In this paper, we address the challenge of formation control for quadrotor unmanned aerial vehicles (UAVs) operating under wind disturbances. The quadrotor platform is widely used due to its agility and versatility, but external factors like wind can significantly degrade performance. We propose an adaptive bearing rigid formation control strategy that leverages potential energy functions and parameter estimation to achieve robust formation shapes. The quadrotor system’s dynamics are modeled with consideration of spatial kinematics and disturbances, ensuring that the formation maintains desired relative bearings despite uncertainties. Our approach focuses on distributed control, where each quadrotor adjusts its position and orientation based on local information, promoting scalability and resilience in multi-quadrotor systems.
The core of our method involves designing a control law that minimizes a potential energy function defined by bearing errors. For a group of quadrotors, the relative position between agents i and j is given by $r_{ij} = r_i – r_j$, and the bearing vector is $g_{ij} = \frac{r_{ij}}{\|r_{ij}\|}$. The desired bearing is denoted as $g_{ij}^*$, and the control objective is to ensure that $\lim_{t \to \infty} (g_{ij}(t) – g_{ij}^*(t)) = 0$ and $\lim_{t \to \infty} (\theta_i(t) – \theta_i^*) = 0$, where $\theta_i$ is the orientation angle of quadrotor i. To handle wind disturbances, we incorporate adaptive estimation for unknown parameters, such as disturbance bounds, enhancing the quadrotor formation’s adaptability.
We begin by modeling the quadrotor dynamics. The position and orientation of each quadrotor are described by the following equations, which account for disturbances:
$$\dot{r}_i = R_i (v_i + dv_i(t))$$
$$\dot{\theta}_i = \omega_i + d\omega_i(t)$$
Here, $r_i = [x_i, y_i, z_i]^T$ represents the position of quadrotor i in 3D space, $R_i$ is the rotation matrix defined as:
$$R_i = \begin{bmatrix} \cos\theta_i & -\sin\theta_i & 0 \\ \sin\theta_i & \cos\theta_i & 0 \\ 0 & 0 & 1 \end{bmatrix}$$
$v_i$ is the nominal linear velocity, $dv_i(t)$ is the disturbance in linear velocity with $|dv_{ik}(t)| \leq D_{v_i}$ for $k \in \{x, y, z\}$, $\omega_i$ is the nominal angular velocity, and $d\omega_i(t)$ is the angular disturbance with $|d\omega_i(t)| \leq D_{\omega_i}$. The control input for quadrotor i is $u_i = v_i + dv_i$, which combines the nominal and disturbance components. This model captures the essential kinematics of a quadrotor under wind effects, allowing us to design a control strategy that compensates for uncertainties.
The communication topology among quadrotors is represented by a graph $\mathcal{G} = (\mathcal{V}, \mathcal{E})$, where $\mathcal{V} = \{1, 2, \dots, n\}$ is the set of nodes (quadrotors), and $\mathcal{E}$ is the set of edges indicating communication links. For each quadrotor i, the set of neighbors is $\mathcal{N}_i$. The bearing-based formation control relies on this graph to ensure that relative bearings converge to desired values. The potential energy function for quadrotor i is defined as:
$$P_i = \sum_{j \in \mathcal{N}_i} P_{ij}(r_{ij})$$
where $P_{ij} = \frac{\|g_{ij} – g_{ij}^*\|^2}{\|r_{ij}\|}$ is the pairwise potential energy. The gradient of this function with respect to $r_{ij}$ is:
$$\beta_{ij} = \frac{\partial P_{ij}}{\partial r_{ij}} = 2(g_{ij} – g_{ij}^*)$$
This gradient drives the control action, guiding each quadrotor to adjust its position to minimize bearing errors. The overall control law for linear velocity and angular velocity of quadrotor i is designed as:
$$v_i = -l_i \cdot R_i^T \sum_{j \in \mathcal{N}_i} (g_{ij} – g_{ij}^*) – S_i \hat{D}_{v_i}$$
$$\omega_i = -k_i (\theta_i – \theta_i^*) – \hat{D}_{\omega_i} \text{sgn}(\theta_i – \theta_i^*)$$
Here, $l_i$ and $k_i$ are positive control gains, $\hat{D}_{v_i}$ and $\hat{D}_{\omega_i}$ are estimates of the disturbance bounds, and $S_i = [\text{sgn}(\phi_{ix}), \text{sgn}(\phi_{iy}), \text{sgn}(\phi_{iz})]^T$ with $\phi_i = \sum_{j \in \mathcal{N}_i} (g_{ij} – g_{ij}^*)^T R_i$. The adaptive laws for updating these estimates are:
$$\dot{\hat{D}}_{v_i} = 4\lambda_i \sum_{j \in \mathcal{N}_i} (g_{ij} – g_{ij}^*)^T R_i S_i$$
$$\dot{\hat{D}}_{\omega_i} = \mu_i |\theta_i – \theta_i^*|$$
where $\lambda_i$ and $\mu_i$ are adaptive gains. These laws enable the quadrotor system to continuously adjust to wind disturbances, ensuring that the formation remains stable and converges to the desired configuration.
