# SKooP: Symmetric Koopman Predictions for Faster and More Generalizable Legged Robot Locomotion with Reinforcement Learning

arXiv: https://arxiv.org/abs/2607.11624 · HTML: https://mlsanity.com/paper/2607.11624/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Placed in RL for robot locomotion and manipulation, one of 15 directions in Deep reinforcement learning methods.
- Builds on 21 of its 33 references that are in the map.
- First posted on arXiv 13 Jul 2026.

## Placed in

- Deep reinforcement learning methods / [RL for robot locomotion and manipulation](https://mlsanity.com/deeprl/rl-for-robot-locomotion-and-manipulation/)

## Builds on (most cited first, 21 of 21)

- [Prioritized Experience Replay](https://mlsanity.com/paper/1511.05952/) (2015)
- [Isaac Gym: High Performance GPU-Based Physics Simulation For Robot Learning](https://mlsanity.com/paper/2108.10470/) (2021)
- [Learning agile and dynamic motor skills for legged robots](https://mlsanity.com/paper/1901.08652/) (2019)
- [Deep learning for universal linear embeddings of nonlinear dynamics](https://mlsanity.com/paper/1712.09707/) (2017)
- [Modern Koopman Theory for Dynamical Systems](https://mlsanity.com/paper/2102.12086/) (2021)
- [Asymmetric Actor Critic for Image-Based Robot Learning](https://mlsanity.com/paper/1710.06542/) (2017)
- [Real-World Humanoid Locomotion with Reinforcement Learning](https://mlsanity.com/paper/2303.03381/) (2023)
- [Linear predictors for nonlinear dynamical systems: Koopman operator meets model predictive control](https://mlsanity.com/paper/1611.03537/) (2016)
- [Geometric Deep Learning and Equivariant Neural Networks](https://mlsanity.com/paper/2105.13926/) (2021)
- [Generalizing Koopman Theory to allow for inputs and control](https://mlsanity.com/paper/1602.07647/) (2016)
- [Optimal construction of Koopman eigenfunctions for prediction and control](https://mlsanity.com/paper/1810.08733/) (2018)
- [Symmetry Considerations for Learning Task Symmetric Robot Policies](https://mlsanity.com/paper/2403.04359/) (2024)
- [Koopman Q-learning: Offline Reinforcement Learning via Symmetries of Dynamics](https://mlsanity.com/paper/2111.01365/) (2021)
- [Leveraging Symmetry in RL-based Legged Locomotion Control](https://mlsanity.com/paper/2403.17320/) (2024)
- [Learning Agile Bipedal Motions on a Quadrupedal Robot](https://mlsanity.com/paper/2311.05818/) (2023)
- [Euclidean, Projective, Conformal: Choosing a Geometric Algebra for Equivariant Transformers](https://mlsanity.com/paper/2311.04744/) (2023)
- [Koopman-Assisted Reinforcement Learning](https://mlsanity.com/paper/2403.02290/) (2024)
- [Dynamics Harmonic Analysis of Robotic Systems: Application in Data-Driven Koopman Modelling](https://mlsanity.com/paper/2312.07457/) (2023)
- [Continual Learning and Lifting of Koopman Dynamics for Linear Control of Legged Robots](https://mlsanity.com/paper/2411.14321/) (2024)
- [Sample-Efficient Reinforcement Learning of Koopman eNMPC](https://mlsanity.com/paper/2503.18787/) (2025)
- [KIPPO: Koopman-Inspired Proximal Policy Optimization](https://mlsanity.com/paper/2505.14566/) (2025)

## Abstract (arXiv)

Reinforcement learning (RL) algorithms classically suffer from poor sample efficiency. In robotics, a recent line of work has emerged addressing this problem by encoding physics priors in the learning process. However, most of these approaches are validated on well-defined, low-dimensional benchmark systems rather than high-dimensional robots with complex nonlinear dynamics. In this paper, we introduce textitSKooP (Symmetric Koopman Predictions), an approach combining the advantages of morphological symmetries with those of a Koopman model learned via autoencoder to enhance policy learning. SKooP learns a Koopman model of the system dynamics alongside the policy. The resulting Koopman predictions are used as privileged observations for the critic, allowing the agent to learn based on smoother, more informative features. We also incorporate group symmetries into the actor, critic, encoder and decoder networks to produce a highly equivariant policy. The SKooP approach is validated via in-depth analysis of the learned Koopman models and symmetric policies to showcase how each of these influences the agent's performance. We also show that the learned policies are transferable to different simulation environments. Our results show that SKooP consistently reduces convergence time and increases the learned reward for multiple challenging bipedal locomotion tasks on a quadruped robot. Project page: https://evelyd.github.io/SymmetricKoopmanPredictions