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Neural Lander: Stable Drone Landing Control using Learned Dynamics
Guanya Shi, Xichen Shi, Michael O'Connell, Rose Yu, Kamyar Azizzadenesheli, Animashree Anandkumar, Yisong Yue, Soon-Jo Chung
TL;DR
Near-ground quadrotor control is challenged by complex aerodynamic disturbances that conventional models and controllers do not fully capture. Neural-Lander learns residual dynamics with a spectrally normalized DNN and embeds them in a nonlinear feedback-linearization controller with stability guarantees. It outperforms the baseline in landing and cross-table tracking while generalizing to unseen data.
Problem
Unsteady, coupled aerodynamic interactions near the ground make precise quadrotor landing difficult, while existing theoretical models and controllers have limited precision.
Method
Neural-Lander learns disturbance forces with a spectrally normalized DNN and integrates the learned model into nonlinear feedback linearization.
Results
Neural-Lander outperforms the Baseline Nonlinear Tracking Controller in landing and cross-table tracking, reducing landing z error from 0.13 m to 0 and cross-table z error from 0.153 m to 0.027 m.
Takeaways & Limitations
The controller combines learned coupled unsteady aerodynamics with rigorous stability analysis and empirically generalizes to unseen data outside the training domain.
Abstract
from arXiv · showhide
Precise near-ground trajectory control is difficult for multi-rotor drones, due to the complex aerodynamic effects caused by interactions between multi-rotor airflow and the environment. Conventional control methods often fail to properly account for these complex effects and fall short in accomplishing smooth landing. In this paper, we present a novel deep-learning-based robust nonlinear controller (Neural Lander) that improves control performance of a quadrotor during landing. Our approach combines a nominal dynamics model with a Deep Neural Network (DNN) that learns high-order interactions. We apply spectral normalization (SN) to constrain the Lipschitz constant of the DNN. Leveraging this Lipschitz property, we design a nonlinear feedback linearization controller using the learned model and prove system stability with disturbance rejection. To the best of our knowledge, this is the first DNN-based nonlinear feedback controller with stability guarantees that can utilize arbitrarily large neural nets. Experimental results demonstrate that the proposed controller significantly outperforms a Baseline Nonlinear Tracking Controller in both landing and cross-table trajectory tracking cases. We also empirically show that the DNN generalizes well to unseen data outside the training domain.
I. INTRODUCTION
Near-ground quadrotor control is difficult because rotor–environment aerodynamic interactions create disruptive ground effects and unmodeled disturbances. Neural-Lander combines learned residual dynamics with nonlinear feedback linearization and stability guarantees, improving landing and cross-table tracking.
- Ground effect changes lift and drag near the ground, while multiple rotors exacerbate aerodynamic interactions and flight-instability complications.
- Existing approaches rely on steady-flow models or slow adaptive and integral control, leaving practical unsteady-flow compensation limited.
- DNN-based control must address limited real-world data, potentially unstable feedback outputs, and difficulty obtaining provable stability.
- Neural-Lander learns residual coupled aerodynamics and vehicle dynamics, then integrates the learned model with nonlinear feedback linearization.
- Spectral normalization enables a globally exponentially stable controller under bounded learning errors by exploiting the DNN’s Lipschitz bound.
- 0 m z-axis error and up to 90% lower x/y drift were achieved in landing, while cross-table z error fell from 0.153 m to 0.027 m versus baseline.
III. DYNAMICS LEARNING USING DNN
The dynamics-learning component uses a ReLU DNN to model disturbance forces from observed states and controls. Spectral normalization is introduced to improve training stability and constrain the network’s sensitivity.
- The learned disturbance force fa is represented with a ReLU deep neural network.ReLU applies the element-wise mapping φ(·) = max(·, 0).
- ReLU is described as faster to train, more robust to hyperparameter changes, and less prone to vanishing gradients than sigmoid.
- The network maps input x to output f(x, θ) through compositions of weighted layers and ReLU activations.
