In recent years, the rapid advancement of drone technology has revolutionized various fields, including military operations, surveillance, and disaster response. Unmanned Aerial Vehicle (UAV) systems, particularly multi-UAV formations, have garnered significant attention due to their ability to perform complex tasks through cooperative control. However, achieving high-precision formation tracking within a specified time frame remains a critical challenge. Traditional control methods often focus on task completion rates without emphasizing transient and steady-state performance, leading to potential inefficiencies in time-sensitive applications. To address this, we propose a distributed formation control scheme that integrates prescribed performance control (PPC) with fixed-time theory, ensuring that multiple Unmanned Aerial Vehicles can achieve and maintain a desired formation configuration within a user-defined time. This approach leverages a double closed-loop control strategy, where the outer loop handles formation tracking using fixed-time PPC, and the inner loop manages attitude synchronization via a proportional-like feedback controller. Through comprehensive simulations, we demonstrate the effectiveness of our method in achieving fixed-time convergence with predefined performance bounds, highlighting its potential for enhancing the reliability and efficiency of drone technology in practical scenarios.
The integration of drone technology into modern systems has enabled unprecedented capabilities in autonomous operations. Unmanned Aerial Vehicle formations, in particular, offer advantages such as improved resource utilization, fault tolerance, and mission success rates. However, existing formation control algorithms often neglect time constraints and performance metrics, which are crucial for applications like search and rescue or combat missions. Prescribed performance control provides a framework to enforce transient and steady-state behaviors by constraining tracking errors within predefined bounds. When combined with fixed-time theory, it ensures that convergence occurs within a fixed duration, independent of initial conditions or controller parameters. This paper explores this synergy, presenting a novel control strategy for multiple Unmanned Aerial Vehicles that guarantees fixed-time formation tracking with high accuracy. Our contributions include the development of a distributed control law based on fixed-time PPC and a stability analysis using Lyapunov methods, validated through numerical simulations.
Preliminaries
We consider a multi-UAV system consisting of N follower drones and one virtual leader, all with identical dynamics. The model for each Unmanned Aerial Vehicle is derived under standard assumptions, such as rigidity and symmetry, leading to a decoupled system of position and attitude subsystems. The dynamics are represented as follows for the i-th drone:
$$ \dot{\mathbf{p}}_i = \mathbf{v}_i, \quad \dot{\mathbf{v}}_i = \mathbf{f}_{pi} + \mathbf{b}_{pi} \mathbf{u}_{pi} $$
$$ \dot{\boldsymbol{\delta}}_i = \boldsymbol{\omega}_i, \quad \dot{\boldsymbol{\omega}}_i = \mathbf{f}_{\delta i} + \mathbf{b}_{\delta i} \mathbf{u}_{\delta i} $$
where $\mathbf{p}_i$ and $\mathbf{v}_i$ denote the position and velocity vectors, $\boldsymbol{\delta}_i$ and $\boldsymbol{\omega}_i$ represent the attitude and angular rate vectors, and $\mathbf{u}_{pi}$ and $\mathbf{u}_{\delta i}$ are the control inputs for position and attitude, respectively. The terms $\mathbf{f}_{pi}$, $\mathbf{b}_{pi}$, $\mathbf{f}_{\delta i}$, and $\mathbf{b}_{\delta i}$ are derived from the UAV’s physical parameters, such as mass, inertia, and gravitational effects. This decoupling allows for separate controller design for each subsystem, simplifying the overall control strategy for the drone technology.
Prescribed performance control is employed to enforce desired transient and steady-state behaviors. The tracking error $e(t)$ is constrained within a performance envelope defined by a function $\rho(t)$:
$$ -\rho(t) < e(t) < \rho(t), \quad \forall t \geq 0 $$
where $\rho(t)$ is a strictly positive, monotonically decreasing function that converges to a steady-state value $\rho_\infty > 0$. For fixed-time convergence, we define $\rho(t)$ as:
$$ \rho(t) = \begin{cases}
(\rho_0 – \rho_\infty)(1 – \tau(t))^p + \rho_\infty, & \text{if } t \leq T_f \\
\rho_\infty, & \text{otherwise}
\end{cases} $$
with $\tau(t) = (t/T_f)^\kappa$, where $\kappa \in (0.5,1)$, $T_f$ is the fixed convergence time, and $\rho_0 > \rho_\infty$ is the initial bound. This ensures that the error converges to within $[-\rho_\infty, \rho_\infty]$ by time $T_f$, independent of initial conditions. To handle the constraints, a transformation function is used, such as the tangent function:
$$ \epsilon(t) = \tan\left(\frac{\pi}{2} \chi(t)\right), \quad \chi(t) = \frac{e(t)}{\rho(t)} $$
which converts the constrained error into an unconstrained variable $\epsilon(t)$, facilitating controller design.
