While latent world models enable the proactive predictions required for extreme parkour, their purely data-driven nature forces them to redundantly encode left-right symmetric interactions as independent patterns. This inflates the learning burden and hinders the capture of geometric regularities, restricting the latent space's efficiency for downstream policies. To address this, we propose SWAP, an end-to-end equivariant symmetric world model. This framework embeds symmetry directly into both the world model and the actor-critic networks. In real-world tests, the robot leaps across a 2.13 m gap and climbs a 1.63 m platform, breaking records for quadruped parkour. Furthermore, the framework exhibits robust geometric generalization to unseen mirrored terrains and exceptional zero-shot transferability across diverse outdoor environments. These results demonstrate that symmetry equivariance is an effective structural prior for pushing the physical boundaries of learned legged locomotion.
SWAP embeds physical symmetry directly into both the latent world model and the actor-critic networks for agile locomotion. The framework features a low-frequency Symmetric Equivariant World Model that maps mirrored physical observations to mirrored latent states. Conditioned on this geometry-aware representation, a high-frequency Equivariant Actor generates symmetric equivariant actions, while an Invariant Critic ensures identical value estimations across mirrored states.
@article{lan2026swap,
title={SWAP: Symmetric Equivariant World-Model for Agile Robot Parkour},
author={Lan, Kaixin and Wang, Ze and Li, Hongyi and Jiang, Lei and Fu, Chaojie and Su, Chengkai and Wong, Choi Lam and Jin, Yongbin and Wang, Hongtao},
journal={arXiv preprint arXiv:2606.19928},
year={2026}
}