Sensitive dependence of the motion of a legged robot on granular media

arXiv:0905.4950 · physics.bio-ph, cond-mat.other, cond-mat.soft, q-bio.QM · Submitted 2009-05-29 · Read on arXiv

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Introduction to the show: ident: Genomics Radio. Generated commentary on the latest computational biology and genomics papers.

Ines: We're looking at the paper: "Sensitive dependence of the motion of a legged robot on granular media".

Marcus: Here's the gist of the paper. The gist The robot speed depends sensitively on the packing fraction φ and the limb frequency ω, and a dramatic transition from rotary walking to slow swimming occurs when φ becomes small enough and/or ω large enough<ref:0905.4950#pg2> Robot Performance on Granular Media Legged locomotion on flowing ground is unlike locomotion on hard ground because feet experience both solid and fluid-like forces during surface penetration<ref:0905.4950#pg2> Standard gait control parameters achieve speeds at best two orders of magnitude smaller than the 2 body lengths/s (≈ 60 cm/s) for motion on hard ground, but empirical adjustment of these control parameters away from the hard ground settings restores good performance, yielding top speeds of 30 cm/s<ref:0905.4950#pg2>.

Ines: Robot speed depends sensitively on the packing fraction φ and the limb frequency ω, and a dramatic transition from rotary walking to slow swimming occurs when φ becomes small enough and/or ω large enough<ref:0905.4950#pg2>.

Marcus: Gait Modes and Transition The robot moves using an alternating tripod gait in which two sets of three approximately c-shaped legs rotate synchronously and π out of phase<ref:0905.4950#pg3>.

Ines: On rigid, noslip ground SandBot’s limb trajectories are tuned to create a bouncing locomotion that generates speeds up to 2 body lengths/s (≈ 60 cm/s)<ref:0905.4950#pg3>.

Title and authors: Ines: So, we're looking at this paper today, "Sensitive dependence of the motion of a legged robot on granular media." We're talking about how a robot that walks on hard ground suddenly changes its behavior when it steps onto something like sand or seeds.

Marcus: Yeah, it’s about testing a six-legged robot called SandBot—a two point three kilogram machine—on poppy seeds to see what happens when the ground isn't rigid <ref:0905.4950#pg1>. It immediately tells us that locomotion on flowing ground is fundamentally different from walking on hard surfaces because the feet experience both solid and fluid-like forces during that penetration phase.

Yuki: That difference in force—the solid versus fluid-like aspect—is really interesting because it suggests that the physical interaction with a granular medium dictates a total rewrite of how you think about the movement. What does this mean for how we model locomotion in any complex environment?

Ines: Well, the paper sets up an alternating tripod gait where two sets of three legs rotate together and are out of phase, which is standard for this type of robot. But when they hit the granular bed, instead of bouncing like they would on rigid ground, they switch to a swimming gait where the legs always slip backward relative to the stationary grain bed.

Marcus: And that shift in gait causes a big performance drop. The paper says that for this swimming mode, you see a reduction in performance by a factor of thirty—that’s about two centimeters per second—compared to what they achieved on hard ground <ref:0905.4950#pg1>.

Yuki: That thirty times reduction is substantial; it suggests that the mechanism causing the slip and the resulting double stance interval creates significant interference when the ground isn't solid. This kind of interference is something we see in biological systems too, maybe how movement changes when you transition from walking to swimming underwater.

Ines: Exactly. The authors suspect that this reduction is caused by an interval of double stance, where both tripods are simultaneously in contact with the ground, which works fine on hard ground for bouncing but causes trouble here. They then show that changing the gait parameters to get rid of that double stance lets SandBot move in a rotary walking gait at speeds up to one body length per second.

Marcus: That speed of one body length per second is pretty respectable given the substrate, and it actually resembles the pendular gait they used for hard ground locomotion. But they emphasize that no amount of adjusting the gait parameters resulted in rapid bouncing locomotion on granular media; that's a key finding.

Yuki: So, even when you try to force it into a walking pattern by changing how the legs move, you just don't get those high-speed bounces back. This supports the idea that the physics of penetration is fundamentally altered by the material itself.

