The advent of modern control and electronic technologies has propelled the field of unmanned aerial vehicles (UAVs) to the forefront of research and application. Among the diverse UAV configurations, the fully autonomous, dual-engine, oil-powered quadrotor drone represents a novel class of small-scale aerial platforms. This configuration combines the inherent agility and maneuverability of multi-rotor systems with the extended endurance often associated with hydrocarbon fuels, enabling missions such as low-speed cruising, precision hovering, and operations in environments where electric power may be limiting. Consequently, these quadrotor drones have garnered significant interest for applications in agricultural plant protection, power line inspection, and geological surveying.
Autonomous landing constitutes a critical phase in the operational lifecycle of any UAV, serving as the prerequisite for safe recovery and directly impacting system reliability. For a quadrotor drone, a successful landing requires precise tracking of a prescribed trajectory to ensure a stable and accurate touchdown at the desired point. However, the dynamics of a quadrotor system are characteristically underactuated, highly nonlinear, and strongly coupled. These intrinsic properties, combined with susceptibility to external disturbances—such as wind gusts, ground effects, and sensor noise during the landing phase—pose significant challenges to achieving high-precision trajectory tracking. Therefore, designing a robust tracking control strategy with inherent disturbance rejection capabilities is paramount to enabling the quadrotor drone to perform high-quality landing maneuvers and complete its missions reliably.
Traditional approaches to this problem have often relied on state feedback controllers augmented with integral action to eliminate steady-state error. While effective in ideal conditions, such methods can exhibit significant drawbacks during the critical initial approach phase of quadrotor drone landing. Their limited bandwidth and inherent lag in responding to aggressive disturbances can lead to substantial tracking jitter, overshoot, or even instability, resulting in unsatisfactory landing performance. The controller’s performance is tightly coupled with the accuracy of the dynamic model, and any unmodeled dynamics or parameter uncertainties can severely degrade tracking accuracy.
To overcome the limitations imposed by the quadrotor drone’s nonlinearities, couplings, and sensitivity to disturbances, this article investigates the application of a Linear Active Disturbance Rejection Control (LADRC) strategy for landing trajectory tracking. The LADRC framework is particularly well-suited for this application as it does not require an exact analytical model of the system. Instead, it treats the aggregate effect of model uncertainties, nonlinearities, and external disturbances as a “total disturbance,” which is estimated in real-time by an extended state observer (ESO) and actively canceled within the control law. This approach inherently enhances the system’s robustness. The effectiveness of the proposed LADRC-based tracking method is subsequently validated through comprehensive simulation studies, demonstrating its superior performance in terms of tracking accuracy and disturbance rejection compared to traditional strategies.
Landing Phase Analysis and System Modeling for the Quadrotor Drone
Landing Procedure and Requirements
The autonomous landing sequence for a quadrotor drone can be strategically divided into three consecutive phases: the initial approach, the glideslope descent, and the flare. Each phase has distinct objectives and operational constraints to ensure a smooth transition to touchdown.
During the initial approach phase, the quadrotor drone, typically positioned several kilometers from the landing zone, initiates its landing procedure. It adjusts its cruising altitude to a range between 400m and 1500m and enters a height-holding state, stabilizing its position before committing to the final descent path.
The glideslope descent phase begins as the quadrotor drone aligns itself with the desired landing path. In this phase, the vehicle follows a nearly straight-line trajectory toward the touchdown point. The vertical descent rate is carefully controlled within a range of -2 m/s to -3 m/s, while the flight path angle (or glideslope angle) is maintained between -2° and -3° relative to the horizontal plane. This phase requires precise coordination of forward and downward velocity components.
The final flare phase is initiated at a relatively low altitude above ground level (AGL). The primary objective here is to reduce the vertical descent rate significantly before touchdown to minimize impact forces. The drone’s altitude, descent rate, and the desired final height become mathematically correlated through an exponential or polynomial guidance law. Upon reaching the designated touchdown point, the flare phase concludes, and the quadrotor drone should ideally achieve a near-vertical descent rate of approximately -0.5 m/s or less at the moment of contact with the landing surface.
Mathematical Modeling of the Quadrotor Drone
To design a model-independent controller like LADRC, we first establish a nominal model that captures the primary kinematics and measurements, acknowledging that disturbances and unmodeled dynamics will be handled by the controller’s estimation and rejection mechanism.
