With the rapid advancement of unmanned aerial vehicle (UAV) technology, quadrotors have emerged as a prominent focus in various fields such as agriculture, environmental monitoring, and disaster management. However, single quadrotors face limitations in complex tasks due to their constrained capabilities. To address this, multi-quadrotor formation control has garnered significant attention, enabling collaborative operations through coordinated maneuvers. Traditional formation control methods, including finite-time, fixed-time, and predefined-time approaches, often encounter challenges such as prolonged convergence times, excessive resource consumption, or impractical acceleration demands. This paper proposes a novel event-triggered two-level tandem game-theoretic control algorithm to achieve rapid formation and efficient resource utilization in multi-quadrotor systems.
The core of the proposed method lies in deconstructing the formation task into multiple games structured within a two-level framework. In the first level, quadrotor pairs engage in two parallel games to derive optimal position and velocity strategies, ensuring Nash equilibrium between their states. The second level involves an internal game where each quadrotor optimizes gain coefficients to balance position and velocity errors, leading to an overall optimal control strategy. An event-triggered mechanism dynamically adjusts strategy updates, reducing computational and communication overhead. Simulations and real-world experiments validate the algorithm’s effectiveness in achieving precise formation with minimal resource expenditure.
Introduction
Quadrotors have become indispensable in modern applications due to their agility and versatility. However, individual quadrotors are limited in handling complex missions, necessitating coordinated multi-quadrotor systems. Formation control enables these systems to maintain specific patterns while following a virtual leader, enhancing operational efficiency. Existing methods, such as finite-time control, ensure convergence within a bounded time but depend on initial conditions. Fixed-time control decouples convergence time from initial states but involves complex parameter tuning. Predefined-time control allows explicit time setting but may impose high initial accelerations. Distributed model predictive control and machine learning approaches offer alternatives but often require extensive communication or computational resources.
This paper introduces a game-theoretic approach combined with event-triggering to overcome these limitations. Game theory facilitates strategic interactions among quadrotors, leading to Nash equilibrium where no quadrotor can unilaterally improve its performance. The two-level tandem structure ensures that local interactions (first level) and global error minimization (second level) are efficiently coordinated. The event-triggered mechanism reduces unnecessary updates, conserving onboard resources. Key contributions include:
- Design of a two-level tandem game framework with parallel games at the first level and serial optimization at the second level.
- Adaptive parameter tuning for critical game parameters based on real-time system states.
- Integration of event-triggering to dynamically control strategy updates.
Quadrotor Dynamic Model
