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arxiv: 2606.25704 · v2 · pith:TCW6AWCGnew · submitted 2026-06-24 · ❄️ cond-mat.soft · cond-mat.stat-mech

Asymmetry-Induced Chiral Dynamics in Coupled Self-Propelled Robots: Spinning and Circular Motion

Priyanka , Nitin Kumar , Harsh Soni This is my paper

Pith reviewed 2026-07-01 07:06 UTC · model grok-4.3

classification ❄️ cond-mat.soft cond-mat.stat-mech
keywords self-propelled robotschiral dynamicsactive Brownian particlesdimerrotational motionasymmetrycollective dynamicsmicroswimmers
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The pith

Asymmetry in the propulsion directions of two spring-coupled robots generates a net torque that produces persistent spinning or circular motion.

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

The paper demonstrates that when two self-propelled robots are joined by a spring and their fixed propulsion angles relative to the connecting axis differ, the mismatch creates a torque that prevents straight-line travel. Varying the two angles selects among three regimes: run-and-tumble motion, stable circular orbits, or pure spinning without net translation. Spring stiffness and rotational noise act as additional controls that can stabilize or destroy each regime. A sympathetic reader would see this as a minimal mechanical explanation for the chiral trajectories observed in many microswimmers, arising solely from geometry and coupling rather than from sensing or external fields.

Core claim

Asymmetry in the propulsion directions of the robots generates net torques that induce persistent rotational motion. Depending on the choice of propulsion angles α1 and α2, the system exhibits three distinct dynamical regimes—run-and-tumble motion, circular trajectories, and spinning—with the geometric configuration primarily determining the realized regime. Spring stiffness and rotational noise act as additional tuning parameters governing the stability of these regimes.

What carries the argument

The dimer of two active Brownian particles linked by a central spring, whose propulsion directions are held at fixed but unequal angles α1 and α2 to the dimer axis; the angle difference supplies the torque that drives rotation.

If this is right

  • Certain angle pairs produce closed circular trajectories whose radius is set by the angle mismatch and the spring length.
  • When the angles are chosen so net force cancels but torque remains, the dimer spins in place with zero center-of-mass velocity.
  • Raising spring stiffness suppresses run-and-tumble behavior and favors the circular or spinning states.
  • Above a threshold rotational noise strength the circular and spinning orbits lose stability and revert to diffusive motion.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same fixed-angle mechanism could be scaled to small groups of robots to generate controlled collective rotation without external guidance.
  • Biological microswimmers that maintain roughly constant body angles relative to their flagellar axis may achieve reliable circling by the identical torque route.
  • Adding hydrodynamic interactions between the robots would either amplify or counteract the spring-generated torque, giving an experimental signature that distinguishes the minimal model from real fluid environments.

Load-bearing premise

Propulsion directions stay fixed relative to the line joining the two robots and the only interaction is the central spring force.

What would settle it

Record the steady-state angular velocity of the dimer while systematically changing the difference between the two propulsion angles; if the measured rotation rate does not increase with that angle difference as the torque calculation predicts, the claim is falsified.

Figures

Figures reproduced from arXiv: 2606.25704 by Harsh Soni, Nitin Kumar, Priyanka.

Figure 1
Figure 1. Figure 1: FIG. 1. Schematic diagram of the system: (a) A single active [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Stream plots for two angular configurations and their [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. (a) Heatmap of the average spin angular velocity [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. (a) Steady-state angle [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. (a) Heatmap of the average centroid speed [PITH_FULL_IMAGE:figures/full_fig_p005_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. (a) Autocorrelation function [PITH_FULL_IMAGE:figures/full_fig_p006_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Steady-state probability distribution [PITH_FULL_IMAGE:figures/full_fig_p006_7.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. (a) Orbital angular velocity [PITH_FULL_IMAGE:figures/full_fig_p007_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10. Dependence of the average spin angular velocity [PITH_FULL_IMAGE:figures/full_fig_p007_10.png] view at source ↗
Figure 12
Figure 12. Figure 12: FIG. 12. (a) Orientational autocorrelation function [PITH_FULL_IMAGE:figures/full_fig_p008_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: FIG. 13. Orbital angular velocity [PITH_FULL_IMAGE:figures/full_fig_p008_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: FIG. 14. (a) Average spin angular velocity [PITH_FULL_IMAGE:figures/full_fig_p008_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: FIG. 15. Mean-square displacement for the (a) circular mo [PITH_FULL_IMAGE:figures/full_fig_p009_15.png] view at source ↗
read the original abstract