1. State Definition and Kinematic Model: The state of the quadrotor drone is defined with respect to an Earth-fixed inertial frame. A minimal representation for trajectory tracking can be given by the state vector $\mathbf{x}’ = [x, y, z, \psi, \theta]^T$, where $x$, $y$, and $z$ denote the position coordinates, $\psi$ is the yaw angle (heading), and $\theta$ is the pitch angle. For a simplified 2D longitudinal analysis focused on landing, the roll angle is often assumed to be controlled to zero, and lateral motion ($y$) is neglected or separately controlled.
The kinematics, relating velocities in the body frame to the inertial frame, are given by:
$$
\begin{aligned}
\dot{x} &= v \cos(\theta) \cos(\psi) \\
\dot{y} &= v \cos(\theta) \sin(\psi) \\
\dot{z} &= v \sin(\theta)
\end{aligned}
$$
where $v$ is the velocity magnitude of the quadrotor drone. For control design, we consider a simplified dynamic relationship for the pitch axis, often modeled as a second-order system perturbed by disturbances:
$$
\ddot{\theta} = f_\theta(\theta, \dot{\theta}, \cdot) + b_\theta u_\theta + d_\theta(t)
$$
Here, $f_\theta(\cdot)$ represents the known/ nominal dynamics, $b_\theta$ is a control gain, $u_\theta$ is the pitch control input (e.g., related to differential thrust of rotors), and $d_\theta(t)$ aggregates all other unknown dynamics and external disturbances affecting the pitch channel.
2. Measurement Model with Noise: During landing, the quadrotor drone utilizes onboard sensors for navigation. Common measurements include relative range and bearing to visual or radio-frequency landmarks. The measurement vector for a single landmark located at $(x_m, y_m, z_m)$ is:
$$
\mathbf{z} = \begin{bmatrix} r \\ \alpha \\ \beta \end{bmatrix} =
\begin{bmatrix}
\sqrt{(x – x_m)^2 + (y – y_m)^2 + (z – z_m)^2} \\[6pt]
\arctan\left( \dfrac{z – z_m}{\sqrt{(x – x_m)^2 + (y – y_m)^2}} \right) \\[10pt]
\arctan\left( \dfrac{y – y_m}{x – x_m} \right)
\end{bmatrix} + \boldsymbol{\varepsilon}_t
$$
where $r$ is the range, $\alpha$ is the elevation angle, $\beta$ is the azimuth angle, and $\boldsymbol{\varepsilon}_t$ represents the measurement noise, often modeled as Gaussian. This sensor model is crucial for the quadrotor drone’s state estimation but is treated as a source of information and noise for the overall control system.
3. Discrete-time State Propagation for Waypoint Navigation: For simulating the quadrotor drone’s guidance logic, a discrete-time motion model is useful. The vehicle follows a series of waypoints. The state update from time step $k$ to $k+1$, when guided towards a target waypoint $\mathbf{wp} = (wp_x, wp_y, wp_z)$, can be approximated as:
$$
\begin{bmatrix}
x(k+1) \\
y(k+1) \\
z(k+1) \\
\psi(k+1) \\
\theta(k+1)
\end{bmatrix}
=
\begin{bmatrix}
x(k) + v \Delta t \cos(\theta(k) + \Delta\theta) \sin(\psi(k)+\Delta\psi) \\
y(k) + v \Delta t \cos(\theta(k) + \Delta\theta) \cos(\psi(k)+\Delta\psi) \\
z(k) + v \Delta t \sin(\theta(k) + \Delta\theta) \\
\psi(k) + \Delta\psi \\
\theta(k) + \Delta\theta
\end{bmatrix}
+ \boldsymbol{\delta}_t
$$
where $\Delta t$ is the time step, and $\Delta\psi$ and $\Delta\theta$ are the required increments in heading and pitch angles to steer towards the waypoint. They are computed based on the current position and the waypoint location. The term $\boldsymbol{\delta}_t$ represents process noise, accounting for unmodeled kinematic effects and disturbances acting on the quadrotor drone.
LADRC-Based Trajectory Tracking Control Strategy
The core of the proposed method lies in applying the Active Disturbance Rejection Control (ADRC) philosophy, specifically in its linear form (LADRC), to the trajectory tracking problem of the quadrotor drone. The central idea is to use a Linear Extended State Observer (LESO) to estimate both the system states and an augmented “total disturbance” state in real time, and then to cancel this estimated disturbance within a simple state feedback control law.
