Learning Quadrotor Dynamics Using Neural Network for Flight Control

Somil Bansal; Anayo K. Akametalu; Frank J. Jiang; Forrest Laine; Claire J. Tomlin

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Learning Quadrotor Dynamics Using Neural Network for Flight Control

Somil Bansal, Anayo K. Akametalu, Frank J. Jiang, Forrest Laine, Claire J. Tomlin

arXiv:1610.05863v1Oct 19, 2016eess.SYcs.ROmath.OC

TL;DR

The paper asks whether dynamics learned from trajectory-specific data can support control of more general quadrotor motions. It trains neural-network models on separate translational and rotational trajectories, then uses them to control coupled yaw and position motion. Experiments indicate that even simple networks generalize the dynamics and support effective control beyond the training trajectories.

Problem
Existing learning-based control approaches focus on specific trajectories, leaving unclear whether their training information can synthesize controllers for more general trajectories.
Method
The paper trains feed-forward ReLU neural networks on separate translational and rotational Crazyflie trajectories, then uses the learned dynamics to control simultaneous yaw and position motion.
Results
The trained neural networks generalize beyond the training data, capture nonlinear translational-rotational couplings, and track trajectories they were not trained on.
Takeaways & Limitations
Even simple feed-forward neural networks can learn quadrotor dynamics accurately enough to be exploited for control of complex trajectories.

Abstract

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Traditional learning approaches proposed for controlling quadrotors or helicopters have focused on improving performance for specific trajectories by iteratively improving upon a nominal controller, for example learning from demonstrations, iterative learning, and reinforcement learning. In these schemes, however, it is not clear how the information gathered from the training trajectories can be used to synthesize controllers for more general trajectories. Recently, the efficacy of deep learning in inferring helicopter dynamics has been shown. Motivated by the generalization capability of deep learning, this paper investigates whether a neural network based dynamics model can be employed to synthesize control for trajectories different than those used for training. To test this, we learn a quadrotor dynamics model using only translational and only rotational training trajectories, each of which can be controlled independently, and then use it to simultaneously control the yaw and position of a quadrotor, which is non-trivial because of nonlinear couplings between the two motions. We validate our approach in experiments on a quadrotor testbed.

I. INTRODUCTION

The paper addresses the difficulty of using trajectory-specific learning to control more general quadrotor motions. It proposes learning dynamics with neural networks and testing whether the learned model generalizes to coupled translational and rotational trajectories.

System identification is important because modeling inaccuracies appear to the controller as external disturbances and affect control success.
Traditional learning approaches improve control for specific trajectories, but do not clearly show how training information supports controllers for more general trajectories.
Unobserved states, rather than stochasticity or unstructured noise, can make simple models inaccurate despite repeatable system dynamics.
Neural networks can model highly nonlinear functions and unobserved states from observed data while extending beyond the training data.
The study trains on separate translational or rotational trajectories, then controls simultaneous translational and rotational motion with the learned model.
The claimed contributions include a control-suitable neural dynamics model, accurate learning from current state-input data, and generalization to nonlinear motion couplings.

II. QUADROTOR SYSTEM IDENTIFICATION

The system-identification formulation learns a function mapping quadrotor state and control inputs to state derivatives. The model separates translational and rotational accelerations and estimates their parameters from observed data by minimizing prediction error.

System identification seeks a function f that maps state-control space to the state derivative.
The quadrotor is represented by a twelve-dimensional state containing position, linear velocity, Euler-angle attitude, and body-frame angular velocity.
The control inputs are thrust and rolling, pitching, and yawing moments expressed in the body frame.
The unknown translational and rotational acceleration components, fv and fω, are approximated as functions of state, control, and learned parameters.
The parameters are learned from observed state-input and acceleration data by minimizing mean squared prediction error.
The input signals are generated by varying the angular speeds of the four propellers, which map linearly to the control inputs.