The kinematic and dynamic models of the i-th quadrotor are expressed as:
$$ \dot{P}_i = V_i $$
$$ \ddot{P}_i = \dot{V}_i $$
where \( P_i = [p_{i,x}, p_{i,y}, p_{i,z}]^T \) and \( V_i = [v_{i,x}, v_{i,y}, v_{i,z}]^T \) denote position and velocity, respectively. The dynamics are given by:
$$ \ddot{p}_{i,x} = -\frac{k_{i,x}}{m_i} \dot{p}_{i,x} + \frac{T_i}{m_i} \left( \cos(\phi_i) \sin(\theta_i) \cos(\psi_i) + \sin(\phi_i) \sin(\psi_i) \right) $$
$$ \ddot{p}_{i,y} = -\frac{k_{i,y}}{m_i} \dot{p}_{i,y} + \frac{T_i}{m_i} \left( \cos(\phi_i) \sin(\theta_i) \sin(\psi_i) – \sin(\phi_i) \cos(\psi_i) \right) $$
$$ \ddot{p}_{i,z} = -\frac{k_{i,z}}{m_i} \dot{p}_{i,z} + \frac{T_i}{m_i} \cos(\phi_i) \cos(\theta_i) – g $$
Here, \( m_i \) is mass, \( T_i \) is thrust, \( k_i = [k_{i,x}, k_{i,y}, k_{i,z}]^T \) are aerodynamic damping coefficients, \( \phi_i, \theta_i, \psi_i \) are roll, pitch, and yaw angles, and \( g \) is gravity. The force components \( S_i = [s_{i,x}, s_{i,y}, s_{i,z}]^T \) are:
$$ s_{i,x} = T_i \left( \cos(\phi_i) \sin(\theta_i) \cos(\psi_i) + \sin(\phi_i) \sin(\psi_i) \right) $$
$$ s_{i,y} = T_i \left( \cos(\phi_i) \sin(\theta_i) \sin(\psi_i) – \sin(\phi_i) \cos(\psi_i) \right) $$
$$ s_{i,z} = T_i \cos(\phi_i) \cos(\theta_i) $$
Under a constant strategy \( S_i \), the quadrotor’s motion over time \( t_s \) is:
$$ V_i(t_s) = \kappa_i (S_i – m_i g e_3) + \lambda_i V_i(t_0) $$
$$ P_i(t_s) = P_i(t_0) + \mu_i (S_i – m_i g e_3) – \gamma_i V_i(t_0) $$
where \( \kappa_i, \lambda_i, \mu_i, \gamma_i \) are diagonal matrices derived from system parameters:
$$ \kappa_{i,j} = \frac{1 – e^{-\frac{k_{i,j}}{m_i} t_s}}{k_{i,j}}, \quad \lambda_{i,j} = e^{-\frac{k_{i,j}}{m_i} t_s} $$
$$ \mu_{i,j} = \frac{t_s}{k_{i,j}} + \frac{m_i}{k_{i,j}^2} \left( e^{-\frac{k_{i,j}}{m_i} t_s} – 1 \right), \quad \gamma_{i,j} = \frac{m_i}{k_{i,j}} \left( e^{-\frac{k_{i,j}}{m_i} t_s} – 1 \right) $$
for \( j = x, y, z \), and \( e_3 = [0, 0, 1]^T \).
Two-Level Tandem Game-Theoretic Control
The formation control problem is decomposed into a two-level game structure. The first level involves external games between communicating quadrotor pairs, while the second level focuses on internal optimization of gain coefficients.
First Level: External Strategy Game
Quadrotor pairs engage in two parallel games to compute position and velocity strategies. For quadrotors i and j, the position game aims to minimize the objective function:
$$ J^P_{ij}(S^P_i, S^P_j) = \alpha \| P_i(t_s) – P_j(t_s) – P_{di} + P_{dj} \|^2 + \beta \| P_i(t_s) – P(t_s) – P_{di} \|^2 $$
where \( S^P_i = [s^P_{i,x}, s^P_{i,y}, s^P_{i,z}]^T \) and \( S^P_j \) are position strategies, \( P_{di}, P_{dj} \) are desired positions, and \( \alpha, \beta \in (0,1) \) are weights. The velocity game minimizes:
$$ J^V_{ij}(S^V_i, S^V_j) = \alpha \| V_i(t_s) – V_j(t_s) \|^2 + \beta \| V_i(t_s) – V(t_s) \|^2 $$
with \( S^V_i = [s^V_{i,x}, s^V_{i,y}, s^V_{i,z}]^T \). The Nash equilibrium conditions lead to linear systems:
$$ H^P_x s^P_x = C^P_x, \quad H^V_x s^V_x = C^V_x $$
where \( H^P_x, H^V_x \) are invertible matrices ensuring unique solutions. Each quadrotor i obtains strategy sets \( \mathcal{S}^P_i = \{S^P_{i,1}, \dots, S^P_{i,n_i}\} \) and \( \mathcal{S}^V_i = \{S^V_{i,1}, \dots, S^V_{i,n_i}\} \) from interactions with neighbors in \( Q_i \).
