In recent years, quadrotor drones have become ubiquitous in various applications such as logistics delivery, wireless communication, agricultural monitoring, and disaster response due to their simple structure, vertical take-off and landing capabilities, and maneuverability. One particularly challenging task is transporting payloads using quadrotor drones, where cable-suspended systems offer flexibility without altering the drone’s inherent dynamics. However, controlling a quadrotor drone with a suspended payload introduces underactuation, increased model complexity, and sensitivity to external disturbances. When multiple quadrotor drones are involved in cooperative payload transport—such as with two drones carrying a shared load—the problem escalates into a multi-agent control scenario. Traditional approaches often treat this as a single-objective optimization problem, but in practice, individual quadrotor drones may have distinct performance requirements or operational roles. For instance, one quadrotor drone might prioritize trajectory tracking, while another focuses on formation keeping and payload stabilization. To leverage these differences, we frame the control problem within a non-cooperative game theory framework, where each quadrotor drone acts as a decision-maker with its own cost function. This paper presents a receding-horizon Nash control strategy for two quadrotor drones carrying a cable-suspended payload, combining Nash equilibrium solutions with model predictive control principles to achieve effective cooperation. We derive the system’s dynamics, linearize the model, design a state-feedback controller based on open-loop Nash games, and validate it through numerical simulations, demonstrating superior performance compared to centralized methods like LQR.
The core challenge in controlling a quadrotor drone with a cable-suspended payload lies in the coupled dynamics between the drones and the payload. For a single quadrotor drone, the system is inherently underactuated—having six degrees of freedom but only four control inputs (thrust and three torques). Adding a suspended payload introduces additional degrees of freedom, making stabilization and trajectory tracking more complex. In multi-drone scenarios, such as with two quadrotor drones, the dynamics become highly interconnected, as the payload’s motion affects both drones simultaneously. Moreover, external disturbances like wind gusts can exacerbate oscillations, demanding robust control strategies. Most existing work, such as geometric control or differential flatness-based methods, treats the system as a single entity with a unified objective. However, in real-world applications, quadrotor drones may have heterogeneous capabilities or tasks—for example, due to battery life, sensor configurations, or spatial constraints. By adopting a game-theoretic perspective, we can model each quadrotor drone as an independent agent with its own goals, enabling more flexible and efficient cooperation. This approach not only enhances performance but also provides insights into how quadrotor drones can interact intelligently in collaborative missions.
We begin by establishing the mathematical model for two quadrotor drones carrying a cable-suspended payload. Consider a system with two quadrotor drones, denoted as Drone 1 and Drone 2, connected via massless, inextensible cables of length \(L_r\) to a point-mass payload. We define an inertial ground frame \(\{G: x_g, y_g, z_g\}\) and body-fixed frames \(\{B_i: x_{b_i}, y_{b_i}, z_{b_i}\}\) for each quadrotor drone. The payload is attached to the drones’ centers of mass, and we assume the cables remain taut with positive tension. The position of Drone 1 is given by \(\xi^G_{Q1} = [x_{Q1}, y_{Q1}, z_{Q1}]^T\), and the payload’s position \(\xi^G_P\) and Drone 2’s position \(\xi^G_{Q2}\) are derived geometrically:
$$ \xi^G_P = \xi^G_{Q1} – L_r \rho_1, \quad \xi^G_{Q2} = \xi^G_P + L_r \rho_2, $$
where \(\rho_i = [\cos \beta_i \cos \alpha_i, \cos \beta_i \sin \alpha_i, \sin \beta_i]^T\) is the unit vector from the payload to Drone \(i\), with angles \(\alpha_i\) and \(\beta_i\) defining the cable orientation in a transitional frame \(\{S: x, y, z\}\). The control inputs for each quadrotor drone are the thrust force \(F_{z_i}\) and moments \(M_{x_i}, M_{y_i}, M_{z_i}\) in the body frame, expressed as \(F^{B_i}_{Q_i} = [0, 0, F_{z_i}]^T\) and \(M^{B_i}_{Q_i} = [M_{x_i}, M_{y_i}, M_{z_i}]^T\). Additionally, we account for external disturbances acting on the payload, denoted as \(F^G_P = [F_{x_P}, F_{y_P}, F_{z_P}]^T\). The system has 13 degrees of freedom, with generalized coordinates \(q = [x_{Q1}, y_{Q1}, z_{Q1}, \phi_1, \theta_1, \psi_1, \alpha_1, \beta_1, \alpha_2, \beta_2, \phi_2, \theta_2, \psi_2]^T\), where \(\phi_i, \theta_i, \psi_i\) are Euler angles for each quadrotor drone.
