Aubry-Andr\'e Localization Transition for an Active Undulator

Andrew Zangwill; Christopher J. Pierce; Daniel I. Goldman; Dmitri Kalinin; Tianyu Wang

REVIEW 3 major objections 6 minor 42 references

Aubry-Andr\'e Localization Transition for an Active Undulator

T0 review · 3 major / 6 minor · reviewed 2026-08-06 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read An undulating robot without sensory feedback becomes trapped in a boulder channel when the boulder spacing is sufficiently aperiodic, showing a transition that quantitatively resembles Aubry-André localization of quantum waves.

desk verdict Solid experimental observation of aperiodic-terrain trapping, but the AA localization claim leans on termination rules and a model that assumes the potential; still deserves a serious referee. read the letter →

arxiv 2507.03715 v2 pith:ZQYERJEW submitted 2025-07-04 physics.bio-ph

Christopher J. Pierce , Tianyu Wang , Dmitri Kalinin , Andrew Zangwill , Daniel I. Goldman This is my paper · ORCID

classification physics.bio-ph

keywords localizationtransitionAubry-Andrémodelundulatorylocomotionresistiveforcetheoryquasiperiodicterrainsnake-likerobottorquefluctuationsmeansquareddisplacement

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper claims that a snake-like robot undulating without sensory feedback through a channel lined with hemispherical boulders passes freely when the boulder pattern is periodic, but becomes trapped when the pattern is made sufficiently aperiodic. The trapping transition is quantified by a drop in mean travel distance, a mean-squared-displacement exponent that falls to zero, and travel-distance distributions that change from Gaussian to exponential. These signatures are claimed to agree with experimental and theoretical results for the Aubry-André localization transition in quantum systems. The paper further claims that a generalized resistive force theory with a spatially varying drag anisotropy reproduces the behavior, and that large fluctuations in the torques required to maintain the gait drive the transition. If correct, the result extends condensed-matter localization concepts to the mechanics of self-deforming locomotors.

What carries the argument

The central object is the Aubry-André terrain: a periodic base row of boulders plus a second incommensurate row whose height h tunes the aperiodicity, with the golden-ratio value β=(√5−1)/2 used as the primary irrational period ratio. The carrying mechanism is a generalized resistive force theory in which the drag anisotropy ξ(X) varies in space as ξ(X)=Δ sin(πX/a)+Γ sin(βπX/a)+1, yielding an equation of motion V_cm = (4π²b²/LλT)∫[ξ(X_cm+x)−1]cos²(2π(x/λ−t/T))dx. This theory converts the terrain geometry directly into forward or backward thrust and produces oscillatory trapping. The paper uses the torque spectral power ratio P_broad/P_0, the integrated broad-band torque noise relative to the gait-frequency peak, to connect the torque-fluctuation mechanism in the model with the termination events observed in the physical robot.

What would settle it

Repeat the experiments with motors that have a higher torque limit and with a constraint that prevents out-of-plane flipping, and observe whether robots still stop at h/d≈0.6. If they pass through the channel, the transition is an artifact of the termination rules; if they still trap and oscillate in place, the torque-fluctuation localization mechanism is confirmed.

Watch

Extended reading notes

Core claim

The central claim is that an open-loop undulating robot exhibits a genuine localization transition as the aperiodicity of its terrain increases: for a strictly periodic boulder array it moves ballistically through the channel, while for a sufficiently aperiodic array it stops at a localized position. The transition is measured by the mean squared displacement exponent, which drops from ballistic (b≈2) to localized (b≈0), and by the travel-distance distribution, whose shape changes from Gaussian (α=2) to exponential (α=1) as the perturbing boulder height increases around h/d≈0.6. The paper argues that localization by oscillatory trapping in the resistive force theory model and localization by trial termination in experiments and simulations are the same phenomenon, both arising from large fluctuations in the driving torques required to maintain the serpenoid gait. A generalized resistive force theory with drag anisotropy given by an Aubry-André-type function reproduces the observed trajectories and the commensurability dependence, including localization for certain rational period ratios.