Motivated by the chiral motility of microswimmers, we investigate how geometric asymmetry in a system of two self-propelled active Brownian robots coupled by a spring gives rise to rich collective dynamics. We demonstrate that asymmetry in the propulsion directions of the robots generates net torques that induce persistent rotational motion. Depending on the choice of propulsion angles $\alpha_1$ and $\alpha_2$, the system exhibits three distinct dynamical regimes -- run-and-tumble motion, circular trajectories, and spinning -- with the geometric configuration primarily determining the realized regime. We further show that spring stiffness and rotational noise act as additional tuning parameters governing the stability of these regimes. These results demonstrate how the interplay of mechanical coupling and activity produces diverse self-organized dynamics in simple robotic dimers, providing a bridge between artificial active systems and biological microswimmers such as bacteria, Chlamydomonas reinhardtii, and spermatozoa.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 0 minor

Summary. The paper presents a minimal model of two self-propelled active Brownian robots (dimers) coupled by a central spring, with fixed but asymmetric propulsion angles α1 and α2 relative to the dimer axis. It claims that this geometric asymmetry alone generates net torques via off-axis forces, producing three distinct dynamical regimes—run-and-tumble motion, circular trajectories, and spinning—whose realization is primarily controlled by the choice of α1 and α2, with spring stiffness k and rotational noise strength serving as additional tuning parameters. The regimes are reported to emerge from direct numerical integration of the model equations.

Significance. If the simulation results hold under scrutiny, the work supplies a clean, geometrically transparent demonstration that chiral collective motion can arise in a minimal active dimer solely from asymmetric propulsion directions and central coupling, without built-in individual chirality or hydrodynamic torques. This offers a useful bridge between artificial robotic systems and biological microswimmers and could serve as a testbed for further analytic or machine-checked treatments of torque balance in active matter.

major comments (1)
  1. [Simulation and Results sections] Simulation and Results sections: The central claims rest on the observation that simulations produce the three stated regimes, yet the manuscript supplies no numerical integration method, time-step size, specific parameter values for α1, α2, k and noise strength, criteria used to classify trajectories (e.g., how circular vs. spinning motion is distinguished), or any error analysis/validation checks. Without these details it is impossible to judge whether the reported regimes are robust or reproducible.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their thoughtful review and for highlighting the need for greater methodological transparency. We address the single major comment below and will incorporate the requested details into a revised manuscript.

read point-by-point responses

  1. Referee: [Simulation and Results sections] Simulation and Results sections: The central claims rest on the observation that simulations produce the three stated regimes, yet the manuscript supplies no numerical integration method, time-step size, specific parameter values for α1, α2, k and noise strength, criteria used to classify trajectories (e.g., how circular vs. spinning motion is distinguished), or any error analysis/validation checks. Without these details it is impossible to judge whether the reported regimes are robust or reproducible.

    Authors: We agree that these details are necessary for reproducibility and were omitted from the submitted manuscript. In the revised version we will add a new subsection (e.g., “Numerical Methods”) that specifies: (i) the integration algorithm and time-step size, (ii) the exact parameter sets (α1, α2, k, noise strength) used to realize each regime, (iii) the quantitative classification criteria (e.g., thresholds on mean angular velocity, curvature, and long-time displacement), and (iv) validation checks including time-step convergence, ensemble statistics, and sensitivity to initial conditions. These additions will allow independent verification of the reported dynamical regimes. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper presents an explicit minimal model of two self-propelled robots coupled by a central spring, with propulsion directions fixed at angles α1 and α2 relative to the dimer axis. Net torque and the three dynamical regimes emerge directly from the geometry of off-axis forces and integration of the equations of motion (including noise and stiffness). No fitted parameters are renamed as predictions, no self-citations are load-bearing for the torque claim, and no definitions reduce to outputs by construction. The central result is internal to the stated assumptions and does not rely on external uniqueness theorems or prior ansatzes from the same authors.