The communication topology among drones is modeled as a directed graph $\mathcal{G} = (\mathcal{V}, \mathcal{E}, \mathbf{A})$, where $\mathcal{V} = \{1, 2, \dots, N\}$ is the set of nodes (drones), $\mathcal{E} \subseteq \mathcal{V} \times \mathcal{V}$ is the edge set, and $\mathbf{A} = [a_{ij}]$ is the adjacency matrix. A drone $j$ is a neighbor of drone $i$ if $a_{ij} = 1$, and $b_i = 1$ indicates that drone $i$ can access the leader’s information. The global formation tracking errors are defined as:
$$ \mathbf{e}_{pi} = \sum_{j \in \mathcal{N}_i} a_{ij} (\mathbf{p}_i – \mathbf{p}_j – \boldsymbol{\Delta}_{ij}) + b_i (\mathbf{p}_i – \mathbf{y}_r – \boldsymbol{\Delta}_i) $$
$$ \mathbf{e}_{vi} = \sum_{j \in \mathcal{N}_i} a_{ij} (\mathbf{v}_i – \mathbf{v}_j) + b_i (\mathbf{v}_i – \dot{\mathbf{y}}_r) $$
where $\mathbf{y}_r$ is the leader’s trajectory, and $\boldsymbol{\Delta}_i$ is the desired relative position for drone $i$. Similar definitions apply to attitude errors $\mathbf{e}_{\delta i}$ and $\mathbf{e}_{\omega i}$. This distributed setup ensures that each Unmanned Aerial Vehicle relies only on local information, enhancing scalability and robustness in drone technology applications.
Control Algorithm Design and Analysis
Our control strategy employs a double closed-loop structure. The outer loop, or formation control layer, utilizes fixed-time PPC to achieve precise formation tracking, while the inner loop, or attitude control layer, uses a proportional-like feedback controller for rapid attitude synchronization. This approach ensures that the multiple Unmanned Aerial Vehicles can maintain the desired formation within a fixed time, leveraging the advantages of drone technology for cooperative tasks.
For the position subsystem, we define an auxiliary variable $\mathbf{q}_i = \mathbf{e}_{pi} + k_v \mathbf{e}_{vi}$, where $k_v > 0$ is a design parameter. The prescribed performance constraint is applied to $\mathbf{q}_i$, ensuring that each component $q_{i,k}(t)$ satisfies $-\rho(t) < q_{i,k}(t) < \rho(t)$ for $k=1,2,3$. Using the transformation function, we obtain the unconstrained error $\boldsymbol{\epsilon}_{qi} = \tan\left(\frac{\pi}{2} \boldsymbol{\chi}_i\right)$ with $\boldsymbol{\chi}_i = \mathbf{q}_i / \rho(t)$. The derivative of $\boldsymbol{\epsilon}_{qi}$ is derived as:
$$ \dot{\boldsymbol{\epsilon}}_{qi} = \boldsymbol{\eta}_i \left( \frac{\dot{\mathbf{q}}_i}{\rho(t)} – \frac{\mathbf{q}_i \dot{\rho}(t)}{\rho^2(t)} \right) $$
where $\boldsymbol{\eta}_i = \text{diag}\left( \frac{\pi}{2} \sec^2\left(\frac{\pi}{2} \chi_{i,1}\right), \dots \right)$. Substituting the dynamics, we design the control input $\mathbf{u}_{pi}$ as:
$$ \mathbf{u}_{pi} = -\frac{1}{\sum_{j \in \mathcal{N}_i} a_{ij} + b_i} \left( k_1 \boldsymbol{\epsilon}_{qi} + \sum_{j \in \mathcal{N}_i} a_{ij} \mathbf{u}_{pj} + b_i \mathbf{Y}_i \right) $$
where $k_1 > 0$ is a gain, and $\mathbf{Y}_i$ aggregates the known terms from the dynamics. This controller ensures that the transformed error $\boldsymbol{\epsilon}_{qi}$ remains bounded, enforcing the prescribed performance on $\mathbf{q}_i$.
For the attitude subsystem, we define a similar variable $\mathbf{q}_{ai} = \mathbf{e}_{\delta i} + k_{\omega} \mathbf{e}_{\omega i}$, with $k_{\omega} > 0$. The control input $\mathbf{u}_{\delta i}$ is designed as:
$$ \mathbf{u}_{\delta i} = -\frac{1}{\sum_{j \in \mathcal{N}_i} a_{ij} + b_i} \left( k_2 \mathbf{q}_{ai} + \sum_{j \in \mathcal{N}_i} a_{ij} \mathbf{u}_{\delta j} + b_i \mathbf{Y}_{ai} \right) $$
where $k_2 > 0$ is a gain, and $\mathbf{Y}_{ai}$ includes the attitude dynamics terms. The inner loop controller ensures fast response and synchronization of attitudes among the Unmanned Aerial Vehicles.
