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Data-Driven MPC for Quadrotors
Guillem Torrente, Elia Kaufmann, Philipp Foehn, Davide Scaramuzza
TL;DR
High-speed quadrotor tracking is challenged by aerodynamic effects that are difficult to model accurately and efficiently for real-time control. The paper learns residual aerodynamic dynamics with Gaussian Processes and integrates them into MPC, improving tracking by up to 70% in simulation and real-world tests.
Problem
Accurate high-speed, high-acceleration quadrotor tracking requires modeling complex aerodynamic effects while keeping the model efficient enough for real-time control.
Method
The approach learns residual aerodynamic dynamics with Gaussian Processes and combines them with nominal quadrotor dynamics inside an MPC pipeline.
Results
Up to 70% trajectory-tracking improvement over the nominal counterpart is demonstrated in synthetic and real-world experiments at speeds up to 14m s−1 and accelerations beyond 4g.
Takeaways & Limitations
The GP-augmented controller outperforms methods relying on simplified correction terms and generalizes between different trajectories.
Abstract
from arXiv · showhide
Aerodynamic forces render accurate high-speed trajectory tracking with quadrotors extremely challenging. These complex aerodynamic effects become a significant disturbance at high speeds, introducing large positional tracking errors, and are extremely difficult to model. To fly at high speeds, feedback control must be able to account for these aerodynamic effects in real-time. This necessitates a modelling procedure that is both accurate and efficient to evaluate. Therefore, we present an approach to model aerodynamic effects using Gaussian Processes, which we incorporate into a Model Predictive Controller to achieve efficient and precise real-time feedback control, leading to up to 70% reduction in trajectory tracking error at high speeds. We verify our method by extensive comparison to a state-of-the-art linear drag model in synthetic and real-world experiments at speeds of up to 14m/s and accelerations beyond 4g.
I. INTRODUCTION
High-speed, agile quadrotor flight is difficult because aerodynamic effects are complex, significant, and hard to model within real-time control constraints. The paper learns residual aerodynamic dynamics from data and integrates them into MPC for improved agile-trajectory control.
- Motivation: High-speed and high-acceleration tracking requires accounting for complex aerodynamic effects, including rotor, fuselage, and interaction-induced forces.These effects are difficult to model because they depend on induced airstream velocity and can include turbulent interactions.
- Motivation: Beyond 5 m s−1 and 2g, prior agile-control work generally does not account for aerodynamic effects, while iterative learning control is constrained to single maneuvers.The cited work reports significant errors from neglecting aerodynamics during fast, agile trajectories and limited generalization for iterative learning control.
- Motivation: Real-time control requires a model that balances aerodynamic-model accuracy against computational complexity.Sophisticated computational-fluid-dynamics simulations can require hours on a compute cluster and must be simplified for high-frequency control loops.
- Approach: The proposed approach learns aerodynamic effects as residual dynamics relative to a simplified quadrotor model using Gaussian Processes.Learning only the residual permits a small model described with few inducing points.
- Results: The learned model is combined with MPC, improving control performance on agile trajectories up to 14 m s−1 and beyond 4g in simulation and real-world experiments.The experiments also report generalization between trajectories and outperforming methods based on simplified correction terms.
Contributions
The paper augments nominal quadrotor dynamics with Gaussian-Process residuals inside MPC, targeting aerodynamic effects while retaining computational tractability and full actuation-limit modeling. It positions this approach against fully learned dynamics, linear drag compensation, and disturbance-reactive GP methods.
- Contributions: The MPC pipeline combines nominal quadrotor dynamics with learned Gaussian-Process residual dynamics to model aerodynamic effects.The approach uses three-dimensional GP predictions and a small number of inducing data points for efficient optimization.
- Related work: Fully learned neural dynamics can be expressive, but their optimization is often intractable because of local minima.Sampling-based optimizers can address this challenge but introduce their own computational trade-offs.
- Related work: Combining a nominal model with a learned correction limits learning to effects that are difficult to capture with the nominal system.This formulation can also give the learned dynamics a different dimensionality from the nominal system.
- Related work: Aerodynamic disturbances in quadrotors arise from rotor and fuselage drag and high-speed lift, with significant effects reported from 5 m s−1.Prior work mainly models rotor drag as a lumped linear effect and uses it for PID feedforward compensation.
