MILD: Tractable Terrain Modeling for Learning Improved Bipedal Locomotion on Deformable Surfaces

Zeren Luo; Jiahui Zhang; Zhe Xu; Wanyue Li; Xinqi Li; Xuechao Chen; Zhangguo Yu; Annan Tang

cs.ROAug 20, 2026

MILD: Tractable Terrain Modeling for Learning Improved Bipedal Locomotion on Deformable Surfaces

Authors: Zeren Luo, Jiahui Zhang, Zhe Xu, Wanyue Li, Xinqi Li, Xuechao Chen, Zhangguo Yu, Annan Tang, +1 more

Organizations: Adaptive Robotic Controls Lab (ArcLab), Department of Mechanical Engineering, The University of Hong Kong, Hong Kong · School of Mechanical and Electrical Engineering, Beijing Institute of Technology, Beijing, China · Graduate School of Information Science and Technology, The University of Tokyo, Tokyo, Japan

Abstract

Enabling robots to walk on yielding terrain is vital for applications ranging from disaster response to planetary exploration. While bipedal robots hold immense potential, their locomotion on deformable surfaces remains limited as current simulators fail to capture the spatiotemporal heterogeneity of such yielding substrates. We present MILD, featuring a physics-grounded discrete-element contact solver that accurately simulates spatially varying foot-terrain interactions. Complementing this model, we train a terrain-aware locomotion controller via deep reinforcement learning with latent modulation and proprioceptive estimation. Quantitative comparisons against state-of-the-art methods show our approach generates more diverse and realistic contact scenarios during training, resulting in controllers that exhibit natural adaptation on real deformable surfaces. Through hardware experiments, we demonstrate the system's capability for online terrain identification and adaptation across a wide range of surface stiffness.

Figures & tables

Fig. 1: The bipedal robot as a multi-articulated floating-base system interacting with compliant surfaces through integrated physics-grounded modeling and learning-based control. The locomotion is demonstrated across terrains with parametrically varied stiffness and deformability (d – density of the polyurethane foam. Unit: $kg/m^{3}$ ).

Fig. 2: Eccentric penetration mechanics of granular media (GM) under arbitrary-shaped feet. The footplate is discretized into elements (DEs), each modeling localized GM interactions via independent developing cones (Eq. ( 1 ) - ( 5 )). Deeper intrusion amplifies the cone’s added mass effect. The model requires each element to satisfy individual friction cone constraints while maintaining continuity constraints between element pairs (Eq. ( 11 ) - ( 13 )). Drift-resistive forces are opposite to the element-wise motion and are scaled by the accumulated post-intrusion moving distance.

Fig. 3: Illustration of different penetration stiffness and depth-dependent resistive stress that lead to different foot penetration scenarios. The color grids indicate the terrain surface $h_{0}$ .

Fig. 4: Schematic of the simulation and training pipeline. The contact solver integrates depth-dependent granular dynamics and drifting resistance to model heterogeneous foot-terrain interactions. In the asymmetric actor-critic (AAC) training pipeline, a VAE-based estimator reconstructs privileged states from proprioception, while the modulation net dynamically adjusts latent terrain representations using estimated feet velocity to enable implicit adaptation.

Term	Equation	Weight
Lin. velocity tracking	$\exp\{-5(\bm{v}_{xy}^{cmd}-\bm{v}_{xy})^{2}\}$	1.4
Ang. velocity tracking	$\exp\{-5(\bm{\omega}^{cmd}_{z}-\bm{\omega}_{z})^{2}\}$	1.1
Feet clearance	$\\|\bm{h}^{f,des}_{t}-\bm{h}^{f}_{t}\}\\|\cdot(1-\mathcal{G}_{p})$	2.4
Gait alignment	$1.3\sum_{i=1}^{2}\mathbb{I}(\mathcal{C}_{i}=\mathcal{G}_{p,i})-0.3$	1.4
Feet slippery speed	$\bm{v}^{f}_{t}\cdot\mathcal{C}$	-0.1
Base acceleration	$\exp\{-3\\|\bm{v}^{b}_{t}-\bm{v}^{f}_{t-1}\\|\}$	0.2

TABLE I: Major reward terms for training policy.

(\cdot)^{cmd}

and

(\cdot)^{des}

is the desired and commanded physical values.

\mathcal{C}

represents the feet contact states.

\mathbb{I}

is the indicator function that returns 1 when the condition is true and 0 otherwise.

Fig. 5: The distribution of positional drift versus penetration for different methods and their standard deviations along the x and y directions.

Fig. 6: Performance of ( A ) the passivity-based controller and ( B ) the learning-based controller. ( C ) Center of Mass (CoM) trajectory comparison.

Fig. 7: MILD controller’s feet clearance across all terrain types ( Left ) and policy performance of the compared contact models on 25D foam ( Right ).

Category	Parameters	Range	Unit
Modeling	Shear band angle $\alpha$	[0.698, 1.310]	$Rad$
	Compact rate $\nu$	[0.8, 2.4]	-
	Surr. mass scale $c_{g}$	[-1.4, 4.6]	-
	Inertial drag scale $c_{d}$	[-6.0, 28.0]	-
	Volume fraction $\phi$	[0.50, 0.64]	-
	Friction coefficient $\mu$	[0.2, 1.3]	-

TABLE II: Randomization range of critical parameters

Fig. 8: The t-SNE classification of the encoded variable $\tilde{\bm{z}}_{t}$ across terrains with varying stiffness levels, where the dataset comprises 15-second walking trials on each terrain type.

Fig. 9: Robot motion analysis during sudden transition to foam penetration while moving forward ( A ) and backward ( B ), with corresponding t-SNE visualization of modulated latent space variations ( C - D ). The latent representations show distinct clustering patterns during terrain transitions, demonstrating the controller’s real-time adaptation to varying surface compliance.

Commanded Velocity $\bm{v}_{x}$ (m/s)		0.3	0.6	0.9	1.2
Cost of Transport (Hardware)	MILD	0.82	0.75	0.78	0.83
	HT-2	$1.05$	$0.96$	$1.20$	$1.08$
	Clock	$1.38$	$1.30$	$1.35$	$1.42$
Maximum torque (Nm) (Hardware)	MILD	58.2	63.2	73.3	88.4
	HT-2	$62.9$	$75.8$	$96.0$	$91.6$
	Clock	$67.6$	$82.5$	$102.8$	$115.2$

TABLE III: Performance comparison. Cost of Transport is calculated as

\text{COT}=\text{Power}/(\text{Weight}\cdot\text{Velocity})

with

\text{Power}=\sum_{actuators}\|\tau\dot{q}\|

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