Bipedal-Walking-Dynamics Model on Granular Terrains

Xunjie Chen; Xinyan Huang; Peter Shan; Jingang Yi; Tao Liu

cs.ROApr 13, 2026

Bipedal-Walking-Dynamics Model on Granular Terrains

Authors: Xunjie Chen, Xinyan Huang, Peter Shan, Jingang Yi, Tao Liu

Organizations: Department of Mechanical and Aerospace Engineering, Rutgers University, Piscataway, NJ 08854 USA · Bridgewater Raritan High School, Bridgewater, NJ 08807 USA · State Key Lab of Fluid Power Transmission and Control and the School of Mechanical Engineering, Zhejiang University, Hangzhou 310027, China

Abstract

Bipeds have demonstrated high agility and mobility in unstructured environments such as sand. The yielding of such granular media brings significant sinkage and slip of the bipedal feet, leading to uncertainty and instability of walking locomotion. We present a new dynamics-modeling approach to capture and predict bipedal-walking locomotion on granular media. A dynamic foot-terrain interaction model is integrated to compute the ground reaction force (GRF). The proposed granular dynamic model has three additional degree-of-freedom (DoF) to estimate foot sinkage and slip that are critical to capturing robot-walking kinematics and kinetics such as cost of transport (CoT). Using the new model, we analyze bipedal kinetics, CoT, and foot-terrain rolling and intrusion affects. Experiments are conducted using a biped robotic walker on sand to validate the proposed dynamic model with robot-gait profiles, media-intrusion prediction, and GRF estimations. This new dynamics model can further serve as an enabling tool for locomotion control and optimization of bipedal robots to efficiently walk on granular terrains.

Figures & tables

Panel of Figure 1 — Fig. 1: Schematic of the bipedal walking model with foot sinkage and slip on granular media. (a) Sagittal dynamics. (b) Stance foot rolling and sinkage with a convex foot shape in the sagittal plane. (c) Frontal dynamics.

Fig. 2: Bipedal-walking-dynamics modeling on granular media with the inclusion of foot–terrain interactions, the rolling-slip effect, and the energy efficiency in terms of CoT.

Panel of Figure 3 — Fig. 3: Experimental setup of the bipedal robot walking locomotion on granular media. (a) Schematic of the actuated joint configuration of the bipedal robot on granular media. (b) A bipedal robot on the sand track with a modularized foot.

Panel of Figure 4 — Fig. 4: Comparisons of robot locomotion results on granular media within one stance phase. Shadowed areas represent one standard deviation. (a) The gait comparison of the model prediction and experiment results. Swing leg: $q_{1,2,3}$ , stance leg: $q_{4,5,6}$ . (b) GRFs on the granular media. (c) Stance foot intrusion variables $x_{s}$ , $y_{s}$ and $z_{s}$ .

Panel of Figure 5 — Fig. 6: (a)-(c) Foot orientation angle $\Delta\theta_{r}$ (negative sign presents decreasing values), the effective rolling radius of the stance foot, and the CoT of walking on sand and rigid ground during one stance phase, respectively. Shadowed areas represent the standard deviations. $\bar{v}_{\mathrm{CoM}}^{x}/\sqrt{gh_{\mathrm{CoM}}}=0.1$ . (d) Velocity dependency of locomotion CoT of the bipedal robot and human on sand and rigid ground, respectively. Each CoT value is mean $\pm$ standard-variation. The solid lines are second-order polynomial fitting. CoT of humanoid locomotion are based on the data by [ 24 ] .

Panel of Figure 6 — Fig. 5: Comparisons of RMSE under different forward-velocity commands. (a) Joint angle ${\bm{q}}$ . (b) GRFs. (c) Intrusion variables $x_{s}$ , $y_{s}$ , and $z_{s}$ . Circle dots: the proposed model. Square dots: model with rigid-ground assumptions.

Granular Terrain Locomotion

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Robust bipedal locomotion on flowable slopes via foot-driven terrain manipulation

Deniz Kerimoglu, Junnosuke Kamohara, Jiyeon Maeng, Ziwon Yoon+3

Department of Mechanical Engineering, Georgia Institute of Technology, Atlanta, GA 30332, USA · School of Physics, Georgia Institute of Technology, Atlanta, GA 30332, USA · Department of Electrical and Computer Engineering, Northeastern University, Boston, MA 02115, USA

Humanoid Robot Locomotion Granular Terrain Locomotion Terrain-Aware Robot Locomotion Robot Locomotion

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Bipedal robots are challenging to control because they operate close to instability, where small variations in foot-terrain contact can rapidly destabilize locomotion. On rigid terrain, bipedal robots mitigate this fragility by using well-established contact mechanics and control strategies. On flowable surfaces such as granular slopes, foot contact can induce large surface deformations and solid-fluid-like transitions, coupling terrain effects with robot dynamics, leading to underperformance or failure. This is partly due to the lack of reliable methods to represent the dynamics of flowable terrain, making it difficult to account for terrain effects in locomotion design. Here, we investigate how controlling terrain response can improve bipedal locomotion on granular slopes by studying the terradynamics of cleated feet, thin plates emanating from the foot soles. Systematic studies of a small-scale (1.4 kg) robophysical biped reveal that cleats with sparse and dense spacing lead to excessive terrain yielding and resistance, respectively, degrading performance and leading to failure. An intermediate cleat spacing distributes interaction forces to maintain substrate stresses near (or below) the yield threshold, enabling walking on granular slopes up to 30 degrees. Guided by these principles, we design a foot that actively adjusts cleat depth and accommodates both rigid and granular terrain. We also demonstrate that the principles of effective foot-terrain interaction translate to a larger (15 kg) autonomous biped. Our study presents an alternative to conventional body-centric robot control approaches, which regulate terrain-induced effects through body motion, by instead regulating terrain interactions through limb-centric approach.

Abstract

Granular Terrain Locomotion

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