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SO(2)-Equivariant Downwash Models for Close Proximity Flight

H. Smith, A. Shankar, J. Gielis, J. Blumenkamp, A. Prorok

TL;DR

This paper presents a novel learning-based approach for modelling the downwash forces that exploits the latent geometries (i.e. symmetries) present in the problem and demonstrates that when trained with only 5 minutes of real-world flight data, the model outperforms state-of-the-art baseline models trained with more than 15 minutes of data.

Abstract

Multirotors flying in close proximity induce aerodynamic wake effects on each other through propeller downwash. Conventional methods have fallen short of providing adequate 3D force-based models that can be incorporated into robust control paradigms for deploying dense formations. Thus, learning a model for these downwash patterns presents an attractive solution. In this paper, we present a novel learning-based approach for modelling the downwash forces that exploits the latent geometries (i.e. symmetries) present in the problem. We demonstrate that when trained with only 5 minutes of real-world flight data, our geometry-aware model outperforms state-of-the-art baseline models trained with more than 15 minutes of data. In dense real-world flights with two vehicles, deploying our model online improves 3D trajectory tracking by nearly 36% on average (and vertical tracking by 56%).

SO(2)-Equivariant Downwash Models for Close Proximity Flight

TL;DR

This paper presents a novel learning-based approach for modelling the downwash forces that exploits the latent geometries (i.e. symmetries) present in the problem and demonstrates that when trained with only 5 minutes of real-world flight data, the model outperforms state-of-the-art baseline models trained with more than 15 minutes of data.

Abstract

Multirotors flying in close proximity induce aerodynamic wake effects on each other through propeller downwash. Conventional methods have fallen short of providing adequate 3D force-based models that can be incorporated into robust control paradigms for deploying dense formations. Thus, learning a model for these downwash patterns presents an attractive solution. In this paper, we present a novel learning-based approach for modelling the downwash forces that exploits the latent geometries (i.e. symmetries) present in the problem. We demonstrate that when trained with only 5 minutes of real-world flight data, our geometry-aware model outperforms state-of-the-art baseline models trained with more than 15 minutes of data. In dense real-world flights with two vehicles, deploying our model online improves 3D trajectory tracking by nearly 36% on average (and vertical tracking by 56%).
Paper Structure (16 sections, 1 theorem, 17 equations, 7 figures)

This paper contains 16 sections, 1 theorem, 17 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Related Work
  3. Contributions
  4. Problem Formulation
  5. Establishing Geometric Priors
  6. Geometric Invariance and Equivariance
  7. Geometric Assumptions on ${\boldsymbol{\mathrm{f}}}_{\mathrm{ext}}$
  8. Geometry-Aware Learning
  9. Rotationally Equivariant Model
  10. Proof of Equivariance
  11. Shallow Learning
  12. Real-world Flight Experiments
  13. Sequential Data Collection
  14. Study: Model Training
  15. Real-World Experiments
  16. ...and 1 more sections

Key Result

Theorem 1

The model $F_{\Theta}(\mathbf{x})$ proposed in (equivariantmodel) for ${\boldsymbol{\mathrm{f}}}_{\mathrm{ext}}(\mathbf{x})$ satisfies Assumption assumption:equivariance.

Figures (7)

  • Figure 1: A snapshot demonstrating the improvement in trajectory tracking with (green) and without (red) our downwash model.
  • Figure 2: An illustration of Assumption \ref{['assumption:equivariance']} on the downwash function ${\boldsymbol{\mathrm{f}}}_{\mathrm{ext}}(\mathbf{x})$. On the left, we provide two combinations of $(\Delta \mathbf{p}, \mathbf{v}^{\mathcal{B}})$ that are related under the rotational equivariance property.
  • Figure 3: Sample Efficiency and Accuracy. Top: A visualization of the validation RMSE of the equivariant and non-equivariant models as a function of the training flight time. For each training time, we compute the average validation RMSE across $5$ trials. Bottom: Summary statistics for the equivariant and non-equivariant models. Position and velocity tracking errors are reported for models trained on the full training dataset.
  • Figure 4: Downward Force Predictions. Downward force predictions [m/s^2] made by the equivariant model (top) and deep non-equivariant model (bottom). On the left (top-down view), Alpha is hovering 1m above Bravo at $(\hat{e}_1, \hat{e}_2) = (0,0)$. On the right (sagittal view), Alpha is hovering 0.1m east of Bravo at $(\hat{e}_1, \hat{e}_3) = (0,0)$. In each plot, Bravo is moving with velocity ${\boldsymbol{\mathrm{v}}}^{\mathcal{B}} = [0.5, 0, 0]^\top$.
  • Figure 5: Lateral Forces. Force predictions and errors during a transition under Alpha with a vertical separation of 0.8m. Alpha is hovering at $(\hat{e}_1, \hat{e}_2) = (0,0)$.
  • ...and 2 more figures

Theorems & Definitions (2)

  • Definition 1: Invariance, Equivariance
  • Theorem 1