The autonomous landing of a quadrotor drone represents a critical capability with significant implications for applications ranging from logistics and infrastructure inspection to search-and-rescue operations and aerial vehicle recovery. However, achieving precise and reliable touchdown is notoriously challenging due to a complex and often unpredictable aerodynamic phenomenon: ground effect. As a quadrotor drone descends towards a surface, the airflow generated by its rotors interacts with the ground, creating a cushion of increased pressure. This results in an additional, uncommanded upward force that can severely degrade landing performance, causing oscillations, position drift, or even preventing a stable descent altogether. The core challenge lies in the fact that this ground effect is highly nonlinear, dependent on the drone’s height, attitude, rotor speeds, and the surrounding geometry, making an accurate first-principles model exceptionally difficult to derive, especially for the complex, interacting flow fields of a multi-rotor system.
Traditional control approaches, such as Proportional-Integral-Derivative (PID) or even some nonlinear methods, often struggle with this unmodeled dynamic. While simplified empirical models like the Cheeseman-Bennett model exist, they are primarily derived for large helicopters with constant rotor speeds and do not adequately capture the behavior of small, agile quadrotor drones with rapidly varying thrust commands. Consequently, controllers designed without explicit compensation for ground effect can become unstable or exhibit large steady-state errors during the final stages of landing. This paper addresses this fundamental problem by proposing a novel, hybrid control strategy that merges the powerful pattern recognition capabilities of deep learning with the rigorous stability guarantees of nonlinear robust control theory. The objective is to enable a quadrotor drone to perform precise, rapid, and stable autonomous landings in the presence of significant and unknown ground effect disturbances.
Dynamical Model of a Quadrotor Drone in Landing
The dynamics of a quadrotor drone during landing are described in an inertial world frame. Let $\boldsymbol{P} = [x, y, z]^T$ and $\boldsymbol{v} = [\dot{x}, \dot{y}, \dot{z}]^T$ denote the position and velocity vectors of the drone’s center of mass. The attitude is represented by the rotation matrix $\boldsymbol{R} \in SO(3)$ which transforms vectors from the body frame to the world frame. The angular velocity in the body frame is $\boldsymbol{\omega} = [p, q, r]^T$. The governing equations are:
$$
\begin{aligned}
\dot{\boldsymbol{P}} &= \boldsymbol{v}, \\
m\dot{\boldsymbol{v}} &= m\boldsymbol{g} + \boldsymbol{R}\boldsymbol{f}_u + \boldsymbol{f}_g, \\
\dot{\boldsymbol{R}} &= \boldsymbol{R} S(\boldsymbol{\omega}), \\
\boldsymbol{J}\dot{\boldsymbol{\omega}} &= \boldsymbol{J}\boldsymbol{\omega} \times \boldsymbol{\omega} + \boldsymbol{\tau}_u.
\end{aligned}
$$
Here, $m$ is the mass, $\boldsymbol{g} = [0, 0, -g]^T$ is the gravity vector, and $\boldsymbol{J}$ is the inertia matrix. The control forces and torques generated by the rotors in the body frame are $\boldsymbol{f}_u = [0, 0, T]^T$ and $\boldsymbol{\tau}_u = [\tau_x, \tau_y, \tau_z]^T$, where $T$ is the total thrust. The key disturbance is $\boldsymbol{f}_g = [f_{gx}, f_{gy}, f_{gz}]^T$, which represents the ground effect force acting on the quadrotor drone in the world frame. This term is a complex function $\boldsymbol{f}_g(\boldsymbol{\xi}_g, \boldsymbol{u})$, where $\boldsymbol{\xi}_g$ includes relevant states like altitude and attitude, and $\boldsymbol{u}$ is the control input vector (typically motor speed commands). The control input is related to the forces and torques via $\boldsymbol{\eta} = [T, \tau_x, \tau_y, \tau_z]^T = \boldsymbol{B}\boldsymbol{u}$, where $\boldsymbol{B}$ is a constant matrix defined by the drone’s geometry and motor parameters.
The primary control challenge is that $\boldsymbol{f}_g$ is unknown and resists simple analytical modeling. This work focuses on learning a compensation model for $\boldsymbol{f}_g$, particularly its dominant vertical component $f_{gz}$, to augment a robust position controller.