Strategy Weighting and Second Level Game
Using Algorithm 1, each quadrotor computes weights for the first-level strategies based on position and velocity errors:
$$ e^P_{ik} = |P_i – P_{ik} – P_{di} + P_{dik}|, \quad e^V_{ik} = |V_i – V_{ik}| $$
The weighted strategies for the second level are:
$$ S^v_i = \sum_{k=1}^{n_i} \text{diag}(w^v_{ik}) S^V_{ik}, \quad S^p_i = \sum_{k=1}^{n_i} \text{diag}(w^p_{ik}) S^P_{ik} $$
In the second level, players \( S^v_i \) and \( S^p_i \) represent gain coefficients \( W^v_i = [w^v_{i,x}, w^v_{i,y}, w^v_{i,z}]^T \) and \( W^p_i = [w^p_{i,x}, w^p_{i,y}, w^p_{i,z}]^T \). The actual control strategy is:
$$ S_i = \text{diag}(W^p_i) S^p_i + \text{diag}(W^v_i) S^v_i $$
The objective functions for the second-level players are:
$$ J^v_i(W^v_i, W^p_i) = \| V_i(t_s) – V(t_s) \|^2 + \| \text{diag}(W^v_i) S^v_i – S^v_i \|^2 $$
$$ J^p_i(W^p_i, W^v_i) = \| P_i(t_s) – P(t_s) – P_{di} \|^2 + \| \text{diag}(W^p_i) S^p_i – S^p_i \|^2 $$
Solving for Nash equilibrium yields optimal gain coefficients, ensuring minimal position and velocity errors.
Event-Triggered Mechanism
To reduce resource consumption, an event-triggered condition is defined based on the total error between quadrotors i and j:
$$ e_{ij} = e^P_{ij} + e^V_{ij}, \quad e^P_{ij} = \| P_i – P_j – P_{di} + P_{dj} \| + \| P_i – P – P_{di} \| $$
$$ e^V_{ij} = \| V_i – V_j \| + \| V_i – V \| $$
Games are triggered only if \( e_{ij} \) exceeds a threshold \( E \) or is increasing. This minimizes unnecessary strategy updates.
Adaptive Parameter Tuning
Algorithm 2 dynamically adjusts the critical parameter \( t_s \) based on the maximum position error \( e^P_{\text{max}} \) among quadrotor pairs:
$$ t_s = e^{(e^P_{\text{max}} – e^P_0) / e^P_0} $$
where \( e^P_0 \) is the initial maximum error. This ensures optimal strategy duration.
Simulation and Experimental Results
The proposed algorithm was validated through simulations and real-platform tests involving six quadrotors. The virtual leader’s trajectory was set as:
$$ P = [10 \sin(0.1t), 10 \cos(0.1t) – 10, 0.5t]^T $$
Initial states and desired positions are summarized in Table 1.
| Quadrotor | Initial Position | Initial Velocity | Desired Position |
|---|---|---|---|
| 1 | [1, 2, 0] | [0, 0, 0] | [1, √3, 0] |
| 2 | [4, 3, 0] | [0, 0, 0] | [2, 0, 0] |
| 3 | [4, -1, 0] | [0, 0, 0] | [1, -√3, 0] |
| 4 | [2, -3, 0] | [0, 0, 0] | [-1, -√3, 0] |
| 5 | [-5, -2, 0] | [0, 0, 0] | [-2, 0, 0] |
| 6 | [-2, 5, 0] | [0, 0, 0] | [-1, √3, 0] |
Simulation parameters are listed in Table 2.
| Parameter | Value |
|---|---|
| α | 0.8 |
| β | 0.2 |
| E | 0.05 |
| m_i | 1.121 |
| k_{i,j} | 0.01 |
The formation achieved convergence within 5 seconds, with position and velocity errors nearing zero. The event-triggered mechanism reduced game triggering frequencies to below 45%, as shown in Table 3.
| Quadrotor | Triggering Frequency |
|---|---|
| 1 | 36.18% |
| 2 | 33.58% |
| 3 | 34.49% |
| 4 | 32.78% |
| 5 | 34.25% |
| 6 | 33.88% |
Comparative studies with finite-time formation control demonstrated the superiority of the proposed method, reducing convergence time by 34.59% and improving formation accuracy by 63.34%.
Conclusion
This paper presented an event-triggered two-level tandem game-theoretic control algorithm for multi-quadrotor formation. The framework efficiently coordinates local and global optimizations through strategic games, achieving rapid formation and resource efficiency. Adaptive parameter tuning and event-triggering enhance practicality in real-world scenarios. Future work will explore dynamic communication topologies and robust control under uncertainties.