Axiom & Free-Parameter Ledger

3 free parameters · 2 axioms · 0 invented entities

The central claim rests on numerical integration of standard active-Brownian-particle Langevin equations augmented by a harmonic spring and fixed propulsion angles; no new entities are postulated.

free parameters (3)
  • propulsion angles α1 and α2
    Chosen by the authors to explore different regimes
  • spring stiffness k
    Tuning parameter controlling regime stability
  • rotational noise strength
    Tuning parameter controlling regime stability
axioms (2)
  • domain assumption Each robot obeys active Brownian particle dynamics with constant self-propulsion speed and rotational diffusion
    Standard model invoked for self-propelled particles
  • domain assumption The only inter-particle force is a central harmonic spring
    Modeling choice stated in the setup

pith-pipeline@v0.9.1-grok · 5690 in / 1215 out tokens · 45789 ms · 2026-07-01T07:06:42.939188+00:00 · methodology

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Reference graph

Works this paper leans on

49 extracted references

  1. [1]

    Spinning motion To quantify the spinning motion of the coupled robot, we define its spin angular velocityωs as the angular veloc- ity of the line joining the two robots i.e., the connecting spring. Fig. 3(a) shows the heat map of the average spin angular velocity⟨ω s⟩as a function of the anglesα 1 and α2. Forα 1 =α 2, the propulsion geometry is symmet- ri...

  2. [2]

    Circular motion We now discuss the circular motion executed by the centroid of the system. To quantify this behavior, we analyze the orientation autocorrelation of the centroid velocity vectorv c(t), defined as C(τ) =⟨cos[ζ(t+τ)−ζ(t)]⟩, whereζ(t) is the orientation angle ofv c(t). For config- urations exhibiting circular motion, the velocity vector vc(t) ...

  3. [3]

    Effect of the rotational noise Here, we study the effect of rotational noise of the robots, characterized byD r, on the spinning and cir- cular motion of the system. Fig. 10 shows the aver- age spin angular velocity⟨ω s⟩as a function ofD r for four different configurations, chosen from the parameter range 50 ◦ ≤α 1 ≤90 ◦ alongα 1 +α 2 = 360◦, where the sp...

  4. [4]

    [13], the connection be- tween robots was rigid and inextensible, corresponding to the limitK→ ∞

    Effect of the stiffness of the spring In the rigid-rod model of Ref. [13], the connection be- tween robots was rigid and inextensible, corresponding to the limitK→ ∞. Replacing the rod with a Hookean spring allows us to continuously tune the mechanical compliance from a highly flexible coupling at smallK to the rigid-rod limit at largeK. This compliance- ...

  5. [5]

    15 shows the mean-square displacement (MSD) for different configurations across the two motility regimes

    Mean square displacement Fig. 15 shows the mean-square displacement (MSD) for different configurations across the two motility regimes. In both cases, the MSD grows ballistically, MSD∼t 2, at short times, reflecting coherent self-propulsion before ro- tational noise reorients the robots, and crosses over to normal diffusion, MSD∼t, at long times. The two ...

    2025

  6. [6]

    Lauga and T

    E. Lauga and T. R. Powers, The hydrodynamics of swim- ming microorganisms, Reports on Progress in Physics72, 096601 (2009)

    2009

  7. [7]

    Elgeti, R

    J. Elgeti, R. G. Winkler, and G. Gompper, Physics of microswimmers—single particle motion and collective behavior: A review, Reports on Progress in Physics78, 056601 (2015)

    2015

  8. [8]

    H. C. Berg, The rotary motor of bacterial flagella, Annual Review of Biochemistry72, 19 (2003)

    2003

  9. [9]

    Polin, I

    M. Polin, I. Tuval, K. Drescher, J. P. Gollub, and R. E. Goldstein,Chlamydomonasswims with two “gears” in a eukaryotic version of run-and-tumble locomotion, Science 325, 487 (2009)

    2009

  10. [10]

    Brennen and H

    C. Brennen and H. Winet, Fluid mechanics of propulsion by cilia and flagella, Annual Review of Fluid Mechanics 9, 339 (1977)

    1977

  11. [11]

    K. F. Jarrell and M. J. McBride, The surprisingly diverse ways that prokaryotes move, Nature Reviews Microbiol- ogy6, 466 (2008)

    2008

  12. [12]

    H. Huo, R. He, R. Zhang, and J. Yuan, SwimmingEs- cherichia colicells explore the environment by l´ evy walk, Applied and Environmental Microbiology87, e02429 (2021)

    2021

  13. [13]

