In recent years, multi-rotor unmanned aerial vehicles (UAVs), often referred to as drones, have demonstrated immense potential across various fields, including surveillance, logistics, agriculture, and environmental monitoring. The coordination of multiple UAVs in swarm formations enhances mission efficiency through shared information, dynamic task allocation, and collaborative operations. However, achieving precise and robust formation control for these UAV drones presents significant challenges due to nonlinear dynamics, external disturbances, communication delays, computational complexity, and the need for adaptability in dynamic environments. Traditional control methods, such as sliding mode control, finite-time control, and fixed-time control, offer solutions but often face limitations like chattering, dependence on initial conditions, or high control energy consumption. More critically, conventional Nash equilibrium approaches, which assume simultaneous decision-making among equal agents, may lack the explicit hierarchical guidance required for missions with clear command structures or rapid response needs. To address these issues, this article proposes a novel hierarchical formation control strategy for quadrotor UAV swarms based on Stackelberg game theory. This framework introduces a leader-follower decision-making hierarchy: a designated UAV drone acts as the Stackelberg leader, proactively determining and committing to its future trajectory to guide the entire formation; the remaining UAV drones, as followers, observe this commitment and engage in a distributed Nash equilibrium game to compute their optimal control inputs collaboratively. This approach combines the global guidance of centralized leadership with the flexibility and robustness of distributed coordination among follower drones. Furthermore, an enhanced disturbance compensation mechanism is designed to estimate and counteract external time-varying disturbances acting on the follower UAVs, improving overall system robustness. Comprehensive simulation experiments validate the effectiveness of the proposed strategy in terms of formation accuracy, convergence speed, and disturbance rejection, demonstrating its superiority over traditional methods like fixed-time sliding mode control.
The core innovation of this work lies in integrating Stackelberg game theory into the control architecture for multi-UAV systems. Unlike simultaneous-move games, Stackelberg games model sequential decision-making with a leader who commits to a strategy first, and followers who then respond optimally. This naturally aligns with many real-world UAV operations where a lead drone (e.g., from a ground station or a designated unit) provides mission directives. For a swarm of quadrotor drones, this translates to the leader UAV broadcasting its intended future states (position, velocity), which serve as a reference trajectory for the follower drones. The followers, in turn, optimize their own control actions not only to track this leader but also to maintain desired relative positions with their neighbors, effectively solving a local Nash equilibrium problem. This hierarchical yet distributed structure ensures that the entire UAV formation moves cohesively while being resilient to individual failures or local communication drops. The mathematical formulation involves designing appropriate cost functions for both leader and follower UAVs, deriving analytical solutions for optimal control forces, and implementing real-time disturbance estimators. The result is a control strategy that is both computationally tractable and highly effective for complex, dynamic environments where multiple UAV drones must operate in close coordination.
1. Dynamic Model of Quadrotor UAV Systems
To develop the control strategy, we first establish the dynamic model for each quadrotor UAV in the swarm. Consider a system comprising N quadrotor drones. Each UAV drone is modeled as a rigid body with six degrees of freedom: three translational and three rotational. However, for formation control focusing on positional coordination, we primarily consider the translational dynamics under external disturbances. Let the index i denote the i-th UAV drone. The position of UAV i in the inertial frame is given by $\mathbf{P}_i = [p_{i,x}, p_{i,y}, p_{i,z}]^\top \in \mathbb{R}^3$, and its velocity is $\mathbf{V}_i = [v_{i,x}, v_{i,y}, v_{i,z}]^\top \in \mathbb{R}^3$. The translational dynamics, incorporating aerodynamic drag effects, are described by:
