Swimming locomotion of Soft Robotic Snakes

Isuru S. Godage

REVIEW 5 major objections 5 minor 9 references

Swimming locomotion of Soft Robotic Snakes

T0 review · 5 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read A simulated soft robot snake swims faster when bending waves shift by a phase of pi/8 between modules than by pi/3.

desk verdict A plausible but unvalidated simulation study: the model is a real extension of the author's prior kinematics, but the headline 'optimal' gait claim outruns the evidence when only two gaits are compared under an admittedly non-realistic contact model. read the letter →

arxiv 1908.05250 v1 pith:LK2MIKCJ submitted 2019-08-14 cs.RO

Isuru S. Godage This is my paper · ORCID

classification cs.RO

keywords softroboticssnakerobotswimminglocomotioncontinuumcontactdynamicshydrodynamicdragmodalkinematicsserpentinegait

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 establishes a simulation model for the swimming locomotion of a soft robotic snake made of three pneumatically actuated continuum modules, and uses it to compare two serpentine gaits. The model places 310 discrete points on the robot's skin, treats water resistance as quadratic drag at each point, and adds a spring-damper force so the lightweight body floats rather than submerges. Simulated swimming shows that a phase offset of $\pi/8$ between successive modules produces faster forward motion than the larger offset $\pi/3$, even though the latter creates a more pronounced body wave. The paper takes this as evidence that soft snake robots are viable candidates for marine applications and that gait parameters can be explored in simulation before hardware tests.

What carries the argument

The load-bearing mechanism is the spatially discretized modal kinematic model combined with a recursive integral Lagrangian dynamic formulation. The snake's shape is described by three soft sections whose kinematics come from the modal approach, and two additional homogeneous transforms ($R_z$ rotation about $+Z$, $p_x$ translation along $+X$) place points on the physical skin, yielding the complete chain $T(q_b,q,\xi)=T_b(q_b)\prod_{i=1}^3 T_i$. Dynamics are assembled as $M\ddot{Q}+C\dot{Q}+D\dot{Q}+G = [0,\tau_e]^T + \sum J_{jk}^T S_{jk}$, where the right-hand side includes the discretized drag reactions. The hydrodynamic model is deliberately simple: 310 point contacts, quadratic drag with an assumed 1 cm$^2$ area per point, and a spring-damper balancing force to enforce flotation. This machinery converts a continuous soft body into a tractable simulation and lets the author rank gait phase offsets.

What would settle it

Run the two pressure inputs from the paper on a physical three-module soft snake in a water tank and measure steady-state forward speeds for phase offsets $\pi/8$ and $\pi/3$; if the $\pi/3$ gait is not slower, the model's gait comparison is falsified. A more direct check is to measure the drag on a floating soft cylinder at the simulated speeds and compare it against the sum of $S_{jk}$ over the 310 discretized points.

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Extended reading notes

Core claim

On the paper's own terms, the central discovery is that the floating-base, modal-kinematic model of a soft snake can reproduce both flotation and serpentine swimming, and that the choice of phase offset between modules materially changes swimming speed. With modal kinematics, the snake's continuous deformation is represented by actuator length variables, and a homogeneous transformation matrix $T(q_b,q,\xi)$ maps the body onto 310 skin-point coordinates. Each skin point experiences a quadratic drag force $S_{jk} = \tfrac{1}{2} C_D \rho A v_{jk}^2$, while a spring-damper contact model supplies the balancing forces that keep the robot at the water surface. Under the pressure signal with phase offset $\pi/8$, the robot swims faster than under the $\pi/3$ signal, indicating that smaller inter-module phase delays are more efficient for this soft swimmer. The author concludes that soft robotic snakes have high potential for marine applications.

Load-bearing premise

The simulation rests on the assumption that the robot floats without submerging and that water interaction is faithfully represented by 310 independent point-drag forces, each with an assumed 1 cm$^2$ area, plus a spring-damper balancing force; if the real fluid interaction differs substantially from these point approximations, the simulated speed ordering between the two gaits may not hold.

