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

arXiv: https://arxiv.org/abs/2602.03511 · HTML: https://mlsanity.com/paper/2602.03511/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 39 of its 48 references that are in the map.
- First posted on arXiv 3 Feb 2026.

## Builds on (most cited first, 30 of 39)

- [Proximal Policy Optimization Algorithms](https://mlsanity.com/paper/1707.06347/) (2017)
- [Representation Learning with Contrastive Predictive Coding](https://mlsanity.com/paper/1807.03748/) (2018)
- [Learning Quadrupedal Locomotion over Challenging Terrain](https://mlsanity.com/paper/2010.11251/) (2020)
- [Learning to Walk in Minutes Using Massively Parallel Deep Reinforcement Learning](https://mlsanity.com/paper/2109.11978/) (2021)
- [RMA: Rapid Motor Adaptation for Legged Robots](https://mlsanity.com/paper/2107.04034/) (2021)
- [Learning by Cheating](https://mlsanity.com/paper/1912.12294/) (2019)
- [Efficient and Accurate Estimation of Lipschitz Constants for Deep Neural Networks](https://mlsanity.com/paper/1906.04893/) (2019)
- [OmniH2O: Universal and Dexterous Human-to-Humanoid Whole-Body Teleoperation and Learning](https://mlsanity.com/paper/2406.08858/) (2024)
- [Real-World Humanoid Locomotion with Reinforcement Learning](https://mlsanity.com/paper/2303.03381/) (2023)
- [Rapid Locomotion via Reinforcement Learning](https://mlsanity.com/paper/2205.02824/) (2022)
- [Walk These Ways: Tuning Robot Control for Generalization with Multiplicity of Behavior](https://mlsanity.com/paper/2212.03238/) (2022)
- [Learning Human-to-Humanoid Real-Time Whole-Body Teleoperation](https://mlsanity.com/paper/2403.04436/) (2024)
- [AttnLRP: Attention-Aware Layer-Wise Relevance Propagation for Transformers](https://mlsanity.com/paper/2402.05602/) (2024)
- [Reinforcement Learning for Robust Parameterized Locomotion Control of Bipedal Robots](https://mlsanity.com/paper/2103.14295/) (2021)
- [Humanoid Parkour Learning](https://mlsanity.com/paper/2406.10759/) (2024)
- [Humanoid Locomotion and Manipulation: Current Progress and Challenges in Control, Planning, and Learning](https://mlsanity.com/paper/2501.02116/) (2025)
- [Regularizing Action Policies for Smooth Control with Reinforcement Learning](https://mlsanity.com/paper/2012.06644/) (2020)
- [Learning Humanoid Locomotion with Perceptive Internal Model](https://mlsanity.com/paper/2411.14386/) (2024)
- [Learning Robust Autonomous Navigation and Locomotion for Wheeled-Legged Robots](https://mlsanity.com/paper/2405.01792/) (2024)
- [BeamDojo: Learning Agile Humanoid Locomotion on Sparse Footholds](https://mlsanity.com/paper/2502.10363/) (2025)
- [Attention-Based Map Encoding for Learning Generalized Legged Locomotion](https://mlsanity.com/paper/2506.09588/) (2025)
- [Hybrid Internal Model: Learning Agile Legged Locomotion with Simulated Robot Response](https://mlsanity.com/paper/2312.11460/) (2023)
- [Hierarchical Reinforcement Learning for Precise Soccer Shooting Skills using a Quadrupedal Robot](https://mlsanity.com/paper/2208.01160/) (2022)
- [Stability-certified reinforcement learning: A control-theoretic perspective](https://mlsanity.com/paper/1810.11505/) (2018)
- [Hamilton-Jacobi Deep Q-Learning for Deterministic Continuous-Time Systems with Lipschitz Continuous Controls](https://mlsanity.com/paper/2010.14087/) (2020)
- [Learning Smooth Humanoid Locomotion through Lipschitz-Constrained Policies](https://mlsanity.com/paper/2410.11825/) (2024)
- [L2C2: Locally Lipschitz Continuous Constraint towards Stable and Smooth Reinforcement Learning](https://mlsanity.com/paper/2202.07152/) (2022)
- [Gradient Regularization Improves Accuracy of Discriminative Models](https://mlsanity.com/paper/1712.09936/) (2017)
- [Learning to walk in confined spaces using 3D representation](https://mlsanity.com/paper/2403.00187/) (2024)
- [Model-free Reinforcement Learning for Robust Locomotion using Demonstrations from Trajectory Optimization](https://mlsanity.com/paper/2107.06629/) (2021)

## Abstract (arXiv)

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.