Enforcing Task-Specified Compliance Bounds for Humanoids via Anisotropic Lipschitz-Constrained Policies

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.