$$ \dot{\mathbf{P}}_i = \mathbf{V}_i $$
$$ \begin{aligned}
\dot{v}_{i,x} &= \frac{T_i}{m_i} (\cos \phi_i \sin \theta_i \cos \psi_i + \sin \phi_i \sin \psi_i) – d_{i,x}, \\
\dot{v}_{i,y} &= \frac{T_i}{m_i} (\cos \phi_i \sin \theta_i \sin \psi_i – \sin \phi_i \cos \psi_i) – d_{i,y}, \\
\dot{v}_{i,z} &= \frac{T_i}{m_i} (\cos \phi_i \cos \theta_i) – g – d_{i,z},
\end{aligned} $$
where $m_i$ is the mass of the UAV drone, $T_i$ is the total thrust generated by its rotors, $g$ is gravitational acceleration, and $\mathbf{d}_i = [d_{i,x}, d_{i,y}, d_{i,z}]^\top$ represents external disturbances (e.g., wind gusts). The angles $\phi_i$, $\theta_i$, $\psi_i$ are the roll, pitch, and yaw angles, respectively. For formation control, we often assume an inner-loop attitude controller stabilizes these angles to desired values derived from the translational control inputs. Thus, we can simplify the control input as a virtual force or acceleration. Define the control acceleration for UAV i as $\mathbf{u}_i = [u_{i,x}, u_{i,y}, u_{i,z}]^\top$, which relates to thrust and orientation. The dynamics can be linearized for control design as:
$$ \ddot{\mathbf{P}}_i = \mathbf{u}_i – \mathbf{d}_i. $$
In discrete-time for implementation with sampling period $\Delta t$, the state update equations are:
$$ \mathbf{P}_i(k+1) = \mathbf{P}_i(k) + \mathbf{V}_i(k) \Delta t + \frac{1}{2} (\mathbf{u}_i(k) – \mathbf{d}_i(k)) \Delta t^2, $$
$$ \mathbf{V}_i(k+1) = \mathbf{V}_i(k) + (\mathbf{u}_i(k) – \mathbf{d}_i(k)) \Delta t. $$
The formation objective is for each follower UAV drone to maintain a desired relative position $\mathbf{P}_i^d$ with respect to a virtual leader or a designated leader UAV. The position error for UAV i is defined as $\mathbf{e}_i = \mathbf{P}_i – \mathbf{P}_L – \mathbf{P}_i^d$, where $\mathbf{P}_L$ is the leader’s position. The control goal is to drive $\mathbf{e}_i$ to zero while ensuring collision avoidance and connectivity.
2. Algebraic Graph Theory for Communication Topology
The communication and sensing network among the follower UAV drones is modeled using algebraic graph theory. Consider an undirected graph $\mathcal{G}_F = (\mathcal{V}_F, \mathcal{E}_F, \mathbf{A}_F)$, where $\mathcal{V}_F = \{1, 2, …, N\}$ is the set of nodes (follower drones), $\mathcal{E}_F \subseteq \mathcal{V}_F \times \mathcal{V}_F$ is the set of edges representing communication links, and $\mathbf{A}_F = [a_{ij}] \in \mathbb{R}^{N \times N}$ is the weighted adjacency matrix. The element $a_{ij} > 0$ if UAV drones i and j can communicate, and $a_{ij}=0$ otherwise. We assume $a_{ii}=0$. The degree matrix $\mathbf{D}_F = \text{diag}(d_1, d_2, …, d_N)$ has diagonal entries $d_i = \sum_{j=1}^{N} a_{ij}$. The Laplacian matrix of the follower network is $\mathbf{L}_F = \mathbf{D}_F – \mathbf{A}_F$. For a leader-follower structure, we also define a diagonal matrix $\mathbf{B} = \text{diag}(b_1, b_2, …, b_N)$, where $b_i > 0$ if follower i receives information directly from the leader UAV, and $b_i=0$ otherwise. This matrix captures the connectivity between followers and the leader drone. The overall graph properties influence the stability and convergence of the formation control algorithm.
| Parameter | Symbol | Typical Value/Range |
|---|---|---|
| Mass of UAV drone | $m_i$ | 1.0 kg |
| Maximum thrust | $T_{\text{max}}$ | 15 N |
| Gravity acceleration | $g$ | 9.81 m/s² |
| Sampling period | $\Delta t$ | 0.01 s |
| Communication range | $R_{\text{com}}$ | 50 m |
| Adjacency weight | $a_{ij}$ | 1 if within range |
3. Stackelberg Game Framework for Hierarchical Control
Stackelberg game theory provides a natural framework for hierarchical decision-making in multi-agent systems. In our context, one UAV drone is designated as the Stackelberg leader (index L), and the remaining N UAV drones are followers. The game proceeds in two stages:
- Leader’s Decision Stage: The leader UAV first computes and broadcasts its planned future trajectory (e.g., position and velocity over a horizon) based on its global mission objectives. This commitment is made public to all follower drones.
- Followers’ Response Stage: Each follower UAV, upon observing the leader’s commitment, determines its own control input by solving a local optimization problem that considers both tracking the leader and coordinating with neighboring followers. The followers act simultaneously, leading to a Nash equilibrium among them.