Load-bearing premise

The paper's load-bearing premise is that stopping a trial because the robot flips, wedges, or overloads a motor is the same physical phenomenon as the oscillatory trapping seen in the RFT model, both being responses to the same large fluctuations in required driving torque.

Editorial extensions

If this is right

An undulating locomotor with a fixed gait and no feedback can be stopped by a deterministically quasiperiodic terrain even though the same gait passes through a periodic terrain.
The Gaussian-to-exponential crossover in travel-distance distributions is a practical diagnostic for detecting the onset of localization in locomotion experiments.
The transition does not require three-dimensional failure modes: the one-dimensional RFT model already shows oscillatory trapping, implying that flips, wedges, and motor overloads preempt rather than cause the localization.
Rational incommensurability values such as β=2/3 can also produce localization, matching the behavior of quantum waves in the Aubry-André model when the perturbation ratio has p≠1.
Large torque fluctuations are the proximate cause of the transition, so measuring joint-torque spectra in a heterogeneous terrain can predict where an open-loop undulator will get stuck.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The same organizing principle suggests that any sufficiently long self-deforming locomotor with an inflexible gait and no feedback will trap in terrain whose spatial frequencies are incommensurate with the gait wavelength, a prediction that could be tested with biological undulators such as snakes or worms in engineered obstacle arrays.
The RFT model implies a design rule: to keep an undulating robot moving, make the terrain periodic at the gait wavelength or add feedback; to trap it, add a small-amplitude incommensurate perturbation.
A quantitative test of the quantum analogy would be to extract the localization length from the exponential tail of the final position distribution and compare its dependence on perturbation amplitude with the Aubry-André localization-length scaling; the paper reports exponential distributions but does not provide this scaling comparison.
The torque-noise ratio P_broad/P_0 could serve as a general, terrain-independent observable for detecting the approach to localization before the robot actually stops.

Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Two things worth knowing about this paper. First, the core experiment is well done: a snake-like robot without sensory feedback traverses a channel with a boulder array, and as the aperiodic component grows, the mean travel distance drops sharply and the robot stops. The transition is real, reproducible in simulation, and supported by rational-β controls. Second, the paper overreaches when it calls this an Aubry-André localization transition. The quantitative signatures used to make that case—MSD exponent b≈0 and exponential distance distributions—are computed from trials that mostly end by termination (flip, wedge, motor overload), not by the oscillatory trapping seen in the RFT model. Once a trial is terminated the position is frozen, so averaging over terminated trajectories drives b to zero and fattens the exponential tail regardless of the underlying dynamics. The authors argue that termination preempts trapping and arises from the same torque fluctuations, but the evidence is indirect: the torque spectra are simulation-only, the RFT and simulation Pbroad/P0 curves differ in shape, and there are no experimental torque data. The RFT model also inserts the AA potential directly into the drag anisotropy ξ(x) (Eq. 4) rather than deriving it from contact mechanics, so the model's localization is partly built in. That said, the paper is honest about its own limitations—it explicitly calls the termination rules ad hoc and the agreement qualitative—and the model derivation is in the main text, so the missing-SM concern is unfounded. What is genuinely new is the mapping of a physical terrain to an AA potential for a self-deforming locomotor, extending the group's earlier diffraction and Bragg work to aperiodic landscapes. The generalization of RFT to spatially varying drag anisotropy is a reasonable first step, and the torque-fluctuation picture is a plausible mechanism worth testing. If the termination-trapping equivalence can be strengthened with direct measurements, and a random-disorder control is added, the paper would be considerably more convincing. I'd bring this to reading group. It deserves a serious referee; I would send it to peer review with the request that the authors address the termination issue head-on, add the random control, report experimental trial counts, and show real torque data.

