In recent years, unmanned aerial vehicle (UAV) flight has garnered increasing attention, with autonomy, intelligence, and swarm coordination representing future trends. Among the key technologies for drone formation, formation transformation and maintenance are critical challenges. The process of formation change and hold can be interpreted as individual drones altering their positions relative to a lead aircraft or virtual reference point, with each drone achieving and maintaining desired positions and velocities in the absolute coordinate system upon completion of the transformation.
Existing research on drone formation transformation and maintenance includes approaches based on consensus theory, adaptive control under time-varying disturbances, optimization of controller parameters using hybrid algorithms, and behavior-based methods combined with virtual leader techniques. However, these methods often neglect the impact of acceleration changes in the lead aircraft or virtual reference point on follower drones, failing to address rapid response in formation changes from an acceleration perspective. Some studies incorporate leader acceleration as a disturbance in control laws but do not provide systematic methods for determining key parameters in neural network structures. To address these gaps, we propose an acceleration estimation method for the lead aircraft or virtual reference point in drone formation based on a radial basis function (RBF) neural network, particularly focusing on a novel approach to determine the Gaussian radial basis function centers using an improved intelligent waterdrop algorithm.
The core of our method lies in accurately estimating the acceleration of reference points in drone formation, enabling predictive control for enhanced formation agility. We design an RBF neural network where position and velocity deviations of drones serve as inputs, and the network outputs estimated accelerations. The Gaussian radial basis function centers are optimized via an intelligent waterdrop algorithm that mimics drone dynamics, with waterdrop velocity and acceleration simulating those of drones. This integration allows for real-time adaptation to formation maneuvers, ensuring minimal deviations during transformations.
To formalize, let $x_i = [e_i, \dot{e}_i]^T$ represent the input vector for the $i$-th direction in the absolute coordinate system (where $i=1,2,3$ correspond to x, y, z axes), with $e_i$ as position deviation and $\dot{e}_i$ as velocity deviation. The RBF neural network structure comprises an input layer with $n=2$ neurons per direction, a hidden layer with $p=7$ neurons, and an output layer with $q=3$ neurons for acceleration estimates. The Gaussian radial basis function is defined as:
$$h_{ij} = f(x_i(k)) = \exp\left(-\frac{\sum_{k=1}^{2} \|x_i(k) – c_{\cdot j}(k)\|^2}{2\sigma_i(k)^2}\right)$$
for $i=1,2,3$, $j=1,2,\dots,7$, and $k=1,2$. Here, $c_{\cdot j}$ is the $j$-th column vector of the center matrix $C$, and $\sigma_i(k)$ is the spread constant. The center matrix $C$ is constructed as:
$$C_{2 \times 7} = \begin{bmatrix}
-c_{1\text{max}} & -c_{1\text{max}} \times 0.5 & -c_{1\text{max}} \times 0.2 & 0 & c_{1\text{max}} \times 0.2 & c_{1\text{max}} \times 0.5 & c_{1\text{max}} \\
-c_{2\text{max}} & -c_{2\text{max}} \times 0.5 & -c_{2\text{max}} \times 0.2 & 0 & c_{2\text{max}} \times 0.2 & c_{2\text{max}} \times 0.5 & c_{2\text{max}}
\end{bmatrix}$$
where $c_{1\text{max}}$ and $c_{2\text{max}}$ are obtained through the intelligent waterdrop algorithm. The spread constants are calculated as:
$$\sigma_i(1) = \frac{c_{1\text{max}}}{2p}, \quad \sigma_i(2) = \frac{c_{2\text{max}}}{2p} \quad \text{for } i=1,2,3$$
with $p=7$. The hidden layer output matrix $H_{3 \times 7}$ is formed by evaluating $h_{ij}$ for all inputs. The actual weight vector $\tilde{W}_i^T$ is initialized to zero and updated iteratively:
$$\tilde{W}_i^T = \tilde{W}_i^T + \Delta W_i, \quad \Delta W_i = \alpha_i E_i^T K_i H_i$$
where $\alpha_i$ is the learning rate, $E_i = [e_i, \dot{e}_i]^T$, and $K_i = [k_{pi}, k_{di}]^T$ are preset gains. The estimated acceleration $\tilde{D}_i$ for the reference point in each direction is then:
$$\tilde{D}_i = \tilde{W}_i^T H_i \quad \text{for } i=1,2,3$$
This acceleration estimate is integrated into a formation control law to enhance responsiveness.
The intelligent waterdrop algorithm for determining $c_{1\text{max}}$ and $c_{2\text{max}}$ involves simulating waterdrops that represent drones. The algorithm steps are as follows:
- Define a training trajectory and iteration stop conditions.
- Set up a soil grid with dimensions based on preset maximum values for $c_{1\text{max}}$ and $c_{2\text{max}}$, assigning initial soil content to each grid cell.
- Initialize $M$ waterdrops with random positions on the trajectory, and velocities and accelerations within drone operational ranges.
