In recent years, quadrotor unmanned aerial vehicles (UAVs) have gained significant attention due to their compact size, high maneuverability, and versatility in applications such as search and rescue, terrain mapping, fire investigation, and traffic monitoring. However, the limited capabilities of a single quadrotor often necessitate the use of swarms to accomplish complex tasks efficiently. Formation control is a critical aspect of quadrotor swarm systems, enabling coordinated movements and stable configurations. As the complexity of formation tasks increases and swarm sizes expand, communication links within the quadrotor cluster can become sparse or weakly coupled, leading to reduced convergence speeds and stability issues. This paper addresses these challenges by proposing a fixed-time formation control strategy for quadrotor swarms, leveraging a multi-hop relay communication protocol to enhance robustness and convergence in distributed systems.
The convergence speed of quadrotor formation systems is a key performance metric, reflecting the dynamic efficiency and stability of the swarm. In multi-agent systems, the algebraic connectivity of the communication graph—specifically, the second smallest eigenvalue of the Laplacian matrix—plays a vital role in determining convergence rates. Traditional methods often assume that communication topologies can be arbitrarily modified, which is impractical in real-world scenarios. To overcome this, we introduce a multi-hop relay communication protocol that improves algebraic connectivity without altering the physical topology. This protocol allows intermediate quadrotor nodes to relay state information, thereby accelerating consensus and formation convergence. Moreover, existing research on multi-hop relay topologies primarily focuses on asymptotic stability, which does not guarantee a bounded convergence time. In contrast, fixed-time control strategies ensure that the system reaches the desired formation within a predetermined time, independent of initial conditions, offering superior precision and robustness.
This paper presents a comprehensive framework for fixed-time formation control of quadrotor swarms under external disturbances and sparse communication links. The contributions include: (1) a novel fixed-time state observer based on the multi-hop relay protocol, enabling each quadrotor to estimate the leader’s state efficiently; (2) a method for computing the adjacency matrix under relay communication, which enhances convergence speed; and (3) a cascade-structured fixed-time formation controller that reduces computational complexity and mitigates high-frequency chattering. Theoretical analysis proves the fixed-time convergence of the proposed scheme, and simulations validate its effectiveness in achieving stable formations within a fixed time.
Problem Formulation and Preliminaries
Consider a quadrotor swarm system consisting of N follower quadrotors and one virtual leader quadrotor. The dynamics of the i-th follower quadrotor, subject to external disturbances, can be described as:
$$ \dot{p}_i(t) = v_i(t), $$
$$ \dot{v}_i(t) = u_i(t) + f_i(t), $$
where \( p_i(t) = [p_{x,i}(t), p_{y,i}(t), p_{z,i}(t)]^T \) and \( v_i(t) = [v_{x,i}(t), v_{y,i}(t), v_{z,i}(t)]^T \) represent the position and velocity vectors of the quadrotor in the ground coordinate system, respectively. The control input is \( u_i(t) = [u_{x,i}(t), u_{y,i}(t), u_{z,i}(t)]^T \), and \( f_i(t) \in \mathbb{R}^3 \) denotes the external disturbance. The desired relative distance between quadrotor i and j is \( h_{ij} \in \mathbb{R}^3 \), and between quadrotor i and the leader is \( h_{i0} \in \mathbb{R}^3 \). The virtual leader’s dynamics are given by:
$$ \dot{p}_0(t) = v_0(t), $$
$$ \dot{v}_0(t) = u_0(t), $$
where \( p_0(t) \), \( v_0(t) \), and \( u_0(t) \in \mathbb{R}^3 \) are the leader’s position, velocity, and control input, respectively.
The communication network among quadrotors is represented by an undirected graph \( \bar{G} = (\bar{V}, \bar{\epsilon}, A, B) \), where \( \bar{V} = V \cup \{0\} \) includes the leader node 0 and follower nodes \( V = \{1, 2, \dots, N\} \). The edge set \( \bar{\epsilon} \subseteq \bar{V} \times \bar{V} \) defines communication links. The adjacency matrix \( A = [a_{ij}] \in \mathbb{R}^{N \times N} \) describes inter-quadrotor connections, with \( a_{ij} = 1 \) if quadrotors i and j can communicate directly, and 0 otherwise. The matrix \( B = [b_{10}, b_{20}, \dots, b_{N0}]^T \in \mathbb{R}^N \) indicates connections between followers and the leader, where \( b_{i0} = 1 \) if follower i can access the leader’s information. The degree matrix is \( D = \text{diag}(d_1, d_2, \dots, d_N) \) with \( d_i = \sum_{j=1}^N a_{ij} \), and the Laplacian matrix is \( L = D – A \).