Referee Report

3 major / 6 minor

Summary. The manuscript reports experiments, multibody simulations, and a modified resistive force theory (RFT) study of a snake-like robot with N=9 servomotors undulating through a narrow channel containing two rows of hemispherical boulders. The boulder arrangement is chosen so that the landscape profile approximates the Aubry-André (AA) potential, with a periodic base lattice and an incommensurate perturbation. The authors observe that for zero perturbation height the robot traverses the channel ballistically, while as the perturbing boulder height h increases, the mean travel distance drops sharply near h/d ≈ 0.6, the MSD exponent goes from approximately 2 to approximately 0, and the travel-distance distributions evolve from Gaussian to exponential. They introduce an RFT model with a spatially varying drag anisotropy of the same functional form as the AA potential and show that the model undulator exhibits similar trajectories, including oscillatory trapping. They argue, via torque spectra, that trial termination in experiments and simulations (flips, wedging, motor overload) is equivalent to oscillatory trapping. The paper concludes that the transition resembles quantum AA localization.

Significance. If the quantitative claims hold, the paper would be a striking demonstration that an active, self-deforming classical system can exhibit a localization transition controlled by landscape aperiodicity, with statistical signatures similar to those of AA localization in ultracold atoms. The experimental apparatus is well designed, the control cases (h=0 ballistic, h=d stopped) are clear, and the authors are appropriately cautious in some passages about the speculative quantum-classical analogy. The main value is as a demonstration of a robust qualitative phenomenon and a testbed for generalized RFT. However, the strength of the quantitative claim depends on whether the termination-dominated statistics reflect genuine physical localization rather than hardware or control artifacts, and that equivalence is not established in the present manuscript.

major comments (3)

[Trapping versus Termination; Figs. 2B, 2C, and 3] The equivalence between termination and oscillatory trapping is the load-bearing step for the quantitative localization claim, and it is not established. Fig. 2B shows that the termination probability rises to near unity at large h/d, and the MSD exponent in Fig. 2C and the travel-distance distributions in Fig. 3 are computed from data that include terminated trials. A terminated trial contributes X(t) = const after the stopping event, which by itself drives the long-time MSD exponent toward 0 and biases the distance distribution toward short travel distances. The authors concede that their termination rules are 'somewhat ad hoc' (end of 'Trapping versus Termination'), and the torque-spectrum comparison in Figs. 6C and 6D is qualitative: the curves differ in shape and steepness, and no experimental torque spectra are shown. To support the claim that the observed transition is a physical localization transition rather than a stopping-protocol artifact, the authors should either recompute the MSD and travel-distance distributions using only trials that reach genuine oscillatory trapping, or provide direct evidence (e.g., experimental joint-torque time series at termination, or a systematic variation of the motor torque limit) that termination events are triggered by the same torque-fluctuation mechanism as RFT trapping.
[Eq. (4) and 'Undulating Transport in Periodic and Aperiodic Terrains: Theory'] The RFT model is partially circular: the drag anisotropy ξ(x) in Eq. (4) is chosen to have exactly the form of the AA potential (Δ sin(πx/a) + Γ sin(βπx/a) + 1), and the paper states 'We have not attempted to calculate the spatial variation of the drag anisotropy for any actual terrain.' Consequently, the RFT result that localization occurs for large Γ/Δ is built in by construction and cannot independently validate the AA analogy. This does not invalidate the experimental observation, but it weakens the claim that the theory 'reproduces the behavior we observe' as evidence for a specific connection to the AA model. The authors should explicitly state this limitation and, ideally, test whether the observed phenomenology is specific to the AA functional form by comparing with alternative aperiodic drag functions (e.g., random or chirped potentials).
[Summary and Conclusions; Abstract] The claim of a 'potentially fundamental connection between classical and quantum wave mechanics' is not supported by the presented data. The authors themselves note (citing Ref. 32) that a classical particle can localize in an AA potential, so the observed trapping may be a generic property of quasiperiodic potentials rather than a wave-mechanical analogue. The similarity to the cold-atom experiments is based on visual resemblance of curves (Fig. 2B vs Ref. 34; Fig. 3B vs Ref. 34) rather than a quantitative comparison of exponents or scaling functions. To make the central claim convincing, the authors should either provide a quantitative comparison with the AA transition (e.g., critical exponent, scaling collapse, or a direct measure of the Lyapunov exponent) or soften the language to a qualitative analogy.

minor comments (6)