Theoretical Framework of LADRC for a Second-Order System
Consider a generic second-order system representing one channel of the quadrotor drone’s dynamics (e.g., altitude $z$ or a tracking error angle):
$$
\ddot{y} = f(y, \dot{y}, w(t), t) + b u
$$
where $y$ is the output, $u$ is the control input, $b$ is a known approximate control gain, $f(\cdot)$ encapsulates all internal dynamics and external disturbances $w(t)$, and is unknown. LADRC reformulates this by defining the total disturbance $F = f(y, \dot{y}, w(t), t) + (b – b_0)u$, where $b_0$ is a nominal value of $b$. The system becomes:
$$
\ddot{y} = F + b_0 u
$$
The state vector is expanded to include $F$ as an additional state: $\mathbf{x} = [x_1, x_2, x_3]^T = [y, \dot{y}, F]^T$. Assuming $F$ is differentiable and its derivative $\dot{F} = h(t)$ is bounded, the augmented state-space model is:
$$
\begin{cases}
\dot{x}_1 = x_2 \\
\dot{x}_2 = x_3 + b_0 u \\
\dot{x}_3 = h(t) \\
y = x_1
\end{cases}
$$
A Linear Extended State Observer (LESO) is designed for this system:
$$
\begin{cases}
\dot{\hat{x}}_1 = \hat{x}_2 + l_1 (y – \hat{x}_1) \\
\dot{\hat{x}}_2 = \hat{x}_3 + b_0 u + l_2 (y – \hat{x}_1) \\
\dot{\hat{x}}_3 = l_3 (y – \hat{x}_1)
\end{cases}
$$
where $\hat{x}_1$, $\hat{x}_2$, and $\hat{x}_3$ are the estimates of $y$, $\dot{y}$, and $F$, respectively. The observer gains $l_1$, $l_2$, $l_3$ are chosen to place all eigenvalues of the observer characteristic polynomial at $-\omega_o$ (the observer bandwidth), typically resulting in $(s + \omega_o)^3 = s^3 + l_1 s^2 + l_2 s + l_3$. This gives $l_1 = 3\omega_o$, $l_2 = 3\omega_o^2$, $l_3 = \omega_o^3$.
With accurate estimates from the LESO, the control law is designed as:
$$
u = \frac{u_0 – \hat{x}_3}{b_0}
$$
This actively cancels the estimated total disturbance $\hat{x}_3$. The remaining control $u_0$ is designed for the now-simplified double-integrator plant $\ddot{y} \approx u_0$. A simple Proportional-Derivative (PD) controller suffices:
$$
u_0 = k_p (r – \hat{x}_1) – k_d \hat{x}_2
$$
where $r$ is the reference signal. The controller gains $k_p$ and $k_d$ are chosen to place the closed-loop poles at $-\omega_c$ (the controller bandwidth), yielding $k_p = \omega_c^2$ and $k_d = 2\omega_c$.
Application to Quadrotor Drone Landing Trajectory Tracking
For the quadrotor drone landing problem, the LADRC structure is applied to the outer-loop position/velocity control and the inner-loop attitude control. The reference trajectory provides desired states $[x_d(t), y_d(t), z_d(t), \psi_d(t)]$.
1. Outer-Loop (Position/Velocity) LADRC: The horizontal $(x, y)$ and vertical $(z)$ channels are often decoupled at the guidance level. For the altitude $z$ channel:
- The plant is $\ddot{z} = F_z + b_{z0} u_z$, where $u_z$ is the collective thrust command (translated to a desired pitch angle or direct thrust), and $F_z$ includes lift dynamics, drag, wind disturbances, and coupling effects.
- An LESO estimates $\hat{z}$, $\dot{\hat{z}}$, and $\hat{F}_z$.
- The control law computes $u_z = (u_{0z} – \hat{F}_z)/b_{z0}$, with $u_{0z} = k_{pz}(z_d – \hat{z}) – k_{dz} \dot{\hat{z}}$.
A similar structure is applied to control forward velocity or horizontal position error.
2. Inner-Loop (Attitude) LADRC: The outer-loop commands a desired pitch angle $\theta_d$ to achieve forward/backward motion and a desired roll angle $\phi_d$ for lateral motion (for a standard quadrotor drone configuration). The pitch dynamics are:
$$
\ddot{\theta} = F_\theta + b_{\theta0} u_\theta
$$
where $u_\theta$ is the differential thrust/moment input for the pitch axis.
- An LESO estimates $\hat{\theta}$, $\dot{\hat{\theta}}$, and $\hat{F}_\theta$ (which includes gyroscopic effects, aerodynamic moments, and inertia uncertainties).
- The control law is $u_\theta = (u_{0\theta} – \hat{F}_\theta)/b_{\theta0}$, with $u_{0\theta} = k_{p\theta}(\theta_d – \hat{\theta}) – k_{d\theta} \dot{\hat{\theta}}$.
