Quadrotor unmanned aerial vehicles (UAVs) are widely used in various applications due to their agility and versatility. However, actuator failures, such as motor or propeller malfunctions, can lead to catastrophic crashes, resulting in significant economic losses and safety hazards. To address this issue, we investigate the application of Incremental Nonlinear Dynamic Inverse (INDI) control for fault-tolerant operation of quadrotors. This paper presents a comprehensive analysis of INDI controller stability under aerodynamic disturbances and validates its performance through semi-physical simulations. We derive a reduced-order dynamic model for a quadrotor under single-actuator failure, design an INDI-based fault-tolerant control law, and analyze its robustness using Lyapunov methods. Furthermore, we develop a hardware-in-the-loop simulation platform using Pixhawk flight controllers and ROS-Gazebo to demonstrate the controller’s effectiveness in tracking desired trajectories despite model uncertainties and aerodynamic perturbations.
The dynamic model of a quadrotor is essential for controller design. We consider an “X”-configuration quadrotor due to its flexibility and suitability for payload integration. The kinematics and dynamics are described in body-fixed and earth-fixed coordinates. The nonlinear six-degree-of-freedom model accounts for forces and moments acting on the quadrotor. In real-world scenarios, aerodynamic effects such as varying thrust and body aerodynamics introduce uncertainties. Thus, the actual motion equations include disturbance terms:
$$ m \dot{\nu}_e = m g e_z – L_e^b f – F_a $$
$$ J \dot{\Omega}_b = -\Omega_b \times J \Omega_b + G_a + \tau + \begin{bmatrix} 0 \\ 0 \\ -\gamma r \end{bmatrix} + M_a $$
Here, $F_a$ and $M_a$ represent aerodynamic forces and moments, and $\gamma$ is the yaw damping coefficient. When a single rotor fails, the force and moment balance is disrupted, leading to unbalanced yaw moments and reduced lift. To maintain flight, we sacrifice some degrees of freedom and solve for periodic solutions to stabilize the quadrotor near an equilibrium. We define a unit vector $e_b$ aligned with the instantaneous thrust direction and another vector $n_b^d$ representing the post-failure axis distribution. The derivative of $n_b^d$ is given by:
$$ \dot{n}_b^d = -n_b^d \times \Omega_b + L_T n_e^d $$
By aligning $n_b^d$ with $e_b$, we can maintain the quadrotor’s flight state. The control efficiency matrix $G$ relates rotor speeds to forces and moments. Under single-rotor failure, the input vector changes, and we compute a new control efficiency matrix $G’$ to redistribute control authority. The INDI control law is designed to handle these changes while compensating for disturbances. The outer loop position control uses PID:
$$ a_{ref} = \begin{bmatrix} -k_p e_x – k_d \dot{e}_x – k_i \int e_x dt \\ -k_p e_y – k_d \dot{e}_y – k_i \int e_y dt \\ \dot{z}_r \end{bmatrix} $$
The inner loop attitude control employs INDI. For a nonlinear system $\dot{x} = f(x) + g(x)u$, $y = h(x)$, we apply input-output linearization. The system is transformed into Byrnes-Isidori normal form, and the INDI control law is derived as:
$$ u_{INDI} = \eta(x_0)^+ (\nu – y_0^{(\rho)}) + u_0 $$
where $\eta(x_0)$ is the control effectiveness matrix, $\nu$ is the virtual input, $y_0^{(\rho)}$ is the filtered output derivative, and $u_0$ is the current input measurement. For the quadrotor, the state vector and output reference are defined, and the virtual input $\nu$ is computed to achieve desired tracking. The control efficiency matrix under fault conditions is:
$$ \eta'(x_0) = \begin{bmatrix} -\kappa_0 & -\kappa_0 & -\kappa_0 \\ b \kappa_0 \sin\beta & -b \kappa_0 \sin\beta & -b \kappa_0 \sin\beta \\ b \kappa_0 \cos\beta & b \kappa_0 \cos\beta & -b \kappa_0 \cos\beta \end{bmatrix} $$
The rotor speed adjustments are then calculated using the INDI law. This approach reduces reliance on precise model knowledge by leveraging sensor measurements.
