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

Long Zhao; Xingjue Liao; Feifei Qian

cs.ROOct 2, 2026

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

Authors: Long Zhao, Xingjue Liao, Feifei Qian

Organizations: Department of Electrical and Computer Engineering, University of Southern California, Los Angeles, CA, USA.

Abstract

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.

Figures & tables

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)

Figure 2: Robot locomotion performance under Buehler-clock and constant-angular-velocity actuation on inclined granular surfaces. Total displacement over ten gait cycles is shown as a function of the slow-phase center, $\phi_{0}$ , for surface inclinations of $0^{\circ}$ , $5^{\circ}$ , $10^{\circ}$ , and $15^{\circ}$ (blue, orange, green, and purple, respectively). Solid curves show the mean across three trials with standard-deviation error bars, while dashed horizontal lines indicate the corresponding constant-angular-velocity baselines. The red dashed line highlights the shift of the optimal slow-phase center toward smaller $\phi_{0}$ as surface inclination increases.

Figure 3: Single-leg peak normal-force location and optimal slow-phase center as functions of surface inclination. (a) Representative single-leg normal-force profile as a function of leg angle. The orange star marks the maximum

F_{z}

, and the corresponding leg angle is defined as the peak-force angle,

\phi_{\mathrm{peak}}

; the light vertical dashed line indicates its angular location. (b) Comparison between the mean peak-force angle,

\phi_{\mathrm{peak}}

(solid black line with circular markers), and the optimal slow-phase center,

\phi_{0}^{*}

(black dashed line with square markers), as functions of surface inclination. Error bars for

\phi_{\mathrm{peak}}

indicate the standard deviation across the tested slow-phase centers. Marker colors correspond to

\theta=0^{\circ}

5^{\circ}

10^{\circ}

, and

15^{\circ}

in blue, orange, green, and purple, respectively.

Figure 4: Reconstruction of the combined normal-force profile,

F_{z,\mathrm{tot}}

, over one gait cycle on a

10^{\circ}

granular slope at

\phi_{0}=160^{\circ}

. The blue and red dashed curves show the individual normal-force contributions from the two legs separated by

0.5T

, and the green solid curve shows their sum.

Figure 5: Reconstructed combined normal- and shear-force profiles on a $10^{\circ}$ granular slope for five slow-phase centers, $\phi_{0}$ . For each $\phi_{0}$ , the forces generated by two legs following identical Buehler-clock trajectories with a phase offset of $0.5T$ were reconstructed from the measured single-leg constant-angular-velocity force profiles corresponding to the slow- and fast-phase velocities and then summed. (a) Combined normal force, $F_{z,\mathrm{tot}}$ . (b) Combined shear force, $F_{x,\mathrm{tot}}$ . Curve color denotes the slow-phase center, with $\phi_{0}$ increasing from $150^{\circ}$ to $190^{\circ}$ as indicated by the color bar.

Figure 6: Robot speed, reconstructed granular resistive forces, and corresponding effective propulsion window, for locomotion on a

10^{\circ}

granular slope. Panels (a,c,e) show trials with

\phi_{0}=155^{\circ}

, and panels (b,d,f) show trials with

\phi_{0}=180^{\circ}

. (a, b) Robot CoM upslope velocity

v_{x}

, (c, d) Robot CoM surface-normal velocity,

v_{z}

and (e, f) Reconstructed combined normal force, where the red markers denote the normal force

F_{z}

and the blue markers denote shear force

F_{x}

. The green horizontal line represents the predicted normal-support threshold,

F_{\mathrm{th}}(10^{\circ})

. The vertical dashed lines mark

t_{1}

and

t_{2}

, the onset of upslope acceleration and deceleration, respectively. The interval [

t_{1}

t_{2}

] defines the effective propulsion window.

Figure 7: Predicted robot displacement from the reconstructed shear force during the effective propulsion window. The blue, orange, green, and purple markers correspond to

\theta=0^{\circ}

5^{\circ}

10^{\circ}

, and

15^{\circ}

, respectively. The red dashed line highlights the shift of the highest-displacement prediction toward smaller

\phi_{0}

as surface inclination increases, consistent with the experimentally observed shift in optimal slow-phase center shown in Fig. 2 .

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.

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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

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