SWAP: Symmetric Equivariant World-Model for Agile Robot Parkour

Abstract: 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.

• The paper introduces SWAP, a framework that enforces strict symmetry equivariance across world modeling and policy networks to reduce redundancy and improve coordinated parkour maneuvers.
• It employs group-equivariant encoders like SE-CNNs and SE-MLPs along with a symmetry-equivariant RSSM to capture mirrored state-action dynamics for efficient simulation and real-world deployment.
• Empirical evaluations show that SWAP achieves record performances (e.g., 2.8m gap leaps and 1.63m climbs) and significantly outperforms unconstrained models in extreme quadruped locomotion.

Symmetric Equivariant World Modeling for Extreme Quadruped Parkour

Motivation and Problem Statement

Legged robot locomotion, particularly agile parkour maneuvers on challenging terrains, places acute demands on perception, control, and generalization. The bilateral symmetry inherent in quadruped morphology and the invariant nature of physical laws under mirroring are geometric priors that are underexploited in conventional latent world-model-based reinforcement learning (RL) approaches. Latent world models such as RSSM-driven architectures, while capable in standard tasks, must redundantly encode mirror-symmetric interactions, inflating sample complexity and hindering downstream policy efficiency—especially under extreme agility requirements where only a narrow fraction of state-action space remains feasible.

SWAP Framework: Architectural Design

The Symmetric Equivariant World-Model for Agile Robot Parkour (SWAP) introduces strict symmetry equivariance constraints across perception, latent dynamics modeling, and motor control policy. The world model employs group-equivariant encoders (SE-CNNs for depth images, SE-MLPs for proprioception), a symmetry-equivariant RSSM (SE-GRU), and equivariant stochastic latent states, enforcing homomorphic mirroring in latent representations. The symmetry group actions on the observation, action, and state spaces are embedded directly in the computational graph at all levels, yielding symmetry-preserving data transformations.

Control is realized via a high-frequency symmetric policy module comprising an equivariant actor and an invariant critic. The actor network's topology is SE-MLP parameterized, guaranteeing mirrored actions under mirrored state input, while the critic output is strictly invariant, ensuring value consistency. AMP-based reward regularization also adopts symmetry-equivariant network structures, enforcing invariant discriminator scores under reflection transformations.

Theoretical Implications and Policy Behavior

Constrained by the symmetric MDP formalism, the optimal policy under SWAP strictly satisfies

$\pi^*(F_s(s)) = F_a(\pi^*(s)),$

implying that optimal actions for mirrored states are simply the symmetric counterparts. This reduces redundancy, narrows the exploration space, and promotes convergence to physically consistent, coordinated contact strategies necessary for extreme maneuvers. Joint symmetry constraints across world modeling and policy serve to avoid asymmetric suboptimal convergence, typical in large state-action contact spaces (e.g., box climbing).

Empirical Results

Simulation and Ablation

SWAP was evaluated against baselines including unconstrained world models, partial symmetry constraints (only the world model or actor), and symmetry penalty loss [18]. In symmetric terrain transfer tasks (mirrored box, stairs, gap scenarios), SWAP exhibited superior zero-shot generalization—success rates in the mirrored test environments remained high in the fully equivariant configuration, while unconstrained baselines deteriorated rapidly with increasing difficulty.

Pixel-wise latent state reconstructions confirmed that SWAP's embedding captured genuine symmetry, avoiding the increased reconstruction error suffered by unconstrained models on mirrored terrains.

Extreme Locomotion Capabilities

In high-difficulty simulation curricula, SWAP achieved the highest success rates for both gap leaping (up to 2.8m) and box climbing (up to 1.9m), exhibiting coordinated bilateral contact and impulse-generation strategies. Baselines converged to inefficient or asymmetric behaviors, with unconstrained models degenerating into wall-colliding motions to evade failure penalties.

Real-World Deployment

Zero-shot deployment on Apollo quadruped was realized without environment-specific fine-tuning. SWAP achieved absolute record-breaking parkour metrics: a 2.13m gap leap and a 1.63m platform climb—outperforming prior systems in both normalized and absolute performance. Robust geometric generalization was observed in diverse indoor and outdoor environments, including wet platforms, specular reflections, tall grass, and loose rubble. The policy maintained dynamic stability under substantial visual and contact uncertainty, confirming the practical utility of symmetry equivariance as a structural prior.

Practical and Theoretical Implications

SWAP demonstrates that hierarchical symmetry embedding across world models and policy networks is highly effective in reducing exploratory redundancy and enforcing coordinated action patterns in extreme agile tasks. This validates symmetry equivariance as a critical structural prior for pushing the physical boundaries of learned locomotion. The practical implications extend to more sample-efficient training, superior sim-to-real transfer, and enhanced robustness under broad environmental perturbations.

From a theoretical perspective, retaining symmetry in world modeling may be generalized to other physical systems (e.g., manipulation, humanoid control), and prompts further exploration of group-theoretic regularization in RL.

Future Directions

While SWAP achieves robust performance on structured parkour scenarios, evaluation in environments with more discrete and irregular footholds (e.g., stepping stones, cluttered outdoor terrains) remains open. Future work may integrate richer terrain distributions and extend symmetry constraints to multi-agent or multi-limb systems, further advancing the applicability of equivariant RL frameworks.

Conclusion

SWAP provides a symmetry-equivariant, end-to-end world modeling and policy learning framework capable of extreme quadruped parkour with record-breaking physical performance and robust zero-shot transfer. By enforcing symmetry equivariance across perception, dynamics, and control, the approach efficiently leverages geometric priors, yielding superior performance under extreme agility constraints and facilitating stable, coordinated motor policies with strong generalization across diverse environments (2606.19928).