The effect of limb kinematics on the speed of a legged robot on granular media

Legged locomotion on flowing ground ({\em e.g.} granular media) is unlike locomotion on hard ground because feet experience both solid- and fluid-like forces during surface penetration. Recent bio-inspired legged robots display speed relative to body size on hard ground comparable to high performing organisms like cockroaches but suffer significant performance loss on flowing materials like sand. In laboratory experiments we study the performance (speed) of a small (2.3 kg) six-legged robot, SandBot, as it runs on a bed of granular media (1 mm poppy seeds). For an alternating tripod gait on the granular bed, standard gait control parameters achieve speeds at best two orders of magnitude smaller than the 2 body lengths/s ($\approx 60$ cm/s) for motion on hard ground. However, empirical adjustment of these control parameters away from the hard ground settings, restores good performance, yielding top speeds of 30 cm/s. Robot speed depends sensitively on the packing fraction $φ$ and the limb frequency $ω$, and a dramatic transition from rotary walking to slow swimming occurs when $φ$ becomes small enough and/or $ω$ large enough. We propose a kinematic model of the rotary walking mode based on generic features of penetration and slip of a curved limb in granular media. The model captures the dependence of robot speed on limb frequency and the transition between walking and swimming modes but highlights the need for a deeper understanding of the physics of granular media.

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.

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.