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

Zewen He; Yoshihiko Nakamura

cs.ROMar 17, 2026

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

Authors: Zewen He, Yoshihiko Nakamura

Organizations: Department of Robotics, Mohamed bin Zayed University of Artificial Intelligence, Masdar City, Abu Dhabi, United Arab Emirates

Abstract

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.

Figures & tables

Panel of Figure 1 — Fig. 1: Task-specific compliance. (a) Hand-pulling experiment under the SILC method. (b) Hand impact experiment. A 2.5 kg weight is attached to the robot’s right hand via a rope and released from a certain height to generate an impact. (c) Stepping motion experiment. The robot’s one foot intermittently steps onto a foam pad placed on the ground, thereby introducing disturbances from uneven terrain.

Fig. 2: Control framework in reinforcement learning pipeline.

Fig. 3: Stiffness upper-bound derivation diagram. $\mbox{\boldmath$K$}_{x}^{\max}$ denotes the task-space stiffness upper-bound ellipsoid, illustrated as the pink ellipsoid; $\mbox{\boldmath$K$}_{q}^{\max}$ denotes the joint-space stiffness upper-bound super-ellipsoid, where individual joint stiffnesses are depicted as torsional springs; $\mbox{\boldmath$K$}_{LCP}$ represents the Lipschitz upper-bound constraint. — Fig. 3: Stiffness upper-bound derivation diagram. $\mbox{\boldmath$ K $}_{x}^{\max}$ denotes the task-space stiffness upper-bound ellipsoid, illustrated as the pink ellipsoid; $\mbox{\boldmath$ K $}_{q}^{\max}$ denotes the joint-space stiffness upper-bound super-ellipsoid, where individual joint stiffnesses are depicted as torsional springs; $\mbox{\boldmath$ K $}_{LCP}$ represents the Lipschitz upper-bound constraint.

Mode	CoM Stiffness	Null-space Stiffness
High-stiff	$[1000,1000,20000]$	$100.0$
Compliance	$[200,200,20000]$	$50.0$
Hard-nullspace	$[200,200,20000]$	$500.0$

TABLE I: CoM task and null-space stiffness setting in SILC method.

Fig. 4: The $x$ -direction CoM trajectory receiving an $-x$ -direction impact (150 N in 0.05 s) on pelvis link. The red triangle indicates the time instant at which the impact is applied. Compared with the high-stiff mode—in which the CoM returns to its original state via a damped oscillatory response—the compliance mode exhibits more pronounced compliance, as evidenced by larger oscillation amplitudes and the lack of convergence back to the original state. Meanwhile, scalar LCP also exhibits similarly high stiffness to high-stiff mode.

Fig. 5: Hand displacement under different task-space compliance settings of SILC when a constant load (-40 N) is applied to the left hand. It can be observed that, in both cases, the hand displacement responds within a very short time ( $<$ 0.1 s). After the response settles, we compare the steady-state position with the initial position and, together with the applied external force, approximately estimate the equivalent task-space stiffness to be 400 (soft) and 1000 N/m (hard), respectively.

Panel of Figure 7 — Fig. 6: According to the definition of the Directional Quadratic-form Budget, at each control step we plot the left-hand side (LHS) and right-hand side (RHS) of ( 34 ), respectively (unit of quadratic form: N $\cdot$ m $\cdot$ rad). The yellow curve corresponds to the RHS of the equation, i.e., the quadratic form under the target $\mbox{\boldmath$ K $}_{q}^{\max}$ , whereas the blue curve corresponds to the LHS, i.e., the current quadratic form measured $\mbox{\boldmath$ K $}_{q}(\mbox{\boldmath$ o $})$ . The red markers indicate instances where the measured value exceeds the target. As shown, under the SILC method, the measured value is strictly bounded by the target constraint; in contrast, under scalar LCP and baseline, the measured quadratic form periodically exceeds the target during the stepping gait.