[Significance Statement] The phrase 'Aubrey-André' should be 'Aubry-André'.
[Results and Discussion, first paragraph] The sentence 'It that, self-propulsion ceases almost immediately' contains a typo; it should read 'It is such that' or 'Thus'.
[Materials and Methods, Simulation Parameters and Protocol] The simulation uses a proportional gain Kp=1 and a torque limit of 20 Nm, while the physical robot is reported to have PD gains KP=8.5 and KD=31.2. The paper should justify that these simulation parameters reproduce the physical motor characteristics, since the torque-saturation threshold is directly involved in the termination rules.
[Materials and Methods, Design and Construction] The sentence 'For boulders where Δ > 50.8 mm (0.8D)' uses Δ, but the symbol Δ is already used in the RFT model for the drag-anisotropy amplitude; the boulder height should be denoted h.
[Fig. 3B] The use of 60% confidence intervals for the stretched-exponential exponent α is unusual; please specify the estimation procedure and why 60% was chosen.
[References] Reference 17 (Ground Control Robotics) is a company website with no year; if it is not essential, consider removing it or replacing it with a citable source.

Circularity Check

2 steps flagged · score 6.0 of 10

Two load-bearing links reduce by construction: trial termination is declared 'localization', so the MSD exponent b≈0 and exponential travel-distance distributions are produced by the stopping protocol, and the RFT model inserts the AA potential directly as the drag function ξ(x), so its AA-like trapping is an input rather than a derived prediction.

self definitional [Results, under 'Undulating Transport in Periodic and Aperiodic Terrains: Experiment' (near Fig. 2); also Fig. 2B-2C]
"This happens in one of two ways: (i) the robot becomes dynamically trapped and oscillates around some point in the channel (Fig. 1D); or, much more frequently, (ii) we terminate the trial and localize the robot at that point because it flips out of the lattice, a servomotor overloads, it wedges itself against the channel wall, or the servomotors are otherwise unable to provide the bending torques needed to maintain the prescribed serpenoid shape (Fig. 1E)."

The paper defines 'localize' to include trials stopped by ad hoc termination rules (flip, wedge, motor overload). Terminated trials have X(t)=const after stopping; with termination probability approaching unity at large h/d, the ensemble-averaged long-time MSD plateaus and any power-law fit returns b≈0 by construction. The travel-distance distribution D is likewise just the distribution of stopping distances under the termination protocol. Thus the claimed quantum-AA signatures (b≈0, exponential D) are not independent measurements unless the 'Trapping versus Termination' equivalence is valid, and the paper's only support for that equivalence is the explicitly qualitative torque-spectrum comparison (Fig. 6C vs 6D, 'qualitatively agree').
renaming known result [Summary and Conclusions; Results: Theory, Eq. 4 and Model Derivation in Methods]
"To model the heterogeneity of our terrain, we replaced the drag parameter with a drag function equal to the AA potential function."

The RFT input is the AA potential itself: ξ(x) = Δ sin(πx/a) + Γ sin(βπx/a) + 1 (Eq. 4), and the paper concedes it did not calculate ξ from contact mechanics ('We have not attempted to calculate the spatial variation of the drag anisotropy for any actual terrain'). The localization transition that emerges as Γ/Δ grows is therefore the AA quasiperiodic structure inserted by hand into the drag function, restated in new coordinates; it cannot independently confirm the AA analogy. The experiments and simulations are independent, but the theoretical 'reproduction' is not a first-principles derivation.

full rationale

The paper is not dominated by self-citation: the cold-atom comparison is an external benchmark, and no load-bearing claim rests on the authors' prior work. However, two central links reduce by construction. First, the quantitative localization signatures are built from trials whose stopping is declared 'localization' by ad hoc termination; terminated trajectories force the MSD plateau and shape the travel-distance distribution, so the b≈0 and exponential-D results are artifacts of the protocol unless the paper's qualitative torque-spectrum argument establishes equivalence. Second, the RFT model inserts the AA potential as the drag anisotropy function, so its AA-like localization is an input assumption rather than a derived consequence. The direct experimental observation that the robot fails to exit the channel as h/d increases is genuine and independent, which prevents a higher score; but the theory-as-confirmation and the termination-to-localization mapping are partial circularities. Score 6 reflects one or more predictions that reduce by construction while the central experimental phenomenon retains independent content.

