Efficient, Responsive, and Robust Hopping on Deformable Terrain

TL;DR

This work develops a low-dimensional, template-based framework for legged hopping on deformable terrain by deriving a hop-to-hop energy map from a hybrid monopod-on-ground model with a depth-dependent yield threshold. The map captures how energy injection, ground yielding, and foot mass interact to shape gait energy, fixed points, and basins of attraction, revealing conditions for efficient, agile, and robust locomotion. The authors validate the model with physical experiments on granular substrates and connect findings to global stability criteria and planning implications, including a massless-foot closed-form analysis and insights into terrain-estimation strategies. The results offer practical guidance for planning and controlling legged robots on yielding substrates and motivate future work on sagittal-plane and fully articulated locomotion on deformable ground. The study highlights the tradeoffs between energy efficiency and responsiveness, and shows how a well-characterized energy map can inform mode switching and terrain-adaptive control.

Abstract

Legged robot locomotion is hindered by a mismatch between applications where legs can outperform wheels or treads, most of which feature deformable substrates, and existing tools for planning and control, most of which assume flat, rigid substrates. In this study we focus on the ramifications of plastic terrain deformation on the hop-to-hop energy dynamics of a spring-legged monopedal hopping robot animated by a switched-compliance energy injection controller. From this deliberately simple robot-terrain template, we derive a hop-to-hop energy return map, and we use physical experiments and simulations to validate the hop-to-hop energy map for a real robot hopping on a real deformable substrate. The dynamical properties (fixed points, eigenvalues, basins of attraction) of this map provide insights into efficient, responsive, and robust locomotion on deformable terrain. Specifically, we identify constant-fixed-point surfaces in a controller parameter space that suggest it is possible to tune control parameters for efficiency or responsiveness while targeting a desired gait energy level. We also identify conditions under which fixed points of the energy map are globally stable, and we further characterize the basins of attraction of fixed points when these conditions are not satisfied. We conclude by discussing the implications of this hop-to-hop energy map for planning, control, and estimation for efficient, agile, and robust legged locomotion on deformable terrain.

Figures (13)

• Figure 1: Terrain and robot model, shown with dimensional variables (left) and dimensionless variables (right). See Sections \ref{['sec:terrain_model']} and \ref{['sec:robot_model']} for details on terrain model and robot model, respectively, and Table \ref{['tab:dim_to_nondim']} for dimensionless nomenclature.
• Figure 2: Directed graph illustrating all domains and all possible transitions for the hybrid system representation of the spring-legged monopod hopping vertically on deformable terrain. The table below the graph defines the switching condition and reset map for each event.
• Figure 3: Top: Illustrative plot of nondimensionalized body and foot trajectories $\xi_\mathrm{b}(\tau)$ and $\xi_\mathrm{f}(\tau)$, respectively, for $\varepsilon_\mathrm{inj}$ = 10, $\phi = 1.25$, $\kappa_\mathrm{c} = 0.1$, $\mu = 0.1$, $\lambda_\mathrm{c} = 35$, and initial conditions $\xi_\mathrm{b} = \lambda_\mathrm{c}$, $\xi_\mathrm{f} = 0$, $\dot{\xi}_\mathrm{b} = \dot{\xi}_\mathrm{f} = 0$.Bottom left: Nondimensionalized COM kinetic energy at each TD, $\varepsilon_\mathrm{TD}[k]$, for the monopod kinematics plotted above; $k$ is the hop number. Bottom right: Detailed view of stance-phase nondimensionalized foot kinematics from the simulation above, for hop number $k = 8$.
• Figure 4: Experimental apparatus: vertically-constrained hopping robot mounted on an x-y gantry over a fluidized bed trackway filled with poppy seeds. Inset (from lynch2020SLP): the robot consists of a motor stator (A), motor slider and acrylic foot (B), force/torque sensor (C), lifting mechanism (D), and vertical guiderail (E).
• Figure 5: Left:Experimental measurements of yield stress versus intruder depth for quasistatic intrusions (gray circles, foot area = $2\times 10^{-3}$ m$^2$) show a surface regime (depth less than approximately 1 cm) and a bulk regime (depth greater than approximately 1 cm); in the bulk regime, the best-fit stress-per-unit-depth is approximately $3.36\times 10^{5}$ N/m$^3$.Middle:Measurements of stress versus depth recorded during jumping experiments (immediately prior to the compression-extension transition) agree well with the bulk-regime stress-per-unit depth estimate of $3.36\times 10^{5}$ N/m$^3$.Right:An example stance-phase foot trajectory from a jumping experiment ($\varepsilon_\mathrm{TD} = 3.8$, $\varepsilon_\mathrm{inj} = 20.3$, $\phi = 2$, $\kappa_\mathrm{c} = 0.27$, $\mu = 0.2$) demonstrates the foot-stop and reyield events predicted by our foot-ground interaction model.

Theorems & Definitions (28)

• Definition 1: Template
• Remark
• Remark
• Remark
• Remark
• Definition 2: Efficiency
• Remark
• Definition 3: Energy Stability Margin
• Remark
• Definition 4: Basin of Attraction