Editorial extensions

If this is right

Gait phase offset can be treated as a tunable design parameter, so the same simulation can rank other offsets and wave shapes before building hardware.
Because the model takes a floating-base, skin-point approach, it can be adapted to other soft-bodied swimmers, not just snakes.
The paper's own identified next step is a volume-based contact model for realistic buoyancy and drag, which would put the point-contact results on firmer physical ground.
The model's speed ranking offers a precise, quantitative target for experimental validation of soft snake swimming.

Reading between the lines

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

The $\pi/8$ versus $\pi/3$ speed ordering is likely sensitive to wave frequency, amplitude, and body stiffness; a parameter sweep over these variables would map the gait-speed landscape and test how robust the ordering is.
The assumed 1 cm$^2$ drag area per contact point is a strong simplification; since drag scales linearly with area, a sensitivity analysis would show whether the ordering survives plausible changes in wetted area.
The paper itself flags the uneven contact pattern produced by the discretized model, so a volume-based fluid model is the natural falsification test for whether the $\pi/8$ advantage is an artifact of the discretization.
A physical prototype experiment is the direct next test: the predicted speed ordering gives a concrete, falsifiable claim that could justify the design of a marine soft snake.

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Signed reviews

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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

You should know two things before reading this one. First, the genuinely new bit is small but real: the author takes his own modal kinematics for continuum arms and adds a floating base, plus a point-contact drag model, to simulate a soft snake swimming. That is a legitimate application of known machinery to a new problem, not a new theory. Second, the headline result—that a π/8 phase offset between modules swims faster than π/3—is only as good as the drag model, and the paper itself admits that model is not realistic. Read it that way and you won't be misled.

What the paper does well: the kinematic extension to skin points is sensible, the recursive Lagrangian dynamics follow from earlier published work, and the drop-test at least checks that the contact model reaches a floating equilibrium. The author is honest about the simplifications—uneven contacts, point drag at a sparse set of waterline points, no submergence—and flags volume-based buoyancy as future work. The simulation code ran, and the qualitative difference between the two gaits is plausible.

Where it gets soft, in proportion: (1) The central π/8-over-π/3 ordering rests entirely on the contact model of Eq. (3), which has no experimental grounding. C_D is never reported, the damping matrix coefficient η is never given, and the spring-damper contact parameters are missing, so no one can reproduce the simulation from the paper alone. (2) Calling π/8 “optimal” is an overstatement when only two hand-picked phase offsets were tested. (3) Eq. (1) has an empty citation. (4) The marine-applications sentence in the abstract is unsupported by any hardware or higher-fidelity evidence. None of these are fatal to the modeling contribution, but they are exactly the kind of gaps that block the result from being carried into design decisions.

The math itself looks directionally sound; the missing parameters are the biggest obstacle, not the derivation.

Bottom line: this is a workshop-to-conference-level modeling paper useful to soft-robotics researchers who want a quick simulation scaffold for gait studies. I would not cite it as evidence about real swimming performance, but I would use or adapt the kinematic/dynamic formulation. It deserves a serious referee rather than a desk rejection, with the expectation that the author fills in parameters and moderates the optimality claim.

Referee Report

5 major / 5 minor

Summary. The paper presents a dynamic simulation framework for a three-module soft robotic snake swimming in water. The robot's kinematics are formulated using the author's prior modal-kinematics model extended to generate skin points, and hydrodynamics are approximated by a discretized set of 310 contact points, each subject to quadratic drag, plus a spring-damper balancing force that prevents submersion. The resulting equations of motion are simulated in MATLAB, with a drop test used to check that the floating equilibrium is reached. Two serpentine swimming gaits are then compared, distinguished by the phase offset between successive soft modules: π/8 for a milder wave and π/3 for a larger-amplitude wave. The central claim is that the simulations identify an optimal bending propagation, namely π/8, for efficient swimming, implying that soft robotic snakes have potential for marine propulsion.