The leader anticipates the followers’ reactions when making its decision, thus optimizing its own objective function accordingly. This sequential structure enables explicit global guidance while allowing distributed computation among follower UAVs. The mathematical formulation involves defining cost functions for both leader and follower drones, which incorporate tracking errors, control efforts, and formation maintenance terms.
4. Stackelberg Leader Strategy Design
The leader UAV’s objective is to track a predefined target trajectory $\mathbf{P}_{L,\text{target}}(t)$ and target velocity $\mathbf{V}_{L,\text{target}}(t)$ while minimizing control energy. Let $\mathbf{s}_L(t_k)$ denote the control force (acceleration scaled by mass) that the leader decides at time step $t_k$. The leader’s predicted state at time $t_k + t_s$, where $t_s$ is a look-ahead step, is:
$$ \mathbf{P}_L(\mathbf{s}_L) = \mathbf{P}_L(t_k) + \mathbf{V}_L(t_k) t_s + \frac{\mathbf{s}_L}{2m_L} t_s^2, $$
$$ \mathbf{V}_L(\mathbf{s}_L) = \mathbf{V}_L(t_k) + \frac{\mathbf{s}_L}{m_L} t_s. $$
The leader’s cost function $J_L$ is a quadratic form:
$$ J_L(\mathbf{s}_L) = \omega_{L,P} \| \mathbf{P}_L(\mathbf{s}_L) – \mathbf{P}_{L,\text{target}}(t_k + t_s) \|^2 + \omega_{L,V} \| \mathbf{V}_L(\mathbf{s}_L) – \mathbf{V}_{L,\text{target}}(t_k + t_s) \|^2 + \omega_{L,s} \| \mathbf{s}_L \|^2, $$
where $\omega_{L,P}$, $\omega_{L,V}$, $\omega_{L,s}$ are positive weighting coefficients. The optimal leader control force $\mathbf{s}_L^*$ is found by setting the gradient $\nabla_{\mathbf{s}_L} J_L = 0$:
$$ \frac{\partial J_L}{\partial \mathbf{s}_L} = 2\omega_{L,P} \left( \mathbf{P}_L(\mathbf{s}_L) – \mathbf{P}_{L,\text{target}} \right)^\top \frac{\partial \mathbf{P}_L}{\partial \mathbf{s}_L} + 2\omega_{L,V} \left( \mathbf{V}_L(\mathbf{s}_L) – \mathbf{V}_{L,\text{target}} \right)^\top \frac{\partial \mathbf{V}_L}{\partial \mathbf{s}_L} + 2\omega_{L,s} \mathbf{s}_L^\top = 0. $$
Substituting the derivatives $\frac{\partial \mathbf{P}_L}{\partial \mathbf{s}_L} = \frac{t_s^2}{2m_L} \mathbf{I}$ and $\frac{\partial \mathbf{V}_L}{\partial \mathbf{s}_L} = \frac{t_s}{m_L} \mathbf{I}$, we obtain the closed-form solution:
$$ \mathbf{s}_L^* = \mathbf{C}_1^{-1} \left[ \omega_{L,P} \frac{t_s^2}{2m_L} \left( \mathbf{P}_{L,\text{target}} – \mathbf{P}_L(t_k) – \mathbf{V}_L(t_k) t_s \right) + \omega_{L,V} \frac{t_s}{m_L} \left( \mathbf{V}_{L,\text{target}} – \mathbf{V}_L(t_k) \right) \right], $$
where $\mathbf{C}_1 = \left( \omega_{L,P} \frac{t_s^4}{4m_L^2} + \omega_{L,V} \frac{t_s^2}{m_L^2} + \omega_{L,s} \right) \mathbf{I}$. After saturation to respect actuator limits, the final leader control $\mathbf{s}_{L,\text{sat}}^*$ is applied. The leader then commits its future state at $t_k + t_s$:
$$ \mathbf{P}_{L,\text{commit}} = \mathbf{P}_L(t_k) + \mathbf{V}_L(t_k) t_s + \frac{\mathbf{s}_{L,\text{sat}}^*}{2m_L} t_s^2, \quad \mathbf{V}_{L,\text{commit}} = \mathbf{V}_L(t_k) + \frac{\mathbf{s}_{L,\text{sat}}^*}{m_L} t_s. $$
This committed state is broadcast to all follower UAV drones, providing a reference for the subsequent Nash game.
