,i+1} – p’_{k,i}) \times (p’_{k,i+2} – p’_{k,i+1})}{(p’_{k,i+1} – p’_{k,i}) \cdot (p’_{k,i+2} – p’_{k,i+1})}\right)$$
$$C_{\text{smooth},k}(P_k) = \alpha_1 \sum_{i=1}^{N_k-1} |\phi_{k,i}| + \alpha_2 \sum_{i=1}^{N_k-2} |\psi_{k,i+1} – \psi_{k,i}|$$
Threat avoidance penalizes proximity to cylindrical danger zones:
$$T_t(p_{k,i}) = \begin{cases}
\infty & d_{k,i,t} \leq R_t + D \\
(R_t + D + S) – d_{k,i,t} & R_t + D < d_{k,i,t} \leq R_t + D + S \\
0 & d_{k,i,t} > R_t + D + S
\end{cases}$$
$$C_{\text{threat},k}(P_k) = \sum_{i=1}^{N_k} \sum_{t=1}^{T} T_t(p_{k,i})$$
The integrated multi-objective formulation combines these costs:
$$\min \left( C_1(P_1,\dots,P_M), \dots, C_M(P_1,\dots,P_M) \right)$$
$$C_k(P_k) = w_1 C_{\text{path},k} + w_2 C_{\text{height},k} + w_3 C_{\text{smooth},k} + w_4 C_{\text{threat},k}$$
| Component | Description | Mathematical Formulation |
|---|---|---|
| Initialization | Hybrid LHS and chaotic mapping | $\text{pop}_{\text{LHS}} = L_b + (U_b – L_b) \cdot X_{\text{LHS}}$ |
| Archive Management | Grid-based crowding distance | $P_i = \frac{N_i}{c}, \quad P’_i = \frac{c}{N_i}$ |
| Ocean Current Motion | Elite-guided vector update | $X_i(t+1) = EL_i(t) + \text{trend} \cdot \text{LevyGaussian}(s)$ |
| Internal Motion (Type A) | Local search around elites | $X_i(t+1) = X_i(t) + (EL^*(t) – X_i(t)) \cdot \text{LevyGaussian}(s)$ |
The MOJS algorithm integrates four key innovations: 1) Hybrid initialization combining Latin Hypercube Sampling (LHS) and chaotic mapping for population diversity; 2) External archive with grid-based crowding distance for Pareto front uniformity; 3) Dynamic step-size mechanism blending Lévy flights and Gaussian distribution; 4) Elite-guided position updates. For ocean current motion:
$$X_i(t+1) = EL_i(t) + \text{trend} \cdot \text{LevyGaussian}(s)$$
$$\text{trend} = \frac{\sum EL – X^*(t)}{n} \cdot \text{rand}(0,1)$$
Type B internal motion uses non-dominated comparisons:
$$\text{Direction} = \begin{cases}
EL_j(t) – EL_i(t) & \text{if } EL_j \prec EL_i \\
EL_i(t) – EL_j(t) & \text{otherwise}
\end{cases}$$
| Metric | MOJS | NSGA-II | MOPSO | MOGWO | MODA |
|---|---|---|---|---|---|
| GD (↓) | 0.014 | 0.038 | 0.127 | 0.045 | 0.092 |
| Spread (↓) | 0.412 | 0.683 | 0.398 | 0.521 | 0.587 |
| HV (↑) | 0.851 | 0.723 | 0.694 | 0.768 | 0.632 |
| Coverage (↑) | 0.91 | 0.42 | 0.37 | 0.58 | 0.33 |
Experimental validation utilized 20 benchmark functions (ZDT, DTLZ, UF series) with six evaluation metrics. Statistical analysis via Wilcoxon signed-rank test (α=0.05) confirmed MOJS’s superiority:
$$\begin{array}{c|c|c|c}
\text{Comparison} & \text{Superior Cases} & \text{Inferior Cases} & \text{Equal Cases} \\
\hline
\text{MOJS vs NSGA-II} & 17 & 0 & 3 \\
\text{MOJS vs MOPSO} & 20 & 0 & 0 \\
\text{MOJS vs MOGWO} & 16 & 1 & 3 \\
\end{array}$$
In Unmanned Aerial Vehicle path planning simulations, MOJS generated collision-free trajectories in complex environments with nine threat zones. The Pareto front distribution demonstrated excellent convergence and coverage for dual-drone systems. Key drone technology parameters included:
$$L_{\min} = 50\text{m}, \quad 100\text{m} \leq H \leq 300\text{m}, \quad \Delta\psi_{\max} = 25^\circ$$
The trajectories maintained continuous Line-of-Sight (LoS) connectivity while minimizing:
$$\sum_{k=1}^{M} \left( w_1 C_{\text{path},k} + w_2 C_{\text{height},k} \right) \quad \text{with} \quad \sum w_i = 1$$
Future research will focus on adaptive parameter tuning and distributed architecture development for large-scale drone technology applications. The integration of real-time dynamic obstacle avoidance remains crucial for advancing Unmanned Aerial Vehicle autonomy in complex operational environments.