- Spectral normalization is applied because first-order optimization can be sensitive to objective curvature and unstable during DNN training.
B. Spectral Normalization
Spectral normalization constrains layer weight norms to bound the DNN’s Lipschitz constant. This bound supports both controlled network sensitivity and the paper’s stability analysis.
- Spectral normalization constrains the Lipschitz constant of the DNN, stabilizing training and supporting generalization.
- For differentiable functions, the Lipschitz constant equals the supremum of the gradient’s maximum singular value over the domain.
- The network’s Lipschitz bound follows from multiplying the spectral norms of its layers because ReLU has Lipschitz norm 1.
- During training, spectral normalization is applied to each weight matrix using an intended DNN Lipschitz constant γ.
- Lemma 3.1 bounds the entire multi-layer ReLU network’s Lipschitz constant when its parameters are spectrally normalized.
C. Constrained Training
The controller trains a Lipschitz-bounded DNN on observed disturbance forces and embeds its predictions in nonlinear feedback linearization. The resulting closed-loop position tracking is globally exponential with bounded approximation error.
- C. Constrained Training: Gradient-based training estimates disturbance forces from observed states and control inputs while constraining the ReLU network’s Lipschitz constant.
- C. Constrained Training: The controller uses spectral normalization to obtain stability guarantees and fixed-point iteration to solve for the resulting control input.
- A. Reference Trajectory Tracking: The position error is placed on a composite manifold whose dynamics make tracking converge exponentially toward the desired position.
- C. Constrained Training: The desired rotor force subtracts the DNN-estimated aerodynamic disturbance from the nominal force required by the reference velocity controller.
- C. Constrained Training: Bounded approximation error yields globally exponential position convergence with bounded tracking error.
- A. Reference Trajectory Tracking: The desired thrust, force direction, attitude, and torque are computed sequentially to track the reference trajectory and attitude.
B. Learning-based Discrete-time Nonlinear Controller
The controller relates the desired wrench to control inputs, then solves the resulting non-affine synthesis problem through fixed-point iteration.
- The desired wrench ηd is related to the control formulation through the system dynamics and controller equations.
- Because the learned disturbance model depends on control input, the synthesis problem is non-affine.
- A fixed-point iteration uses current and previous discrete-time control inputs to solve the control synthesis problem.
V. NONLINEAR STABILITY ANALYSIS
The stability analysis treats control allocation as an iterative mapping and establishes convergence when that mapping is contractive.
- The control input is shown to converge to the solution of the fixed-point equation when all states are fixed.
- The iteration defines a mapping uk = F(uk−1), whose contraction yields convergence to a unique fixed-point solution.
- For feasible control inputs in a compact set, the mapping satisfies ∥F(u1) − F(u2)∥2 < α∥u1 − u2∥2 for some α < 1.
B. Stability of Learning-based Nonlinear Controller
The paper states assumptions on desired trajectories, control updates, and learning error, then uses Lipschitz analysis to establish bounded-error stability and convergence.
- The analysis assumes bounded desired position trajectories and a one-step control difference bounded by ρ∥s∥.
- The control-update assumption is motivated by faster attitude and motor loops, allowing several within-update changes to be neglected.
- The learning error of ˆfa is assumed uniformly bounded by ϵm over compact state and control sets.
- Under Assumptions 1–3 and λmin(Kv) > Laρ, the composite variable s exponentially converges to an error ball of radius ϵm/(λmin(Kv) − Laρ).
- The stability proof begins by selecting the Lyapunov function V(s) = 1/2m∥s∥2 and differentiating it under the controller.
- The Lipschitz property of ˆfa and the minimum eigenvalue λmin(Kv) are used in the stability bound.
- The analysis further states finite-gain Lp stability and input-to-state stability, with position error convergence linked to the composite variable.
VI. EXPERIMENTS
Experiments use an instrumented Intel Aero quadrotor in a motion-capture environment to evaluate the learned controller and model.