Significance. The paper contributes a computationally tractable approach to simulating swimming gaits for continuous soft robots, an area where hardware experiments are expensive and analytic models are scarce. The extension of modal kinematics to include a discretized skin surface is a useful idea, and the paper is commendably explicit about the limitations of its point-contact hydrodynamics. If the model were fully specified and the gait-ordering result could be shown robust to the contact approximation, the framework would provide a cheap screening tool for soft-robot swimming gaits. As it stands, the contribution is a preliminary modeling study: the central numerical finding (π/8 faster than π/3) is not backed by any experimental or higher-fidelity validation, and several model parameters are unreported, so the headline claim is not yet supported by the evidence presented.

major comments (5)

[Section II-B3, Eq. (4)] The equation of motion is stated as M¨Q + C˙Q + D˙Q + G = [0; τ_e] + Σ J^T S, with Q ∈ R^12, but the damping matrix is defined as D = ηI ∈ R^{6×6}. As written, D˙Q is dimensionally inconsistent. If D is intended to act only on the floating-base coordinates, that split must be defined explicitly; otherwise the matrix dimensions must be corrected. This is a load-bearing modeling issue because the damping affects the dynamics and hence the simulated swimming speed.
[Section II-B2, Eq. (3)] The entire swimming force rests on S = 1/2 C_D ρ A v^2, with A arbitrarily set to 1 cm^2, yet the numerical value of the drag coefficient C_D is never given, nor are the spring-damper parameters used in the contact model to balance gravity. The damping coefficient η in Eq. (4) is also unspecified. Without these values, the simulation is not reproducible and an independent check of the claimed speed ordering (π/8 vs. π/3) is impossible. The manuscript must report all such parameters and, ideally, justify them with a sensitivity study or a reference to experimental data.
[Section III, first study] The paper explicitly acknowledges that the contact model 'makes uneven contacts..., which is not realistic.' This is a serious concern because the floating configuration and the distribution of drag forces are directly computed from the same point-contact model. The paper offers no comparison with an alternative (e.g., volume-based) hydrodynamics model and no experimental data to show that the relative speeds of the two gaits are robust to the contact approximation. The stated limitation therefore directly undermines the abstract's claim that the model is 'numerically tested to identify the optimal bending propagation.' At minimum, the claim must be reworded to refer to the behavior of this specific simplified model, and the robustness of the π/8-over-π/3 ordering to the contact parameters should be demonstrated.
[Section III, Eqs. (5)-(6)] Only two phase offsets, π/8 and π/3, are simulated. The conclusion that the framework 'identifies the optimal bending propagation' is an overstatement: the simulations show only that π/8 outperforms π/3 in this particular model. The paper itself later says ongoing work is needed to find the optimal wave signal. The abstract and conclusions should be qualified accordingly, and the term 'optimal' should be replaced by the more accurate claim of 'better than the tested alternative' unless an actual search is performed.
[Sections II-A and II-B] The number of actuators per module is inconsistent. The prototype description states each module is 'powered by two pneumatically powered soft McKibben type actuators,' while the system model describes 'three mechanically identical variable length actuators' and the joint-space vector q_i = [l_i1(t), l_i2(t)]^T has only two components. This inconsistency creates an ambiguity in how the input pressures in Eqs. (5)-(6) map to actuator lengths, and it must be resolved (e.g., by explicitly stating whether one actuator is collinear with the neutral axis or otherwise dependent) for the model to be reproducible.

minor comments (5)

[Abstract] The phrase 'a distributed contact modal' should be 'a distributed contact model.'
[Section II-B1, Eq. (1)] The sentence 'In addition to the previous results in []' contains an empty citation; the applicable prior work should be cited, or the phrase removed.
[Section II-B1, Eq. (1)] The symbols σ_i and r_i in the two appended homogeneous transforms are not explicitly defined; the text should state that σ_i is the angular position around the module's cross-section and r_i is the radius of the skin surface.
[Section III, paragraph after Eq. (4)] The text says 'The period of the gait is 1/2 rads−1,' but 1/2 rad/s is an angular frequency, not a period; the wording should be corrected (e.g., 'the angular frequency is 1/2 rad/s').
[Section I] There are several typographical errors, such as 'tot he base' and 'the robot lightweight' (should be 'the robot is lightweight'). A thorough language edit is recommended.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the gait comparison is an emergent simulation output; self-cited prior models are reused, not defined in terms of the target result.