Learning Ground Effect with Deep Neural Networks
To tackle the unmodeled dynamics, we employ a Deep Neural Network (DNN) as a universal function approximator. A DNN can learn the intricate mapping from the quadrotor drone’s state and control input to the ground effect force from experimental data. We construct a dataset $\mathcal{D} = \{(\boldsymbol{\xi}_g^{(i)}, \boldsymbol{u}^{(i)}), \boldsymbol{f}_g^{(i)}\}_{i=1}^{N}$ by executing various landing maneuvers and estimating the instantaneous ground effect using the following inversion of the dynamics:
$$
\boldsymbol{f}_g = m\boldsymbol{a} + m\boldsymbol{g} – \boldsymbol{R}\boldsymbol{f}_u.
$$
Acceleration $\boldsymbol{a}$ is obtained from an inertial measurement unit (IMU), attitude $\boldsymbol{R}$ from a state estimator, and thrust $\boldsymbol{f}_u$ from the commanded motor signals. We use a feedforward network with Rectified Linear Unit (ReLU) activation functions due to their favorable convergence properties and robustness:
$$
\hat{\boldsymbol{f}}_g(\boldsymbol{\xi}_g, \boldsymbol{u}; \boldsymbol{\theta}) = \boldsymbol{W}^{L+1} \phi(\boldsymbol{W}^L \phi( \dots \phi(\boldsymbol{W}^1 \boldsymbol{x}) \dots )),
$$
where $\boldsymbol{x} = [\boldsymbol{\xi}_g^T, \boldsymbol{u}^T]^T$ is the input, $\phi(\cdot) = \max(0, \cdot)$ is the ReLU function, $\boldsymbol{\theta} = \{\boldsymbol{W}^1, \dots, \boldsymbol{W}^{L+1}\}$ are the weight matrices, and $\hat{\boldsymbol{f}}_g$ is the network’s estimate of the true ground effect.
Ensuring Stability via Spectral Normalization
Simply injecting a learned network into a feedback loop can jeopardize stability, as the network’s response to novel inputs is not guaranteed to be bounded. To mitigate this, we apply Spectral Normalization (SN) during training. SN constrains the Lipschitz constant of the network, which bounds how much the output can change for a given change in input. The Lipschitz constant $L_f$ of a function $f$ satisfies $\|f(\boldsymbol{x}_1) – f(\boldsymbol{x}_2)\| \leq L_f \|\boldsymbol{x}_1 – \boldsymbol{x}_2\|$. For a deep network, its Lipschitz constant is bounded by the product of the spectral norms (largest singular values) of its weight matrices: $L_f \leq \prod_{l=1}^{L+1} \sigma(\boldsymbol{W}^l)$.
Spectral Normalization rescales each weight matrix as follows:
$$
\bar{\boldsymbol{W}}^l = \frac{\boldsymbol{W}^l}{\sigma(\boldsymbol{W}^l)} \cdot \gamma^{\frac{1}{L+1}},
$$
where $\gamma > 0$ is a user-defined constant. This ensures that the spectral norm of each normalized layer is $\gamma^{\frac{1}{L+1}}$, and consequently, the Lipschitz constant of the entire network is bounded by $\gamma$:
$$
\| \hat{\boldsymbol{f}}_g(\cdot; \bar{\boldsymbol{\theta}}) \|_{\text{Lip}} \leq \gamma.
$$
This property is crucial for the subsequent stability analysis of the closed-loop system involving the quadrotor drone. The training process thus becomes a constrained optimization:
$$
\min_{\boldsymbol{\theta}} \frac{1}{N} \sum_{i=1}^{N} \| \boldsymbol{f}_g^{(i)} – \hat{\boldsymbol{f}}_g(\boldsymbol{\xi}_g^{(i)}, \boldsymbol{u}^{(i)}; \boldsymbol{\theta}) \|^2 \quad \text{s.t.} \quad \| \hat{\boldsymbol{f}}_g(\cdot; \boldsymbol{\theta}) \|_{\text{Lip}} \leq \gamma.
$$
Super-Twisting Robust Controller with DNN Compensation
We design a nonlinear controller for the quadrotor drone’s translational dynamics. Let $\boldsymbol{P}_d(t)$ be a smooth desired landing trajectory. The position and velocity tracking errors are $\boldsymbol{e} = \boldsymbol{P}_d – \boldsymbol{P}$ and $\dot{\boldsymbol{e}} = \dot{\boldsymbol{P}}_d – \boldsymbol{v}$. We define a sliding surface variable $\boldsymbol{s} = \dot{\boldsymbol{e}} + \boldsymbol{\Lambda} \boldsymbol{e}$, where $\boldsymbol{\Lambda} > 0$ is a diagonal gain matrix.
