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Sensitive dependence of the motion of a legged robot on granular media
Chen Li, Paul B. Umbanhowar, Haldun Komsuoglu, Daniel E. Koditschek, Daniel I. Goldman
TL;DR
Legged robots can lose substantial speed on granular ground, where substrate packing and limb frequency shape locomotion. The authors test SandBot on poppy seeds and model rotary walking through limb penetration and slip. Adjusting the gait restores speeds up to 30 cm/s, while increasing frequency or reducing packing can shift motion into slow swimming.
Problem
Legged robots that perform well on hard ground can lose substantial speed on flowing granular materials, where feet experience both solid- and fluid-like forces.
Source: Results and Discussion · p. 2Method
The authors measure SandBot speed across packing fractions and limb frequencies, then model rotary walking using limb penetration and step length.
Source: Results and Discussion · p. 5Results
30 cm/s is attainable with adjusted gait parameters, compared with approximately 2 cm/s using the hard-ground settings on granular media.
Source: Results and Discussion · p. 2Takeaways & Limitations
SandBot’s speed is sensitive to packing fraction and limb frequency, with lower packing or higher frequency associated with transition from rotary walking to slow swimming.
Source: Results and Discussion · p. 4The model assumes the two tripods act independently and that the robot underside rests on the surface at initial limb contact.
Source: Results and Discussion · pp. 5–6
Abstract
from arXiv · showhide
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 $\phi$ and the limb frequency $\omega$, and a dramatic transition from rotary walking to slow swimming occurs when $\phi$ becomes small enough and/or $\omega$ 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.
Results and Discussion
SandBot’s granular-ground speed depends strongly on packing fraction and limb frequency, and its gait shifts from rotary walking to slow swimming. A two-parameter model captures key speed trends through penetration depth and step length, while relying on simplifying assumptions about tripod contact.
Results and Discussion
- 30 cm/s is attainable after gait adjustment, versus about 2 cm/s with the hard-ground settings and 60 cm/s on hard ground.The adjusted gait uses rotary walking; the tested settings did not produce rapid bouncing on granular media.Source: Results and Discussion · p. 2
- At ω = 16 rad/s, changing φ from 0.600 to 0.633 raised average speed from approximately 2 cm/s to 20 cm/s.For fixed ω, speed is nearly constant above a critical packing fraction and close to 2 cm/s below it.Source: Results and Discussion · p. 4
- Rotary walking dominates at low ω and high φ; increasing ω drives deeper penetration until step length vanishes and swimming supplies the remaining forward motion.The slower swimming mode occurs across all tested packing fractions for ω ≥ 28 rad/s.Source: Results and Discussion · p. 4
- For fixed φ, speed rises sub-linearly with ω to a maximum, then rapidly falls to approximately 2 cm/s as the robot swims.The loss is more sudden for φ ⪆ 0.6; both the critical frequency and maximum speed change around φ ≈ 0.6.Source: Results and Discussion · p. 5, Figure 2 · p. 5
- The model links speed to step length and penetration depth: deeper penetration shortens each step, explaining sub-linear speed growth and performance loss above ωc.It treats tripods independently, models one c-leg supporting one-third of the robot mass, and assumes the robot underside initially rests on the surface.Source: Results and Discussion · pp. 5–7
- In the model, average speed is ωs/π because each clock period contains two leg rotations; penetration depth is set by granular resistance, gravity, and acceleration.At low ω, the model gives minimum penetration d = mg/k; finite ω requires greater penetration to accelerate the robot body.Source: Results and Discussion · p. 6
- The walking-to-swimming transition is gradual below φ ≈ 0.6 but abrupt above it, where steps can reach previously disturbed, looser material and shrink further.The authors hypothesize that disturbed material has lower packing fraction and penetration resistance; the critical step size is near s/R = 1.Source: Results and Discussion · p. 7
- At high swimming frequencies, limb-driven excavation can reduce thrust on subsequent strokes, while faster grain motion can also increase inertial thrust.The competition between excavation, depression refill, and rate-dependent thrust remains relevant to understanding stranding and recovery.Source: Results and Discussion · p. 8
Conclusions
Experiments varying granular volume fraction and limb frequency show that SandBot’s performance is highly sensitive to both. A kinematic model links this sensitivity to limb penetration, while the paper concludes that better granular-physics understanding is needed for further design and control advances.
- Changes in φ of less than 1% can result in either rapid motion or failure to move, and slight kinematic changes have a similar effect.Source: Conclusions · p. 8
- A kinematic model attributes sublinear speed growth with ω and rapid failure at small φ or large ω to increased limb penetration and changes in local φ.Source: Conclusions · p. 8
- Further advances in performance and limb design require deeper understanding of penetration, drag, and crater formation and collapse, especially their dependence on φ.Source: Conclusions · p. 8
- Physical models such as SandBot may inform locomotion biology, while granular-interaction theory could guide limb-design and control strategies.Source: Conclusions · p. 8
Materials and Methods
The methods control granular packing and gait timing, measure SandBot’s motion, and model the forces and kinematics underlying rotary walking.
Materials and Methods
- SandBot’s clock signal uses three gait parameters, and the study uses the soft-ground setting after the hard-ground signal produced about 1 cm/s on granular media.The hard-ground signal caused simultaneous stance phases of both tripods; encoder comparisons showed trajectory error of a few percent.Source: Materials and Methods · pp. 8–9
- A fluidized-bed trackway sets poppy-seed volume fraction from φ = 0.580 to φ = 0.633 using controlled air pulses.The bed is initially fluidized, then short pulses pack the grains; φ is calculated from grain mass, bed volume, and seed density.Source: Materials and Methods · p. 9
- Synchronized 100-frame/s dorsal and lateral videos track the robot and yield v_x(t), with three trials for each tested (φ, ω) combination.An LED synchronizes video and motor-encoder data, and a rail-pulley system limits cable drag.Source: Materials and Methods · p. 9
- The full and simple models fit measured speed for v_x ≤ v*_x and agree when step size is below the critical value s = R.Figure 4 compares the fitted speed curves; the full and simple models use different Δt and α parameters.Source: Figure 4 · p. 10, Materials and Methods · p. 10
- For fixed material strength, penetration depth increases with limb frequency until step size falls below a critical value or granular force cannot support and accelerate the robot.In Figure 4a, intersections of required-force and granular-force curves determine penetration depth and therefore step size.Source: Figure 4 · p. 10, Materials and Methods · p. 10
- The model generalizes the simple k z grain-force approximation to a depth-dependent isotropic yield stress αz over the curved limb.The vertical force is integrated over limb segments; the full model uses an exact body-acceleration expression when the limbs gain purchase.Source: Materials and Methods · pp. 9–10
Supporting movies
The supporting materials list a compaction-control item and four movies showing SandBot’s gait and operation on granular media.
Supporting movies
- The supplementary list includes a demonstration of granular-media compaction control using an air-fluidized bed.This is listed as item 1.Source: Supporting movies · p. 12
- One movie shows SandBot stuck with its solid-ground-tuned gait, while another shows motion with a granular-media-tuned gait.These are listed as Movies S1 and S2.Source: Supporting movies · p. 12
- The remaining movies cover different granular-media compactions and SandBot’s walking and swimming modes.These are listed as Movies S3–S5.Source: Supporting movies · p. 12