- Compute the probability $p(u,v)_k$ for a waterdrop at grid $u$ to move to grid $v$ at step $k$:
$$p(u,v)_k = \begin{cases}
\frac{\chi_s f(\text{soil}(u,v)_k)}{\sum_{\phi} f(\text{soil}(u,v)_k)} + \frac{\chi_d \left( \frac{1}{\sum_{m=1}^M \|X_{m,k}^v\|} \right)}{\sum_{\phi} \left( \frac{1}{\sum_{m=1}^M \|X_{m,k}^v\|} \right)}, & v \in \phi \\
0, & v \notin \phi
\end{cases}$$
where $\chi_s + \chi_d = 1$, $\phi$ is the set of feasible next grids, $f(\text{soil}(u,v)_k) = 1/(\epsilon_s + g(\text{soil}(u,v)_k))$ with $\epsilon_s$ as a small positive constant, and $g(\cdot)$ ensures non-negative soil content. The term $\|X_{m,k}^v\|$ is the norm of the matrix formed by position and velocity deviations for waterdrop $m$ at grid $v$.
- Select the next grid $w$ based on computed probabilities.
- Update the soil content change $\delta \text{soil}(u,w)_k$:
$$\delta \text{soil}(u,w)_k = \alpha_s \sum_{m=1}^M \|X_{m,k}^w\|$$
where $\alpha_s$ is a preset parameter.
- Update the soil content at grid $w$:
$$\text{soil}(u,w)_k = (1 – \rho) \text{soil}(u,w)_{k-1} – \rho \delta \text{soil}(u,w)_k$$
with $\rho = 0.01$.
- Repeat until grid boundaries are reached or stop conditions (e.g., iteration count) are met. Upon termination, output $(c_{1\text{max}}, c_{2\text{max}})$ from the grid with minimal soil content.
For our simulations, we used a training trajectory of $y=0.5x$ with acceleration bounds of $(-5, 5) \, \text{m/s}^2$, minimum speed $V_{\text{min}} = 35 \, \text{m/s}$, and maximum speed $V_{\text{max}} = 80 \, \text{m/s}$. The algorithm yielded $c_{1\text{max}} = 24$ and $c_{2\text{max}} = 5$, leading to the center matrix and spread constants as defined earlier.
We conducted simulation experiments with a drone formation of six follower drones in a hexagonal pattern around a virtual reference point. The control law incorporating the estimated acceleration is:
$$N = B^{-1} (-K_d \dot{e} – K_p e – G – \tilde{D} + a)$$
where $N = [n_1, n_2, n_3]^T$ is the overload vector in the body coordinate system, $B$ is the rotation matrix to absolute coordinates, $K_d = \text{diag}(k_{d1}, k_{d2}, k_{d3})$ and $K_p = \text{diag}(k_{p1}, k_{p2}, k_{p3})$ are gain matrices, $G = [0, 0, g]^T$ with $g$ as gravitational acceleration, $\tilde{D} = [\tilde{D}_1, \tilde{D}_2, \tilde{D}_3]^T$ is the estimated acceleration vector, and $a$ represents wind disturbance loads. We set $\alpha = [2, 2, 2]$, $K_i = [0.65, 0.18]^T$, and initial conditions as shown in Table 1.
| Drone ID | Position (m) | Velocity (m/s) | Path Angle (°) | Heading Angle (°) |
|---|---|---|---|---|
| UAV1 | (93.3, 75, 50) | 50 | 0 | 0 |
| UAV2 | (50, 100, 50) | 50 | 0 | 0 |
| UAV3 | (6.7, 75, 50) | 50 | 0 | 0 |
| UAV4 | (6.7, 25, 50) | 50 | 0 | 0 |
| UAV5 | (50, 0, 50) | 50 | 0 | 0 |
| UAV6 | (93.3, 25, 50) | 50 | 0 | 0 |
The target region was a hexagon centered at $(-763, 850, 50)$. To test robustness, we introduced wind disturbances during flights, with wind models detailed in Figure 2 (simulated as time-varying vectors). The drone formation path involved multiple turns and wind exposure, as depicted in Figure 3. Our results show that the formation maintained cohesion despite disturbances, with accelerations converging to zero near the target (Figures 4 and 5).
| Drone ID | x-position Error (m) | y-position Error (m) | x-velocity Error (m/s) | y-velocity Error (m/s) |
|---|---|---|---|---|
| UAV1 | -0.0366 | -0.0077 | 0.0001 | -0.0004 |
| UAV2 | -0.0205 | -0.0633 | 0.0026 | 0.0048 |
| UAV3 | -0.0354 | 0.0138 | 0.0004 | 0.0002 |
| UAV4 | -0.0363 | -0.0637 | 0.0001 | 0.0055 |
| UAV5 | -0.0354 | -0.0123 | 0.0003 | -0.0014 |
| UAV6 | -0.0356 | -0.0163 | 0.0003 | 0.0015 |
Position and velocity deviations over time are shown in Figures 6-9. Deviations peaked during wind gusts and turns but rapidly diminished post-disturbance, confirming the effectiveness of our acceleration estimation. In the target region, deviations were centimeter-level for position and negligible for velocity (Table 2), demonstrating precise formation hold.
The advantages of our method for drone formation are threefold. First, it provides a systematic way to determine Gaussian radial basis function centers without reliance on empirical estimates or exhaustive searches. Second, accurate acceleration estimation enables precise prediction of flight movements, facilitating rapid response in drone formation maneuvers. Third, integration into control laws minimizes position and velocity deviations, meeting stringent formation-keeping requirements even under dynamic conditions.
In conclusion, we have developed a robust acceleration estimation method for reference points in drone formation using an RBF neural network optimized by an intelligent waterdrop algorithm. This approach enhances formation agility and stability, as validated through simulations with wind disturbances and complex flight paths. Future work may extend this to larger swarms or real-time implementations, further advancing autonomous drone formation capabilities.