The control objective is to design a fully distributed, fixed-time formation control protocol such that the position error \( e_{pi}(t) = p_i(t) – p_0(t) – h_{i0} \) converges to zero within a fixed time \( t_f \), i.e.,
$$ \lim_{t \to t_f} \| e_{pi}(t) \|_2 = 0, \quad t \in [t_f, \infty). $$
Assumptions include: (1) the leader’s input \( u_0(t) \) is bounded, (2) disturbances \( f_i(t) \) have a known upper bound, and (3) the communication topology is connected and undirected, with at least one follower having direct access to the leader’s information.
Multi-hop Relay Communication Protocol
The multi-hop relay communication protocol enhances the algebraic connectivity of the quadrotor swarm without modifying the physical topology. Let \( A_h \) denote the adjacency matrix for the h-th hop relay, and \( A’_h \) the cumulative adjacency matrix after h hops. The elements of \( A_h \) are computed recursively. For the first hop, \( A_1 = A \). For subsequent hops, the adjacency gain is calculated as:
$$ a_{h,ij} = \sum_{l=1}^N a_{1,il} a_{h-1,lj} = \alpha’_{1,i} \alpha_{h-1,j}, \quad i \neq j, $$
where \( \alpha_{h,i} \) and \( \alpha’_{h,i} \) are the i-th column and row vectors of \( A_h \), respectively. Thus, the h-hop adjacency matrix is:
$$ A_h = A_1 A_{h-1} – \text{diag}(\alpha’_{1,1} \alpha_{h-1,1}, \dots, \alpha’_{1,N} \alpha_{h-1,N}). $$
The cumulative adjacency matrix after h hops is \( A’_h = \sum_{i=1}^h A_i \). Similarly, the cumulative leader-follower matrix is \( B’_h = \sum_{i=1}^h B_i \), where \( B_i = A_1 B_{i-1} \). The degree matrix after h hops is \( D_h = \sum_{i=1}^h D_i \), and the Laplacian matrix is \( L’_h = D_h – A’_h = \sum_{i=1}^h L_i \). The matrix \( W_h = L’_h + \text{diag}(B’_h) \) is positive definite, and its minimum eigenvalue \( \lambda_{W_h,\min} \) increases with the number of hops, enhancing convergence speed.
Table 1 summarizes the notation used in the relay communication protocol.
| Symbol | Description |
|---|---|
| \( A_h \) | Adjacency matrix for the h-th hop |
| \( A’_h \) | Cumulative adjacency matrix after h hops |
| \( B’_h \) | Cumulative leader-follower matrix after h hops |
| \( L’_h \) | Cumulative Laplacian matrix after h hops |
| \( \lambda_{W_h,\min} \) | Minimum eigenvalue of \( W_h \) |
Fixed-Time State Observer
To address the challenge of limited leader information accessibility, a fixed-time state observer is designed for each follower quadrotor. The observer estimates the leader’s state using multi-hop relayed information. Let \( \hat{p}_{i0}(t) \) and \( \hat{v}_{i0}(t) \) denote the estimated position and velocity of the leader by quadrotor i. The observer dynamics are:
$$ \dot{\hat{v}}_{i0}(t) = -c_{v1} \text{sig}\left( \sum_{j=1}^N a’_{h,ij} (\hat{v}_{i0}(t) – \hat{v}_{j0}(t)) + b’_{h,i} (\hat{v}_{i0}(t) – v_0(t)) \right)^{\kappa_1} – c_{v2} \text{sig}\left( \sum_{j=1}^N a’_{h,ij} (\hat{v}_{i0}(t) – \hat{v}_{j0}(t)) + b’_{h,i} (\hat{v}_{i0}(t) – v_0(t)) \right)^{\kappa_2} – u_{\max} \text{sign}\left( \sum_{j=1}^N a’_{h,ij} (\hat{v}_{i0}(t) – \hat{v}_{j0}(t)) + b’_{h,i} (\hat{v}_{i0}(t) – v_0(t)) \right), $$