5. Stackelberg Follower Strategy Design via Nash Equilibrium
Each follower UAV i, upon receiving the leader’s commitment, aims to compute its control force $\mathbf{s}_i$ to minimize a local cost function $J_i$. This cost function balances several objectives: (i) tracking the leader’s committed state, (ii) maintaining desired relative positions with neighboring followers, and (iii) minimizing control effort. Let $\mathbf{P}_i(\mathbf{s}_i)$ and $\mathbf{V}_i(\mathbf{s}_i)$ denote the predicted position and velocity of follower i after applying control $\mathbf{s}_i$, analogous to the leader’s prediction equations. The cost function for follower i is:
$$ \begin{aligned}
J_i(\mathbf{s}_i, \mathbf{s}_{-i}) &= \alpha \sum_{j \in \mathcal{N}_i} a_{ij} \| (\mathbf{P}_i(\mathbf{s}_i) – \mathbf{P}_j(\mathbf{s}_j)) – (\mathbf{P}_i^d – \mathbf{P}_j^d) \|^2 \\
&\quad + \beta \sum_{j \in \mathcal{N}_i} a_{ij} \| \mathbf{V}_i(\mathbf{s}_i) – \mathbf{V}_j(\mathbf{s}_j) \|^2 \\
&\quad + \eta \left[ \alpha_L \| \mathbf{P}_i(\mathbf{s}_i) – \mathbf{P}_{L,\text{commit}} – \mathbf{P}_i^d \|^2 + \beta_L \| \mathbf{V}_i(\mathbf{s}_i) – \mathbf{V}_{L,\text{commit}} \|^2 \right] \\
&\quad + \omega_{i,s} \| \mathbf{s}_i \|^2,
\end{aligned} $$
where $\mathbf{s}_{-i}$ denotes the control forces of all followers except i, $\mathcal{N}_i$ is the set of neighbors of UAV i, $\mathbf{P}_i^d$ is the desired relative position with respect to the leader, and $\alpha, \beta, \alpha_L, \beta_L, \eta, \omega_{i,s}$ are positive weights. The first term penalizes formation shape errors, the second term encourages velocity consensus, the third term ensures tracking of the leader, and the last term penalizes control effort.
Since all followers decide simultaneously given the leader’s commitment, their interactions form a non-cooperative game. A Nash equilibrium is reached when no follower can unilaterally reduce its cost by changing its control force. Mathematically, the set of control forces $\{\mathbf{s}_i^*\}$ satisfies:
$$ \mathbf{s}_i^* = \arg \min_{\mathbf{s}_i} J_i(\mathbf{s}_i, \mathbf{s}_{-i}^*) \quad \forall i. $$
Given the quadratic nature of $J_i$, the Nash equilibrium can be found by solving the system of linear equations obtained from the first-order conditions $\frac{\partial J_i}{\partial \mathbf{s}_i} = 0$ for all i. Stacking the control vectors into $\mathbf{S} = [\mathbf{s}_1^\top, \mathbf{s}_2^\top, …, \mathbf{s}_N^\top]^\top \in \mathbb{R}^{3N}$, we have:
$$ \mathbf{H} \mathbf{S} = \mathbf{h}, $$
where $\mathbf{H}$ is a block matrix that depends on the graph Laplacian $\mathbf{L}_F$, matrix $\mathbf{B}$, weights, and system parameters, and $\mathbf{h}$ is a vector involving the leader’s committed state, current states, and desired formations. The explicit derivation yields:
$$ \mathbf{H} = \left( \alpha \mathbf{L}_F \otimes \mathbf{\Gamma}_P + \beta \mathbf{L}_F \otimes \mathbf{\Gamma}_V + \eta (\alpha_L \mathbf{B} \otimes \mathbf{\Gamma}_P + \beta_L \mathbf{B} \otimes \mathbf{\Gamma}_V) + \omega_s \mathbf{I}_{3N} \right), $$
where $\otimes$ denotes Kronecker product, $\mathbf{\Gamma}_P = \frac{t_s^4}{4m_i^2} \mathbf{I}_3$, $\mathbf{\Gamma}_V = \frac{t_s^2}{m_i^2} \mathbf{I}_3$ (assuming equal mass $m_i$ for simplicity), and $\omega_s$ is a common control weight. The vector $\mathbf{h}$ is:
$$ \mathbf{h} = – \left( \alpha \mathbf{L}_F \otimes \frac{t_s^2}{2m_i} \mathbf{I}_3 \right) \mathbf{P}_{\text{rel}} – \left( \beta \mathbf{L}_F \otimes \frac{t_s}{m_i} \mathbf{I}_3 \right) \mathbf{V}_{\text{cur}} – \eta \left( \alpha_L \mathbf{B} \otimes \frac{t_s^2}{2m_i} \mathbf{I}_3 \right) (\mathbf{P}_{\text{cur}} – \mathbf{P}_{L,\text{commit}} – \mathbf{P}^d) – \eta \left( \beta_L \mathbf{B} \otimes \frac{t_s}{m_i} \mathbf{I}_3 \right) (\mathbf{V}_{\text{cur}} – \mathbf{V}_{L,\text{commit}}), $$
with $\mathbf{P}_{\text{rel}}$, $\mathbf{V}_{\text{cur}}$ being stacked vectors of relative positions and current velocities. Solving $\mathbf{S} = \mathbf{H}^{-1} \mathbf{h}$ gives the Nash equilibrium control forces for all follower UAV drones. This solution is distributed in the sense that each follower only needs information from its neighbors and the leader, and the computation can be parallelized.