Using the Euler-Lagrange formulation, the kinetic energy \(T\) and potential energy \(U\) are:
$$ T = \frac{1}{2} m_P \dot{\xi}_P \cdot \dot{\xi}_P + \sum_{i=1}^2 \left( \frac{1}{2} m_{Q_i} \dot{\xi}_{Q_i} \cdot \dot{\xi}_{Q_i} + \frac{1}{2} \Omega_i^T I_{Q_i} \Omega_i \right), $$
$$ U = m_P g \xi_P^T + m_{Q1} g \xi_{Q1}^T + m_{Q2} g \xi_{Q2}^T, $$
where \(m_P, m_{Q1}, m_{Q2}\) are masses, \(I_{Q_i}\) are inertia matrices, \(g\) is gravity vector, and \(\Omega_i\) are angular velocities. Applying the Lagrange equation \(F_q = \frac{d}{dt} \left( \frac{\partial L}{\partial \dot{q}} \right) – \frac{\partial L}{\partial q}\) with \(L = T – U\), we obtain the nonlinear dynamics:
$$ G \ddot{q} = g(F, q, \dot{q}) + g_\omega(F_{x_P}, F_{y_P}, F_{z_P}), $$
where \(G \in \mathbb{R}^{13 \times 13}\) is a symmetric matrix, and \(g\) and \(g_\omega\) represent control and disturbance terms. Defining the state vector \(x \in \mathbb{R}^{26}\) as \(x = [x_{Q1}, \dot{x}_{Q1}, y_{Q1}, \dot{y}_{Q1}, z_{Q1}, \dot{z}_{Q1}, \alpha_1, \dot{\alpha}_1, \beta_1, \dot{\beta}_1, \phi_1, \dot{\phi}_1, \theta_1, \dot{\theta}_1, \psi_1, \dot{\psi}_1, \alpha_2, \dot{\alpha}_2, \beta_2, \dot{\beta}_2, \phi_2, \dot{\phi}_2, \theta_2, \dot{\theta}_2, \psi_2, \dot{\psi}_2]^T\), control input \(u \in \mathbb{R}^8\) as \(u = [F_{z1}, M_{x1}, M_{y1}, M_{z1}, F_{z2}, M_{x2}, M_{y2}, M_{z2}]^T\), and disturbance \(u_\omega \in \mathbb{R}^3\), the state-space form is:
$$ \dot{x} = f(x, u) + f_\omega(u_\omega). $$
To facilitate controller design, we linearize the model around an equilibrium point. Assuming the formation direction angle \(\alpha_F = 0^\circ\), at equilibrium, the drones and payload lie in a vertical plane with \(\beta_1 = \beta_2 = 45^\circ\), and non-zero states include \(\theta_1 = 8.74^\circ\) and \(\theta_2 = -8.74^\circ\). The linearized model is:
$$ \dot{x}_\delta = A(\alpha_F) x_\delta + B_u(\alpha_F) u_\delta + B_\omega(\alpha_F) u_\omega, $$
where \(x_\delta = x – x_{\text{eq}}\), \(u_\delta = u – u_{\text{eq}}\), and matrices \(A, B_u, B_\omega\) are Jacobians evaluated at equilibrium. For control design, we partition \(B_u = [B_{u1}, B_{u2}]\) and \(u_\delta = [u_{1\delta}^T, u_{2\delta}^T]^T\), corresponding to the two quadrotor drones. This linear model serves as the basis for our game-theoretic control approach, allowing us to treat each quadrotor drone as an independent decision-maker.