$$ \dot{\hat{p}}_{i0}(t) = -c_{p1} \text{sig}\left( \sum_{j=1}^N a’_{h,ij} (\hat{p}_{i0}(t) – \hat{p}_{j0}(t)) + b’_{h,i} (\hat{p}_{i0}(t) – p_0(t)) \right)^{\kappa_1} – c_{p2} \text{sig}\left( \sum_{j=1}^N a’_{h,ij} (\hat{p}_{i0}(t) – \hat{p}_{j0}(t)) + b’_{h,i} (\hat{p}_{i0}(t) – p_0(t)) \right)^{\kappa_2} + \hat{v}_{i0}(t), $$
where \( c_{v1}, c_{v2}, c_{p1}, c_{p2} > 0 \) are gain constants, \( 0 < \kappa_1 < 1 \), \( \kappa_2 > 1 \), and \( \text{sig}(x)^\alpha = \text{sign}(x) |x|^\alpha \). The estimation errors are defined as \( \tilde{v}_{i0}(t) = \hat{v}_{i0}(t) – v_0(t) \) and \( \tilde{p}_{i0}(t) = \hat{p}_{i0}(t) – p_0(t) \). The global error vectors are \( \tilde{v}_0(t) = [\tilde{v}_{10}(t)^T, \dots, \tilde{v}_{N0}(t)^T]^T \) and \( \tilde{p}_0(t) = [\tilde{p}_{10}(t)^T, \dots, \tilde{p}_{N0}(t)^T]^T \). The error dynamics can be written as:
$$ \dot{\tilde{v}}_0(t) = -c_{v1} \text{sig}(\Xi \tilde{v}_0(t))^{\kappa_1} – c_{v2} \text{sig}(\Xi \tilde{v}_0(t))^{\kappa_2} – \mathbf{1}_N \otimes u_0(t) – u_{\max} \text{sign}(\Xi \tilde{v}_0(t)), $$
$$ \dot{\tilde{p}}_0(t) = -c_{p1} \text{sig}(\Xi \tilde{p}_0(t))^{\kappa_1} – c_{p2} \text{sig}(\Xi \tilde{p}_0(t))^{\kappa_2} + \tilde{v}_0(t), $$
where \( \Xi = W_h \otimes I_n \). Using Lyapunov analysis, it can be shown that the estimation errors converge to zero in fixed time \( T_l = T_{l,v} + T_{l,p} \), with:
$$ T_{l,v} = \frac{2^{\frac{1-\kappa_1}{2}}}{c_{v1}(1-\kappa_1) \lambda_{W_h,\min}^{\frac{1+\kappa_1}{2}}} + \frac{2^{\frac{1-\kappa_2}{2}} n^{-\kappa_2} N^{\frac{\kappa_2-1}{2}}}{c_{v2}(\kappa_2-1) \lambda_{W_h,\min}^{\frac{1+\kappa_2}{2}}}, $$
$$ T_{l,p} = \frac{2^{\frac{1-\kappa_1}{2}}}{c_{p1}(1-\kappa_1) \lambda_{W_h,\min}^{\frac{1+\kappa_1}{2}}} + \frac{2^{\frac{1-\kappa_2}{2}} n^{-\kappa_2} N^{\frac{\kappa_2-1}{2}}}{c_{p2}(\kappa_2-1) \lambda_{W_h,\min}^{\frac{1+\kappa_2}{2}}}. $$
The multi-hop relay protocol increases \( \lambda_{W_h,\min} \), thereby reducing the convergence time of the observer.
Fixed-Time Formation Controller
After the state observer converges, the estimated states are used in the formation controller. The dynamics of the quadrotor are rewritten using virtual inputs. Define the virtual velocity input \( v^*_i(t) \) and virtual control input \( u^*_i(t) \), with the velocity tracking error \( e_{vi}(t) = v_i(t) – v^*_i(t) \). The system becomes:
$$ \dot{p}_i(t) = e_{vi}(t) + v^*_i(t), $$
$$ \dot{e}_{vi}(t) = u^*_i(t) + f_i(t), $$
where \( u^*_i(t) = u_i(t) – \dot{v}^*_i(t) \). The formation error is \( e_{pi}(t) = p_i(t) – \hat{p}_{i0}(t) – h_{i0} \). The fixed-time control laws are designed as follows.
The virtual velocity control law is:
$$ u^*_i(t) = -k_1 e_{vi}(t) – k_2 \left( e_{vi}(t) \right)^{\frac{q_1}{p_1}} + \text{sign}\left( \| e_{vi}(t) \|_1 – 1 \right) – \delta_f \text{sign}(e_{vi}(t)), $$
where \( k_1, k_2 > 0 \), \( p_1 \) and \( q_1 \) are positive integers with \( 1 < \frac{q_1}{p_1} < 2 \), and \( \delta_f \) is the disturbance bound. The position tracking controller is:
$$ v^*_i(t) = -k_3 e_{pi}(t) – k_4 \left( e_{pi}(t) \right)^{\frac{q_2}{p_2}} + \text{sign}\left( \| e_{pi}(t) \|_1 – 1 \right) + \dot{\hat{v}}_{i0}(t), $$
with \( k_3, k_4 > 0 \), and \( 1 < \frac{q_2}{p_2} < 2 \). The actual control input is \( u_i(t) = \dot{v}^*_i(t) + u^*_i(t) \).
