Proprioceptive Force Estimation for Quadruped Locomotion and Human-Robot Interaction

Run Wang; Xu Yang; Alapati Tuerxun; Yilin Mo

cs.ROSep 28, 2026

Proprioceptive Force Estimation for Quadruped Locomotion and Human-Robot Interaction

Authors: Run Wang, Xu Yang, Alapati Tuerxun, Yilin Mo

Organizations: Department of Automation, Tsinghua University, Beijing, China

Abstract

Payload forces must be accommodated during locomotion, while leash forces can specify desired motion. We investigate whether a shared three-dimensional force estimate in newtons, inferred from proprioceptive history under sustained loading, can support both tasks. An estimator and locomotion policy are jointly trained with supervised force and velocity outputs and learned latent context. The estimated force conditions locomotion and additionally generates planar-velocity and yaw-rate commands for leash guidance through an analytical map. In sustained-force simulation sweeps, temporal means of componentwise force root mean square error range from 1.44 to 2.83,N. Compared with a domain-randomized baseline, the framework reduces velocity-tracking and base-orientation error scores by 21.6% and 46.5%, respectively, and increases mean survival from 68.29% to 94.60% in separate sustained-force tests. Unitree Go1 experiments demonstrate stationary vertical and horizontal force estimation, locomotion with an 8.5,kg payload whose weight exceeds the 70,N training force limit, and leash guidance using the same force-estimation interface.

Figures & tables

Fig. 1: Go1 demonstrations of the external-force interface: (a) payload-force estimation; (b) payload locomotion using online force estimates; (c,d) horizontal-force estimation; and (e,f) leash-guided locomotion. Arrows indicate applied forces.

Fig. 2: Shared force-estimation interface. Encoder $\mathcal{E}_{\psi}$ and policy $\pi_{\theta}$ are jointly trained with privileged critic $V_{\phi}$ ; normalization is omitted here. Force estimates condition locomotion in both tasks and additionally generate leash commands through $\mathcal{M}$ . Payload commands are supplied externally. Deployed weights remain fixed.

Method	Online representation	Locomotion role	Guidance / compliance
Locomotion adaptation
RMA [ 7 ]	Latent context	Policy conditioning	—
BAS [ 13 ]	Mass, CoM shift, friction	Policy + safety	—
Beyond Robustness [ 14 ]	Load state and properties	Load stabilization	—
Force-based interaction
DeFazio et al. [ 10 ]	Tug estimate ( $\Delta v$ proxy)	Policy conditioning	Discrete turn selection

TABLE I: Main deployment representations and their uses in locomotion and force interaction. Auxiliary velocity and latent outputs are omitted for force estimators.

Task rewards	Expression ( $r_{j}$ )	Weight ( $w_{j}$ )
Tracking lin. vel.	$\exp\left\{-4\left\\|\bm{v}_{xy}^{\text{cmd}}-\bm{v}_{xy}\right\\|_{2}^{2}\right\}$	$1.0$
Tracking ang. vel.	$\exp\left\{-4\left(\omega_{z}^{\text{cmd}}-\omega_{z}\right)^{2}\right\}$	$0.5$
Regularization	Expression ( $r_{j}$ )	Weight ( $w_{j}$ )
Alive	$1.0$	$0.5$
Base height	$\left(h_{b}^{\text{nominal}}-h_{b}\right)^{2}$	$-50.0$
Orientation	$\left\\|\bm{g}_{xy}\right\\|_{2}^{2}$	$-5.0$

TABLE II: Reward Function Components.

Quantity	Setting
Friction coefficient	$\mathcal{U}(0.5,1.25)$
Velocity-push interval	$15$ s
Push velocity magnitude	$\mathcal{U}(0,1)$ m/s
Continuous-force resampling	Every $11$ s
Continuous-force magnitude	$\mathcal{U}(0,70)$ N
Continuous-force direction	Uniform on $\mathbb{S}^{2}$

TABLE III: Domain randomization shared by all compared controllers.

Fig. 4: Stationary hardware force estimation. A 5 kg dumbbell is placed on the robot and removed. The curves show estimated force components, with payload gravity serving as the reference for the settled vertical value.

Fig. 5: Stationary horizontal-force estimation. Spring-scale magnitudes are 41 N at $t=11$ s for the x-axis pull and 45 N at $t=24$ s for the y-axis pull. The corresponding marked estimates are $-39.7$ N and $-50.1$ N; signs indicate the negative base-axis directions.