| Parameter | Role | Typical Value |
|---|---|---|
| $\omega_{L,P}$ | Leader position tracking weight | 1.8 |
| $\omega_{L,V}$ | Leader velocity tracking weight | 0.9 |
| $\omega_{L,s}$ | Leader control effort weight | 0.002 |
| $\alpha$ | Follower formation position weight | 0.92 |
| $\beta$ | Follower formation velocity weight | 0.08 |
| $\alpha_L$ | Follower leader position tracking weight | 1.0 |
| $\beta_L$ | Follower leader velocity tracking weight | 1.0 |
| $\eta$ | Leader-follower tracking balance | 20 |
| $\omega_{i,s}$ | Follower control effort weight | 0.01 |
6. Disturbance Compensation Mechanism for Enhanced Robustness
External disturbances, such as wind, can significantly degrade the performance of UAV formation control. To enhance robustness, we design a disturbance compensator for each follower UAV drone. The idea is to estimate the net disturbance acting on the drone by comparing the actual acceleration experienced with the expected acceleration from the control input. Let $\mathbf{u}_{i,\text{nominal}}(t)$ be the nominal control acceleration computed from the Nash equilibrium solution (i.e., $\mathbf{s}_i^*/m_i$). At each small time step $\Delta t$, the actual acceleration applied $\mathbf{u}_{i,\text{applied}}(t)$ may differ due to disturbances. A raw disturbance estimate $\hat{\mathbf{d}}_{i,\text{raw}}(t)$ is:
$$ \hat{\mathbf{d}}_{i,\text{raw}}(t) = \mathbf{u}_{i,\text{applied}}(t – \Delta t) – \mathbf{u}_{i,\text{nominal}}(t) – \mathbf{u}_{i,\text{comp}}(t – \Delta t), $$
where $\mathbf{u}_{i,\text{comp}}$ is the compensation acceleration from the previous step. To smooth the estimate and reduce noise, a first-order low-pass filter is applied:
$$ \mathbf{u}_{i,\text{comp}}(t) = (1 – a_{\text{filt}}) \mathbf{u}_{i,\text{comp}}(t – \Delta t) + a_{\text{filt}} g_{\text{comp}} \hat{\mathbf{d}}_{i,\text{raw}}(t), $$
with filter coefficient $a_{\text{filt}} \in (0,1)$ and gain $g_{\text{comp}} \in (0,1)$. The total control acceleration finally applied to follower UAV i is:
$$ \mathbf{u}_{i,\text{final}}(t) = \mathbf{u}_{i,\text{nominal}}(t) + \mathbf{u}_{i,\text{comp}}(t). $$
This compensator effectively reduces the impact of slow-varying disturbances, ensuring that the formation control remains accurate even in gusty conditions. The parameters $a_{\text{filt}}$ and $g_{\text{comp}}$ can be tuned based on the expected disturbance characteristics; for example, $a_{\text{filt}}=0.4$ and $g_{\text{comp}}=0.9$ work well for moderate wind gusts.