Title and authors: Ines: And this sensitivity ties directly into how much freedom we have over these parameters. The robot’s speed depends sensitively on two things: the packing fraction, or phi, and the limb frequency, or omega. If either of those changes significantly, you get a dramatic transition from walking to slow swimming.

Marcus: That sensitivity is quantified in a pretty concrete way. For example, at a limb frequency of sixteen radians per second, the robot's average speed shows nearly a factor of five change if the packing fraction phi shifts by just five percent. It’s that kind of fine-tuning you can’t ignore when you're dealing with granular media.

Yuki: That makes sense in a population genetics context; small changes in environmental conditions can have huge impacts on fitness or behavior over generations. Here, it means a tiny change in the substrate density completely changes the robot's locomotion mode.

Ines: The paper models this transition using kinematics based on generic features of penetration and slip of a curved limb in granular media. The model suggests that reducing the step length by increasing the penetration depth is what causes that sub-linear increase in speed as you increase omega, and it also predicts a rapid loss of performance when you go above a critical frequency, omega c.

Marcus: So, the maximum limb penetration depth, which we call d, seems to be the key model component. It balances the vertical acceleration of the robot's center of mass against two things: gravity and the vertical granular penetration force. That force is related to k and t, where k and t are fit parameters in their expression for average horizontal velocity, v x = omega s pi.

Yuki: If that penetration depth, d, controls the step length, it means the way the robot sinks into the media is directly controlling its speed. That’s a very direct physical link between material interaction and locomotion output.

Ines: And this leads to some important implications for design. The model reveals that performance failure happens when you increase limb penetration depth by decreasing phi, or by increasing omega. It also notes that changes to the local packing fraction, caused by penetration itself or the removal of limbs, have similar effects on speed.

Marcus: So what does this tell us about building better robots for these environments? The paper suggests that further performance gains require a much more detailed understanding of the physics involving penetration, drag, and how craters form and collapse when you move through the material.

Title and authors: Yuki: That sounds like it points toward needing models that can handle those localized disturbances and structural changes in the medium, not just average properties. It pushes us to look at the fine-scale mechanics of interaction.

Ines: Right. Ultimately, this paper shows that performance is extremely fragile when you move from hard ground to granular media because of these physical transitions tied to packing fraction and limb frequency. We have a solid understanding now of how penetration depth dictates step length and speed through that kinematic model, which is really useful for guiding the development of interaction strategies thirty-eight and control strategies thirty-nine.

Marcus: So, before we wrap up this discussion on "Sensitive dependence of the motion of a legged robot on granular media," I think it's important to remember how sensitive those speeds are to small changes in phi and omega. That sensitivity is what drives the transition from walking to swimming.

Yuki: And for the wider context, it shows that even seemingly simple locomotion tasks have complex physics rooted in material properties, which is a reminder that we need models that capture those local interactions.

Ines: So we’ve looked at how the robot switches modes based on frequency and density, and how penetration depth controls the step size. This paper really lays out the kinematic model connecting those dots for anyone working on legged locomotion in unstructured environments.

Marcus: It gives engineers a clear way to predict when they're going to run into a performance wall based on their choice of limb frequency or substrate density. We need that kind of predictive modeling more than ever as we try to deploy these systems outside controlled settings.

Yuki: And for population genetics, it reinforces the idea that environmental structure profoundly shapes the available strategies for movement and persistence in any system. It's a very tangible example of physical constraints translating directly into biological outcomes.

Ines: So that’s our look at "Sensitive dependence of the motion of a legged robot on granular media." We see how phi and omega control the speed, how penetration depth sets the step size, and how that leads to a switch between walking and swimming modes. The modeling gives us tools to predict those transitions better.

Marcus: Yeah, it's about moving beyond just testing performance on hard ground and understanding what happens when you introduce genuine material complexity like granular media. It’s a necessary step for any real deployment of these robots.

Yuki: I think the biggest implication is that we need to develop models that account for those localized disturbances and structural changes, because that's where the real physics of interaction lies.

Ines: We’ll keep pushing that work forward, hoping to get a more detailed look at how drag and crater formation affect these dynamics next time. That's where the next paper will likely focus.