Fig. 6: Hardware locomotion with an 8.5 kg payload secured beneath the torso. Ours uses online force estimates from the proprioceptive estimator to condition the locomotion policy. Both controllers receive the same velocity commands.

Fig. 7: Recorded training rewards. Solid curves use Gaussian smoothing with standard deviation $\sigma_{\mathrm{smooth}}=3$ recorded samples; lighter traces are the unsmoothed recorded rewards. Oracle uses ground-truth conditioning.

Fig. 10: Hardware leash guidance using online force estimates. The sequence includes forward motion, a $180^{\circ}$ turn, and small-radius turning. The heading reference is reconstructed offline from force estimates and measured yaw.

Method	X: 80 N	Y: 60 N	Z: 80 N
Oracle	100.00%	100.00%	84.08%
Ours	100.00%	100.00%	83.79%
IA	52.25%	84.77%	69.04%
DR	94.73%	44.04%	66.11%

TABLE IV: Simulation survival rates at the specified force conditions. Oracle is a privileged reference; bold marks the best among Ours, IA, and DR.

Explore similar work

QuietWalk: Physics-Informed Reinforcement Learning for Ground Reaction Force-Aware Humanoid Locomotion Under Diverse Footwear
Apr 26, 2026Hanze Hu, Luying Feng, Silu Chen +6Ground Reaction Forces Inverse Dynamics
SixthSense: Task-Agnostic Proprioception-Only Whole-Body Wrench Estimation for Humanoids
May 2, 2026Xingzhou Chen, Xiayan Xu, Yan Ning +8Proprioception Contact Wrench Estimation
The Hong Kong University of Science and Technology, Hong Kong SAR, China · Zhejiang University, Hangzhou, China · Tencent Robotics X, Shenzhen, China
TRACE: Learned Proprioceptive Odometry for Legged Robots under Unreliable Contact Conditions
Aug 6, 2026Taehyeon Kong, Woojin Kim, Jemin HwangboOdometry Proprioception
Korea Advanced Institute of Science and Technology (KAIST), Yuseong-gu, Daejeon 34141, Republic of Korea

Apr 26, 2026, cs.ROJ/K move · Enter open · S save

QuietWalk: Physics-Informed Reinforcement Learning for Ground Reaction Force-Aware Humanoid Locomotion Under Diverse Footwear

Hanze Hu, Luying Feng, Silu Chen, Tianjiang Zheng+5

Ground Reaction Forces Inverse Dynamics Physics-Informed Learning

Open paper AskPDF arXiv

Humanoid robots operating in human-centered environments (e.g., homes, hospitals, and offices) must mitigate foot--ground impact transients, as impact-induced vibration and noise degrade user experience and repeated impacts accelerate hardware wear. However, existing low-noise locomotion training often relies on kinematic proxy objectives or fragile force sensors, and footwear-induced changes in contact dynamics introduce distribution shifts that hinder policy generalization.We present QuietWalk, a physics-informed reinforcement learning framework for ground-reaction-force-aware humanoid locomotion under diverse footwear conditions. QuietWalk employs an inverse-dynamics-constrained physics-informed neural network (PINN) to estimate per-foot vertical ground reaction forces (GRFs) from proprioceptive signals, and integrates the frozen predictor into the RL training loop to penalize predicted impact forces without requiring force sensors at deployment.On a held-out real-robot dataset, enforcing inverse-dynamics consistency reduces vertical GRF prediction errors by 82%-86% compared with a purely supervised predictor and improves the coefficient of determination from 0.39/0.67 to 0.99/0.99 for the left/right feet. On hardware at 1.2 m/s (barefoot; averaged over four floor materials), QuietWalk reduces mean A-weighted noise level by 7.17 dB and peak noise level by 4.98 dB under a consistent recording setup. Cross-footwear experiments (barefoot, skate shoes, athletic sneakers, and high heels) across multiple surfaces further demonstrate robust adaptation to footwear-induced contact variations.

Abstract

Explore similar work

QuietWalk: Physics-Informed Reinforcement Learning for Ground Reaction Force-Aware Humanoid Locomotion Under Diverse Footwear

SixthSense: Task-Agnostic Proprioception-Only Whole-Body Wrench Estimation for Humanoids

TRACE: Learned Proprioceptive Odometry for Legged Robots under Unreliable Contact Conditions