    H. C. Berg,Random Walks in Biology(Princeton Uni- versity Press, Princeton, NJ, 1993)

    1993

  14. [14]

    M. E. Cates, Diffusive transport without detailed bal- ance in motile bacteria: Does microbiology need statisti- cal physics?, Reports on Progress in Physics75, 042601 (2012)

    2012

  15. [15]

    G. Fier, D. Hansmann, and R. C. Buceta, A stochastic model for directional changes of swimming bacteria, Soft Matter13, 3385 (2017)

    2017

  16. [16]

    H. C. Berg and D. A. Brown, Chemotaxis inEscherichia colianalysed by three-dimensional tracking, Nature239, 500 (1972)

    1972

  17. [17]

    Tailleur and M

    J. Tailleur and M. E. Cates, Statistical mechanics of in- teracting run-and-tumble bacteria, Physical Review Let- ters100, 218103 (2008)

    2008

  18. [18]

    Paramanick, U

    S. Paramanick, U. Pardhi, H. Soni, and N. Kumar, Spon- taneous emergence of run-and-tumble-like dynamics in a robotic analog of chlamydomonas: Experiment and the- ory, Physical Review Letters135, 168301 (2025)

    2025

  19. [19]

    Bechinger, R

    C. Bechinger, R. Di Leonardo, H. L¨ owen, C. Reichhardt, G. Volpe, and G. Volpe, Active particles in complex and crowded environments, Reviews of Modern Physics88, 045006 (2016)

    2016

  20. [20]

    N. C. Darnton, L. Turner, S. Rojevsky, and H. C. Berg, On torque and tumbling in swimmingEscherichia coli, Journal of Bacteriology189, 1756 (2007)

    2007

  21. [21]

    Lauga, W

    E. Lauga, W. R. DiLuzio, G. M. Whitesides, and H. A. Stone, Swimming in circles: motion of bacteria near solid boundaries, Biophysical Journal90, 400 (2006)

    2006

  22. [22]

    W. R. DiLuzio, L. Turner, M. Mayer, P. Garstecki, D. B. Weibel, H. C. Berg, and G. M. Whitesides,Es- 11 cherichia coliswim on the right-hand side, Nature435, 1271 (2005)

    2005

  23. [23]

    R¨ uffer and W

    U. R¨ uffer and W. Nultsch, High-speed cinematographic analysis of the movement ofChlamydomonas, Cell Motil- ity5, 251 (1985)

    1985

  24. [24]

    P. V. Bayly, B. L. Lewis, P. S. Kemp, R. B. Pless, and S. K. Dutcher, Efficient spatiotemporal analysis of the flagellar waveform of chlamydomonas reinhardtii, Cy- toskeleton67, 56 (2010)

    2010

  25. [25]

    Cortese and K

    D. Cortese and K. Y. Wan, Control of helical navigation by three-dimensional flagellar beating, Physical Review Letters126, 088003 (2021)

    2021

  26. [26]

    S. K. Dutcher, Asymmetries in the cilia of chlamy- domonas, Philosophical Transactions of the Royal Soci- ety B374, 20190153 (2019)

    2019

  27. [27]

    Z. Wang, S. A. Bentley, J. Li, K. Y. Wan, and A. C. H. Tsang, Light-dependent switching of circling handedness in microswimmer navigation, Physical Review Letters 136, 078301 (2026)

    2026

  28. [28]

    B. M. Friedrich and F. J¨ ulicher, Flagellar synchroniza- tion independent of hydrodynamic interactions, Physical Review Letters109, 138102 (2012)

    2012

  29. [29]

    H. Xin, N. Zhao, Y. Wang, X. Zhao, T. Pan, Y. Shi, and B. Li, Optically controlled living micromotors for the manipulation and disruption of biological targets, Nano Letters20, 7177 (2020)

    2020

  30. [30]

    Gadˆ elha, P

    H. Gadˆ elha, P. Hern´ andez-Herrera, F. Montoya, A. Darszon, and G. Corkidi, Human sperm uses asym- metric and anisotropic flagellar controls to regulate swim- ming symmetry and cell steering, Science Advances6, eaba5168 (2020)

    2020

  31. [31]

    Kamiya and E

    R. Kamiya and E. Hasegawa, Intrinsic difference in beat frequency between the two flagella ofChlamydomonas reinhardtii, Experimental Cell Research173, 299 (1987)

