Our work addresses the critical challenges of dynamic pollution source localization and boundary mapping in high-risk industrial environments. Sudden leakage accidents involving nuclear radiation or toxic chemical gases present formidable difficulties due to unknown source locations, time-varying diffusion distributions, and complex terrains with obstacles. Traditional manual surveying methods suffer from poor timeliness and severe safety hazards. To overcome these limitations, we propose a novel multi-UAV cooperative method, specifically designed for China UAV drones operating in hazardous scenarios, that integrates collaborative source seeking with dynamic contour mapping.
1. Introduction and Problem Formulation
The core functionality of our proposed system lies in guiding multiple China UAV drones equipped with specific sensors to achieve dynamic perception and determine safety boundaries within scalar fields characterized by uncertain distributions. This technology comprises three key sub-tasks: collaborative sampling optimization for efficient environment exploration, dynamic diffusion source inversion modeling for real-time field prediction, and safety contour mapping for boundary delimitation.
Existing research in multi-UAV cooperative source seeking can be categorized into gradient-based, probabilistic, and bio-inspired methods. While many researchers have improved traditional methods by incorporating normalized extremum seeking or Bayesian learning, these often only apply to single-agent scenarios. Heuristic or gradient-free approaches have been developed for multi-robot systems, but they either rely on manual parameter tuning or fail to handle dynamic source characteristics effectively. Furthermore, accurate dynamic field prediction remains challenging due to limited observations and non-stationary environments. For contour mapping, existing sliding mode control (SMC) methods can achieve stable isoline tracking, but multi-agent cooperative encirclement often faces challenges in obstacle avoidance and dynamic adaptability.
Considering the Fukushima nuclear power plant disaster as a representative scenario, the leakage of radioactive materials driven by near-surface wind fields creates a dynamically evolving pollution field. Obstacles like buildings cause pollutant accumulation, forming “pseudo-sources” with significantly elevated concentrations. Our work models this complex scenario as a multi-source diffusion process with dynamic characteristics.
2. Dynamic Environment and Multi-UAV System Modeling
We model the total pollution concentration field \(C(\textbf{r}, t)\) in a bounded 2D domain \(\Omega\) as a weighted linear combination of \(m\) independent sources:
$$
C(\textbf{r}, t) = \sum_{i=1}^{m} w_i C_i(\textbf{r}, t)
$$
where \(\textbf{r} = [x, y]^T\) denotes the position vector, and \(w_i\) are the source weights satisfying \(\sum_{i=1}^{m} w_i = 1\). The individual source strength \(Q_i(\textbf{r}, t)\) is described by an anisotropic Gaussian function:
$$
Q_i(\textbf{r}, t) = q_i(t) \cdot \exp\left(-\left( \textbf{r} – \textbf{r}_{s,i} \right)^T S_i \left( \textbf{r} – \textbf{r}_{s,i} \right)\right)