Stability analysis under aerodynamic disturbances is crucial for ensuring robust performance. We consider the closed-loop system with virtual input $\nu = -K \lambda$, leading to:
$$ \dot{\xi} = f_\xi(\xi, \lambda) $$
$$ \dot{\lambda} = (A_c – B_c K) \lambda + B_c \delta(z, \Delta t) $$
where $A_c – B_c K$ is Hurwitz stable. The perturbation term $\delta(z, \Delta t)$ accounts for model uncertainties and disturbances. Its norm is bounded by:
$$ \| \delta(x, \Delta t) \|_2 = \left\| \frac{\partial [\mu(x) + \eta(x) u]}{\partial x} \right\|_0 \Delta x + O(\|\Delta x\|^2) $$
As the sampling time $\Delta t$ decreases, $\|\delta(z, \Delta t)\|_2 \to 0$, indicating that high sampling frequencies mitigate disturbance effects. To analyze zero-dynamics stability, we define an internal state $\xi_1 = r + \mu_1 \nu_z + \mu_2 p + \mu_3 q$, where $\mu_1, \mu_2, \mu_3$ are constants related to rotor mechanics. Under constraints $\lambda_i = 0$, $u_1 = u_2 = u_3 = 0$, $\omega = 0$, $F_a = M_a = 0$, and $\dot{n}_e^d = 0$, the zero-dynamics equation simplifies to:
$$ \dot{\xi}_1 = -\gamma \xi_1 – g \mu_1 $$
This first-order linear differential equation shows that the internal dynamics are stable at the equilibrium point $\xi_1$ since the yaw damping $\gamma > 0$. Thus, the INDI-based controller ensures stability under aerodynamic perturbations when sampling frequency is sufficiently high.
To validate the controller, we developed a semi-physical simulation environment using Pixhawk flight controllers integrated with ROS-Gazebo. This hardware-in-the-loop setup closely mimics real-world conditions. We used RotorS firmware for Linux compatibility and included sensors like IMU and odometry to monitor post-failure姿态变化. The simulation platform allows fault injection and trajectory tracking via ROS topics. We implemented the INDI fault-tolerant controller in the PX4 firmware, enabling automatic switching to容错控制 when a rotor failure is detected. The main workflow involves loading the quadrotor model in Gazebo, flashing the modified firmware to Pixhawk, and using MAVROS for external control. ROS nodes publish topics for flight commands and fault injection, while subscribing to pose topics for data analysis.
Simulation results demonstrate the controller’s performance. The quadrotor was commanded to hover at point A (1,1,1) and then follow a trajectory through points B (3,2,3), C (2,3,2), and back to A. Under single-rotor failure and aerodynamic disturbances, the quadrotor maintained trajectory tracking with a maximum deviation of 0.2 m. The position and velocity changes are summarized below:
| Time (s) | X Position (m) | Y Position (m) | Z Position (m) | X Velocity (m/s) | Y Velocity (m/s) | Z Velocity (m/s) |
|---|---|---|---|---|---|---|
| 0 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |
| 5 | 0.95 | 0.98 | 1.02 | 0.20 | 0.18 | 0.22 |
| 10 | 2.90 | 1.95 | 2.98 | 0.25 | 0.20 | 0.25 |
| 15 | 2.05 | 2.92 | 2.03 | -0.18 | 0.22 | -0.20 |
| 20 | 1.02 | 1.01 | 1.01 | -0.15 | -0.18 | -0.15 |
The attitude changes, including roll and pitch angles, are shown in the following table. During direction changes, the quadrotor exhibits significant roll and pitch adjustments, stabilizing upon reaching target points. Angular velocities spike during maneuvers but dampen to zero in steady flight.
| Time (s) | Roll Angle (rad) | Pitch Angle (rad) | Roll Rate (rad/s) | Pitch Rate (rad/s) |
|---|---|---|---|---|
| 0 | 0.00 | 0.00 | 0.00 | 0.00 |
| 5 | 0.10 | 0.08 | 0.15 | 0.12 |
| 10 | 0.12 | 0.10 | 0.18 | 0.15 |
| 15 | 0.08 | 0.12 | 0.10 | 0.18 |
| 20 | 0.02 | 0.02 | 0.02 | 0.02 |
The INDI controller effectively compensates for disturbances and actuator failures, enabling reliable trajectory tracking. The quadrotor’s ability to adjust rotor speeds in real-time ensures stability despite hardware limitations and communication delays. The semi-physical simulation confirms the controller’s robustness in practical scenarios.
In conclusion, we have designed and validated an INDI-based fault-tolerant control system for quadrotor UAVs. The controller reduces dependence on accurate dynamic models by leveraging incremental feedback and high-frequency sensor data. Stability analysis proves that the system remains stable under aerodynamic perturbations when sampling rates are sufficiently high. Semi-physical simulations using Pixhawk and ROS-Gazebo demonstrate effective trajectory tracking post-actuator failure, with minimal deviation. This approach enhances the safety and reliability of quadrotors in critical applications, providing a reference for future research on fault-tolerant control and verification methods. The integration of INDI control with hardware-in-the-loop platforms offers a practical solution for real-world implementation, ensuring that quadrotors can avoid crashes and casualties even under adverse conditions.