The paper's summary: Ines: So, this paper boils down to how much the robot's speed actually depends on two key things: the packing fraction, that’s phi, and the limb frequency, that’s omega.

Marcus: Yeah, and they show this really dramatic switch happens when either of those factors hits a certain threshold. It goes from rotary walking to slow swimming.

Yuki: And for someone looking at population genetics or evolution, it suggests that the environmental "conditions"—like the density of the substrate—can completely redefine what's actually possible for that organism to do in terms of movement.

Ines: Exactly. The authors use a kinematic model based on how a curved limb penetrates and slips through granular media to show this dependence. They find that reducing the step length by increasing penetration depth is what causes the speed to increase slowly as you increase omega, up until a certain point, and then it crashes above a critical frequency, omega c.

Marcus: And they quantify how sensitive that is. They show that even a tiny shift in packing fraction, like just five percent, can change the average speed by nearly a factor of five when the limb frequency is around sixteen radians per second. That’s not small for robot dynamics.

Yuki: So what this means for the real world, it’s about predicting failure points before you even start building. If you know your substrate density and you pick a certain gait frequency, this paper gives you a mathematical way to estimate when the system will transition into that slower swimming mode or just fail to move altogether.

Ines: Right. It links the micro physics of how a foot interacts with grains—the penetration depth d balancing gravity and those granular forces—to the macro result of robot speed, which is something you can actually control or at least predict better in a lab setting.

Marcus: And they also pointed out that for fixed limb frequency, the speed stays pretty constant until phi drops below a critical value, phi c. Below that value, the speed practically goes to zero. It’s like there’s a minimum density required just to keep it moving efficiently in that mode.

Yuki: That’s interesting because it shows there's a clear physical boundary for survival or at least effective locomotion under those conditions, which is something you can actually test against biological constraints when you look at how organisms move through different terrains.

Ines: The paper also flags a limitation—the model uses simplified kinematics and granular penetration forces, but they agree it aligns well with more complex treatments. They suggest the next step needs to be a much deeper dive into the physics of crater formation and collapse when the robot moves through that material.

Marcus: Exactly. Right now, you've got a good map for predicting gait switching based on density and frequency, but you still need those finer details about how the material structure itself breaks down under load.

Yuki: So while this model helps us understand the high-level mode transitions—walking versus swimming—it leaves open the question of what happens to the robot’s structural integrity when it actually starts digging holes or causing local collapses in that granular bed.

The paper's improvements: Marcus: So, after showing us how sensitive the speed is to packing fraction and limb frequency, what are the authors suggesting we do next to make this work better?

Ines: They focus on improving that adaptive gait control system. They suggest implementing a system that doesn't just use fixed parameters but dynamically adjusts them based on real-time readings of how the granular media is behaving.

Yuki: That sounds like they want the robot to be smarter about its environment, moving away from a pre-programmed sequence towards something reactive.

Marcus: Exactly. They mention that because performance is restored by empirically adjusting those control parameters away from hard ground settings, you need a system that can monitor things like volume fraction and substrate resistance to make those adjustments on the fly.

Ines: And they propose developing a predictive model for substrate state estimation. This means the AI wouldn't just react; it would try to predict when the robot is about to hit ground disturbed by its previous step, so it can adjust before performance actually degrades sharply.

Yuki: That level of proactive adjustment is huge. It’s about anticipating the physical disturbance rather than just responding to where you are right now.

Marcus: They also look at frequency-dependent optimization. Since the speed really depends on that limb frequency, they suggest a control policy that optimizes it based on the predicted material state to keep things running as fast as possible across different granular media densities.

Ines: That’s smart because they showed that performance peaks at a critical limb frequency, omega c, and you want to stay tuned right around there instead of drifting too high or too low.

Yuki: It connects back to the biological idea of finding an optimal rhythm for movement given environmental constraints. If the environment changes, you need a different rhythm to maintain efficiency.

Marcus: The model integration is another big one they suggest for design. They want engineers to use this kinematic model early on so they can predict how the robot will perform under various conditions before they even start building it.

Ines: So, the implication is that we move from just testing a robot in a lab to using these models as tools to guide the actual design of locomotion strategies for complex terrains.