- Novelty: Unlike disturbance-reactive GP methods, this work predicts aerodynamic effects from fast ego-motion and integrates them tightly into full-state MPC down to motor inputs.The formulation accounts for the platform’s true actuation limits.
III. METHODOLOGY
The methodology models the quadrotor as a 13-dimensional rigid-body system driven by individual rotor thrusts. Continuous nominal dynamics are integrated in discrete time with fourth-order Runge–Kutta integration.
- Nominal Quadrotor Dynamics Model: The quadrotor is modeled as a 6-degree-of-freedom rigid body with mass m and diagonal inertia matrix J.The nominal state space has 13 dimensions and uses individual rotor thrusts as control inputs.
- Nominal Quadrotor Dynamics Model: The nominal dynamics include collective thrust, gravity, and body torque effects in the translational and rotational equations.Body torque is formed from rotor thrusts, rotor displacements, and the rotor drag torque constant.
- Nominal Quadrotor Dynamics Model: Rotor thrusts determine the body torque through the rotor geometry and drag-torque constant cτ.The rotor displacements are denoted dx and dy.
- Nominal Quadrotor Dynamics Model: The continuous dynamics are discretized using an explicit fourth-order Runge–Kutta method with state xk, input uk, and integration step δt.The resulting discrete transition is represented by xk+1 = f RK4(xk, uk, δt).
C. Gaussian Process-Augmented Dynamics
The method augments nominal quadrotor dynamics with Gaussian Processes that learn and correct dynamics errors for MPC prediction. It assumes Gaussian noise and uses separate one-dimensional output GPs for each state dimension.
- Gaussian Process-Augmented Dynamics: Gaussian Processes predict quadrotor dynamics errors and correct the nominal model at each time instance.The corrected model is propagated forward within MPC using Runge-Kutta integration.
- Gaussian Process-Augmented Dynamics: Gaussian noise with time-invariant diagonal covariance permits independent treatment of each output dimension through separate one-dimensional GPs.
- Gaussian Process-Augmented Dynamics: The model uses an RBF kernel with diagonal length scales, data and prior noise variances, and feature vectors as its inputs.
- Gaussian Process-Augmented Dynamics: GP predictions are computed from training and query feature samples through the GP mean and covariance.
D. MPC Formulation
The MPC tracks a reference by minimizing a cost subject to nominal or corrected dynamics and system constraints. The implementation uses multiple shooting and real-time sequential quadratic programming.
- MPC Formulation: MPC minimizes a cost while stabilizing the system along reference state and input trajectories.
- MPC Formulation: The optimization enforces system dynamics, an initial condition, equality constraints, and inequality constraints such as input limits.
- MPC Formulation: The application uses a quadratic tracking cost over N discretization steps across a horizon T, with 0 ≤u ≤umax constraints.
- MPC Formulation: The system dynamics can use the GP-corrected model through an extended Runge-Kutta integration function.
- MPC Formulation: The quadratic optimization is constructed with multiple shooting and solved by SQP in a real-time iteration scheme using ACADOS and CasADi.
E. Practical Implementation
The learned dynamics implementation maps body-frame velocities to body-frame acceleration disturbances while targeting low computational cost. The mapping is expressed componentwise across velocity axes.
- Practical Implementation: The implementation seeks to maximize GP-MPC performance while minimizing computational cost added to optimization.
- Practical Implementation: Aerodynamic effects are represented in the body reference frame rather than an unspecified coordinate frame.
- Practical Implementation: The learning problem identifies a mapping from body-frame velocities Bv to body-frame acceleration disturbances Bae, written as Bae = µ(Bv).
- Practical Implementation: The learned mapping is represented through separate velocity components µvx(Bvxk), µvy(Bvyk), and µvz(Bvzk).
F. Data Collection and Model Learning
The GPs are trained from real-world flight data by converting velocity prediction discrepancies into acceleration errors. Kernel hyperparameters are optimized, while regular subsampling keeps real-time MPC tractable.
- Data Collection and Model Learning: Real-world flight data records actual and nominally predicted next-sample velocities together with each timestep.
- Data Collection and Model Learning: Time-normalized velocity errors are computed as acceleration errors for GP fitting.
- Data Collection and Model Learning: Kernel hyperparameters l, σn, and σf are selected by maximum likelihood optimization on the collected dataset.
- Data Collection and Model Learning: A small set of inducing points is sampled at regular intervals because full-dataset GP regression would make real-time MPC optimization intractable.