Using Lyapunov functions \( V_1(t) = \| e_{vi}(t) \|_1 \) and \( V_2(t) = \| e_{pi}(t) \|_1 \), it can be proven that the errors converge to zero in fixed time \( T_{f,\max} = T_{v,\max} + T_{p,\max} \), where:
$$ T_{v,\max} = \frac{p_1}{k_1(q_1 – 2p_1)} \ln \left( \frac{k_2}{k_2 + k_1} \right) – \frac{p_1}{k_1 q_1} \ln \left( \frac{k_2}{k_2 + k_1} n^{\frac{q_1}{p_1}} \right), $$
$$ T_{p,\max} = \frac{p_2}{k_3(q_2 – 2p_2)} \ln \left( \frac{k_4}{k_4 + k_3} \right) – \frac{p_2}{k_3 q_2} \ln \left( \frac{k_4}{k_4 + k_3} n^{\frac{q_2}{p_2}} \right). $$
This controller reduces computational burden and chattering compared to traditional fixed-time methods, as it avoids complex power functions and frequent sign switches.
Table 2 provides a comparison of controller characteristics.
| Feature | Proposed Controller | Traditional Fixed-Time Controller |
|---|---|---|
| Computational Complexity | Low | High |
| Chattering Suppression | Effective | Limited |
| Convergence Time | Fixed and Bounded | Dependent on Initial Conditions |
| Communication Requirements | Multi-hop Relay | Direct Links |
Numerical Simulations
To validate the proposed method, simulations were conducted with a quadrotor swarm of one leader and five followers. The communication topology is shown in Figure 1, with a maximum relay hop count of 4. The leader’s initial position is \( p_0(0) = [5, 5, 6]^T \) and velocity is \( v_0(0) = [\pi/2, 0, 0.2]^T \). The leader’s control input is \( u_0(t) = [-0.05\pi^2 \sin(0.05\pi t), -0.05\pi^2 \cos(0.05\pi t), 0]^T \). External disturbances on followers are \( f_i(t) = [0.6 \sin(0.1 i t), -0.6 \sin(0.1 i t), 0.6 \cos(0.1 i t)]^T \). The desired formation is defined by relative distances \( h_{i0} \), and initial states are given in Table 3.
| Quadrotor | Desired Distance \( h_{i0} \) (m) | Initial Position \( p_i(0) \) (m) | Initial Velocity \( v_i(0) \) (m/s) |
|---|---|---|---|
| 1 | [-2, 0, 0]^T | [2, -2, 2]^T | [1, 0, 1]^T |
| 2 | [-2, -2, 0]^T | [2, 2, 0]^T | [0, 1, 0]^T |
| 3 | [-2, 2, 0]^T | [2, 1, 1]^T | [0, -0.5, 0]^T |
| 4 | [-4, -2, 0]^T | [1, 2, 0]^T | [0, 0.3, 1]^T |
| 5 | [-4, 2, 0]^T | [0, 0.3, 1]^T | [0, 0.3, 1]^T |
The observer parameters are \( c_{v1} = 15 \), \( c_{v2} = 9 \), \( c_{p1} = 8 \), \( c_{p2} = 12 \), \( \kappa_1 = 0.5 \), \( \kappa_2 = 2 \), and \( u_{\max} = 0.8 \). The controller parameters are \( k_1 = 5.8 \), \( k_2 = 5.2 \), \( k_3 = 2 \), \( k_4 = 3 \), \( q_1 = q_2 = 1 \), \( p_1 = p_2 = 2 \), and \( \delta_f = 0.6 \). The simulation duration is 50 s with a step size of 0.001 s.
The results demonstrate that the observer errors converge within 0.63 s for velocity and 0.775 s for position, which are below the theoretical bounds. The formation errors converge to zero in 1.675 s, also within the theoretical limit. The control inputs show reduced chattering compared to traditional methods, validating the effectiveness of the proposed approach.
Conclusion
This paper presents a fixed-time formation control strategy for quadrotor swarms using a multi-hop relay communication protocol. The proposed fixed-time state observer enables efficient estimation of the leader’s state, while the fixed-time controller ensures rapid convergence to the desired formation. The multi-hop relay enhances algebraic connectivity, reducing convergence time without altering the physical topology. Theoretical analysis and simulations confirm the fixed-time stability and robustness of the system. Future work will explore adaptive relay protocols and applications in dynamic environments.