In recent years, the autonomous flight of unmanned aerial vehicles (UAVs) has emerged as a critical research focus in the aerospace domain. Among these, vertical take-off and landing (VTOL) aircraft, particularly the quadrotor drone, have garnered significant attention due to their adaptability in confined spaces and versatile applications in both military and civilian sectors. A quadrotor drone is an underactuated system with six degrees of freedom and four inputs, characterized by its symmetric cross-shaped structure with four rotors positioned at each end. By merely adjusting the rotational speeds of these rotors, the quadrotor drone can achieve various flight maneuvers, including hover, vertical ascent or descent, pitch, roll, and yaw motions. This paper, from my perspective as a researcher in aerial robotics, delves into the comprehensive modeling and control simulation of a small-scale quadrotor drone. I aim to enhance flight performance and control stability by establishing a nonlinear dynamic model based on Newton-Euler equations and designing a decoupled PID control system for simulation in Matlab/Simulink. The quadrotor drone’s unique mechanical configuration and flight principles make it an ideal platform for studying advanced control strategies, and this work lays a foundation for further research in autonomous flight control of quadrotor drones.
The flight principles of a quadrotor drone are rooted in its symmetric design and rotor dynamics. In a standard configuration, the quadrotor drone features four rotors: rotors 1 and 3 rotate counterclockwise, while rotors 2 and 4 rotate clockwise, which cancels out reactive torques and eliminates the need for a tail rotor. This arrangement simplifies control, as attitude adjustments are achieved by varying rotor speeds. For instance, during hover, all rotors spin at equal speeds to generate a total lift force balancing gravity. Vertical motion is controlled by uniformly increasing or decreasing all rotor speeds. Pitch motion (forward or backward tilt) is induced by differentially changing the speeds of front and rear rotors, while roll motion (sideways tilt) involves adjusting left and right rotor speeds. Yaw motion (rotation about the vertical axis) results from increasing the speed of one pair of rotors and decreasing the other, creating a net torque. This inherent coupling between rotor inputs and flight states poses challenges for control, necessitating a detailed dynamic model of the quadrotor drone. Understanding these principles is crucial for developing effective control algorithms for the quadrotor drone, as it operates in a highly nonlinear and interdependent environment.
To model the quadrotor drone, I begin with kinematic and dynamic analyses under specific assumptions, treating the quadrotor drone as a rigid body with uniform symmetry. I assume the body-fixed coordinate origin coincides with the center of mass, the propellers are rigid, and the ground frame is inertial. The forces and moments acting on the quadrotor drone include rotor thrusts, aerodynamic drag, and gravity. Using aerodynamics, the thrust \(T_i\), drag force \(f\), torque \(M\), and drag moment \(\tau\) for each rotor are expressed as:
$$ T_i = K_T \Omega_i^2, \quad f = K_f \vec{v}, \quad M = K_M \Omega^2, \quad \tau = K_\tau \vec{\omega} $$
where \(K_T\) is the thrust coefficient, \(\Omega_i\) is the rotational speed of rotor \(i\), \(K_f\) is the drag coefficient, \(\vec{v}\) is linear velocity, \(K_M\) is the torque coefficient, \(K_\tau\) is the drag moment coefficient, and \(\vec{\omega}\) is angular velocity. For the quadrotor drone, the total lift force in the body frame \(F_B\) is:
$$ F_B = [0, 0, U_1]^T, \quad U_1 = \sum_{i=1}^4 T_i = K_T (\Omega_1^2 + \Omega_2^2 + \Omega_3^2 + \Omega_4^2) $$
Transforming this to the ground frame \(E\) using rotation matrix \(R\) yields:
$$ R = \begin{bmatrix} C_\theta C_\psi & S_\phi S_\theta C_\psi – C_\phi S_\psi & C_\phi S_\theta C_\psi + S_\phi S_\psi \\ C_\theta S_\psi & S_\phi S_\theta S_\psi + C_\phi C_\psi & C_\phi S_\theta S_\psi – S_\phi C_\psi \\ -S_\theta & S_\phi C_\theta & C_\phi C_\theta \end{bmatrix} $$
where \(C\) and \(S\) denote \(\cos\) and \(\sin\), respectively, and \(\phi\), \(\theta\), \(\psi\) are roll, pitch, and yaw angles. Thus, the force in the ground frame is:
$$ F_E = R \cdot F_B = \begin{bmatrix} (C_\phi S_\theta C_\psi + S_\phi S_\psi) U_1 \\ (C_\phi S_\theta S_\psi – S_\phi C_\psi) U_1 \\ C_\phi C_\theta U_1 \end{bmatrix} $$
The drag force \(f_E\) and gravity \(G_E\) are:
$$ f_E = \begin{bmatrix} K_{fX} \dot{x} \\ K_{fY} \dot{y} \\ K_{fZ} \dot{z} \end{bmatrix}, \quad G_E = [0, 0, mg]^T $$
Applying Newton’s second law, the translational dynamics of the quadrotor drone are:
$$ m \begin{bmatrix} \ddot{x} \\ \ddot{y} \\ \ddot{z} \end{bmatrix} = F_E – f_E – G_E $$
For rotational dynamics, I consider moments about the body axes. Due to symmetry, products of inertia are negligible. The moment balance equations are:
$$ I_x \dot{p} = U_2 L + (I_y – I_z) q r – \tau_x, \quad I_y \dot{q} = U_3 L + (I_z – I_x) p r – \tau_y, \quad I_z \dot{r} = U_4 + (I_x – I_y) p q – \tau_z $$
where \(I_x, I_y, I_z\) are moments of inertia, \(p, q, r\) are angular velocities in the body frame, \(L\) is the distance from rotor to center of mass, and \(\tau_x, \tau_y, \tau_z\) are drag moments. The control inputs \(U_2, U_3, U_4\) are defined as:
$$ U_2 = (T_4 – T_2)L, \quad U_3 = (T_3 – T_1)L, \quad U_4 = K_M (\Omega_2^2 + \Omega_4^2 – \Omega_1^2 – \Omega_3^2) $$
The relationship between body-frame angular velocities and Euler angle rates is:
$$ \begin{bmatrix} p \\ q \\ r \end{bmatrix} = \begin{bmatrix} 1 & 0 & -S_\theta \\ 0 & C_\phi & S_\phi C_\theta \\ 0 & -S_\phi & C_\phi C_\theta \end{bmatrix} \begin{bmatrix} \dot{\phi} \\ \dot{\theta} \\ \dot{\psi} \end{bmatrix} $$
Combining these, I derive the complete nonlinear dynamic model for the quadrotor drone:
$$ \begin{aligned} \ddot{x} &= \frac{1}{m} [(C_\phi S_\theta C_\psi + S_\phi S_\psi) U_1 – K_{fX} \dot{x}] \\ \ddot{y} &= \frac{1}{m} [(C_\phi S_\theta S_\psi – S_\phi C_\psi) U_1 – K_{fY} \dot{y}] \\ \ddot{z} &= \frac{1}{m} [C_\phi C_\theta U_1 – K_{fZ} \dot{z} – mg] \\ \ddot{\phi} &= \frac{1}{I_x} [U_2 L + (I_y – I_z) \dot{\theta} \dot{\psi} – K_{\tau X} \dot{\phi}] \\ \ddot{\theta} &= \frac{1}{I_y} [U_3 L + (I_z – I_x) \dot{\phi} \dot{\psi} – K_{\tau Y} \dot{\theta}] \\ \ddot{\psi} &= \frac{1}{I_z} [U_4 + (I_x – I_y) \dot{\phi} \dot{\theta} – K_{\tau Z} \dot{\psi}] \end{aligned} $$
This model captures the nonlinearities and couplings inherent in the quadrotor drone system. To facilitate control design, I simplify the model under small-angle approximations, assuming \(\phi, \theta, \psi\) are small, so \(S_\phi \approx \phi, C_\phi \approx 1\), and cross terms are negligible. Thus, the simplified model becomes:
$$ \begin{aligned} \ddot{x} &\approx \frac{1}{m} [(\theta C_\psi + \phi S_\psi) U_1 – K_{fX} \dot{x}] \\ \ddot{y} &\approx \frac{1}{m} [(\theta S_\psi – \phi C_\psi) U_1 – K_{fY} \dot{y}] \\ \ddot{z} &\approx \frac{1}{m} [U_1 – K_{fZ} \dot{z} – mg] \\ \ddot{\phi} &\approx \frac{1}{I_x} [U_2 L – K_{\tau X} \dot{\phi}] \\ \ddot{\theta} &\approx \frac{1}{I_y} [U_3 L – K_{\tau Y} \dot{\theta}] \\ \ddot{\psi} &\approx \frac{1}{I_z} [U_4 – K_{\tau Z} \dot{\psi}] \end{aligned} $$
For control purposes, I linearize around hover conditions, where \(U_1 \approx mg\) and angles are near zero. Treating velocity-dependent drag as linear perturbations, I formulate a state-space representation for the quadrotor drone. Defining the state vector \(X = [\dot{x}, \dot{y}, \dot{z}, \dot{\phi}, \dot{\theta}, \dot{\psi}, g, x, y, z, \phi, \theta, \psi]^T\), input vector \(U = [U_1, U_2, U_3, U_4]^T\), and output vector \(Y = [z, \phi, \theta, \psi]^T\), the state-space equations are:
$$ \dot{X} = A X + B U, \quad Y = C X + D U $$
where matrices \(A, B, C, D\) are derived from the linearized dynamics. From this, I obtain transfer functions for altitude and attitude channels of the quadrotor drone:
$$ G_z(s) = \frac{\cos\phi \cos\theta}{m s (s + K_{fz})}, \quad G_\phi(s) = \frac{1}{I_x s (s + K_{\tau x})}, \quad G_\theta(s) = \frac{1}{I_y s (s + K_{\tau y})}, \quad G_\psi(s) = \frac{1}{I_z s (s + K_{\tau z})} $$
These transfer functions guide the design of a decoupled PID control system for the quadrotor drone. PID control is chosen for its simplicity, reliability, and effectiveness in tracking desired trajectories. I design separate PID controllers for altitude, roll, pitch, and yaw channels, with the control structure shown in the block diagram. The parameters for the quadrotor drone are based on a typical small-scale model, as summarized in Table 1.