The control law for the total force $\boldsymbol{f}_d = \boldsymbol{R}\boldsymbol{f}_u$ (expressed in the world frame) is synthesized as:
$$
\boldsymbol{f}_d = m \left( \ddot{\boldsymbol{P}}_d – \boldsymbol{g} – \frac{1}{m}\hat{\boldsymbol{f}}_g(\boldsymbol{\xi}_g, \boldsymbol{u}) + \boldsymbol{\Lambda} \dot{\boldsymbol{e}} + \boldsymbol{K}_1 \boldsymbol{S}_{k1} – \boldsymbol{Y} \right).
$$
Here, $\hat{\boldsymbol{f}}_g(\boldsymbol{\xi}_g, \boldsymbol{u})$ is the DNN estimate of the ground effect. The term $\boldsymbol{S}_{k1}$ and the auxiliary variable $\boldsymbol{Y}$ are components of the super-twisting algorithm, a second-order sliding mode technique known for its robustness and reduced chattering:
$$
\boldsymbol{S}_{k1} = \begin{bmatrix} |s_x|^{1/2} \text{sgn}(s_x) \\ |s_y|^{1/2} \text{sgn}(s_y) \\ |s_z|^{1/2} \text{sgn}(s_z) \end{bmatrix}, \quad \dot{\boldsymbol{Y}} = -\boldsymbol{K}_2 \begin{bmatrix} \text{sgn}(s_x) \\ \text{sgn}(s_y) \\ \text{sgn}(s_z) \end{bmatrix}.
$$
$\boldsymbol{K}_1 = \text{diag}(k_{1x}, k_{1y}, k_{1z})$ and $\boldsymbol{K}_2 = \text{diag}(k_{2x}, k_{2y}, k_{2z})$ are positive diagonal gain matrices. The commanded total thrust $T_d$ and desired attitude $\boldsymbol{R}_d$ are extracted from $\boldsymbol{f}_d$ and a desired yaw angle. A fast inner-loop attitude controller tracks $\boldsymbol{R}_d$. Since the control input $\boldsymbol{u}$ appears inside the DNN estimate $\hat{\boldsymbol{f}}_g(\boldsymbol{\xi}_g, \boldsymbol{u})$, solving for $\boldsymbol{u}$ requires handling a non-affine structure. This is done iteratively in discrete time: $\boldsymbol{u}_k = \boldsymbol{B}^{-1} [ (\boldsymbol{f}_d(\hat{\boldsymbol{f}}_g(\boldsymbol{\xi}_g, \boldsymbol{u}_{k-1})) \cdot \hat{\boldsymbol{k}})^T, \boldsymbol{\tau}_d^T ]^T$, where $\hat{\boldsymbol{k}}$ is the unit vector of the body-fixed z-axis.
Stability and Finite-Time Convergence Analysis
The closed-loop stability is proven using Lyapunov theory. Consider the dynamics along one axis (e.g., $z$). The error dynamics can be written as:
$$
\dot{s}_z = -k_{1z} |s_z|^{1/2} \text{sgn}(s_z) + Y_z + \frac{1}{m} \left( \hat{f}_{gz}(\xi_g, u_{k-1}) – f_{gz}(\xi_g, u_k) \right),
$$
$$
\dot{Y}_z = -k_{2z} \text{sgn}(s_z).
$$
Define the learning error as $\epsilon_z = \hat{f}_{gz} – f_{gz}$ and assume it is bounded. Define a state vector $\boldsymbol{\zeta}_z = [|s_z|^{1/2} \text{sgn}(s_z), Y_z]^T$. Consider the Lyapunov candidate function $V_z = \boldsymbol{\zeta}_z^T \boldsymbol{P} \boldsymbol{\zeta}_z$, where $\boldsymbol{P}$ is a symmetric positive definite matrix. After detailed derivation involving the spectral normalization property $\|\hat{f}_{gz}(\cdot, u_1) – \hat{f}_{gz}(\cdot, u_2)\| \leq L_a \|u_1 – u_2\|$ (where $L_a \leq \gamma$) and the contraction mapping property of the iterative input solver, the time derivative of $V_z$ can be bounded as:
$$
\dot{V}_z \leq -\frac{\kappa \Upsilon}{\lambda_{\text{max}}^{1/2}(\boldsymbol{P})} V_z^{1/2}, \quad \forall \|\boldsymbol{\zeta}_z\| \geq \varrho,
$$
for some positive constants $\kappa, \Upsilon, \varrho$. This inequality satisfies the conditions for finite-time stability. Therefore, the sliding variable $s_z$ converges to zero in finite time, which implies that the position tracking error $e_z$ also converges to zero in finite time. The same analysis holds for the $x$ and $y$ axes. Thus, the proposed controller ensures that the quadrotor drone’s landing error converges to zero within a finite time despite the presence of unknown ground effects, provided the controller gains are selected according to conditions derived from the matrix $\boldsymbol{P}$ and the network’s Lipschitz bound $\gamma$.