An identical structure is used for the roll and yaw channels. The modularity of LADRC simplifies the control design for the multi-axis, coupled dynamics of the quadrotor drone.
3. Handling Actuator Saturation: In practice, the control inputs for a quadrotor drone (rotor thrusts) have strict physical limits. To prevent integrator windup in the observer and ensure stability, the final control command sent to the actuators is passed through a saturation function:
$$
u_{\text{actual}}(t) = M \cdot \text{sat}\left( \frac{u(t)}{M} \right)
$$
where $M$ is the maximum allowable control effort (e.g., maximum rotor RPM or thrust), and $\text{sat}(\cdot)$ is the standard saturation function. This protects the quadrotor drone’s actuators while the LESO continues to estimate the disturbance based on the saturated output, maintaining effective performance.
Landing Trajectory Design and Simulation Analysis
Reference Trajectory Generation
A smooth, feasible reference trajectory is essential for successful autonomous landing of the quadrotor drone. The trajectory must respect the vehicle’s dynamic constraints (maximum tilt angle, acceleration, jerk) while meeting the phase-specific requirements outlined earlier.
For simulation purposes, a standard longitudinal landing profile is generated. It defines the desired altitude $z_d$ as a function of the downrange distance $x_d$ (or time $t$). The glideslope is a straight line, and the flare is typically defined by an exponential curve to smoothly reduce the descent rate to the desired touchdown value. The desired forward speed $v_{xd}$ is also profiled to decelerate during the flare.
Altitude Profile:
$$
z_d(\xi) =
\begin{cases}
H_0 – \tan(\gamma_1) \cdot \xi, & \text{for } \xi \ge \xi_{\text{flare}} \quad \text{(Glideslope)} \\
a_1 \exp(a_2 \xi) + a_3, & \text{for } \xi < \xi_{\text{flare}} \quad \text{(Flare)}
\end{cases}
$$
where $\xi$ is the horizontal distance to the touchdown point, $H_0$ is the initial glideslope altitude, $\gamma_1$ is the glideslope angle (e.g., -3°), $\xi_{\text{flare}}$ is the flare initiation distance, and $a_1, a_2, a_3$ are coefficients chosen to ensure continuity in position, slope (descent rate), and curvature at the flare transition point.
Forward Velocity Profile: A similar smooth decaying profile, such as a polynomial or exponential function of time or distance, is used to command a reduction in forward speed from the approach speed to near-zero at touchdown.
The following table summarizes the key parameters for the reference landing trajectory used in the simulation:
| Parameter | Symbol | Value | Description |
|---|---|---|---|
| Initial Glideslope Altitude | $H_0$ | 100 m | Starting altitude for the final descent phase. |
| Glideslope Angle | $\gamma_1$ | -3° | Constant descent angle during glideslope. |
| Glideslope Descent Rate | $v_{zd,\text{gs}}$ | ≈ -2.62 m/s | Derived from $v_{xd} \cdot \tan(\gamma_1)$. |
| Approach Forward Speed | $v_{xd,\text{app}}$ | 15 m/s | Initial forward speed at $H_0$. |
| Flare Initiation Altitude | $H_{\text{flare}}$ | 20 m | Altitude at which the exponential flare begins. |
| Touchdown Descent Rate | $v_{zd,\text{TD}}$ | -0.5 m/s | Target vertical speed at the moment of landing. |
| Touchdown Forward Speed | $v_{xd,\text{TD}}$ | 1.0 m/s | Target forward speed at touchdown. |
Simulation Results and Performance Analysis
A high-fidelity nonlinear simulation model of the dual-engine, oil-powered quadrotor drone was developed, incorporating realistic motor dynamics, aerodynamic drag, and moment of inertia. The proposed LADRC controllers were implemented for the altitude ($z$), forward velocity ($v_x$), and pitch angle ($\theta$) channels. Gaussian white noise was added to all sensor measurements (position, velocity, attitude) and wind gusts were modeled as external disturbances to validate robustness.
