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Energy Prediction on Sloping Ground for Quadruped Robots

Mohamed Ounally, Cyrille Pierre, Johann Laconte

Abstract

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.

Energy Prediction on Sloping Ground for Quadruped Robots

Abstract

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.
Paper Structure (12 sections, 6 equations, 5 figures)

This paper contains 12 sections, 6 equations, 5 figures.

Table of Contents

  1. INTRODUCTION
  2. RELATED WORK
  3. THEORY
  4. Notations and Assumptions
  5. Energy of a given path
  6. EXPERIMENTS
  7. Experimental Setup
  8. Data Preprocessing
  9. Calibration and Evaluation Protocol
  10. Superposition and Path Equivalences
  11. Force-equivalent cost versus slope angle
  12. CONCLUSION

Figures (5)

  • Figure 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.
  • Figure 2: Coordinate system on a sloped terrain. The robot moves with velocity $\bm{\varpi}(t)$ on a slope of angle $\alpha$, oriented with a heading $\gamma$ with respect to the slope direction.
  • Figure 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.
  • Figure 4: Measured force-equivalent cost as a function of slope angle $\alpha$ for uphill (left) and downhill (right) runs. Red and blue points correspond to two velocity headings, $\bm{\varpi}(t) = [1,0,0]^T$ (forward) and $\bm{\varpi}(t) = [0,1,0]^T$ (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.
  • Figure 5: Estimated yaw torque as a function of slope angle $\alpha$ for pure rotation movements. The trend suggests an increase of rotational cost with slope.