Assumptions & free parameters 5 free parameters · 5 assumptions · 0 invented entities

The central claim rests on four main assumptions: the validity of overdamped RFT for the robot, the ad hoc choice of the AA form for the drag anisotropy, the equivalence of termination and trapping, and the transfer of quantum AA localization criteria to a classical contact-driven system. No genuinely new physical entity is introduced; the 'AA terrain' is a descriptive name for a boulder layout.

free parameters (5)

drag anisotropy amplitude Δ (RFT) = not fitted; varied in model
In Eq. (4), ξ(x)=Δ sin(πx/a)+Γ sin(βπx/a)+1; the value of Δ is chosen to be 'large enough' for ballistic thrust, but no absolute value is calibrated to experiment. The model comparison is qualitative.
perturbing anisotropy amplitude Γ (RFT) = not fitted; Γ/Δ varied from 0 to 1
Controls the aperiodic component of the drag landscape. The RFT transition occurs at an unstated critical Γ/Δ, and no mapping to experimental h/d is provided.
simulation friction coefficient μ = not stated
Bullet's isotropic Coulomb friction is 'tuned to approximate the high-dissipation, low-inertia regime observed in experiments'; the numerical value is not reported.
vertical flip termination threshold = 15 cm
Simulations terminate if any body point exceeds 15 cm in the z-direction; this threshold directly determines the termination probability and hence the measured D distributions.
joint torque saturation limit = 20 Nm
Simulated motor torque limit beyond which actuation saturates. Motor overload is one of the termination causes in both experiments and simulations.

assumptions (5)

domain assumption Resistive force theory applies to the robot in the overdamped, small-amplitude regime
Eq. (3) assumes a slender filament with small lateral displacement, negligible inertia, and drag-dominated dynamics. The physical robot has finite thickness, inertia, and 3D contact, so the mapping to RFT is an approximation.
ad hoc to paper The terrain's effect on the robot is fully captured by a position-dependent drag anisotropy ξ(x) of the same functional form as the AA potential
Eq. (4) is asserted as a minimal model without derivation from contact mechanics. Localization in the RFT model is guaranteed by the imposed aperiodic form of ξ(x).
ad hoc to paper Termination events in experiments and simulations are equivalent to oscillatory trapping in RFT
The authors state 'despite the somewhat ad hoc rules we developed to initiate termination, the systematics ... qualitatively agree'; this equivalence bridges the termination-dominated statistics to the RFT localization transition.
domain assumption The robot's gait is exactly a sinusoidal traveling wave and the body does not deform along the direction of motion
Eq. (2) and the relation X = X0 + x are used in the derivation of Eq. (12). Real robots have finite segment lengths and small deviations from a perfect serpenoid shape.
domain assumption Quantum AA model localization criteria for rational β transfer to a classical contact-driven system
The interpretation that β=2/3 should localize for p≠1 is imported from numerical studies of the quantum AA model (ref. 31) and applied to the robot's classical dynamics.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Aubry-Andr\'e Localization Transition for an Active Undulator." pith.science (2026). https://pith.science/paper/ZQYERJEW

@misc{pith2026250703715,
  author       = {Pith},
  title        = {Pith review of: Aubry-Andr\'e Localization Transition for an Active Undulator},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ZQYERJEW}},
  note         = {Machine review of arXiv:2507.03715}
}

