Impact of Leg Stiffness on Energy Efficiency in One Legged Hopping

TL;DR

This work addresses energy efficiency in legged locomotion by examining how leg stiffness $k_l$ influences energy cost across speeds for a planar monopedal hopper. It employs a direct-collocation-based optimal control framework to map average speed $v_{avg}$ and $k_l$ to cost of transport, comparing variable stiffness against a fixed baseline. The key finding is that variable stiffness can reduce $CoT$ by up to ~20% at certain speeds, but the average gain is about 6%, with fixed stiffness achieving similar performance across most speeds. The study provides guidance on the practical value of stiffness adaptation in legged robots and introduces a grid-based method to explore stiffness-speed-energy trade-offs, highlighting the balance between control complexity and energy savings.

Abstract

In the fields of robotics and biomechanics, the integration of elastic elements such as springs and tendons in legged systems has long been recognized for enabling energy-efficient locomotion. Yet, a significant challenge persists: designing a robotic leg that perform consistently across diverse operating conditions, especially varying average forward speeds. It remains unclear whether, for such a range of operating conditions, the stiffness of the elastic elements needs to be varied or if a similar performance can be obtained by changing the motion and actuation while keeping the stiffness fixed. This work explores the influence of the leg stiffness on the energy efficiency of a monopedal robot through an extensive parametric study of its periodic hopping motion. To this end, we formulate an optimal control problem parameterized by average forward speed and leg stiffness, solving it numerically using direct collocation. Our findings indicate that, compared to the use of a fixed stiffness, employing variable stiffness in legged systems improves energy efficiency by 20 % maximally and by 6.8 % on average across a range of speeds.

Figures (5)

• Figure 1: A monopedal robot with parallel elastic actuation in the leg and hip joint.
• Figure 2: Visualization of an optimal hopping gait $\mathsf{g}_{0}$ at an average speed of $v_\mathrm{avg} = 1\,\sqrt{l_\circ g}$, and optimal leg stiffness $k_\mathrm{l}^\star = 4.44\, m_\circ g / l_\circ$. The gait begins in Stance, when the monoped’s foot touches the ground, followed by a flight phase that starts when the foot loses contact with the ground. During Flight, ground contact is assumed impossible.
• Figure 3: Top view of the map of optimal gaits generated using the grid based exploration in the $(v_\mathrm{avg}, k_\mathrm{l})$ parameter space. Each point represents an optimal gait, with its color indicating the corresponding cost of transport. The families of gaits $\mathcal{A}$ (optimal leg stiffness) and $\mathcal{C}$ (constant leg stiffness) are shown respectively in red and orange.
• Figure 4: Comparison of the cost of transport for the family of gaits $\mathcal{A}$ with optimal stiffness and the family of gaits $\mathcal{C}$ with constant stiffness over various average forward speeds.
• Figure 5: Zoomed-in side view of the map of optimal gaits, showing different slices of constant $v_\mathrm{avg}$.