Experimental Validation and Comparative Results
We implemented the proposed controller on a custom-built quadrotor drone platform (mass: 0.91 kg, diagonal wheelbase: 35 cm) within a motion capture system for high-precision state estimation. A dataset of over 48,000 samples was collected during manual landing maneuvers to train the DNN. The network had an architecture of 5-20-30-30-20-1 nodes, with inputs being the four motor PWM signals and the drone’s altitude. Spectral normalization was applied with $\gamma=1$.
Training and Generalization Comparison
The DNN’s performance was compared against a standard 5th-order polynomial fitted via least squares (LS). The key metric is generalization to altitudes outside the dense training range. The results, summarized qualitatively below, show the DNN’s superior ability to provide a physically plausible extrapolation.
| Method | Performance on Training Data (0-200 cm) | Performance on Extrapolation (200-250 cm) | Suitability for Control |
|---|---|---|---|
| Least Squares (5th Order Poly) | Good fit | Poor, erratic prediction | Low (poor generalization) |
| DNN with Spectral Normalization | Good fit | Good, smooth and plausible prediction | High (stable generalization) |
Landing Flight Experiments
Three autonomous landing experiments from 1 meter height were conducted:
- No Compensation: The baseline super-twisting controller without ground effect compensation.
- Cheeseman-Bennett Model Compensation: The super-twisting controller augmented with the standard empirical model $T_{IGE}/T_{OGE} = 1 / (1 – (R/(4z))^2)$.
- Proposed DNN Compensation: The super-twisting controller augmented with our spectrally normalized DNN model.
The key results are quantified in the following table and described thereafter.
| Controller Type | Successful Landing? | Time from Descent Start to Touchdown (s) | Final Landing Error Characteristics | Observations During Descent |
|---|---|---|---|---|
| No Compensation | No | N/A (Did not land) | Stuck ~9 cm above ground | Severe ground effect caused a sustained hover, preventing descent. |
| Cheeseman-Bennett Comp. | Yes | ~25.4 | Large initial error bulge, slow convergence | Visible influence of ground effect, sluggish response. |
| Proposed DNN Comp. | Yes | ~5.5 | Fast, smooth, and direct convergence | Ground effect visibly compensated; descent profile matched ascent profile. |
Experiment 1 (No Compensation): The quadrotor drone initiated descent but became trapped approximately 9 cm above the ground. The integrator in the controller could not overcome the persistent upward disturbance from the ground effect, demonstrating the critical need for explicit compensation.
Experiment 2 (Cheeseman-Bennett Compensation): The quadrotor drone landed successfully but took over 25 seconds. The descent curve showed a significant deviation or “bulge” due to the model’s inaccuracy, requiring the robust part of the controller to slowly correct the resulting error. This highlights the inadequacy of the simple model for the quadrotor drone.
Experiment 3 (Proposed DNN Compensation): The quadrotor drone performed a rapid and smooth landing in just 5.5 seconds. The descent trajectory was nearly symmetric to the ascent trajectory, indicating that the learned DNN model effectively canceled the ground effect disturbance. The finite-time convergence properties of the controller were clearly observed.
Conclusion and Future Work
This paper presented a novel and effective solution to the challenging problem of ground effect during quadrotor drone landings. By integrating a spectrally normalized deep neural network for disturbance learning with a super-twisting robust controller, we achieved precise, fast, and stable autonomous landing. The spectral normalization provided a crucial link between the learning component and control theory, enabling a formal stability proof with finite-time convergence guarantees. Real-time flight experiments conclusively demonstrated the superiority of the proposed method over both uncompensated control and compensation based on a classical empirical model. The quadrotor drone equipped with our controller reliably touched down without oscillatory behavior or performance degradation near the ground.
Future work will focus on extending the learned model to account for more complex scenarios. This includes incorporating horizontal position and terrain geometry into the DNN’s input space to enable safe landing on sloped or uneven surfaces. Furthermore, investigating online adaptation mechanisms for the DNN could allow a single quadrotor drone to adjust to changing environmental conditions or different landing platforms, enhancing the overall robustness and versatility of the autonomous landing system.