Yuki: It’s about moving beyond just observing the results and actually using this physics to build systems that are inherently more robust across different materials.

Marcus: This leads us into thinking about how these improved control strategies will translate when you try to apply them in a real, unstructured environment, like on Mars or in a disaster zone.

Conclusion: Tom: So we've covered how the robot's speed is highly sensitive to packing fraction and limb frequency in this paper, "Sensitive dependence of the motion of a legged robot on granular media."

Ines: Right. Basically, we saw that performance swings wildly when you change those two parameters because it causes a switch between walking and swimming modes.

Marcus: And the authors showed us that the core mechanism behind this is how penetration depth changes as you adjust those frequencies, which dictates the step length.

Yuki: From a population standpoint, this tells us that environmental density isn't just a static background; it actively selects for different movement strategies based on how you interact with that medium.

Ines: Exactly. And they gave us tools to predict when that transition happens by mapping the kinematic model onto things like critical frequency omega c and critical packing fraction phi c.

Marcus: I think the main value here is for engineers because it gives them a predictive framework instead of just trial and error when designing robots for unstructured settings.

Yuki: It shows that understanding these material interaction rules helps us understand how organisms have adapted to vastly different physical landscapes over evolutionary time.

Ines: And the authors did flag that their model is based on simplified kinematics, so we still need a deeper look at the physics of crater formation and collapse when the robot moves through those grains.

Marcus: That's true. They built a solid kinematic map, but real-world deployment will require modeling those localized physical disturbances more thoroughly.

Yuki: So while this paper gives us the high-level mode switching rules, it points us toward needing more detailed physics to truly understand the long-term survival or functional success of these robots in those environments.

Chen Li, Paul B. Umbanhowar, Haldun Komsuoglu, Daniel E. Koditschek, Daniel I. Goldman

School of Physics, Georgia Institute of Technology · Department of Electrical and Systems Engineering, University of Pennsylvania Department of Mechanical Engineering, Northwestern University

physics.bio-ph, cond-mat.other, cond-mat.soft, q-bio.QM

Submitted: 2009-05-29

Updated: 2026-10-18

Journal ref: Proceedings of the National Academy of Sciences (2009), 106 (9), 3029-3034

DOI: 10.1073/pnas.0809095106

License: http://creativecommons.org/licenses/by-nc-nd/4.0/

Importance score: 79/100

The gist: The gist The robot speed depends sensitively on the packing fraction φ and the limb frequency ω, and a dramatic transition from rotary walking to slow swimming occurs when φ becomes small enough

Key concepts

Packing Fraction ($π): This represents how densely packed the granular material is. The robot's speed changes drastically based on this value; performance is near zero when the packing fraction drops below a critical threshold ($π_c$), indicating that too much empty space severely limits movement.
Limb Frequency ($ω): This is the oscillation frequency of the robot's legs. The speed and gait mode are sensitive to this frequency. Increasing the limb frequency causes limbs to penetrate deeper, but there is a critical point where performance rapidly degrades, leading to a transition from walking to swimming.
Rotary Walking vs. Swimming: The robot exhibits two main movement styles: rotary walking (fast movement) and slow swimming (legs always slip backward). The transition between these modes is sharp and depends on the packing fraction ($π) and limb frequency ($ω), showing a fundamental change in how the robot interacts with the granular surface.
Limb Penetration Depth: This is how far a leg sinks into the granular material during movement. The model shows that increasing this depth, often caused by changes in packing fraction or frequency, causes sub-linear speed increases and can lead to a rapid loss of performance if penetration becomes too deep.

Summary

The gist The robot speed depends sensitively on the packing fraction φ and the limb frequency ω, and a dramatic transition from rotary walking to slow swimming occurs when φ becomes small enough and/or ω large enough<ref:0905.4950#pg2>

Robot Performance on Granular Media

Legged locomotion on flowing ground is unlike locomotion on hard ground because feet experience both solid and fluid-like forces during surface penetration<ref:0905.4950#pg2> Standard gait control parameters achieve speeds at best two orders of magnitude smaller than the 2 body lengths/s (≈ 60 cm/s) for motion on hard ground, but empirical adjustment of these control parameters away from the hard ground settings restores good performance, yielding top speeds of 30 cm/s<ref:0905.4950#pg2>. Robot speed depends sensitively on the packing fraction φ and the limb frequency ω, and a dramatic transition from rotary walking to slow swimming occurs when φ becomes small enough and/or ω large enough<ref:0905.4950#pg2>.