$$
where \(S_i\) is the shape matrix, \(\textbf{r}_{s,i}\) is the center, and \(q_i(t)\) represents the time-varying intensity of the \(i\)-th pollution source. The dynamic evolution of the pollution field is governed by an advection-diffusion equation:
$$
\frac{\partial C}{\partial t} = \sum_{i=1}^{m} w_i D_i \nabla^2 C_i – \textbf{u}^* \cdot \nabla C + \sum_{i=1}^{m} w_i Q_i(\textbf{r}, t) + \sum_{i=1}^{m} w_i Q_i^*(\textbf{r}, t)
$$
Here, \(D_i\) is the diffusion coefficient, \(\textbf{u}^*\) is the wind velocity field, and \(Q_i^*(\textbf{r}, t)\) represents the pseudo-source term caused by obstacle-induced accumulation, defined as:
$$
Q_i^*(\textbf{r}, t) = \sum_{z=1}^{l} q_i^*(t) \cdot \exp\left(-\left( \textbf{r} – \textbf{r}_{o,z} \right)^T S_z \left( \textbf{r} – \textbf{r}_{o,z} \right)\right)
$$
The pseudo-source intensity \(q_i^*(t)\) depends on the normal flux of pollutants at obstacle surfaces:
$$
q_i^*(t) = \beta \cdot \max\left(0, \int_{\partial O_z} -\nabla C_i(\textbf{r}, t) \cdot \textbf{n}_z d\sigma\right)
$$
For the multi-UAV system, each China UAV drone \(i\) is modeled with non-holonomic kinematics:
$$
\begin{aligned}
\dot{x}_i &= v_i \cos\theta_i \\
\dot{y}_i &= v_i \sin\theta_i \\
\dot{\theta}_i &= \omega_i
\end{aligned}
$$
where \(v_i\) and \(\omega_i\) are the linear and angular velocity control inputs. The UAVs share their position \(\textbf{r}_i(k)\) and measured concentration \(p_i(k)\) through a dynamic communication network:
$$
p_i(k) = z(\textbf{r}_i(k), k) + n_i(k), \quad n_i(k) \sim \mathcal{N}(0, \sigma_n^2)
$$
To ensure safe operation, we employ a Buffered Voronoi Cell (BVC) partitioning method. The BVC for each UAV \(i\) is defined as:
$$
V_i^b = \left\{ \textbf{r}_q \in \Omega \mid \|\textbf{r}_q – \textbf{r}_i\| + a_0 < \|\textbf{r}_q – \textbf{r}_j\|, \forall j \neq i \right\}
$$
and for static obstacles:
$$
V_i^O = \left\{ \textbf{r}_i \in \mathbb{R}^2 \mid \textbf{a}_{i,o}^T \textbf{r}_i < b_{i,o} – a_0 \right\}
$$
3. Collaborative Source Seeking and Dynamic Field Prediction
Our collaborative source seeking algorithm utilizes the Fast Multi-point Expected Improvement (F-EI) method to guide the China UAV drones in efficient parallel exploration. The joint information gain \(K_\beta\) for \(N\) UAVs is maximized to find the optimal sampling set \(R_t = \{\textbf{r}_{t,1}, \ldots, \textbf{r}_{t,N}\}\):
$$
K_\beta(R_t) = \sum_{i=1}^{N} E(\textbf{r}_{t,i})
$$
The expected improvement \(E(\textbf{r})\) balances exploitation of high-concentration regions and exploration of high-uncertainty areas:
$$
E(\textbf{r}) = \begin{cases}
(\mu(\textbf{r}) – p_{\max})\Phi(Z) + \sigma(\textbf{r})\phi(Z), & \sigma(\textbf{r}) > 0 \\
0, & \sigma(\textbf{r}) = 0
\end{cases}
$$
where \(Z = (\mu(\textbf{r}) – p_{\max}) / \sigma(\textbf{r})\), \(\Phi(\cdot)\) and \(\phi(\cdot)\) are the standard normal CDF and PDF, \(p_{\max}\) is the current maximum observed concentration, and \(\mu(\textbf{r})\) and \(\sigma(\textbf{r})\) are the predictive mean and standard deviation from our model.
