# Pioneering SE(2)-Equivariant Trajectory Planning for Automated Driving

arXiv: https://arxiv.org/abs/2403.11304 · HTML: https://mlsanity.com/paper/2403.11304/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Placed in End-to-end system architecture, one of 9 directions in End-to-end autonomous driving.
- Builds on 47 of its 51 references that are in the map.
- First posted on arXiv 17 Mar 2024.

## Placed in

- End-to-end autonomous driving / [End-to-end system architecture](https://mlsanity.com/driving/end-to-end-system-architecture/)

## Builds on (most cited first, 30 of 47)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Semi-Supervised Classification with Graph Convolutional Networks](https://mlsanity.com/paper/1609.02907/) (2016)
- [nuScenes: A multimodal dataset for autonomous driving](https://mlsanity.com/paper/1903.11027/) (2019)
- [Group Equivariant Convolutional Networks](https://mlsanity.com/paper/1602.07576/) (2016)
- [Planning-oriented Autonomous Driving](https://mlsanity.com/paper/2212.10156/) (2022)
- [E(n) Equivariant Graph Neural Networks](https://mlsanity.com/paper/2102.09844/) (2021)
- [Learning to Simulate Complex Physics with Graph Networks](https://mlsanity.com/paper/2002.09405/) (2020)
- [Tensor field networks: Rotation- and translation-equivariant neural networks for 3D point clouds](https://mlsanity.com/paper/1802.08219/) (2018)
- [VAD: Vectorized Scene Representation for Efficient Autonomous Driving](https://mlsanity.com/paper/2303.12077/) (2023)
- [SE(3)-Transformers: 3D Roto-Translation Equivariant Attention Networks](https://mlsanity.com/paper/2006.10503/) (2020)
- [VectorNet: Encoding HD Maps and Agent Dynamics from Vectorized Representation](https://mlsanity.com/paper/2005.04259/) (2020)
- [General E(2)-Equivariant Steerable CNNs](https://mlsanity.com/paper/1911.08251/) (2019)
- [ST-P3: End-to-end Vision-based Autonomous Driving via Spatial-Temporal Feature Learning](https://mlsanity.com/paper/2207.07601/) (2022)
- [3D Steerable CNNs: Learning Rotationally Equivariant Features in Volumetric Data](https://mlsanity.com/paper/1807.02547/) (2018)
- [Learning from Protein Structure with Geometric Vector Perceptrons](https://mlsanity.com/paper/2009.01411/) (2020)
- [Vector Neurons: A General Framework for SO(3)-Equivariant Networks](https://mlsanity.com/paper/2104.12229/) (2021)
- [Harmonic Networks: Deep Translation and Rotation Equivariance](https://mlsanity.com/paper/1612.04642/) (2016)
- [Multimodal Trajectory Predictions for Autonomous Driving using Deep Convolutional Networks](https://mlsanity.com/paper/1809.10732/) (2018)
- [End-to-end Interpretable Neural Motion Planner](https://mlsanity.com/paper/2101.06679/) (2021)
- [Parting with Misconceptions about Learning-based Vehicle Motion Planning](https://mlsanity.com/paper/2306.07962/) (2023)
- [Is Ego Status All You Need for Open-Loop End-to-End Autonomous Driving?](https://mlsanity.com/paper/2312.03031/) (2023)
- [Learning Steerable Filters for Rotation Equivariant CNNs](https://mlsanity.com/paper/1711.07289/) (2017)
- [Rethinking the Open-Loop Evaluation of End-to-End Autonomous Driving in nuScenes](https://mlsanity.com/paper/2305.10430/) (2023)
- [PRECOG: PREdiction Conditioned On Goals in Visual Multi-Agent Settings](https://mlsanity.com/paper/1905.01296/) (2019)
- [GameFormer: Game-theoretic Modeling and Learning of Transformer-based Interactive Prediction and Planning for Autonomous Driving](https://mlsanity.com/paper/2303.05760/) (2023)
- [Incorporating Symmetry into Deep Dynamics Models for Improved Generalization](https://mlsanity.com/paper/2002.03061/) (2020)
- [THOMAS: Trajectory Heatmap Output with learned Multi-Agent Sampling](https://mlsanity.com/paper/2110.06607/) (2021)
- [Rotation equivariant vector field networks](https://mlsanity.com/paper/1612.09346/) (2016)
- [EqMotion: Equivariant Multi-agent Motion Prediction with Invariant Interaction Reasoning](https://mlsanity.com/paper/2303.10876/) (2023)
- [LookOut: Diverse Multi-Future Prediction and Planning for Self-Driving](https://mlsanity.com/paper/2101.06547/) (2021)

## Built on by (most cited first, 4 of 4)

- [LaneCPP: Continuous 3D Lane Detection using Physical Priors](https://mlsanity.com/paper/2406.08381/) (2024)
- [When Planners Meet Reality: How Learned, Reactive Traffic Agents Shift nuPlan Benchmarks](https://mlsanity.com/paper/2510.14677/) (2025)
- [Learning Through Retrospection: Improving Trajectory Prediction for Automated Driving with Error Feedback](https://mlsanity.com/paper/2504.13785/) (2025)
- [Dynamic Intent Queries for Motion Transformer-based Trajectory Prediction](https://mlsanity.com/paper/2504.15766/) (2025)

## Abstract (arXiv)

Planning the trajectory of the controlled ego vehicle is a key challenge in automated driving. As for human drivers, predicting the motions of surrounding vehicles is important to plan the own actions. Recent motion prediction methods utilize equivariant neural networks to exploit geometric symmetries in the scene. However, no existing method combines motion prediction and trajectory planning in a joint step while guaranteeing equivariance under roto-translations of the input space. We address this gap by proposing a lightweight equivariant planning model that generates multi-modal joint predictions for all vehicles and selects one mode as the ego plan. The equivariant network design improves sample efficiency, guarantees output stability, and reduces model parameters. We further propose equivariant route attraction to guide the ego vehicle along a high-level route provided by an off-the-shelf GPS navigation system. This module creates a momentum from embedded vehicle positions toward the route in latent space while keeping the equivariance property. Route attraction enables goal-oriented behavior without forcing the vehicle to stick to the exact route. We conduct experiments on the challenging nuScenes dataset to investigate the capability of our planner. The results show that the planned trajectory is stable under roto-translations of the input scene which demonstrates the equivariance of our model. Despite using only a small split of the dataset for training, our method improves L2 distance at 3 s by 20.6 % and surpasses the state of the art.