III. NEURAL NETWORK MODEL

The paper uses supervised learning with a simple two-layer ReLU network to predict quadrotor dynamics from current state and input. The architecture is chosen to balance modeling capability with controller design requirements.

The neural network is trained with supervised learning to predict the next system state from the current state and input.
The model uses a hidden ReLU layer and a linear output layer to represent the unknown linear acceleration component fv.
The network input is the current state and control, while its hidden layer has N units and its output layer has three units.
Training determines α1 by minimizing mean squared error between predicted and observed translational acceleration, with a similar architecture used for fω.
The architecture parameters α1 are the weights and biases (W,w,B,b), and a similar parameterization is used for α2.
Only current state-input information is provided because including histories could complicate feedback-controller design despite potentially improving model complexity or accuracy.

IV. CONTROL DESIGN

The control design targets complex trajectories combining rotational and translational motion, including sinusoid-yaw trajectories. Because exact tracking may be infeasible, the approach uses a dynamically feasible reference trajectory for stabilization.

The controller is designed to stabilize quadrotor trajectories that combine rotational and translational motions, such as sinusoid-yaw trajectories.
Exact tracking may be impossible when system constraints or dynamic infeasibility prevent the desired trajectory from being realized.
The method first computes a reference trajectory satisfying the system dynamics and then tracks that reference.

A. Computation of a Feasible Reference

The feasible-reference stage finds a trajectory and control sequence that stay close to the desired trajectory while satisfying the learned system dynamics. Because the resulting optimization is non-convex, it is solved using sequential convex optimization, although even simple neural networks make the problem challenging.

The full quadrotor model is formed after the neural networks learn the translational and angular acceleration components.
The reference problem uses a discrete-time system model with state, input, sampling interval, and learned neural-network parameters.
The optimization seeks a dynamically feasible trajectory as close as possible to the desired trajectory, together with the control that produces it.
Because the neural-network dynamics are nonlinear, the optimization is non-convex and is solved with sequential convex optimization by repeatedly constructing convex local approximations.
Even relatively simple neural-network structures make this optimization very challenging, motivating the choice of a simple network.

B. Linear Quadratic Regulator (LQR)

The LQR stage stabilizes the dynamically feasible reference while compensating for mismatch between the modeled and actual quadrotor dynamics. It uses a near-hover linear model with reference-frame rotation to handle non-zero yaw, but its feedback law has scope limitations.

Applying the optimized open-loop control can deviate from the planned trajectory because of model mismatch and unmodeled disturbances, so feedback is added.
The feedback controller uses LQR designed from a linear near-hover model and applies a reference rotation to correct for non-zero yaw.
The LQR formulation minimizes a quadratic cost over the error dynamics, with positive definite state and control weighting matrices.
Dynamic programming produces a state-feedback matrix, using the feasible reference and nominal control obtained from sequential convex optimization.
The control law is not optimal when roll and pitch change significantly because it assumes near-hover motion and only yaw rotation.
Tracking quality depends on the accuracy of the open-loop state and system dynamics model, including learned translational-rotational couplings for sinusoid-yaw trajectories.

C. Crazyflie 2.0 and On-board PD Controller

The Crazyflie testbed combines slower ground-station feedback with faster on-board attitude control. VICON and IMU measurements provide the state, while LQR and PD control together stabilize the vehicle around reference trajectories.

State feedback is limited by the slowest sensor, motivating a low-level controller between feedback updates.
The control block diagram uses ground-station LQR at 100Hz and an on-board PD controller at 250Hz.
Crazyflie 2.0 provides the experimental platform for collecting training data and conducting sinusoid-yaw experiments.
VICON at 100Hz and the on-board IMU at 250Hz jointly provide the quadrotor’s 12-dimensional state.
The controller uses an augmented input containing thrust, desired attitude, and desired angular rates for the reference trajectory and feedback law.

V. EXPERIMENTS

The experiments section presents the data collection and neural-network training results used to evaluate quadrotor dynamics learning and control.