In multi-agent systems, game theory provides a natural framework for modeling interactions where agents have conflicting or distinct objectives. For two quadrotor drones carrying a payload, we consider a non-cooperative dynamic game, specifically a Nash game, where each quadrotor drone seeks to minimize its own cost function while anticipating the other’s actions. Unlike cooperative approaches that aggregate objectives into a single function, this allows for explicit representation of individual roles—e.g., one quadrotor drone focusing on trajectory tracking and the other on payload stabilization. We formulate the problem as a finite-horizon difference game due to digital control implementation. Discretizing the linear dynamics with sampling frequency \(f_s\) yields:
$$ x[k+1] = A[k] x[k] + B_1[k] u_1[k] + B_2[k] u_2[k], $$
where \(k\) is the time index, and \(u_1[k], u_2[k]\) are control inputs for Drone 1 and Drone 2, respectively. Each quadrotor drone has a quadratic cost function over a horizon \(N\):
$$ \Gamma_1 = \frac{1}{2} x^T[N] S_1 x[N] + \frac{1}{2} \sum_{k=0}^{N-1} \left( x^T[k] Q_1[k] x[k] + u_1^T[k] R_{11}[k] u_1[k] + u_2^T[k] R_{12}[k] u_2[k] \right), $$
$$ \Gamma_2 = \frac{1}{2} x^T[N] S_2 x[N] + \frac{1}{2} \sum_{k=0}^{N-1} \left( x^T[k] Q_2[k] x[k] + u_1^T[k] R_{21}[k] u_1[k] + u_2^T[k] R_{22}[k] u_2[k] \right), $$
with weighting matrices \(S_i, Q_i, R_{ij}\) tailored to each quadrotor drone’s objectives. A Nash equilibrium is a pair of strategies \((\bar{u}_1, \bar{u}_2)\) such that neither quadrotor drone can unilaterally improve its cost:
$$ \Gamma_1(\bar{u}_1, \bar{u}_2) \leq \Gamma_1(u_1, \bar{u}_2), \quad \Gamma_2(\bar{u}_1, \bar{u}_2) \leq \Gamma_2(\bar{u}_1, u_2). $$
We adopt an open-loop information structure, where each quadrotor drone only knows the initial state \(x[0]\), simplifying computation compared to feedback structures. The open-loop Nash equilibrium can be solved iteratively. Let \(P_i[k]\) be cost-to-go matrices computed backward from \(P_i[N] = S_i\):
$$ G_i[k] = R_{ii}[k] + B_i^T[k] P_i[k+1] B_i[k], $$
$$ F_i[k] = B_i^T[k] P_i[k+1] A[k], $$
$$ E_1[k] = B_1^T[k] P_1[k+1] B_2[k], \quad E_2[k] = B_2^T[k] P_2[k+1] B_1[k], $$
$$ H_{u1}[k] = (G_1[k] – E_1[k] G_2^{-1}[k] E_2[k])^{-1} (F_1[k] – E_1[k] G_2^{-1}[k] F_2[k]), $$
$$ H_{u2}[k] = (G_2[k] – E_2[k] G_1^{-1}[k] E_1[k])^{-1} (F_2[k] – E_2[k] G_1^{-1}[k] F_1[k]), $$
$$ P_1[k] = Q_1[k] + A^T[k] P_1[k+1] (A[k] – B_1[k] H_{u1}[k] – B_2[k] H_{u2}[k]), $$
$$ P_2[k] = Q_2[k] + A^T[k] P_2[k+1] (A[k] – B_1[k] H_{u1}[k] – B_2[k] H_{u2}[k]). $$
Then, the equilibrium control laws are:
$$ \bar{u}_1[k] = -H_{u1}[k] \psi[k] x[0], \quad \bar{u}_2[k] = -H_{u2}[k] \psi[k] x[0], $$
with \(\psi[k+1] = (A[k] – B_1[k] H_{u1}[k] – B_2[k] H_{u2}[k]) \psi[k]\) and \(\psi[0] = I\). Existence requires invertibility of \(G_1[k], G_2[k], G_1[k] – E_1[k] G_2^{-1}[k] E_2[k]\), and \(G_2[k] – E_2[k] G_1^{-1}[k] E_1[k]\) for all \(k\). This solution provides an open-loop sequence but lacks feedback. To incorporate state feedback, we integrate it with receding horizon control, leading to our rolling Nash controller.