    1987

  32. [32]

    D. Wei, G. Quaranta, M.-E. Aubin-Tam, and D. S. W. Tam, The younger flagellum sets the beat forChlamy- domonas reinhardtii, eLife13, e86102 (2024)

    2024

  33. [33]

    K¨ ummel, B

    F. K¨ ummel, B. ten Hagen, R. Wittkowski, I. Buttinoni, R. Eichhorn, G. Volpe, H. L¨ owen, and C. Bechinger, Circular motion of asymmetric self-propelling particles, Physical Review Letters110, 198302 (2013)

    2013

  34. [34]

    Liebchen and D

    B. Liebchen and D. Levis, Chiral active matter, EPL (Eu- rophysics Letters)139, 67001 (2022)

    2022

  35. [35]

    Paramanick, A

    S. Paramanick, A. Pal, H. Soni, and N. Kumar, Program- ming tunable active dynamics in a self-propelled robot, European Physical Journal E47, 34 (2024)

    2024

  36. [36]

    E. M. Purcell, Life at low Reynolds number, American Journal of Physics45, 3 (1977)

    1977

  37. [37]

    Y. Xia, Z. Hu, D. Wei, K. Chen, Y. Peng, and M. Yang, Biomimetic synchronization in biciliated robots, Physical Review Letters133, 048302 (2024)

    2024

  38. [38]

    K. Y. Wan and R. E. Goldstein, Coordinated beating of algal flagella is mediated by basal coupling, Proceedings of the National Academy of Sciences113, E2784 (2016)

    2016

  39. [39]

    V. F. Geyer, F. J¨ ulicher, J. Howard, and B. M. Friedrich, Cell-body rocking is a dominant mechanism for flagellar synchronization in a swimming alga, Proceedings of the National Academy of Sciences110, 18058 (2013)

    2013

  40. [40]

    Bianchi, F

    S. Bianchi, F. Saglimbeni, G. Frangipane, D. Dell’Arciprete, and R. Di Leonardo, Flagellar elasticity and the multiple swimming modes of inter- facial bacteria, Physical Review Research4, L022044 (2022)

    2022

  41. [41]

    Y. Liu, R. Claydon, M. Polin, and D. R. Brumley, Tran- sitions in synchronization states of model cilia through basal-connection coupling, Physical Review E97, 052416 (2018)

    2018

  42. [42]

    Zheng, H

    Z. Zheng, H. Bie, X. Mao, and K. Bhattacharya, Self- oscillation and synchronization transitions in elastoactive structures, Physical Review Letters130, 178202 (2023)

    2023

  43. [43]

    L¨ owen, Chirality in microswimmer motion: From cir- cle swimmers to active turbulence, European Physical Journal Special Topics225, 2319 (2016)

    H. L¨ owen, Chirality in microswimmer motion: From cir- cle swimmers to active turbulence, European Physical Journal Special Topics225, 2319 (2016)

    2016

  44. [44]

    Caprini and U

    L. Caprini and U. Marini Bettolo Marconi, The role of disorder in the motion of chiral active particles in the presence of obstacles, Soft Matter18, 7358 (2022)

    2022

  45. [45]

    Caprini and U

    L. Caprini and U. Marini Bettolo Marconi, Chiral ac- tive matter in external potentials, Soft Matter19, 7988 (2023)

    2023

  46. [46]

    Caprini, A

    L. Caprini, A. Petrini, and U. Marini Bettolo Marconi, Modeling chiral active particles: from circular motion to odd interactions, Journal of Physics A: Mathematical and Theoretical (2026), review article

    2026

  47. [47]

    C. W. Chan, D. Wu, K. Qiao, K. L. Fong, Z. Yang, Y. Han, and R. Zhang, Chiral active particles are sensi- tive reporters to environmental geometry, Nature Com- munications15, 1406 (2024)

    2024

  48. [48]

    G. S. Klindt, C. Ruloff, C. Wagner, and B. M. Friedrich, In-phase and anti-phase flagellar synchronization by waveform compliance and basal coupling, New Journal of Physics19, 113052 (2017)

    2017

  49. [49]

    R. Chen, Z. Hu, Y. Liu,et al., Differential bending stiff- ness of the bacterial flagellar hook under counterclock- wise and clockwise rotations, Physical Review Letters 130, 138401 (2023)

    2023