full rationale

The paper's derivation chain is self-contained in the relevant sense: Eq. (3) defines point drag from velocity and fluid properties, Eq. (4) assembles the floating-base dynamics, and the pi/8-versus-pi/3 speed comparison is an output of integrating those equations rather than an input. The stiffness and damping values (1900 N/m and 90 N/m-s) are experimentally identified inputs, but they are not renamed as predictions; the simulated gait ordering is not statistically forced by those fits. The self-citations to [2], [3], and [6] reuse the author's prior kinematic/dynamic formulations and parameter-identification procedure. Under the stated rules, those citations are real evidence: they do not contain the swimming-speed result, they are not invoked as a uniqueness theorem, and they do not smuggle in the pi/8 conclusion by ansatz. The paper itself flags the main weakness in Section III: the point-contact model 'makes uneven contacts..., which is not realistic,' and it notes ongoing work on a volume-based contact model. That is a correctness or validation limitation, not circularity: the model could be inaccurate without being equivalent to its own output. Similarly, calling pi/8 'optimal' after testing only two phase offsets is an evidentiary overreach, but it is a search or generalization gap, not a definitional reduction. The only circularity-adjacent feature is ordinary reuse of prior self-cited modeling machinery, which does not make the central simulation claim an input in disguise. Score 2 reflects minor self-citation context rather than any circular derivation.

Assumptions & free parameters 6 free parameters · 6 assumptions · 0 invented entities

The central claim rests on the author's prior kinematic and dynamic formulations for continuum arms, with actuator parameters borrowed from earlier experimental identification, plus a simplified surface-floating and point-drag contact model. No independent experimental validation is offered, so the simulation outcomes are conditional on these modeling choices.

free parameters (6)

Actuator stiffness = 1900 N/m (rounded)
Used in the dynamic model; reported as approximately identified in prior experimental work [6], rounded to the nearest 100.
Actuator damping coefficient = 90 N/(m/s) (rounded)
Used in the dynamic model; reported as approximately identified via an experimental procedure similar to [6], rounded to the nearest 10.
Drag coefficient C_D = not specified
Appears in Eq. (3) for the drag force at each contact point; no numerical value is given in the paper, so the simulations are not fully reproducible.
Contact point area A = 1 cm^2 (assumed)
Stated in Sec. II-B2 as an assumed area per contact point.
Damping matrix coefficient eta = not specified
D = eta I in Eq. (4); no value or identification procedure is given.
Spring-damper contact model parameters = not specified
Used to balance gravity at the water surface in Sec. II-B2; stiffness and damping values are not listed.

assumptions (6)

domain assumption Modal kinematics from [3]
The kinematic model in Eq. (1) extends the author's prior modal kinematics for multisection continuum arms; the paper assumes this model transfers to a floating-base snake with skin points.
domain assumption Floating-base rigid body representation
The base of the snake is modeled with a floating coordinate system T_b in SE3 with 6 parameters, allowing free translation and rotation relative to the world frame.
domain assumption No submergence / surface floating assumption
Sec. II-B2 states the robot is lightweight and does not submerge; a spring-damper contact model artificially balances gravity at the water surface.
domain assumption Quadratic drag model
Hydrodynamic forces are modeled as S_jk = 1/2 C_D rho A v^2 at discrete points (Eq. 3); fluid displacement, lifting forces, and wave effects are ignored.
domain assumption Linear spring-damper actuator model
Actuator elasticity and damping are represented by stiffness and damping values identified in [6]; hysteresis is neglected.
standard math Recursive integral Lagrangian formulation
The equations of motion (Eq. 4) are derived using an extended recursive integral Lagrangian approach; this is a standard analytical mechanics framework.