read the original abstract

The transport of deformable self-propelling objects like bacteria, worms, snakes, and robots through heterogeneous environments is poorly understood. In this paper, we use experiment, simulation, and theory to study a snake-like robot as it undulates without sensory feedback through a narrow channel containing a linear array of boulder-like hemispherical obstacles. The profile of the boulder landscape approximates a one-dimensional potential introduced by Aubry and Andr\'e (AA) to study wave function localization in aperiodic lattices. The AA model provides a deterministically disordered alternative to the better-known phenomenon of Anderson localization, which occurs in truly random disordered lattices. When the boulder landscape is strictly periodic, the robot can pass completely through the channel. But if the landscape is sufficiently aperiodic, the robot becomes trapped and fails to exit the channel. The metrics we use to quantify this transition -- including exponential distributions of robot position when localized -- agree well with earlier experimental and theoretical work on a localization transition that occurs when quantum waves interact with the AA potential. A theoretical treatment of the robot's motion using resistive force theory modified to include spatially varying drag forces reproduces the behavior we observe. Further, our results indicate that the transition is generated by large fluctuations in the driving torques required for self-propulsion. These results point to a potentially fundamental connection between classical and quantum wave mechanics and the locomotion of undulators. Our study illustrates how analogies with models from condensed matter physics and wave optics can lead to the discovery of principles of self-propulsion in non-periodic landscapes.

Figures

Figures reproduced from arXiv: 2507.03715 by the authors.

**Figure 1.** A snake-like robot and its self-propelled motion through a narrow channel lined with rigid hemispherical boulders. (A) A robot made from N = 9 servo-motors connected serially at hinged joints interacts with an obstacle course of 3D-printed truncated spheres (one wall of the channel has been removed for visibility); (B1) The angle αm at each of N hinged joints is specified as a function time to produce bending waves … view at source ↗

**Figure 3.** Distributions of robot travel distance. (A) Solid curves are the experimental data for h = 0.5d (green), h = 0.65d (blue), and h = d (red); dashed curves are fits to the stretched exponential function A exp [−(D − D0) α]. (B) The variation of the fitting parameter α with h/d. Open circles are experimental results; closed circles are simulation results. Error bars represent 60% confidence intervals. demonstrates that… view at source ↗

**Figure 4.** Resistive Force Theory modified to include a spatially varying drag anisotropy. (A) Side (upper panel) and top (lower panel) views of a thin rod making contact with a curved frictional surface. See text for discussion of the forces shown. (B) The net force pattern (red) associated with a periodic drag landscape (upper panel) always produces net thrust to the right, while the force pattern associated with an aperiodi… view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

42 extracted references · 40 canonical work pages

[1]

serpenoid

Results and Discussion Undulating Transport in Periodic and Aperiodic Terrains: Experiment. Our snake-like robot was constructed from a set ofN = 9 servomotors (Dynamixel XC330-M288-T) connected by hinged joints (Fig. 1A). To mimic the undulatory gaits observed for certain snakes, nematode worms, and sandfish lizards ( 12), we imposed a “serpenoid” ( 25) ...
[2]

Nature 245, 380–382 (1973)

HC Berg, RA Anderson, Bacteria swim by rotating their flagellar filaments. Nature 245, 380–382 (1973)

work page 1973
[3]

E Lauga, Bacterial hydrodynamics. Annu. Rev. Fluid Mech . 48, 105–130 (2016)

work page 2016
[4]

Z Guo, L Mahadevan, Limbless undulatory propulsion on land. Proc. Natl. Acad. Sci . 105, 3179–3184 (2008)

work page 2008
[5]

Science 346, 224–229 (2014)

H Marvi, et al., Sidewinding with minimal slip: Snake and robot ascent of sandy slopes. Science 346, 224–229 (2014)

work page 2014
[6]

Copeia pp

BC Jayne, Kinematics of terrestrial snake locomotion. Copeia pp. 915–927 (1986)

work page 1986
[7]

C Li, PB Umbanhowar, H Komsuoglu, DE Koditschek, DI Goldman, Sensitive dependence of the motion of a legged robot on granular media. Proc. Natl. Acad. Sci . 106, 3029–3034 (2009)

work page 2009
[8]

GI Taylor, Analysis of the swimming of long and narrow animals. Proc. Royal Soc. London. Ser. A. Math. Phys. Sci . 214, 158–183 (1952)

work page 1952

Show all 42 references

[9]

J Gray, G Hancock, The propulsion of sea-urchin spermatozoa. J. Exp. Biol . 32, 802–814 (1955)

1955
[10]

A Shapere, F Wilczek, Geometry of self-propulsion at low reynolds number. J. Fluid Mech . 198, 557–585 (1989)

1989
[11]