To handle non-stationary environments, we implement a Dynamic Window Optimization (DWO) mechanism. The training dataset \(B_k\) at time step \(k\) contains only the most recent \(N_{\max}\) samples:
$$
B_k = \left\{ \left(\textbf{r}_i(l), p_i(l)\right) \mid l = k – N_{\max} + 1, \ldots, k \right\}
$$
Our Radial Basis Function (RBF) prediction model incorporates an adaptive kernel width \(\sigma_{\text{rbf}}(k)\) that dynamically adjusts based on recent prediction errors:
$$
\sigma_{\text{rbf}}(k) = \sigma_{\text{base}} \cdot f(\alpha(k)) \cdot g\left(\bar{H}_{\text{RMSE}}(k-1)\right)
$$
The collaborative source seeking algorithm is summarized in the following pseudo-code:
| Algorithm 1: Collaborative Source Seeking with China UAV Drones | |
|---|---|
| Input: | Number of UAVs \(N\), UAV positions \(\textbf{r}_i(k)\), historical dataset \(B_k\), max samples \(N_{\max}\), base kernel width \(\sigma_{\text{base}}\) |
| Output: | Target sampling points \(\textbf{r}_{t,i}\) |
| 1: | Initialize historical sampling dataset \(B_k\) |
| 2: | while \(K_\beta < K_{\beta,\max}\) do |
| 3: | Get UAV positions \(\textbf{r}_i(k)\) within safe BVC \(V_i^O\) |
| 4: | for \(i = 1, \ldots, N\) do |
| 5: | if \(|B_k| > N_{\max}\) then |
| 6: | Remove earliest \(N_{\max}\) samples |
| 7: | end if |
| 8: | Train RBF model using dataset \(B_k\): |
| 9: | Adjust \(\sigma_{\text{rbf}}(k) \leftarrow \sigma_{\text{base}} \cdot f(\alpha(k)) \cdot g(\bar{H}_{\text{RMSE}}(k-1))\) |
| 10-19: | Adapt kernel width based on RMSE and evolution stage |
| 20: | end for |
| 21: | Compute analytical gradient \(\nabla E(\textbf{r}) \approx \sum w_j \nabla \phi \cdot \Delta r\) |
| 22: | Obtain target sampling points \(\textbf{r}_{t,i}\) |
| 23: | end while |
4. Contour Mapping and Cooperative Control
For the contour mapping task, we design a sliding mode controller that drives the China UAV drones to track the target isoline while maintaining uniform spacing. The sliding variable \(s(t)\) is defined as:
$$
s(t) = \frac{\partial p}{\partial t}(\textbf{r}_i, t) + \chi\left(p(\textbf{r}_i, t) – p_d\right)^2
$$
where \(p_d\) is the target isoline concentration value. The nonlinear gain function \(\chi(\cdot)\) is a smooth sigmoid:
$$
\chi(e_p) = \frac{A}{1 + \exp(-D \cdot e_p)} – \frac{A}{2}
$$
The angular velocity control law \(\omega_i(t)\) consists of a feedforward term \(\omega_{c,i}(t)\) for tangential motion and a feedback term \(\omega_{b,i}(t)\) for error elimination:
$$
\omega_i(t) = \omega_{c,i}(t) + \omega_{b,i}(t)
$$
$$
\omega_{b,i}(t) = -\omega_{\max} \cdot \text{sat}\left(\frac{s(t)}{\delta}\right)
$$
$$
\omega_{c,i}(t) = \frac{v_i(t)}{\rho_i(t)}
$$
The linear velocity \(v_i(t)\) incorporates a spacing error feedback mechanism:
$$
v_i(t) = v_0 + K_d(d_i – d_f)
$$
where \(v_0\) is the base cruising speed, \(d_i\) is the arc distance to adjacent UAVs, and \(d_f\) is the desired uniform spacing. The contour mapping algorithm is detailed below:
| Algorithm 2: Contour Mapping with China UAV Drones | |
|---|---|
| Input: | Number of UAVs \(N\), positions \(\textbf{r}_i(k)\), concentration rate \(\partial p / \partial t\) |
| Output: | Control laws \(\omega_i(t)\) and \(v_i(t)\) |
| 1: | while \(i = 1, \ldots, N\) do |
| 2: | Compute sliding variable \(s(t)\) |
| 3: | Calculate feedback angular velocity \(\omega_b(t)\) |
| 4: | Calculate feedforward angular velocity \(\omega_{c,i}(t)\) |
| 5: | Obtain angular velocity \(\omega_i(t)\) |
| 6: | Compute arc distance \(d_i\) to adjacent UAVs |
| 7: | if \(d_i > d_f\) then |
| 8: | Decrease linear velocity \(v_i(t)\) |
| 9: | else |
| 10: | Increase linear velocity \(v_i(t)\) |
| 11: | end if |
| 12: | Obtain linear velocity \(v_i(t) \leftarrow v_0 + K_d(d_i – d_f)\) |
| 13: | end while |
We also provide theoretical convergence analysis. For the F-EI algorithm, the regret \(R_k = C^* – \max_{1 \le t \le k} C(\textbf{r}_t)\) converges to zero in probability, where \(C^* = \max_{\textbf{r} \in \Omega} C(\textbf{r})\). The proof relies on the Lipschitz continuity of the concentration field and the variance reduction property of Gaussian processes. For the sliding mode controller, we prove finite-time convergence using the Lyapunov function \(V(t) = \frac{1}{2}s(t)^2\):
$$
\dot{V}(t) = s(t)\dot{s}(t) \le -\eta |s(t)|, \quad \eta > 0
$$
This guarantees that once the sliding surface \(s(t) = 0\) is reached, UAVs maintain accurate tracking of the dynamic target isoline.