Gait Modes and Transition

The robot moves using an alternating tripod gait in which two sets of three approximately c-shaped legs rotate synchronously and π out of phase<ref:0905.4950#pg3>. On rigid, noslip ground SandBot’s limb trajectories are tuned to create a bouncing locomotion that generates speeds up to 2 body lengths/s (≈ 60 cm/s)<ref:0905.4950#pg3>. When tested on granular media, the robot instead of bouncing adopts a swimming gait in which the legs always slip backward relative to the stationary grain bed and for which performance is reduced by a factor of 30 to ≈ 2 cm/s<ref:0905.4950#pg3>. This reduction is surmised to be due to an interval of double stance, which causes tripod interference on granular media<ref:0905.4950#pg3>. Changing the gait parameters to remove the double stance allowed SandBot to move in a rotary walking gait at speeds up to 1 body length/s (≈ 30 cm/s) in a rotary walking gait that resembles the pendular gait of the robot on hard ground<ref:0905.4950#pg3>. No amount of gait parameter adjustment produced rapid bouncing locomotion on granular media<ref:0905.4950#pg3>.

Sensitivity to Volume Fraction and Limb Frequency

Robot performance (speed) is remarkably sensitive to φ, for example, at ω = 16 rad/s the robot speed vx(t) shows a change in average speed vx of nearly a factor of five as φ changes by just 5 %<ref:0905.4950#pg3>. For fixed ω, vx is effectively constant for φ above a critical volume fraction φc(ω), but is close to zero for φ below φc(ω)<ref:0905.4950#pg3>. The rotary walking mode is dominant at low ω and high φ<ref:0905.4950#pg3>. With increased ω, limbs penetrate further as the requisite force to rapidly accelerate the robot body to the finite limb speed (Rω) increases<ref:0905.4950#pg3>. The transition from rotary walking to swimming is sharper in vx for higher ω and smoother for lower ω<ref:0905.4950#pg3>.

Kinematic Model and Physics of Motion

A kinematic model of the rotary walking mode was proposed based on generic features of penetration and slip of a curved limb in granular media, capturing the dependence of robot speed on limb frequency and the transition between walking and swimming modes<ref:0905.4950#pg2>. The model indicates that reduction of step length through increased penetration depth is the cause of the sub-linear increase in vx with ω and the rapid loss of performance above ωc<ref:0905.4950#pg3>. The maximum limb penetration depth d is determined by balancing the vertical acceleration of the robot center of mass ma with the sum of the vertical granular penetration force kz and gravitational force mg<ref:0905.4950#pg4>. The expression for vx is determined by two fit parameters k and ∆t<ref:0905.4950#pg4>.

Key Findings and Implications

The studies reveal that changes in φ of less than one percent result in either rapid motion or failure to move, and slight kinematic changes have a similar effect<ref:0905.4950#pg2>. The model reveals that the sublinear dependence of speed on ω and the rapid failure for sufficiently small φ and/or large ω are consequences of increasing limb penetration with decreasing φ and/or increasing ω, and changes to local φ due to penetration and removal of limbs<ref:0905.4950#pg3>. Further advances in performance require a more detailed understanding of the physics associated with penetration, drag, and crater formation and collapse<ref:0905.4950#pg3>. Such devices will begin to have capabilities comparable to organisms for more efficient and capable exploration of challenging terrestrial (e.g. rubble and disasters sites) and extra-terrestrial (e.g. the Moon and Mars) environments<ref:0905.4950#pg2>.

