In recent years, the coordination control of multi-drone systems has garnered significant attention due to its robustness, flexibility, scalability, and economic benefits. As a fundamental control method in this domain, the consensus problem for multiple drones remains a hot research topic in control theory. Consensus control involves drones utilizing onboard sensors to perceive information from neighboring drones, enabling interaction and information sharing within the swarm. Numerous studies have addressed drone formation control, but many simplify the drone dynamics to first- or second-order systems, overlooking nonlinear factors. Additionally, research often focuses on planar formations, with limited exploration of three-dimensional space. In this work, we propose a novel approach to drone formation control by integrating cascade system theory with an input-constrained consensus algorithm. Our method avoids oversimplification of the drone model and effectively handles input constraints, ensuring practical applicability in real-world scenarios.
The core of our approach lies in transforming the complex vertical take-off and landing (VTOL) drone model into a cascade form using cascade system theory. This transformation preserves the nonlinear characteristics of the original system, unlike previous works that reduce it to lower-order dynamics. We then design a consensus algorithm with input constraints by incorporating the bounded hyperbolic tangent function, which limits control inputs to feasible ranges. Based on this algorithm, we develop a formation control strategy that enables drones to achieve and maintain desired formations in three-dimensional space. Our simulations, conducted on the Matlab platform, validate the effectiveness of our method, showing convergence to consensus and successful formation flight.
The general model for a VTOL drone system, derived from Newton-Euler principles, is given by:
$$
\begin{align}
m\ddot{\xi} &= -T R e_3 + F_e(\dot{\xi}, d(t)) + R \Sigma_R \Gamma, \\
\dot{R} &= R S(\omega), \\
I \dot{\omega} &= -S(\omega) I \omega + \Gamma,
\end{align}
$$
where \( m \) is the mass, \( I \) the inertia matrix, \( T \) the thrust, \( \Gamma \) the torque vector, \( \xi \in \mathbb{R}^3 \) the position in inertial coordinates, \( R \in SO(3) \) the rotation matrix, \( d(t) \) external disturbances, \( \omega \in \mathbb{R}^3 \) the angular velocity, \( F_e \) the external forces, \( \Sigma_R \) a constant coupling matrix, and \( e_3 = (0,0,1)^T \). The operator \( S(\cdot) \) denotes a skew-symmetric matrix. For a system of \( N \) identical drones, the model extends accordingly.
Using cascade system theory, we transform this model into a cascade form to simplify analysis and control design. Define state variables \( x_{i1} = m_i \xi_i \), \( x_{i2} = m_i \dot{\xi}_i \), \( x_{i3} = -T_i R_i e_3 + m_i g e_3 \), and \( x_{i4} = -R_i \delta \) with \( \delta = (T_i \omega_{i2}, -T_i \omega_{i1}, \dot{T}_i)^T \). After derivations, the cascade model is:
$$
\begin{align}
\dot{x}_{i1} &= x_{i2}, \\
\dot{x}_{i2} &= x_{i3} + u_{i1}, \\
\dot{x}_{i3} &= x_{i4}, \\
\dot{x}_{i4} &= u_{i2},
\end{align}
$$
where \( u_{i1} = R_i \Sigma_R \Gamma_i \) and \( u_{i2} \) is a function of thrust and torque derivatives. This representation avoids neglecting nonlinear terms and provides a structured framework for control design.
The consensus problem for drone formation aims to ensure all drone states converge to a common value. For the cascade model, we design input-constrained control laws using the hyperbolic tangent function, which is bounded and ensures practical input limits. The control laws are:
$$
\begin{align}
u_{i1} &= -k_1 \sum_{j \in N_i} a_{ij} \tanh(x_{i1} – x_{j1}) – k_2 \sum_{j \in N_i} a_{ij} \tanh(x_{i2} – x_{j2}), \\
u_{i2} &= -k_3 \sum_{j \in N_i} a_{ij} \tanh(x_{i3} – x_{j3}) – k_4 \sum_{j \in N_i} a_{ij} \tanh(x_{i4} – x_{j4}),
\end{align}
$$
where \( k_1, k_2, k_3, k_4 > 0 \) are constants, \( N_i \) is the set of neighbors of drone \( i \), and \( a_{ij} \) are adjacency matrix elements representing communication links. The use of \( \tanh(\cdot) \) bounds the inputs, preventing saturation and enhancing robustness in real-world drone formation scenarios.