The experiments evaluate neural-network dynamics learning and control using collected quadrotor flight data.

A. Data Collection

Crazyflie data were collected from translational, rotational, and unstructured flights, then processed to train feed-forward networks for acceleration modeling. The networks use carefully selected state and input representations and achieve similar training and testing errors.

Data collection: Crazyflie data were collected from XY, XZ, and YZ sinusoidal flights, fixed-position yaw rotations, and manually unstructured flights.The recordings contain state and input data from autonomous and manual flights.
Network training: The training procedure uses two neural networks to learn linear and angular acceleration components from collected data.The networks minimize mean squared error after preprocessing the recorded inputs and states.
Input preprocessing: Control-independent inputs are derived from the recorded PD-augmented inputs before being supplied to the neural networks.This preprocessing is intended to make the learned dynamics independent of the control scheme.
Input preprocessing: Sines and cosines replace orientation angles, while position is omitted because the modeled accelerations should be position independent.The trigonometric representation avoids distinguishing equivalent angles at 0 and 2π radians.
Training protocol: 60% of the collected data was used for training, 25% for validation, and the remainder for testing.Training used resilient backpropagation with MATLAB’s Neural Network Toolbox.
Training results: 0.134 and 0.135 are the normalized training and testing MSEs for fv, while 0.341 and 0.344 are the corresponding values for fω.The close training and testing values indicate accurate prediction on unseen data without apparent overfitting.

C. Sinusoid-yaw Trajectory Tracking Using NN Models

The learned neural-network model is used to track a trajectory combining sinusoidal translation and yaw rotation, despite training on separate translational or rotational motions. It tracks closely and yields lower error than a model-free controller, supporting generalization to coupled motion.

Trajectory design: The evaluation trajectory combines sinusoidal XY-plane motion with simultaneous yaw rotation, requiring the model to capture nonlinear translational-rotational couplings.The trajectory therefore differs from the separately translational or rotational training motions.
Control setup: The full learned model is used to compute a dynamically feasible reference, which is flown with near-hover LQR and yaw rotation.The resulting experiment is labeled the NN model trajectory.
Generalization: The NN model trajectory tracks the desired combined-motion trajectory closely, demonstrating generalization beyond the training data.The trained networks capture nonlinear couplings and track trajectories on which they were not explicitly trained.
Model-free comparison: The model-free comparison removes open-loop control derived from the neural-network model, leaving the LQR controller and desired trajectory as the reference.This comparison tests how much tracking performance depends on the learned model’s open-loop signal.
Tracking performance: The NN model has significantly lower tracking error than the model-free trajectory, especially because its dynamics model supplies improved open-loop control.The paper links small tracking error to accurate dynamics modeling around the desired trajectory.
Overall finding: The experiments indicate that deep neural networks can learn quadrotor dynamics accurately and represent behavior beyond the data used for training.With appropriate architecture and inputs, the learned model can also be used for control.
Scope: The paper notes that a physics-based quadrotor model can also achieve good tracking, so the experiment tests neural-network efficacy rather than model superiority.It suggests combining physics-based modeling with neural networks to learn incremental unmodeled dynamics as future work.

VI. CONCLUSION

The conclusion reports that simple feed-forward neural networks can learn quadrotor dynamics accurately, generalize beyond training trajectories, and support effective control. It also identifies hybrid physics-based and neural-network modeling as a future direction.

Simple feed-forward neural networks generalize beyond training trajectories and learn quadrotor dynamics to good accuracy.The conclusion emphasizes that the learned function can be exploited for control, not merely approximation.
The experiments demonstrate that learned dynamics can be used effectively to control the quadrotor.
Neural networks are presented as an alternative for identifying complex quadrotor dynamics when deriving a physics-based model is difficult.
The paper does not compare neural-network models against physics-based models, although combining both may improve control performance.The proposed hybrid approach would use a neural network to learn incremental unmodeled dynamics on top of a physics-based model.