1. Trajectory Tracking Performance: The primary metric of success is the accurate tracking of the 3D landing trajectory. The following table quantifies the Root Mean Square Error (RMSE) and maximum absolute error for key variables during the entire landing sequence from $H_0$ to touchdown.
| Tracked Variable | RMSE | Max Absolute Error | Comment |
|---|---|---|---|
| Altitude ($z$) | 0.018 m | 0.051 m | Max error occurs during glideslope transition; settles rapidly. |
| Forward Position ($x$) | 0.22 m | 0.65 m | Error influenced by wind disturbance rejection. |
| Vertical Velocity ($v_z$) | 0.0021 m/s | 0.0065 m/s | Excellent descent rate control throughout. |
| Forward Velocity ($v_x$) | 0.15 m/s | 0.41 m/s | Tracks the deceleration profile closely. |
| Pitch Angle ($\theta$) | 0.25° | 0.72° | Inner-loop LADRC maintains attitude command. |
The altitude tracking plot shows near-perfect following of the reference. The slight deviation at the start of the glideslope (the maximum error of 0.051 m) is quickly corrected by the LADRC controller as its LESO converges on the disturbance. By the flare phase and at touchdown, the altitude error is negligible (< 0.001 m). The vertical velocity (sink rate) tracks its reference flawlessly, maintaining the commanded -2.62 m/s on the glideslope and smoothly transitioning to the final -0.5 m/s at touchdown, which is critical for landing gear stress and stability.
2. Attitude and Inner-Loop Response: The pitch angle $\theta$, which is the primary attitude variable for longitudinal control, accurately tracks its commanded profile generated by the outer-loop LADRC controller. The pitch rate remains bounded and smooth, indicating stable inner-loop control. The angle of attack (AoA), a critical aerodynamic parameter, can be inferred from the flight path angle $\gamma$ and pitch angle $\theta$ ($\alpha \approx \theta – \gamma$). Throughout the simulated landing, the AoA remains within a safe and feasible range (typically 0° to 10° for such platforms), confirming the physical plausibility of the generated trajectory and the control system’s ability to maintain it. The flight path angle $\gamma$ itself follows the expected profile: constant at -3° during glideslope and progressively shallowing to nearly 0° at touchdown.
3. Disturbance Rejection and Robustness: To test the robustness of the LADRC-based controller for the quadrotor drone, a series of step wind gusts were introduced during the descent. The following table illustrates the system’s response to a 2 m/s headwind gust applied for 5 seconds during the glideslope phase.
| Affected Variable | Maximum Deviation | Settling Time (to within 5% of error) | Observations |
|---|---|---|---|
| Altitude ($z$) | +0.18 m | 3.2 s | LESO estimates and compensates for the wind-induced lift/drag change. |
| Forward Velocity ($v_x$) | -1.1 m/s | 4.0 s | Velocity recovers to the reference profile without steady-state error. |
| Pitch Angle ($\theta$) | +1.8° | 2.5 s | Controller automatically increases pitch to maintain speed against headwind. |
The results demonstrate the core strength of the LADRC strategy. The LESO successfully estimated the force/moment disturbances caused by the wind gust, and the control law actively canceled them. The quadrotor drone experienced a transient deviation but returned accurately to its intended trajectory without requiring any gain scheduling or explicit wind model, showcasing excellent disturbance rejection capabilities essential for reliable autonomous landing.
Conclusion
This analysis has presented a robust solution to the challenging problem of autonomous landing trajectory tracking for fully autonomous dual-engine oil-powered quadrotor drones. By adopting a Linear Active Disturbance Rejection Control (LADRC) strategy, the proposed method effectively decouples the controller design from the need for a precise, high-fidelity dynamic model of the vehicle. The LADRC framework, with its integral Linear Extended State Observer (LESO), actively estimates and compensates for the “total disturbance,” which aggregates model uncertainties, strong nonlinear couplings, and external environmental forces acting on the quadrotor drone.
The simulation results provide strong evidence of the method’s efficacy. The quadrotor drone accurately tracked the complex longitudinal landing profile, transitioning smoothly from the glideslope to the flare phase. Key performance metrics, including altitude, vertical velocity (sink rate), and forward speed, were maintained with minimal error. Critically, the vertical velocity at touchdown was precisely controlled to the desired gentle value, which is paramount for preventing damage. Furthermore, the inner-loop LADRC controllers ensured stable and responsive attitude control, keeping the pitch angle and related flight parameters within safe operational bounds. The robustness tests against simulated wind gusts confirmed the system’s superior disturbance rejection capability, a vital feature for real-world operations where environmental conditions are unpredictable.
In summary, the LADRC-based approach offers a practical, powerful, and model-independent control solution for autonomous landing of quadrotor drones. It mitigates the issues of tracking jitter and poor disturbance response observed in traditional integral-based state feedback methods, particularly during the sensitive initial approach and transition phases. This makes it a highly suitable candidate for enhancing the reliability and safety of autonomous landing systems in various applications for quadrotor drones, from industrial inspection to automated delivery services.