For formation control, we extend the consensus algorithm to include desired formation offsets. Let \( h^d_{i1}, h^d_{i2}, h^d_{i3}, h^d_{i4} \) denote formation distances for each state. The formation control laws are:
$$
\begin{align}
u_{i1} &= -k^*_1 \sum_{j \in N_i} a_{ij} \tanh(x_{i1} – x_{j1} – h^d_{i1}) – k^*_2 \sum_{j \in N_i} a_{ij} \tanh(x_{i2} – x_{j2} – h^d_{i2}), \\
u_{i2} &= -k^*_3 \sum_{j \in N_i} a_{ij} \tanh(x_{i3} – x_{j3} – h^d_{i3}) – k^*_4 \sum_{j \in N_i} a_{ij} \tanh(x_{i4} – x_{j4} – h^d_{i4}),
\end{align}
$$
where \( k^*_1, k^*_2, k^*_3, k^*_4 > 0 \). These laws enable drones to achieve specified formations while respecting input constraints, crucial for stable drone formation flight in applications like surveillance or light shows.
We analyze the stability of the system using Lyapunov theory and cascade system principles. Assume the communication topology is a fixed undirected connected graph. Define vectors \( Z_1 = [x_{11}, x_{21}, \dots, x_{N1}]^T \), \( Z_2 = [x_{12}, x_{22}, \dots, x_{N2}]^T \), \( Z_3 = [x_{13}, x_{23}, \dots, x_{N3}]^T \), and \( Z_4 = [x_{14}, x_{24}, \dots, x_{N4}]^T \). The closed-loop system under consensus laws becomes:
$$
\begin{align}
\dot{Z}_1 &= Z_2, \\
\dot{Z}_2 &= -k_1 \tanh(L Z_1) – k_2 \tanh(L Z_2) + Z_3, \\
\dot{Z}_3 &= Z_4, \\
\dot{Z}_4 &= -k_3 \tanh(L Z_3) – k_4 \tanh(L Z_4),
\end{align}
$$
where \( L \) is the Laplacian matrix. We prove global uniform stability through the following theorem.
Theorem 1: For the system above with \( k_1, k_2 > 0 \) and an undirected connected graph, the subsystem \( \dot{Z}_1 = Z_2 \), \( \dot{Z}_2 = -k_1 \tanh(L Z_1) – k_2 \tanh(L Z_2) \) is globally uniformly stable.
Proof: Consider the Lyapunov function \( V_1 = \frac{k_1}{2} \sum_{i} \sum_{j \in N_i} a_{ij} \ln[\cosh(x_{i1} – x_{j1})] + \frac{k_2}{2} \sum_{i} \sum_{j \in N_i} a_{ij} \ln[\cosh(x_{i2} – x_{j2})] \). Its derivative yields \( \dot{V}_1 \leq 0 \), ensuring stability. Invoking invariant set theorems, convergence to consensus is established.
Theorem 2: The cascade system under control laws (5) and (6) achieves global uniform stability for an undirected connected graph.
Proof: The system can be viewed as a cascade of two subsystems. Using Lemma 2 from cascade system theory, since the first subsystem is stable and the coupling term satisfies bounded conditions, global stability follows.
Theorem 3: The formation control laws (7) and (8) enable the system to converge to desired formation offsets, achieving drone formation flight.
Proof: Similar to Theorem 1, by shifting coordinates to formation errors, stability is proven via Lyapunov methods.