7. Simulation Experiments and Performance Analysis
To validate the proposed Stackelberg game-based hierarchical control strategy, we conduct numerical simulations using MATLAB/Simulink. The scenario involves one leader UAV and four follower UAV drones. The leader’s target trajectory is a 3D Lissajous curve: $\mathbf{P}_{L,\text{target}}(t) = [2\sin(0.25t), 2\cos(0.25t) – 2, 0.3t]^\top$ meters. The desired formation is a square in the horizontal plane with offsets $\mathbf{P}_1^d = [-0.5, 0, -0.5]^\top$, $\mathbf{P}_2^d = [-0.5, 0, 0.5]^\top$, $\mathbf{P}_3^d = [0.5, 0, 0.5]^\top$, $\mathbf{P}_4^d = [0.5, 0, -0.5]^\top$ meters relative to the leader. The communication graph among followers is a line topology with Laplacian $\mathbf{L}_F$ as in Equation (21), and $\mathbf{B} = \text{diag}(1,0,0,0)$ indicating only follower 1 directly receives the leader’s commitment. External disturbance is set to $\mathbf{d}_i(t) = [0.5\sin(0.5t), 0.5\cos(0.5t), 0.02]^\top$ m/s² for all UAV drones. Simulation parameters are as in Tables 1 and 2, with total time $T=25$ s.
The simulation results demonstrate excellent formation tracking. The leader UAV accurately follows the target trajectory, and the follower UAV drones converge to the desired formation within approximately 3 seconds, maintaining it throughout the flight. The position errors for followers, defined as $\|\mathbf{e}_i\|$, rapidly decrease to less than 0.01 m. The control inputs are smooth and within saturation limits. The disturbance compensator effectively reduces the error caused by the sinusoidal wind, as seen in the comparison of trajectories with and without compensation. Specifically, the compensated system shows a 70% reduction in root-mean-square tracking error compared to the uncompensated case. Moreover, the formation remains stable even when communication links are intermittently dropped, showcasing the robustness of the Nash equilibrium solution.
To highlight the advantages of the Stackelberg approach, we compare it with a fixed-time sliding mode control (FTSMC) method applied to the same formation problem. The FTSMC controller is designed for each follower UAV to track a virtual leader derived from the actual leader’s trajectory. The comparison metrics include convergence time, steady-state error, control effort, and robustness to disturbances. As summarized in Table 3, the Stackelberg game-based controller achieves faster convergence (3 s vs. 8 s), lower steady-state error (0.008 m vs. 0.05 m), and smoother control actions without chattering. The FTSMC, while robust, exhibits noticeable chattering and higher overshoot during initial formation acquisition. This comparison underscores the benefits of the hierarchical game-theoretic framework for multi-UAV coordination.
| Performance Metric | Stackelberg Game Control | Fixed-Time Sliding Mode Control |
|---|---|---|
| Convergence Time (to 1 cm error) | 3.0 s | 8.2 s |
| Steady-State Position Error (RMS) | 0.008 m | 0.05 m |
| Control Effort (Average norm) | 2.1 N | 3.5 N |
| Chattering | None | Noticeable |
| Disturbance Rejection | Excellent (compensated) | Good but with steady error |
| Scalability to More UAV Drones | High (distributed computation) | Medium (centralized reference) |
Furthermore, we test scalability by increasing the number of follower UAV drones to 10 in a hexagonal formation. The Stackelberg framework maintains stable control with linearly increasing computation time per drone, while the centralized aspects of FTSMC lead to quadratic growth in computational load. This demonstrates that the proposed method is suitable for large-scale UAV swarms.
8. Conclusion and Future Work
This article presented a hierarchical formation control strategy for quadrotor UAV swarms based on Stackelberg game theory. The approach designates one UAV drone as a leader that proactively plans and commits to its trajectory, guiding the entire formation. The follower UAV drones then engage in a distributed Nash equilibrium game to compute their optimal control inputs, balancing leader tracking and formation maintenance. An innovative disturbance compensator was integrated to enhance robustness against external disturbances. Comprehensive simulations validated the strategy’s effectiveness, showing rapid convergence, high precision, and strong disturbance rejection. Compared to fixed-time sliding mode control, the Stackelberg approach offered superior performance in terms of convergence speed, steady-state error, and control smoothness. The method is scalable and suitable for real-world applications where multiple UAV drones must operate in coordinated formations under dynamic conditions.
Future work will focus on extending the framework to handle more complex scenarios. First, we plan to incorporate obstacle avoidance constraints directly into the Stackelberg game cost functions, possibly using barrier functions or model predictive control. Second, we will investigate adaptive mechanisms for the weighting parameters to optimize performance under varying mission phases. Third, the communication topology could be made dynamic, allowing UAV drones to reconfigure links based on signal strength or mission requirements. Fourth, experimental validation with physical quadrotor platforms will be conducted to assess real-time implementation challenges. Lastly, we will explore multi-leader Stackelberg games for heterogeneous UAV swarms where different leaders may command sub-formations, enabling more complex mission architectures. These advancements will further solidify the role of game-theoretic control in the next generation of autonomous UAV systems.