IV. EXPERIMENTS AND RESULTS
The evaluation compares nominal, linear-drag, and GP-augmented MPC across simulated and real quadrotor trajectories, while examining generalization and computational design choices.
- The evaluation addresses closed-loop contribution, comparison with linear aerodynamic compensation, generalization to unseen trajectories, and ablation of design choices.
- Three trajectories—Random, Circle, and Lemniscate—are used to assess tracking performance with nominal and data-augmented MPC models.
- Simulation training data are collected from aggressive random polynomial trajectories, after which Nominal, RDRv, and GP-MPC controllers are tested without retraining.
- Real-world experiments use nominal runs on circle and lemniscate trajectories for model fitting, followed by RDRv and GP-MPC rollouts in different permutations.
- The experiments consider obstacle-free scenarios focused on aerodynamic-disturbance tracking error, without using predicted GP covariance.
B. Experiments in Simulation
Simulation experiments separate simplified and Gazebo settings to study computation, model size, tracking across speeds, and behavior under simulated aerodynamic effects.
- Simplified Simulation: Simplified simulation uses fourth-order Runge-Kutta integration with 0.5 ms steps, perfect odometry, ideal thrust tracking, and instantaneous MPC computation.
- Simplified Simulation: The GP-size study examines the trade-off between predictive performance and optimization time, then evaluates different GP sizes in closed-loop tracking.
- Simplified Simulation: Closed-loop tests compare Nominal, RDRv, and GP-MPC on circle and lemniscate trajectories at varying speeds, using aggressive random-trajectory data up to 16 m s−1 axis-wise.
- Simplified Simulation: 15–25 inducing samples provide the identified optimal range, with 4–5 ms optimization time, because performance barely increases beyond 20 samples.
- Simplified Simulation: Both RDRv and GP-MPC significantly improve over Nominal, but RDRv fails above 4 m s−1 while GP-MPC improves consistently across tested speeds.
- Gazebo Simulation: Gazebo experiments use the AscTec Hummingbird with RotorS, ground-truth odometry, and training velocities from −12 to 12 m s−1.
- Gazebo Simulation: In Gazebo, RDRv and GP-MPC perform similarly because RotorS models only rotor drag, which follows a linear mapping with body-frame velocity.
C. Experiments in the Real World
Real-world experiments compare GP-MPC with Nominal and RDRv controllers on fast circle and lemniscate trajectories, including a modified platform with asymmetric aerodynamic disturbance. GP-MPC slightly outperforms RDRv on the compact platform and significantly outperforms it when a drag board introduces stronger nonlinear effects.
- Experimental setup: The real-world platform weighs 0.8 kg, has a 5:1 thrust-to-weight ratio, and receives control commands at 50 Hz.Pose estimates are provided at 100 Hz by an optical tracking system.
- Default platform: Up to 50%, GP-MPC and RDRv improve tracking performance in the default real-world setting, with GP-MPC slightly outperforming RDRv.The models are trained on circle and lemniscate trajectories and tested across their permutations without retraining.
- Default platform: Rotor drag dominates disturbance on the compact, powerful platform, making the linear RDRv augmentation effective in the default setting.GP-MPC’s slight advantage is attributed to its ability to account for imperfect thrust mappings.
- Drag-board platform: With a vertical drag board, GP-MPC significantly outperforms RDRv because the linear model fails to capture the full nonlinearity of the aerodynamic effects.The corresponding linear fit produces significant bias in the measured aerodynamic effects.
V. CONCLUSION
The paper augments nominal quadrotor dynamics with Gaussian-Process residual models inside an MPC controller to compensate for aerodynamic effects. Experiments in simulation and the real world show improved tracking and performance beyond a state-of-the-art linear drag model.
- Conclusion: Gaussian Processes augment nominal quadrotor dynamics to compensate for aerodynamic effects in a Model Predictive Controller.The GP models predict nominal-model acceleration error from current body-frame velocity using previously recorded flight data.
- Conclusion: The GP-augmented controller significantly improves positional tracking error in simulation and on a real quadrotor.The conclusion reports this improvement across the evaluated experiments.
- Conclusion: The approach outperforms a state-of-the-art linear drag model in extensive simulation and real-world experiments.The authors identify online adaptation to changing wind or battery conditions as a future direction enabled by fast GP fitting.