To analyze stability, we use a Lyapunov function candidate:
$$V = \sum_{i=1}^n P_i + \frac{1}{2} \sum_{i=1}^n (\theta_i – \theta_i^*)^2 + \frac{1}{\lambda_i} \tilde{D}_{v_i}^2 + \frac{1}{\mu_i} \tilde{D}_{\omega_i}^2$$
where $\tilde{D}_{v_i} = D_{v_i} – \hat{D}_{v_i}$ and $\tilde{D}_{\omega_i} = D_{\omega_i} – \hat{D}_{\omega_i}$ are estimation errors. The time derivative of V is computed as:
$$\dot{V} = 2 \sum_{i=1}^n \dot{r}_i^T \frac{\partial P_i}{\partial r_i} + \sum_{i=1}^n (\theta_i – \theta_i^*) \dot{\theta}_i – \frac{1}{\lambda_i} \tilde{D}_{v_i} \dot{\hat{D}}_{v_i} – \frac{1}{\mu_i} \tilde{D}_{\omega_i} \dot{\hat{D}}_{\omega_i}$$
Substituting the dynamics and control laws, we obtain:
$$\dot{V} \leq -4l_i \sum_{i=1}^n \sum_{j \in \mathcal{N}_i} (g_{ij} – g_{ij}^*)^T R_i R_i^T \sum_{j \in \mathcal{N}_i} (g_{ij} – g_{ij}^*) – \sum_{i=1}^n k_i (\theta_i – \theta_i^*)^2 \leq 0$$
This shows that V is non-increasing, and by Barbalat’s lemma, we conclude that $\lim_{t \to \infty} (g_{ij}(t) – g_{ij}^*(t)) = 0$ and $\lim_{t \to \infty} (\theta_i(t) – \theta_i^*) = 0$, proving global asymptotic stability of the quadrotor formation system. The adaptive mechanisms ensure that even with bounded wind disturbances, the formation achieves the desired bearing rigidity.
For simulation, we consider a system of 8 quadrotors with initial positions in a 3D space. The quadrotor dynamics include disturbances modeled as $dv_i(t) = 1.2 \cos(v_i)$ and $d\omega_i(t) = 0.1 \cos(\theta_i)$. The control parameters are set as $k_i = 2$, $\lambda_i = 0.2$, and $\theta_i^* = 1$. The initial positions are chosen to test formation convergence under wind effects. The table below summarizes the initial conditions and key parameters for the quadrotor system:
| Quadrotor ID | Initial Position (x, y, z) | Control Gain $l_i$ | Adaptive Gain $\lambda_i$ |
|---|---|---|---|
| 1 | (1, 0.2, 0) | 1.0 | 0.2 |
| 2 | (1, 0.4, 0) | 1.0 | 0.2 |
| 3 | (1, 0.6, 0) | 1.0 | 0.2 |
| 4 | (1, 0.8, 0) | 1.0 | 0.2 |
| 5 | (1, 1.0, 0) | 1.0 | 0.2 |
| 6 | (1, 1.2, 0) | 1.0 | 0.2 |
| 7 | (1, 1.4, 0) | 1.0 | 0.2 |
| 8 | (1, 1.6, 0) | 1.0 | 0.2 |
The simulation results demonstrate that the quadrotor formation converges to the desired bearing configuration within 5 seconds, despite the presence of wind disturbances. The bearing errors decrease asymptotically, and the orientation angles approach their desired values. The adaptive estimates $\hat{D}_{v_i}$ and $\hat{D}_{\omega_i}$ converge to true disturbance bounds, validating the effectiveness of our control strategy for quadrotor systems. The use of potential energy functions ensures that the formation maintains rigidity, while the distributed nature of the control allows each quadrotor to operate based on local information, reducing communication overhead.
In terms of performance metrics, we evaluate the formation error defined as $E(t) = \sum_{i=1}^n \sum_{j \in \mathcal{N}_i} \|g_{ij}(t) – g_{ij}^*\|^2$. The table below shows the convergence of this error over time for the quadrotor system:
| Time (s) | Formation Error E(t) | Average Bearing Error |
|---|---|---|
| 0 | 2.5 | 0.35 |
| 1 | 1.8 | 0.25 |
| 2 | 1.2 | 0.18 |
| 3 | 0.7 | 0.12 |
| 4 | 0.3 | 0.07 |
| 5 | 0.1 | 0.03 |
The results indicate that the formation error decreases rapidly, highlighting the robustness of the adaptive bearing rigid control for quadrotor UAVs. The quadrotor system’s ability to compensate for wind disturbances is crucial in real-world applications, such as surveillance or payload delivery, where precise formation flying is required. The adaptive laws ensure that the control parameters adjust dynamically, making the quadrotor formation resilient to environmental changes.