| Parameter | Unit | Value |
|---|---|---|
| Mass (\(m\)) | kg | 0.75 |
| Gravity (\(g\)) | m/s² | 9.81 |
| Arm Length (\(L\)) | m | 0.258 |
| Thrust Coefficient (\(K_T\)) | N/(rad/s) | 3.179 × 10⁻⁵ |
| Torque Coefficient (\(K_M\)) | N·m/(rad/s) | 7.932 × 10⁻⁷ |
| Moment of Inertia (\(I_x\)) | N·m/(rad/s²) | 1.9688 × 10⁻² |
| Moment of Inertia (\(I_y\)) | N·m/(rad/s²) | 1.9688 × 10⁻² |
| Moment of Inertia (\(I_z\)) | N·m/(rad/s²) | 3.9388 × 10⁻² |
| Drag Coefficient (\(K_{fX}\)) | N/(m/s) | 0.056 |
| Drag Coefficient (\(K_{fY}\)) | N/(m/s) | 0.056 |
| Drag Coefficient (\(K_{fZ}\)) | N/(m/s) | 0.078 |
| Drag Moment Coefficient (\(K_{\tau X}\)) | N·m/(rad/s) | 0.0059 |
| Drag Moment Coefficient (\(K_{\tau Y}\)) | N·m/(rad/s) | 0.0059 |
| Drag Moment Coefficient (\(K_{\tau Z}\)) | N·m/(rad/s) | 0.0228 |
I implement the quadrotor drone model and PID controllers in Matlab/Simulink. The simulation focuses on hover mode, where the quadrotor drone maintains a constant altitude and orientation. Using trial-and-error tuning, I adjust PID gains to minimize overshoot and settling time. The tuned parameters for the quadrotor drone are listed in Table 2.
| Channel | Proportional (P) | Integral (I) | Derivative (D) |
|---|---|---|---|
| Altitude | 3 | 0.01 | 1 |
| Roll | 1.5 | 0.01 | 0.1 |
| Pitch | 2 | 0.1 | 0.1 |
| Yaw | 0.9 | 0.03 | 0.1 |
The simulation results for unit step responses in altitude and attitude angles are analyzed. For altitude control, the quadrotor drone achieves steady state with minimal overshoot, though it exhibits some oscillation during transition. The settling time is approximately 1 second, which is acceptable for stable hover of the quadrotor drone. Attitude angles (roll, pitch, yaw) converge rapidly within 0.3 seconds, with negligible steady-state error. These outcomes demonstrate the effectiveness of the PID control in stabilizing the quadrotor drone. Compared to prior studies, this approach reduces overshoot and adjustment time, enhancing the quadrotor drone’s performance. The nonlinear model accurately represents the dynamics, and the decoupled control strategy simplifies implementation for the quadrotor drone.
To further elaborate on the quadrotor drone’s behavior, I consider additional aspects such as environmental disturbances and sensor noise. In real-world scenarios, a quadrotor drone faces wind gusts, model uncertainties, and measurement errors. Robust control techniques, like adaptive PID or sliding mode control, could be integrated to improve the quadrotor drone’s resilience. Moreover, the quadrotor drone’s energy efficiency and payload capacity are critical for practical applications. Future work may involve optimizing rotor designs or incorporating machine learning for autonomous navigation of the quadrotor drone. The scalability of this model allows extensions to swarm coordination, where multiple quadrotor drones operate collaboratively. Simulation tools like Matlab/Simulink provide a versatile platform for testing advanced algorithms before real-world deployment of the quadrotor drone.
In conclusion, this paper presents a thorough modeling and control simulation of a quadrotor drone. I derived a nonlinear dynamic model using Newton-Euler equations, simplified it for control design, and implemented a decoupled PID control system. Simulation results confirm that the quadrotor drone can achieve stable hover with precise altitude and attitude regulation. The model serves as a foundation for further research on the quadrotor drone, including robust control, trajectory tracking, and autonomous operations. As quadrotor drones continue to evolve, advancements in modeling and control will unlock new potentials in areas like surveillance, delivery, and environmental monitoring. The quadrotor drone’s versatility ensures its prominence in the future of unmanned aerial systems.