To validate our approach, we conducted simulations for both consensus and formation control. For consensus, we considered a system of 6 drones with communication topology as a ring graph. The Laplacian matrix for this topology is:
$$
L = \begin{bmatrix}
2 & -1 & 0 & 0 & 0 & -1 \\
-1 & 2 & -1 & 0 & 0 & 0 \\
0 & -1 & 2 & -1 & 0 & 0 \\
0 & 0 & -1 & 2 & -1 & 0 \\
0 & 0 & 0 & -1 & 2 & -1 \\
-1 & 0 & 0 & 0 & -1 & 2
\end{bmatrix}.
$$
Initial conditions for states were set randomly. We compared unconstrained inputs (without tanh) and input-constrained cases. The results show that with input constraints, all states converge to consensus smoothly, whereas unconstrained inputs lead to divergence due to saturation. For drone formation control, we simulated 3 drones in a line formation. The desired formation distances were set as offsets in 3D space. The drones successfully achieved and maintained the formation, as shown in trajectory plots.
Table 1 summarizes key parameters used in simulations.
| Parameter | Description | Value |
|---|---|---|
| \( N \) | Number of drones | 6 (consensus), 3 (formation) |
| \( k_1, k_2, k_3, k_4 \) | Control gains for consensus | 1.0, 1.0, 0.5, 0.5 |
| \( k^*_1, k^*_2, k^*_3, k^*_4 \) | Control gains for formation | 1.2, 1.2, 0.6, 0.6 |
| Communication | Topology type | Undirected connected graph |
| Simulation time | Duration | 40 seconds |
The simulation results highlight the importance of input constraints in drone formation control. Without constraints, control inputs may exceed physical limits, causing instability. Our method ensures bounded inputs through the hyperbolic tangent function, enabling practical implementation. For example, in drone formation light shows, where precise coordination is essential, input constraints prevent actuator saturation and ensure smooth maneuvers.
Further analysis involves the effect of communication delays or switching topologies on drone formation stability. Future work could extend our approach to handle time-varying delays or dynamic networks, common in real-world drone swarms. Additionally, integrating obstacle avoidance into the formation control framework would enhance autonomy. The cascade system theory provides a robust foundation for such extensions, as it decouples complex dynamics into manageable subsystems.
In conclusion, we have presented a comprehensive method for drone formation control based on cascade system theory and input-constrained consensus algorithms. By transforming the VTOL drone model into a cascade form, we preserve nonlinearities and design bounded control laws using hyperbolic tangent functions. Our stability analysis proves global convergence for both consensus and formation control. Simulations confirm the effectiveness, showing that drones achieve consensus and maintain formations in 3D space. This work advances drone formation control by addressing input constraints and model accuracy, with potential applications in aerial displays, surveillance, and cooperative transport. Future directions include adaptive gain tuning, robustness to disturbances, and experimental validation with physical drones.
The mathematical formulations and control designs are generalizable to other multi-agent systems. For instance, the consensus algorithm with input constraints can be applied to robot swarms or autonomous vehicles. The key insight is the use of bounded functions to enforce practical limits while ensuring stability. In drone formation scenarios, this approach balances performance and feasibility, making it suitable for large-scale deployments.
To further illustrate, consider the energy consumption during drone formation flight. Input constraints naturally limit control efforts, reducing energy usage and extending flight time. This is critical for applications requiring prolonged operations. Our method implicitly optimizes energy by preventing excessive inputs, a desirable feature in sustainable drone technologies.
Another aspect is scalability. The distributed nature of our control laws allows the system to scale to hundreds of drones without central coordination. Each drone relies only on local neighbor information, reducing communication overhead. This aligns with trends in swarm robotics, where decentralization enhances robustness and flexibility.
In summary, drone formation control via input-constrained consensus and cascade theory offers a practical and theoretical framework for coordinating multi-drone systems. The integration of bounded control functions ensures real-world applicability, while cascade simplifications facilitate analysis and design. As drone technologies evolve, such methods will play a pivotal role in enabling complex autonomous behaviors.