CMR: Contractive Mapping Embeddings for Robust Humanoid Locomotion on Unstructured Terrains

Robust disturbance rejection remains a longstanding challenge in humanoid locomotion, particularly on unstructured terrains where sensing is unreliable and model mismatch is pronounced. While perception information, such as height map, enhances terrain awareness, sensor noise and sim-to-real gaps can destabilize policies in practice. In this work, we provide theoretical analysis that bounds the return gap under observation noise, when the induced latent dynamics are contractive. Furthermore, we present Contractive Mapping for Robustness (CMR) framework that maps high-dimensional, disturbance-prone observations into a latent space, where local perturbations are attenuated over time. Specifically, this approach couples contrastive representation learning with Lipschitz regularization to preserve task-relevant geometry while explicitly controlling sensitivity. Notably, the formulation can be incorporated into modern deep reinforcement learning pipelines as an auxiliary loss term with minimal additional technical effort required. Further, our extensive humanoid experiments show that CMR potently outperforms other locomotion algorithms under increased noise.