# Task-Specified Compliance Bounds for Humanoids via Lipschitz-Constrained Policies

arXiv: https://arxiv.org/abs/2603.16180 · HTML: https://mlsanity.com/paper/2603.16180/

## 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 14 of its 22 references that are in the map.
- First posted on arXiv 17 Mar 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, 14 of 14)

- [Proximal Policy Optimization Algorithms](https://mlsanity.com/paper/1707.06347/) (2017)
- [AMASS: Archive of Motion Capture as Surface Shapes](https://mlsanity.com/paper/1904.03278/) (2019)
- [Learning to Walk in Minutes Using Massively Parallel Deep Reinforcement Learning](https://mlsanity.com/paper/2109.11978/) (2021)
- [Real-World Humanoid Locomotion with Reinforcement Learning](https://mlsanity.com/paper/2303.03381/) (2023)
- [Learning Human-to-Humanoid Real-Time Whole-Body Teleoperation](https://mlsanity.com/paper/2403.04436/) (2024)
- [Reinforcement Learning for Robust Parameterized Locomotion Control of Bipedal Robots](https://mlsanity.com/paper/2103.14295/) (2021)
- [Humanoid Locomotion and Manipulation: Current Progress and Challenges in Control, Planning, and Learning](https://mlsanity.com/paper/2501.02116/) (2025)
- [ResMimic: From General Motion Tracking to Humanoid Whole-body Loco-Manipulation via Residual Learning](https://mlsanity.com/paper/2510.05070/) (2025)
- [FALCON: Learning Force-Adaptive Humanoid Loco-Manipulation](https://mlsanity.com/paper/2505.06776/) (2025)
- [Learning Smooth Humanoid Locomotion through Lipschitz-Constrained Policies](https://mlsanity.com/paper/2410.11825/) (2024)
- [FACET: Force-Adaptive Control via Impedance Reference Tracking for Legged Robots](https://mlsanity.com/paper/2505.06883/) (2025)
- [HMC: Learning Heterogeneous Meta-Control for Contact-Rich Loco-Manipulation](https://mlsanity.com/paper/2511.14756/) (2025)
- [Spectral Normalization for Lipschitz-Constrained Policies on Learning Humanoid Locomotion](https://mlsanity.com/paper/2504.08246/) (2025)
- [CoTaP: Compliant Task Pipeline and Reinforcement Learning of Its Controller with Compliance Modulation](https://mlsanity.com/paper/2509.25443/) (2025)

## Abstract (arXiv)

Reinforcement learning (RL) has demonstrated substantial potential for humanoid bipedal locomotion and the control of complex motions. To cope with oscillations and impacts induced by environmental interactions, compliant control is widely regarded as an effective remedy. However, the model-free nature of RL makes it difficult to impose task-specified and quantitatively verifiable compliance objectives, and classical model-based stiffness designs are not directly applicable. Lipschitz-Constrained Policies (LCP), which regularize the local sensitivity of a policy via gradient penalties, have recently been used to smooth humanoid motions. Nevertheless, existing LCP-based methods typically employ a single scalar Lipschitz budget and lack an explicit connection to physically meaningful compliance specifications in real-world systems. In this study, we propose an anisotropic Lipschitz-constrained policy (ALCP) that maps a task-space stiffness upper bound to a state-dependent Lipschitz-style constraint on the policy Jacobian. The resulting constraint is enforced during RL training via a hinge-squared spectral-norm penalty, preserving physical interpretability while enabling direction-dependent compliance. Experiments on humanoid robots show that ALCP improves locomotion stability and impact robustness, while reducing oscillations and energy usage.