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Terrain-Dependent Intra-Cycle Leg Timing for Effective Locomotion on Granular Slopes

Terrain-Dependent Intra-Cycle Leg Timing for Effective Locomotion on Granular Slopes

Long Zhao, Xingjue Liao, Feifei Qian · N/A · 2026

locomotion reinforcement-learning
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Summary

Locomotion on granular slopes such as sand dunes remains a fundamental challenge for legged robots due to reduced shear strength and gravity-induced anisotropic yielding of granular media.

Abstract Summary

Locomotion on granular slopes such as sand dunes remains a fundamental challenge for legged robots due to reduced shear strength and gravity-induced anisotropic yielding of granular media. Using a hexapedal robot on a tiltable granular bed, we systematically measure locomotion speed together with slope-dependent normal and shear granular resistive forces. While normal penetration resistance remains nearly unchanged with inclination, shear resistance decreases substantially as slope angle increases. Guided by these measurements, we develop a simple robot-terrain interaction model that predicts anchoring timing, step length, and resulting robot speed, as functions of terrain strength and slope angle. The model reveals that slope-induced performance loss is primarily governed by delayed anchoring and increased backward slip rather than excessive sinkage. By extending the model to generalized terrain conditions, we construct failure phase diagrams that identify sinkage- and slippage-induced failure regimes, enabling quantitative risk estimation for locomotion on granular slopes. This physics-informed framework provides predictive insight into terrain-dependent failure mechanisms and offers guidance for safer and more robust robot operation on deformable inclines.

Key Points

  • Locomotion on granular slopes such as sand dunes remains a fundamental challenge for legged robots due to reduced shear strength and gravity-induced anisotropic yielding of granular media.
  • Using a hexapedal robot on a tiltable granular bed, we systematically measure locomotion speed together with slope-dependent normal and shear granular resistive forces.
  • While normal penetration resistance remains nearly unchanged with inclination, shear resistance decreases substantially as slope angle increases.
  • Guided by these measurements, we develop a simple robot-terrain interaction model that predicts anchoring timing, step length, and resulting robot speed, as functions of terrain strength and slope angle.
  • The model reveals that slope-induced performance loss is primarily governed by delayed anchoring and increased backward slip rather than excessive sinkage.

Materials and Methods

A small RHex-type [ 18 ] robot (Fig. 1 a) was used to investigate locomotion on level and inclined granular surfaces [ 18 ] . The robot body measured 22 cm in length and 10 cm in width, with a total mass of 1.5 kg. It was equipped with six C-shaped rotary legs, each with a radius, $R$, of 3 cm, a width of 0.8 cm, and an arc span of $225^{\circ}$.

The angular positions of the robot legs, $\phi$, were controlled using a microcontroller (Arduino Uno), and actuated by servo motors (Dynamixel XL330).

Materials and Methods — Terrain-Dependent Intra-Cycle Leg Timing for Effective Locomotion on Granular Slopes
Figure 1: Robot and experimental setup. (a) RHex-type robot used in this study. The blue and red legs indicate the two alternating tripod groups. (b) Single-leg force-profile measurement setup. The force components $F_{x}$, $F_{y}$, and $F_{z}$ were measured along the fore-aft, lateral, and upward directions, respectively. (c) Buehler-clock trajectories for the two tripod groups over one gait cycle, $t/T$. The blue and red curves show the leg angular positions of the two tripod groups, with a phase offset of $0.5T$ between them. The slow phase is defined by its angular range $\phi_{s}$, angular center $\phi_{0}$, and duty factor $d_{c}$, which specifies the fraction of the gait period occupied by the slow phase. The vertical black dashed line indicates the instant illustrated in the inset, which shows the corresponding angular configuration of the two tripod groups. (d) A side-view diagram of the robot leg and kinematics. Here $h$ is the hip height, $d$ is the leg penetration depth, $\phi$ is the leg angular position, measured counterclockwise from the $+z$ direction to the line connecting the leg rotation axis to the center of curvature of the C-shaped leg; $\omega$ is the leg angular velocity, and $\theta$ is the surface inclination. (e) Tiltable fluidized-bed setup used for robot locomotion experiments. The robot traveled in the upslope direction with velocity $v$, while three motion-capture cameras recorded its position in the coordinate frame $(x,y,z)$.

Model generalization and broader applicability

While the analysis in Sec. 3.4 explains the observed locomotion performance, its reliance on measured force profiles and robot kinematics limits its application to conditions that have already been tested.

To achieve a more general framework that predicts locomotion performance directly from the robot, gait, and terrain parameters, we use granular resistive force theory (RFT) to predict the leg-terrain force profiles, and then use the predicted forces to determine the effective propulsion window and estimate the resulting step length.

Model generalization and broader applicability — Terrain-Dependent Intra-Cycle Leg Timing for Effective Locomotion on Granular Slopes
Figure 8: Predicted effect of leg radius on locomotion performance and optimal Buehler-clock timing. (a) Predicted step length, $s_{p}$, as a function of the slow-phase center $\phi_{0}$ for leg radii $R=3$ cm (solid lines) and $R=4$ cm (dashed lines). The blue, orange, green, and purple curves correspond to $\theta=0^{\circ}$, $5^{\circ}$, $10^{\circ}$, and $15^{\circ}$, respectively. (b) Optimal slow-phase center, $\phi_{0}^{*}$, as a function of surface inclination for $R=3$ cm (solid line) and $R=4$ cm (dashed line). For $R=3$ cm at $\theta=15^{\circ}$, the model predicts zero displacement for all tested values of $\phi_{0}$, so no optimal $\phi_{0}^{*}$ is reported.

Sources and Demonstrations

Figures are reproduced from Zhao et al. See the full paper for experimental details and the project page, when the paper links one, for demonstrations and videos.

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