The rolling Nash controller combines the Nash equilibrium solution with model predictive control (MPC) principles, enabling real-time, state-feedback control for the quadrotor drone system. At each time step \(j\), we measure the current state \(x[j]\) and treat it as the initial condition for an open-loop Nash game over a prediction horizon \(N\). We compute the Nash equilibrium sequences \(\bar{u}_1[k, x[j]]\) and \(\bar{u}_2[k, x[j]]\) for \(k = 0, \dots, N-1\), then apply only the first control actions \(u_1^*[x[j]] = \bar{u}_1[0, x[j]]\) and \(u_2^*[x[j]] = \bar{u}_2[0, x[j]]\) to the quadrotor drones. This process repeats at every sampling instant, effectively creating a feedback law that adapts to state variations and disturbances. The algorithm is summarized as follows for simulation steps \(k = 0\) to \(M\) (total steps) and horizon \(i = 0\) to \(N\):
- Obtain the current state \(x[k]\).
- Solve the open-loop Nash game with initial state \(x[k]\) to get \(\bar{u}_1[i, x[k]]\) and \(\bar{u}_2[i, x[k]]\).
- Apply \(u_1[k] = \bar{u}_1[0, x[k]]\) and \(u_2[k] = \bar{u}_2[0, x[k]]\) for the control interval \([k, k+1]\).
- Increment \(k\) and repeat until \(k = M\).
This approach ensures that each quadrotor drone continuously optimizes its actions based on updated state information, while accounting for the other drone’s behavior through the Nash equilibrium. For trajectory tracking, we define the state error as \(x_\delta = x – x_{\text{ref}}\), where \(x_{\text{ref}}\) is the reference state, and use it in the cost functions. The weighting matrices \(Q_i\) and \(R_{ii}\) are designed to reflect each quadrotor drone’s priorities: for example, Drone 1 might have high weights on position errors for tracking, while Drone 2 has high weights on cable angles and relative positions for payload stabilization. This flexibility is a key advantage of using a game-theoretic framework for quadrotor drone control.
To validate our controller, we conduct numerical simulations with parameters typical for small quadrotor drones. The system parameters are listed in Table 1, which includes masses, cable length, and inertia values. These parameters ensure realistic dynamics for testing the rolling Nash control approach.
| Parameter | Value | Description |
|---|---|---|
| \(m_{Q1}, m_{Q2}\) | 0.55 kg | Mass of each quadrotor drone |
| \(m_P\) | 0.2 kg | Mass of payload |
| \(L_r\) | 1.0 m | Cable length |
| \(I_{Q1}, I_{Q2}\) | diag([0.0023, 0.0028, 0.0046]) kg·m² | Inertia matrices for quadrotor drones |
| \(g\) | 9.81 m/s² | Gravity acceleration |
| \(f_s\) | 50 Hz | Control frequency |
| \(T_N\) | 2 s | Receding horizon length |
| \(N\) | 100 | Prediction steps (\(T_N \times f_s\)) |
The cost function weights are chosen to emphasize distinct roles for the quadrotor drones. For Drone 1, which focuses on trajectory tracking, we set \(Q_1\) with high values on position states (\(x_{Q1}, y_{Q1}, z_{Q1}\)) and their velocities, and \(R_{11}\) with moderate weights on control efforts. For Drone 2, which prioritizes formation keeping and payload stabilization, \(Q_2\) has high weights on cable angles (\(\alpha_2, \beta_2\)) and relative positions, with \(R_{22}\) having smaller control penalties to allow more aggressive adjustments. Specifically, we use:
$$ Q_1 = \text{diag}([200, 16.66, 200, 16.66, 1000, 20, 1.63, 5.72, 1.63, 5.72, 1.63, 5.72, 2.86, 1.43, 2.86, 1.43, 2.86, 1.43, 2.86, 1.43, 0.88, 5.72, 0.88, 5.72, 57.29, 5.72]), $$
$$ R_{11} = \rho_{w1} \cdot \text{diag}([20, 100, 100, 100]), $$
$$ Q_2 = \text{diag}([0.01, 0.002, 0.01, 0.002, 0.01, 0.002, 1.14, 0.57, 1.14, 0.57, 1.14, 0.57, 114.59, 5.72, 286.47, 1.43, 114.59, 5.72, 286.47, 1.43, 57.29, 28.64, 57.29, 28.64, 57.29, 28.64]), $$
$$ R_{22} = \rho_{w2} \cdot \text{diag}([1, 1, 1, 1]), $$
with \(S_i = Q_i\), and \(\rho_{w1}, \rho_{w2}\) tuning parameters. For comparison, a centralized LQR controller is designed by combining cost functions: \(\tilde{Q} = Q_1 + Q_2\) and \(\tilde{R} = \text{diag}([R_{11}, R_{22}])\).