The control inputs for the x and y directions, $u_{x,i}$ and $u_{y,i}$, are derived from the desired roll and pitch angles $\phi_{d,i}$ and $\theta_{d,i}$ using the relations:
$$ \phi_{d,i} = \arcsin\left( \frac{u_{x,i} \sin(\psi_{d,i}) – u_{y,i} \cos(\psi_{d,i})}{\sqrt{u_{x,i}^2 + u_{y,i}^2}} \right) $$
$$ \theta_{d,i} = \arcsin\left( \frac{u_{x,i} \cos(\psi_{d,i}) + u_{y,i} \sin(\psi_{d,i})}{\cos(\phi_{d,i}) \sqrt{u_{x,i}^2 + u_{y,i}^2}} \right) $$
where $\psi_{d,i}$ is the desired yaw angle. This completes the double closed-loop control structure for the drone technology.
Stability Analysis
To analyze the closed-loop stability, we consider Lyapunov functions for both subsystems. For the position subsystem, define $V_i = \frac{1}{2} \boldsymbol{\epsilon}_{qi}^T \boldsymbol{\epsilon}_{qi} + \frac{1}{2} \mathbf{q}_{ai}^T \mathbf{q}_{ai}$. Taking the derivative and substituting the control laws, we obtain:
$$ \dot{V}_i = \boldsymbol{\epsilon}_{qi}^T \dot{\boldsymbol{\epsilon}}_{qi} + \mathbf{q}_{ai}^T \dot{\mathbf{q}}_{ai} = -k_1 \boldsymbol{\epsilon}_{qi}^T \boldsymbol{\eta}_i \boldsymbol{\epsilon}_{qi} – k_2 \mathbf{q}_{ai}^T \mathbf{q}_{ai} < 0 $$
since $\boldsymbol{\eta}_i > 0$ and $k_1, k_2 > 0$. This proves that $\boldsymbol{\epsilon}_{qi}$ and $\mathbf{q}_{ai}$ are bounded and converge to zero, implying that the original errors $\mathbf{e}_{pi}$ and $\mathbf{e}_{vi}$ satisfy the prescribed performance bounds and converge within the fixed time $T_f$. Similarly, for the attitude subsystem, boundedness of $\mathbf{q}_{ai}$ ensures convergence of $\mathbf{e}_{\delta i}$ and $\mathbf{e}_{\omega i}$. Thus, the multi-UAV system achieves fixed-time formation tracking with guaranteed performance, demonstrating the robustness of our approach in drone technology.
Simulation Results and Analysis
We conducted simulations with a system of six follower drones and one virtual leader, using the parameters listed in Table 1. The initial positions of the followers were set to various points in 3D space, and the leader’s trajectory was defined as $\mathbf{y}_r = [0.5t, -0.5t, 0]^T$. The desired formation was a regular hexagon with relative positions $\boldsymbol{\Delta}_i$ as specified. The communication topology followed a directed graph where each drone communicates with specific neighbors and the leader.
| Parameter | Value |
|---|---|
| Mass (m) [kg] | 2.0 |
| Gravity (g) [m/s²] | 9.8 |
| Moment of Inertia (Ixx) [kg·m²] | 0.55 |
| Moment of Inertia (Iyy) [kg·m²] | 0.51 |
| Moment of Inertia (Izz) [kg·m²] | 0.96 |
| Arm Length (l) [m] | 0.1 |
| Propeller Inertia (Jp) [m] | 0.01 |
The controller parameters were chosen as $k_1 = 0.1$, $k_v = 6$, $\rho_0 = 50$, $\rho_\infty = 0.1$, $\kappa = 0.55$, and $T_f = 35$ seconds. The simulation results show that the global formation errors $\mathbf{e}_{pi}$ and $\mathbf{e}_{vi}$ converge to within the prescribed bounds by $T_f$, as illustrated in the error plots. The relative distances between neighboring drones stabilize to the desired values, confirming the hexagon formation. Attitude errors $\mathbf{e}_{\delta i}$ and $\mathbf{e}_{\omega i}$ also converge rapidly, ensuring synchronized orientation. The position evolution over time demonstrates that all Unmanned Aerial Vehicles achieve and maintain the formation, with the final positions matching the target configuration. These results validate the effectiveness of our fixed-time PPC approach in drone technology for high-precision formation control.
Conclusion
In this paper, we presented a fixed-time formation control scheme for multiple Unmanned Aerial Vehicles using prescribed performance control. The double closed-loop strategy enables precise tracking and attitude synchronization within a user-specified time, enhancing the capabilities of drone technology in time-critical applications. The stability analysis confirms the convergence of all errors, and simulations demonstrate the algorithm’s effectiveness. Future work will address collision avoidance, connectivity maintenance, and obstacle avoidance within the PPC framework, further advancing the reliability of multi-UAV systems. This research underscores the potential of integrating fixed-time theory with performance constraints to overcome limitations in traditional drone control methods, paving the way for more autonomous and efficient Unmanned Aerial Vehicle operations.