	Action Jitter ( $\times 10^{4}$ ) $\downarrow$	DoF Velocity Jitter [rad/s] ( $\times 10^{5}$ ) $\downarrow$	Torque Jitter [Nm] ( $\times 10^{5}$ ) $\downarrow$	Energy [J] $\downarrow$
High-stiff SILC	$3.08\pm 0.19$	$1.97\pm 0.17$	$4.89\pm 0.34$	$4.77\pm 3.07$
Compliance SILC	$3.19\pm 0.28$	$2.03\pm 0.28$	$5.04\pm 0.38$	$4.99\pm 5.17$
Hard-nullspace SILC	$3.03\pm 0.17$	$1.84\pm 0.16$	$4.80\pm 0.22$	$4.86\pm 4.47$
Scalar LCP ( $\lambda=0.5$ )	$3.52\pm 7.00$	$1.74\pm 1.96$	$4.47\pm 5.01$	$5.09\pm 4.76$
Baseline	$4.13\pm 1.94$	$1.68\pm 0.51$	$5.35\pm 1.13$	$7.58\pm 10.74$

TABLE II: Comparison of walking motion under random velocity command. (environment number: 1,024)

	Settling time [s] $\downarrow$	High-frequency energy [J] $\downarrow$
SILC (soft)	$1.2$	$1.01\times 10^{-8}$
Baseline	$2.4$	$10.0\times 10^{-8}$

TABLE III: Metrics of the right hand

z

-direction trajectory oscillations under external impacts.

		RMS $\downarrow$	Peak-to-Peak (P95-P5) $\downarrow$
Compliance	Gyro.	$0.247$	$0.726$
SILC	Acc. ( $x$ and $y$ )	$3.885$	$12.66$
Baseline	Gyro.	$0.472$	$1.433$
	Acc. ( $x$ and $y$ )	$4.859$	$13.80$

TABLE IV: IMU measurement in the stepping experiment.

Compliant Robot Control

See all 15 papers

Smoothness as a Constraint for Stable Humanoid Locomotion
Sep 21, 2026Utsav Panchal, Denis Kleyko, Unal Artan +1Robotic Control Humanoid Robot Locomotion
AI, Robotics and Cybersecurity Center (ARC) and Department of Computer Science, Örebro University, Sweden
LAC: Linear and Angular Compliance for Humanoid Whole-body Control
Aug 26, 2026Yang Liu, Zhongkai Gu, Wei Zhu +1Humanoid Robot Control Whole-Body Control
Neuro-Robotics Laboratory, Tohoku University, Japan
CompliantWBC: Whole-Body Compliance for Heavy Humanoids via Force Latent Estimation and Residual Impedance Targets
Sep 27, 2026Tan-Dzung Do, Cuc T. Trinh, Tuan Dat Phuong +4RL for Robotics Humanoid Robot Control
VinRobotics · National University of Singapore · VinUniversity +1

Sep 21, 2026J/K move · Enter open · S save

Smoothness as a Constraint for Stable Humanoid Locomotion

Utsav Panchal, Denis Kleyko, Unal Artan, Amy Loutfi

AI, Robotics and Cybersecurity Center (ARC) and Department of Computer Science, Örebro University, Sweden

Robotic Control Humanoid Robot Locomotion Humanoid Robot Control Constrained RL

Open paper AskPDF arXiv

Embodied AI systems, particularly humanoid robots deployed in real world scenarios require whole-body control policies that are both task-responsive and physically smooth. However, smoothness is not uniform across the body: lower body must remain sufficiently reactive, while the upper body must be tightly regulated to preserve stability. Existing reinforcement learning approaches typically impose smoothness through auxiliary terms in the reward function, which compete with task objectives, treating the body as uniform and provide no direct control over the physical quantities responsible for smooth behavior. We introduce DeCap (Decoupled Constraint-aware policy), a constrained reinforcement learning algorithm that decouples whole-body smoothness into separate upper- and lower-body constraint groups, each formulates smoothness as explicit constraints on physical motion limits. To improve constraint satisfaction near feasibility boundaries, DeCap incorporates a bounded barrier penalty that activates proactively as limits are approached and remains bounded at the constraint limit. On real-world humanoid whole-body control task, DeCap reduces upper-body action rate by 2.50x and acceleration by 2.18x relative to reward-based smoothness policies, while also improving lower-body smoothness and reducing transient motion. We demonstrate that a fixed set of smoothness constraints transfers across diverse terrains, alleviating the need of extensive reward tuning.

Abstract

Compliant Robot Control

Explore similar work

Smoothness as a Constraint for Stable Humanoid Locomotion

LAC: Linear and Angular Compliance for Humanoid Whole-body Control

CompliantWBC: Whole-Body Compliance for Heavy Humanoids via Force Latent Estimation and Residual Impedance Targets