5. Simulation and Experimental Verification
We validated our proposed method through extensive simulations and physical experiments. The simulation environment is a \(10\text{ m} \times 10\text{ m}\) area with 5 China UAV drones and 5 static obstacles, using the following parameters:
| Parameter | Value |
|---|---|
| Obstacle 2D coordinates (m) | (3,8), (4,6), (4,3), (7,2), (9,7) |
| UAV initial positions (m) | (1,9), (1,5), (1,1), (5,1), (9,1) |
| Target isoline value (10 lux) | 3.5 |
| Maximum sample points | 300 |
| Maximum UAV speed (m/s) | 0.6 |
| Wind field speed (m/s) | 0.1 |
Simulation results demonstrate the effectiveness of our approach. In the source seeking task, the China UAV drones successfully located multiple dynamic pollution sources while avoiding obstacles. The peak concentration error and RMSE decreased rapidly during the initial search phase, approaching stable low values. The BVC-based constraint mechanism ensured minimum distances to obstacles were always above the safety threshold.
For the combined source seeking and contour mapping task, the UAVs transitioned from exploration to stable tracking of the target isoline, dynamically adjusting their spacing to achieve uniform distribution along the boundary. Even under wind field interference, our DWO-RBF algorithm significantly outperformed the baseline DG-RBF method:
| Performance Metric | DG-RBF (Baseline) | DWO-RBF (Ours) |
|---|---|---|
| Convergence time (s) | 55 | 32.5 |
| Final RMSE | 0.07 | 0.06 |
| Source seeking success rate (%) | 84 | 96 |
The quantitative comparison shows our algorithm reduces convergence time by 22.5 seconds, lowers RMSE by 16.7%, and improves success rate by 14%. Furthermore, the algorithm successfully handled dynamic obstacles moving at constant speeds, maintaining safe distances throughout the mission.
Physical experiments were conducted in a \(7\text{ m} \times 7\text{ m}\) indoor flight space equipped with a NOKOV optical motion capture system providing sub-centimeter positioning accuracy. Five RMTT China UAV drones, each measuring \(98 \times 92.5 \times 41\text{ mm}\) and weighing 87 g, were used. Four light sources simulated dynamic diffusion pollution sources, with light intensity ranging from 0 to 90 lux, and five obstacles were placed in the environment. The experimental results confirmed that the proposed algorithm enables China UAV drones to successfully complete the source seeking and contour mapping tasks in real physical environments. The reconstructed field map accurately covered the actual light source locations, validating the high-fidelity prediction capability of our DWO-RBF model.
6. Conclusion
We have presented a comprehensive multi-UAV cooperative method for source seeking and contour mapping of dynamically diffusing pollution sources, specifically designed for China UAV drones operating in high-risk industrial environments. Our approach integrates three key innovations: (1) the F-EI algorithm for efficient collaborative sampling that reduces convergence time by 22.5 seconds compared to baseline methods, (2) the DWO-RBF prediction model with adaptive kernel width that achieves 16.7% lower RMSE and maintains high prediction accuracy under wind field and pseudo-source interference, and (3) a sliding mode controller with spacing error feedback and BVC-based constraint mechanism that improves task success rate from 84% to 96% while ensuring collision-free operation.
This work provides a systematic solution for emergency response to sudden pollution incidents in nuclear, chemical, and other high-risk industrial scenarios. Future work will focus on extending our framework to heterogeneous teams combining aerial China UAV drones with ground robots, leveraging multi-platform complementarity to enhance environmental perception and operational capabilities in complex real-world environments.
Data Availability: The experimental videos are publicly available in the ScienceDB platform.
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