Furthermore, we analyze the impact of different wind disturbance levels on the quadrotor formation. The disturbance bounds $D_{v_i}$ and $D_{\omega_i}$ are varied, and the convergence time is recorded. The table below summarizes the results for a quadrotor system with 8 agents:
| Disturbance Level | Max $D_{v_i}$ | Max $D_{\omega_i}$ | Convergence Time (s) |
|---|---|---|---|
| Low | 0.5 | 0.05 | 4.2 |
| Medium | 1.0 | 0.1 | 5.0 |
| High | 1.5 | 0.15 | 6.1 |
As expected, higher disturbance levels lead to longer convergence times, but the adaptive control strategy still ensures formation stability for the quadrotor system. This demonstrates the scalability of our approach for multi-quadrotor applications, where each quadrotor can independently handle local disturbances without global knowledge.
In conclusion, we have presented an adaptive bearing rigid formation control strategy for quadrotor UAVs that effectively handles wind disturbances. The use of potential energy functions and adaptive estimation allows the quadrotor system to achieve desired formation shapes with global stability guarantees. Simulation results confirm the efficacy of the method, showing rapid convergence and robustness in various scenarios. Future work could extend this to more complex quadrotor formations or integrate it with obstacle avoidance techniques. The quadrotor platform’s flexibility makes it ideal for such advanced control strategies, paving the way for reliable autonomous operations in dynamic environments.
The mathematical formulation of our control strategy ensures that each quadrotor contributes to the overall formation stability. For instance, the collective potential energy $P = \sum_{i=1}^n P_i$ serves as a Lyapunov function, and its minimization drives the system toward the desired state. The adaptive laws are derived from gradient descent principles, ensuring that parameter estimates converge to true values over time. This approach is particularly beneficial for quadrotor systems operating in windy conditions, as it reduces the need for precise disturbance models.
Additionally, we consider the communication topology’s role in the quadrotor formation. The graph $\mathcal{G}$ is assumed to be connected, meaning that there is a path between any two quadrotors. This connectivity ensures that bearing information propagates through the network, enabling global formation control. The edge set $\mathcal{E}$ defines which quadrotors communicate, and for each edge (i,j), the bearing error $g_{ij} – g_{ij}^*$ is used in the control law. The rigidity of the formation is maintained as long as the graph is bearing rigid, which is a key property for achieving unique formation shapes in quadrotor systems.
To further illustrate the control design, we derive the error dynamics for the quadrotor system. Let $e_{ij} = g_{ij} – g_{ij}^*$ be the bearing error. Then, the time derivative is $\dot{e}_{ij} = \frac{\partial g_{ij}}{\partial r_{ij}} \dot{r}_{ij}$. Using the chain rule, we have $\dot{e}_{ij} = \frac{I – g_{ij} g_{ij}^T}{\|r_{ij}\|} \dot{r}_{ij}$, where I is the identity matrix. Substituting the dynamics, we get:
$$\dot{e}_{ij} = \frac{I – g_{ij} g_{ij}^T}{\|r_{ij}\|} R_i (v_i + dv_i(t)) – \frac{I – g_{ji} g_{ji}^T}{\|r_{ji}\|} R_j (v_j + dv_j(t))$$
This error dynamics is used in the Lyapunov analysis to show convergence. The control law for $v_i$ is designed to cancel out the disturbance terms through adaptive estimation, ensuring that $\dot{e}_{ij}$ approaches zero over time. Similarly, for orientation, the error $\tilde{\theta}_i = \theta_i – \theta_i^*$ has dynamics $\dot{\tilde{\theta}}_i = \omega_i + d\omega_i(t) – \dot{\theta}_i^*$. Since $\dot{\theta}_i^*$ is typically zero for constant desired orientations, the control law for $\omega_i$ drives $\tilde{\theta}_i$ to zero.
The adaptive gains $\lambda_i$ and $\mu_i$ are chosen based on the expected disturbance magnitudes. For practical quadrotor applications, these gains can be tuned experimentally to balance responsiveness and stability. The table below provides recommended gain values for different quadrotor sizes:
| Quadrotor Size | $\lambda_i$ Range | $\mu_i$ Range | Typical Wind Conditions |
|---|---|---|---|
| Small | 0.1-0.3 | 0.05-0.15 | Light breezes |
| Medium | 0.2-0.5 | 0.1-0.2 | Moderate winds |
| Large | 0.4-0.8 | 0.15-0.3 | Strong gusts |
These values ensure that the quadrotor system adapts efficiently to environmental changes. In summary, our adaptive bearing rigid formation control offers a robust solution for multi-quadrotor systems, leveraging local interactions and adaptive mechanisms to maintain formation integrity under wind disturbances. The quadrotor’s dynamics are fully accounted for, making this approach suitable for real-world deployment.
Finally, we emphasize that the quadrotor formation control strategy is distributed, meaning each quadrotor computes its own control input based on local neighbor information. This reduces computational load and enhances scalability for large-scale quadrotor networks. The use of bearing measurements instead of absolute positions makes the system more resilient to sensor errors and environmental variations. Overall, this work contributes to the advancement of autonomous quadrotor technologies, enabling reliable formation flying in challenging conditions.