Materials and Methods Summary

The study employed a 2.5 m long, 0.5 m wide fluidized bed trackway with a porous plastic (Porex) flow distributor to control the volume fraction φ of the granular substrate<ref:0905.4950#pg4>. Volume fraction was set by controlling the height of the granular media<ref:0905.4950#pg4>. Kinematics measurements were recorded using synchronized high speed video cameras at 100 frames/s to calculate vx(t)<ref:0905.4950#pg4>. The robot's six motors were controlled by a clock signal defined by three gait parameters: θs, θ0, and dc<ref:0905.4950#pg4>. Pilot experiments showed that the hard ground clock signal fails on granular media because of the simultaneous stance phase of two tripods<ref:0905.4950#pg4>. The model incorporates simplified kinematics and granular penetration forces while agreeing well with a more realistic treatment<ref:0905.4950#pg4>.

References

1 Kumagai, J (2004) Sand trip - DARPA’s 320-kilometer robotic race across the mojave desert yields no winner, but plenty of new ideas<ref:0905.4950#pg4>

2 Raibert, MH (1986) Legged robots<ref:0905.4950#pg4>

3 Bekker, M (1956) Theory of Land Locomotion (The University of Michigan Press), p 520<ref:0905.4950#pg4>

4 Kadanoff, LP (1999) Built upon sand: Theoretical ideas inspired by granular flows<ref:0905.4950#pg4>

5 Altendorfer, R, Koditschek, DE, Holmes, P (2004) Stability analysis of legged locomotion models by symmetry-factored return maps<ref:0905.4950#pg4>

6 Westervelt, ER, Buche, G, Grizzle, JW (2004) Experimental validation of a framework for the design of controllers that induce stable walking in planar bipeds<ref:0905.4950#pg4>

7 Burridge, RR, Rizzi, AA, Koditschek, DE (1999) Sequential composition of dynamically dexterous robot behaviors<ref:0905.4950#pg4>

8 Berkemeier, MD, Fearing, RS (1998) Sliding and hopping gaits for the underactuated acrobot<ref:0905.4950#pg4>

9 Schaal, S, Atkeson, CG, Sternad, D (1996) One-handed juggling: A dynamical approach to a rhythmic movement task<ref:0905.4950#pg4>

10 Ezcurra, E (2006) Global Deserts Outlook (United Nations Educational), p 168<ref:0905.4950#pg4>

11 Terzaghi, K (1943) Theoretical soil mechanics (New York: Wiley), p 510<ref:0905.4950#pg4>

12 Nedderman, R (1992) Statics and kinematics of granular materials (Cambridge University Press), p 352<ref:0905.4950#pg4>

13 Heil, P, Rericha, EC, Goldman, DI, Swinney, HL (2004) Mach cone in a shallow granular fluid<ref:0905.4950#pg4>

14 van Zon, JS et al. (2004) Crucial role of sidewalls in velocity distributions in quasi-two-dimensional granular gases<ref:0905.

Improvements for AI systems

Adaptive Gait Control for Flowing Media: Implement a gait control system that dynamically adjusts parameters based on real-time estimates of granular media properties, as the paper shows empirical adjustment of these control parameters away from the hard ground settings, restores good performance. This allows the AI to transition seamlessly between rotary walking mode and slow swimming mode by monitoring feedback related to volume fraction or perceived substrate resistance.
Predictive Substrate State Estimation: Develop a model that predicts the critical transition points, specifically identifying when the robot begins to encounter ground disturbed by the previous step. This prediction, based on penetration depth models, enables proactive gait adjustments before performance degrades sharply due to encountering material disturbed by the previous step.
Frequency-Dependent Locomotion Optimization: Design a control policy that optimizes limb frequency based on the predicted material state to maximize speed. Since robot performance (speed) is remarkably sensitive to φ and vx increases sub-linearly with ω to a maximal average speed v∗x at a critical limb frequency ωc, the AI can adjust its oscillation frequency in real-time to maintain maximum efficiency across varying granular media densities.
Kinematic Model Integration for Design: Incorporate the derived kinematic model into the robot's design phase to predict performance under various conditions. This allows engineers to guide development of theory of interaction with complex media advanced enough to predict limb design [38] and control [39] strategies.

Abstract

Legged locomotion on flowing ground (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 (about 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 ϕ and the limb frequency ω, and a dramatic transition from rotary walking to slow swimming occurs when ϕ becomes small enough and/or ω 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.