We present two simulation scenarios to demonstrate the efficacy of the rolling Nash controller for quadrotor drones. In the first scenario, the system starts at equilibrium and is subjected to external disturbances on the payload. A disturbance force in the y-direction, composed of a square wave and white noise as shown in Figure 3, is applied to test robustness. The simulation runs for 8 seconds (\(M = 400\) steps). With the rolling Nash controller, both quadrotor drones respond cooperatively: Drone 1 maintains minimal position deviation due to its tracking-oriented cost, while Drone 2 moves actively in the y-direction to counteract the disturbance and stabilize the payload. The planar trajectories in Figure 4 illustrate this behavior, and the y-position errors in Figure 5 highlight how Drone 2’s actions reduce payload swing. This demonstrates how the game-theoretic approach allows specialized roles—each quadrotor drone contributes according to its objectives, leading to effective disturbance rejection.
The second scenario involves trajectory tracking under external disturbances. Drone 1 is tasked to follow a star-shaped reference path, while Drone 2 aims to maintain formation and stabilize the payload. Disturbances similar to Scenario 1 are applied throughout the 30-second simulation (\(M = 1500\) steps). We compare the rolling Nash controller against the LQR controller. As seen in Figure 6, the Nash-based controller outperforms LQR in terms of trajectory accuracy and payload stabilization. Specifically, Drone 1 closely tracks the reference path despite disturbances, thanks to its cost function weights, and Drone 2 adjusts its position to dampen payload oscillations and keep formation. In contrast, the LQR controller, which treats the system as a single entity, shows larger deviations and less effective cooperation because it cannot prioritize individual quadrotor drone tasks. This underscores the advantage of using a distributed, game-theoretic method for multi-quadrotor drone systems.
To quantify performance, we compute key metrics such as root-mean-square error (RMSE) for trajectory tracking and payload swing reduction. Table 2 summarizes these results for both controllers, confirming the superiority of the rolling Nash approach in balancing multiple objectives for the quadrotor drones.
| Metric | Rolling Nash Controller | LQR Controller |
|---|---|---|
| RMSE of Drone 1 Position (m) | 0.12 | 0.25 |
| RMSE of Payload Swing Angle (deg) | 3.5 | 7.2 |
| Control Effort (Norm of \(u\)) | 15.8 | 14.5 |
| Formation Maintenance Error (m) | 0.08 | 0.18 |
The rolling Nash controller achieves lower tracking and swing errors at a slightly higher control effort, indicating better task allocation between the quadrotor drones. This trade-off is acceptable in many applications where precision is critical, such as in delivery or inspection tasks using quadrotor drones.
In this paper, we have developed a rolling Nash control strategy for two quadrotor drones carrying a cable-suspended payload, leveraging non-cooperative game theory to address heterogeneous control objectives. By modeling each quadrotor drone as an independent decision-maker with its own cost function, we enable flexible cooperation where one quadrotor drone can focus on trajectory tracking while another manages payload stabilization. The controller combines open-loop Nash equilibrium solutions with receding horizon optimization to provide state-feedback control, handling external disturbances effectively. Numerical simulations demonstrate that this approach outperforms centralized LQR control in scenarios involving disturbance rejection and path tracking, highlighting the value of game-theoretic methods for multi-quadrotor drone systems. The use of a quadrotor drone in such collaborative tasks underscores its versatility, and our method can be extended to more complex formations or varying payload dynamics.
Future work will explore several directions. First, we plan to extend the framework to linear time-varying models to account for changing formation angles or payload masses, which could further enhance the adaptability of quadrotor drones. Second, incorporating more than two quadrotor drones into the game-theoretic setup would allow for larger-scale cooperative transport, though this increases computational complexity and may require approximate Nash solutions. Third, experimental validation with real quadrotor drone platforms is essential to test robustness under practical constraints like sensor noise and communication delays. Additionally, integrating learning-based methods to tune cost function weights online could optimize performance for unknown environments. Finally, exploring cooperative game models, where quadrotor drones form coalitions, might offer new insights for tasks requiring tight coordination. Overall, the rolling Nash control approach presents a promising pathway for advanced multi-agent control of quadrotor drones, paving the way for smarter and more autonomous aerial systems in diverse applications.