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

arXiv: https://arxiv.org/abs/2305.18983 · HTML: https://mlsanity.com/paper/2305.18983/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- Builds on 28 of its 49 references that are in the map.
- First posted on arXiv 30 May 2023.

## Builds on (most cited first, 28 of 28)

- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Geometric Deep Learning: Grids, Groups, Graphs, Geodesics, and Gauges](https://mlsanity.com/paper/2104.13478/) (2021)
- [Group Equivariant Convolutional Networks](https://mlsanity.com/paper/1602.07576/) (2016)
- [Geometric deep learning: going beyond Euclidean data](https://mlsanity.com/paper/1611.08097/) (2016)
- [E(3)-Equivariant Graph Neural Networks for Data-Efficient and Accurate Interatomic Potentials](https://mlsanity.com/paper/2101.03164/) (2021)
- [Directional Message Passing for Molecular Graphs](https://mlsanity.com/paper/2003.03123/) (2020)
- [MACE: Higher Order Equivariant Message Passing Neural Networks for Fast and Accurate Force Fields](https://mlsanity.com/paper/2206.07697/) (2022)
- [Spherical CNNs](https://mlsanity.com/paper/1801.10130/) (2018)
- [General E(2)-Equivariant Steerable CNNs](https://mlsanity.com/paper/1911.08251/) (2019)
- [Steerable CNNs](https://mlsanity.com/paper/1612.08498/) (2016)
- [Learning SO(3) Equivariant Representations with Spherical CNNs](https://mlsanity.com/paper/1711.06721/) (2017)
- [A General Theory of Equivariant CNNs on Homogeneous Spaces](https://mlsanity.com/paper/1811.02017/) (2018)
- [Geometric and Physical Quantities Improve E(3) Equivariant Message Passing](https://mlsanity.com/paper/2110.02905/) (2021)
- [Clebsch-Gordan Nets: a Fully Fourier Space Spherical Convolutional Neural Network](https://mlsanity.com/paper/1806.09231/) (2018)
- [Neural Lander: Stable Drone Landing Control using Learned Dynamics](https://mlsanity.com/paper/1811.08027/) (2018)
- [SO(2)-Equivariant Reinforcement Learning](https://mlsanity.com/paper/2203.04439/) (2022)
- [Equivariant Q Learning in Spatial Action Spaces](https://mlsanity.com/paper/2110.15443/) (2021)
- [Sample Efficient Grasp Learning Using Equivariant Models](https://mlsanity.com/paper/2202.09468/) (2022)
- [On-Robot Learning With Equivariant Models](https://mlsanity.com/paper/2203.04923/) (2022)
- [Safe Learning of Quadrotor Dynamics Using Barrier Certificates](https://mlsanity.com/paper/1710.05472/) (2017)
- [The Surprising Effectiveness of Equivariant Models in Domains with Latent Symmetry](https://mlsanity.com/paper/2211.09231/) (2022)
- [KNODE-MPC: A Knowledge-based Data-driven Predictive Control Framework for Aerial Robots](https://mlsanity.com/paper/2109.04821/) (2021)
- [Theoretical Aspects of Group Equivariant Neural Networks](https://mlsanity.com/paper/2004.05154/) (2020)
- [Neural-Swarm: Decentralized Close-Proximity Multirotor Control Using Learned Interactions](https://mlsanity.com/paper/2003.02992/) (2020)
- [Neural-Swarm2: Planning and Control of Heterogeneous Multirotor Swarms using Learned Interactions](https://mlsanity.com/paper/2012.05457/) (2020)
- [EGO-Swarm: A Fully Autonomous and Decentralized Quadrotor Swarm System in Cluttered Environments](https://mlsanity.com/paper/2011.04183/) (2020)
- [Equivariant Reinforcement Learning for Quadrotor UAV](https://mlsanity.com/paper/2206.01233/) (2022)
- [Nonlinear MPC for Quadrotors in Close-Proximity Flight with Neural Network Downwash Prediction](https://mlsanity.com/paper/2304.07794/) (2023)

## Built on by (most cited first, 7 of 7)

- [Morphological-Symmetry-Equivariant Heterogeneous Graph Neural Network for Robotic Dynamics Learning](https://mlsanity.com/paper/2412.01297/) (2024)
- [ProxFly: Robust Control for Close Proximity Quadcopter Flight via Residual Reinforcement Learning](https://mlsanity.com/paper/2409.13193/) (2024)
- [Docking Multirotors in Close Proximity using Learnt Downwash Models](https://mlsanity.com/paper/2311.13988/) (2023)
- [Flying Quadrotors in Tight Formations using Learning-based Model Predictive Control](https://mlsanity.com/paper/2410.09727/) (2024)
- [Symmetries-enhanced Multi-Agent Reinforcement Learning](https://mlsanity.com/paper/2501.01136/) (2025)
- [Multi-Task Reinforcement Learning of Drone Aerobatics by Exploiting Geometric Symmetries](https://mlsanity.com/paper/2602.10997/) (2026)
- [Wake Up to the Past: Using Memory to Model Fluid Wake Effects on Robots](https://mlsanity.com/paper/2603.22472/) (2026)

## Abstract (arXiv)

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%).