Reports on progress physics 72, 096601 (2009)

E Lauga, TR Powers, The hydrodynamics of swimming microorganisms. Reports on progress physics 72, 096601 (2009)

2009
[12]

T Zhang, DI Goldman, The effectiveness of resistive force theory in granular locomotion. Phys. Fluids 26, 101308 (2014). Pierce et al. PNAS — August 25, 2025 — vol. XXX — no. XX — 9 DRAFT

2014
[13]

JM Rieser, et al., Geometric phase predicts locomotion performance in undulating living systems across scales. Proc. Natl. Acad. Sci. U. S. A. 121, e2320517121 (2024)

2024
[14]

(MIT press), (2017)

RR Murphy, Disaster robotics. (MIT press), (2017)

2017
[15]

(Springer), pp

M Maimone, A Johnson, Y Cheng, R Willson, L Matthies, Autonomous navigation results from the mars exploration rover (mer) mission in Experimental Robotics IX: The 9th International Symposium on Experimental Robotics . (Springer), pp. 3–13 (2006)

2006
[16]

S Shrivastava, et al., Material remodeling and unconventional gaits facilitate locomotion of a robophysical rover over granular terrain. Sci. robotics 5, eaba3499 (2020)

2020
[17]

A Bechar, C Vigneault, Agricultural robots for field operations: Concepts and components. Biosyst. engineering 149, 94–111 (2016)

2016
[18]

I Ground Control Robotics, Ground control robotics, inc.. (year?)
[19]

PloS one 3, e2550 (2008)

S Park, et al., Enhanced caenorhabditis elegans locomotion in a structured microfluidic environment. PloS one 3, e2550 (2008)

2008
[20]

T Majmudar, EE Keaveny, J Zhang, MJ Shelley, Experiments and theory of undulatory locomotion in a simple structured medium. J. Royal Soc. Interface 9, 1809–1823 (2012)

2012
[21]

JM Rieser, et al., Dynamics of scattering in undulatory active collisions. Phys. Rev. E 99, 022606 (2019)

2019
[22]

PE Schiebel, et al., Mechanical diffraction reveals the role of passive dynamics in a slithering snake. Proc. Natl. Acad. Sci . 116, 4798–4803 (2019)

2019
[23]

T Wang, et al., Mechanical intelligence simplifies control in terrestrial limbless locomotion. Sci. Robotics 8, eadi2243 (2023)

2023
[24]

IEEE T ransactions on Robotics 24, 88–104 (2008)

AA Transeth, RI Leine, C Glocker, KY Pettersen, P Liljeb¨ack, Snake robot obstacle-aided locomotion: Modeling, simulations, and experiments. IEEE T ransactions on Robotics 24, 88–104 (2008)

2008
[25]

(IEEE), pp

T Wang, J Whitman, M Travers, H Choset, Directional compliance in obstacle-aided navigation for snake robots in 2020 American Control Conference (ACC) . (IEEE), pp. 2458–2463 (2020)

2020
[26]

Snake-Like Locomotors Manip

S Hirose, Biologically inspired robots. Snake-Like Locomotors Manip. (1993)

1993
[27]

of The International Conference on Intelligent Robots and Systems (IROS)

E Rohmer, SPN Singh, M Freese, Coppeliasim (formerly v-rep): a versatile and scalable robot simulation framework in Proc. of The International Conference on Intelligent Robots and Systems (IROS). (2013)

2013
[28]

DL Hu, J Nirody, T Scott, MJ Shelley, The mechanics of slithering locomotion. Proc. Natl. Acad. Sci. 106, 10081–10085 (2009)

2009
[29]

S Aubry, G Andr ´e, Analyticity breaking and anderson localization in incommensurate lattices. Ann. Isr. Phys. Soc 3, 133 (1980)

1980
[30]

G Dom ´ınguez-Castro, R Paredes, The aubry–andr´e model as a hobbyhorse for understanding the localization phenomenon. Eur. J. Phys. 40, 045403 (2019)

2019
[31]

PW Anderson, Absence of diffusion in certain random lattices. Phys. review 109, 1492 (1958)

1958
[32]