- The experimental setup includes a 17-camera motion-capture system, WiFi communication, and an onboard Linux computer.
- The Intel Aero drone weighs 1.47 kg and uses a 2.56 GHz Intel Atom x7 processor with 4 GB DDR3 RAM.
- Eight reflective infrared markers provide position, attitude, and velocity estimates at 100Hz.
A. Bench Test
The study identifies aerodynamic disturbance forces from flight data and compares a learned DNN ground-effect model with theoretical and bench-test references. Spectral normalization produces smoother predictions that generalize beyond the training velocity range.
- Training and nominal model: The training data combine state estimates and control inputs from manually piloted flights at varying heights to estimate disturbance force fa.The force is computed from measured acceleration, gravity, and nominal propeller thrust.
- DNN model: The DNN models fa using height, velocity, attitude, and control input, with four fully connected ReLU hidden layers and spectral normalization.The network has 12 input dimensions and 3 output dimensions.
- Ground-effect estimation: The learned DNN estimates near-ground aerodynamic force more accurately than the theoretical steady ground-effect model across height.The comparison uses hovering data at different heights.
- Bench-test comparison: The learned vertical disturbance-force trend with rotation speed is consistent with bench-test measurements of CT.The comparison is performed at a fixed height and vertical velocity.
- Generalization: Spectral normalization yields smoother predictions that generalize to velocity domains outside training, while preserving observed ground-effect and air-drag trends.Training covered vz from −1 m/s to 1 m/s, whereas the tested heatmaps also include −2 m/s to −1 m/s.
D. Baseline Controller
The controller evaluation compares Neural-Lander with a baseline that omits learned aerodynamic disturbance forces. Neural-Lander enables smoother, more accurate landing, while reduced-capacity models and integral control show important performance limitations.
- Integral-controller variation: Integral control can cancel steady-state set-point error, but flight performance is sensitive to the integral gain, especially during trajectory tracking.The sensitivity is reported from flight results and is illustrated in the demonstration video.
- Controller comparison: Neural-Lander precisely and smoothly reaches the ground, whereas the baseline struggles to achieve zero terminal height because of ground effect.The take-off and landing test commands position from (0, 0, 0) to (0, 0, 1) and back with zero desired velocity.
- Controller comparison: Neural-Lander mitigates horizontal x−y drift by learning additional aerodynamics such as air drag.The reported landing evaluation attributes the horizontal improvement to learned aerodynamic effects.
- DNN capacity: A one-layer model reduces z error but cannot land the drone, while a zero-layer model produces significant take-off error.The baseline corresponds to ˆfa ≡ 0, and the capacity comparison is shown in Fig. 4.
- Spectral normalization: Without spectral normalization, Neural-Lander can produce unexpected controller outputs that lead to a crash in experiments.This observation empirically supports using spectral normalization during DNN training and controller design.
F. Trajectory Tracking Performance
Neural-Lander is evaluated on an elliptic trajectory near a table, where spatially varying ground effects challenge physics-based modeling. It outperforms the Baseline Controller across position axes and achieves lower height variance by capturing table-edge effects.
- Task design: The elliptic trajectory has a 10-second period, passes partly over a table, and transitions sharply to free space at the table edge.The task is designed to expose dynamics that are difficult to model with physics-based approaches.
- Modeling: A new ReLU DNN incorporates x-y positions as additional input features to model complex dynamics near the table.The model is trained on data collected by manually flying the drone near the table.
- Modeling: Only the spectrally-normalized DNN exhibits a clear table boundary in the learned ground-effect heatmap.The comparison fixes other inputs while varying x and y.
- Tracking results: Neural-Lander outperforms the Baseline Controller for desired-position tracking in all x, y, and z axes.It also shows lower height variance at the table edge by capturing changes in ground effects over the table.
- Scope: The reported benefits extend to multiple ground-interaction scenarios, while further generalization to unseen state and disturbance domains remains future work.The paper specifically identifies wind-fan-array disturbances as an example for future evaluation.