Energy management is a fundamental challenge for legged robots in outdoor environments. Endurance directly constrains mission success, while efficient resource use reduces ecological impact. This paper investigates how terrain slope and heading orientation influence the energetic cost of quadruped locomotion. We introduce a simple energy model that relies solely on standard onboard sensors, avoids specialized instrumentation, and remains applicable in previously unexplored environments. The model is identified from field runs on a commercial quadruped and expressed as a compact function of slope angle and heading. Field validation on natural terrain shows near-linear trends of force-equivalent cost with slope angle, consistently higher lateral costs, and additive behavior across trajectory segments, supporting path-level energy prediction for planning-oriented evaluation.
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Fig. 1 : For legged robots, the complexity of their dynamics makes energy consumption difficult to predict. The energy consumption of the two illustrated paths (blue: direct uphill, red: indirect detour) is not necessarily proportional to distance: on sloped terrain, longer but smoother trajectories can require less energy than shorter, steeper ones.
Fig. 2 : Coordinate system on a sloped terrain. The robot moves with velocity ϖ(t) on a slope of angle α , oriented with a heading γ with respect to the slope direction.
Fig. 3 : Energy superposition: measured energy of composite paths compared with the sum of their parts. Close agreement supports the additivity assumption used by the model.
Fig. 4 : Measured force-equivalent cost as a function of slope angle α for uphill (left) and downhill (right) runs. Red and blue points correspond to two velocity headings, ϖ(t)=[1,0,0]T (forward) and ϖ(t)=[0,1,0]T (lateral). The uphill panel shows increasing cost with slope angle, while the downhill panel shows a decrease only for forward motion. By contrast, lateral motion remains more expensive and tends to worsen with slope. These opposite trends indicate that energetic cost depends not only on terrain inclination but also on robot orientation with respect to the slope.
Fig. 5 : Estimated yaw torque as a function of slope angle α for pure rotation movements. The trend suggests an increase of rotational cost with slope.
Energy-efficient robot locomotion studies how robot mechanics and control can reduce power use and cost of transport across terrains and movement modes.
Learning-based quadrupedal locomotion typically relies on complex reward formulations that entangle task specification, operational limits, gait preference, and terrain adaptation within a single optimization objective. We instead treat these functions through distinct mechanisms: rewards for task specification, constraints for operational limits, energy minimization for gait preference, and exteroceptive perception for adapting energy use to terrain difficulty. We show that these components jointly enable efficient, terrain-adaptive locomotion, and that removing each component exposes a distinct failure mode. Our formulation removes explicit gait priors (including air-time, contact-count, and foot-clearance targets) in favor of emergent behavior. Compared to a conventional complex-reward baseline, our formulation achieves comparable terrain traversal while reducing cost of transport by 56% and operational-limit violations by 96%. The resulting policies transfer zero-shot to a physical Unitree Go2 using LiDAR-based elevation mapping. Project website with videos: https://tinyurl.com/locomposition.
Loukas Kordos, Leonard T. Franz, Simon Rappenecker +4
Technical University of Munich. · University of Tübingen. · Hertie Institute for Clinical Brain Research & Center for Integrative Neuroscience. +1
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.
Long Zhao, Xingjue Liao, Feifei Qian
Department of Electrical and Computer Engineering, University of Southern California, Los Angeles, CA, USA.
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.