G Modugno, Exponential localization in one-dimensional quasi-periodic optical latticess. New J. Phys. 11, 033023 (2009)

2009
[33]

M Albert, P Leboeuf, Localization by bichromatic potentials versus anderson localization. Phys. Rev. A 81, 013614 (2010)

2010
[34]

Z Mokhtarin, A Zippelius, Dynamics of active filamenents in porous media. Phys. Rev. Lett. 123, 028001 (2019)

2019
[35]

Nature 453, 895–898 (2008)

G Roati, et al., Anderson localization of a non-interacting bose-einstein condensate. Nature 453, 895–898 (2008)

2008
[36]

A Aspect, M Inguscio, Anderson localization of ultracold atoms. Phys. T oday62, 30 (2009)

2009
[37]

Elife 9, e51412 (2020)

PE Schiebel, et al., Mitigating memory effects during undulatory locomotion on hysteretic materials. Elife 9, e51412 (2020)

2020
[38]

JM Rieser, TD Li, JL Tingle, DI Goldman, JR Mendelson III, Functional consequences of convergently evolved microscopic skin features on snake locomotion. Proc. Natl. Acad. Sci . 118, e2018264118 (2021)

2021
[39]

elegans crawling locomotion

Y Rabets, M Backholm, K Dalnoki-Veress, WS Ryu, Direct measurements of drag forces in c. elegans crawling locomotion. Biophys. journal 107, 1980–1987 (2014)

2014
[40]

arXiv preprint arXiv:2210.09570 (2022)

K Diaz, B Chong, S Tarr, E Erickson, DI Goldman, Water surface swimming dynamics in lightweight centipedes. arXiv preprint arXiv:2210.09570 (2022)

2022 arXiv
[41]

X Zhang, N Naughton, T Parthasarathy, M Gazzola, Friction modulation in limbless, three-dimensional gaits and heterogeneous terrains. Nat. communications 12, 6076 (2021)

2021
[42]

10 — www.pnas.org/cgi/doi/10.1073/pnas.XXXXXXXXXX Pierce et al

E Coumans, Y Bai, Pybullet, a python module for physics simulation for games, robotics and machine learning (http://pybullet.org) (2016–2021). 10 — www.pnas.org/cgi/doi/10.1073/pnas.XXXXXXXXXX Pierce et al

2016 doi

Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.

[1] [1]

serpenoid

Results and Discussion Undulating Transport in Periodic and Aperiodic Terrains: Experiment. Our snake-like robot was constructed from a set ofN = 9 servomotors (Dynamixel XC330-M288-T) connected by hinged joints (Fig. 1A). To mimic the undulatory gaits observed for certain snakes, nematode worms, and sandfish lizards ( 12), we imposed a “serpenoid” ( 25) ...

[2] [2]

Nature 245, 380–382 (1973)

HC Berg, RA Anderson, Bacteria swim by rotating their flagellar filaments. Nature 245, 380–382 (1973)

work page 1973

[3] [3]

E Lauga, Bacterial hydrodynamics. Annu. Rev. Fluid Mech . 48, 105–130 (2016)

work page 2016

[4] [4]

Z Guo, L Mahadevan, Limbless undulatory propulsion on land. Proc. Natl. Acad. Sci . 105, 3179–3184 (2008)

work page 2008

[5] [5]

Science 346, 224–229 (2014)

H Marvi, et al., Sidewinding with minimal slip: Snake and robot ascent of sandy slopes. Science 346, 224–229 (2014)

work page 2014

[6] [6]

Copeia pp

BC Jayne, Kinematics of terrestrial snake locomotion. Copeia pp. 915–927 (1986)

work page 1986

[7] [7]

C Li, PB Umbanhowar, H Komsuoglu, DE Koditschek, DI Goldman, Sensitive dependence of the motion of a legged robot on granular media. Proc. Natl. Acad. Sci . 106, 3029–3034 (2009)

work page 2009

[8] [8]

GI Taylor, Analysis of the swimming of long and narrow animals. Proc. Royal Soc. London. Ser. A. Math. Phys. Sci . 214, 158–183 (1952)

work page 1952