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Cite this review

Pith. "Pith review of Swimming locomotion of Soft Robotic Snakes." pith.science (2026). https://pith.science/paper/LK2MIKCJ

@misc{pith2026190805250,
  author       = {Pith},
  title        = {Pith review of: Swimming locomotion of Soft Robotic Snakes},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/LK2MIKCJ}},
  note         = {Machine review of arXiv:1908.05250}
}

read the original abstract

Bioinspired snake robotics has been a highly active area of research over the years and resulted in many prototypes. Much of these prototypes takes the form of serially jointed-rigid bodies. The emergence of soft robotics contributed to a new type of snake robots made from compliant and structurally deformable modules. Leveraging the controllable large bending, these robots can naturally generate various snake locomotion gaits. Here, we investigate the swimming locomotion of soft robotic snakes. A numerically efficient dynamic model of the robot is first derived. Then, a distributed contact modal is augmented to incorporate hydrodynamic forces. The model is then numerically tested to identify the optimal bending propagation for efficient swimming. Results show that the soft robotic snakes have high potential to be used in marine applications.

Figures

Figures reproduced from arXiv: 1908.05250 by the authors.

**Figure 2.** Grided contact points on the soft robot snake highlighting the contact [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗

**Figure 3.** The snake swimming locomotion resulting from the input signals [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

9 extracted references · 9 canonical work pages

[1]

Autonomous undulatory serpentine locomotion utilizing body dynamics of a ﬂuidic soft robot

Onal, Cagdas D., and Daniela Rus. "Autonomous undulatory serpentine locomotion utilizing body dynamics of a ﬂuidic soft robot." Bioinspiration & biomimetics 8, no. 2 (2013): 026003

work page 2013
[2]

Dynamics for vari- able length multisection continuum arms

Godage, Isuru S., Gustavo A. Medrano-Cerda, David T. Branson, Emanuele Guglielmino, and Darwin G. Caldwell. "Dynamics for vari- able length multisection continuum arms." The International Journal of Robotics Research 35, no. 6 (2016): 695-722

work page 2016
[3]

Modal kinematics for multisection continuum arms

Godage, Isuru S., Gustavo A. Medrano-Cerda, David T. Branson, Emanuele Guglielmino, and Darwin G. Caldwell. "Modal kinematics for multisection continuum arms." Bioinspiration & biomimetics 10, no. 3 (2015): 035002

work page 2015
[4]

Simulation of contact using a nonlinear damping model

Marhefka, Duane W., and David E. Orin. "Simulation of contact using a nonlinear damping model." In Proceedings of IEEE international conference on robotics and automation, vol. 2, pp. 1662-1668. IEEE, 1996

work page 1996
[5]

Shape function-based kinematics and dynamics for variable length continuum robotic arms

Godage, Isuru S., David T. Branson, Emanuele Guglielmino, Gustavo A. Medrano-Cerda, and Darwin G. Caldwell. "Shape function-based kinematics and dynamics for variable length continuum robotic arms." In 2011 IEEE International Conference on Robotics and Automation, pp. 452-457. IEEE, 2011

work page 2011
[6]

Pneumatic muscle actuated continuum arms: Modelling and experimental assessment

Godage, Isuru S., David T. Branson, Emanuele Guglielmino, and Darwin G. Caldwell. "Pneumatic muscle actuated continuum arms: Modelling and experimental assessment." In 2012 IEEE International Conference on Robotics and Automation, pp. 4980-4985. IEEE, 2012

work page 2012
[7]

A snake-like swimming robot using IPMC actuator/sensor

Kamamichi, Norihiro, Masaki Yamakita, Kinji Asaka, and Zhi-Wei Luo. "A snake-like swimming robot using IPMC actuator/sensor." In Proceed- ings 2006 IEEE International Conference on Robotics and Automation,

work page 2006
[8]

Theoretical modeling and experimental analysis of a pressure-operated soft robotic snake

Luo, Ming, Mahdi Agheli, and Cagdas D. Onal. "Theoretical modeling and experimental analysis of a pressure-operated soft robotic snake." Soft Robotics 1, no. 2 (2014): 136-146

work page 2014

Show all 9 references

[2006]

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

Autonomous undulatory serpentine locomotion utilizing body dynamics of a ﬂuidic soft robot

Onal, Cagdas D., and Daniela Rus. "Autonomous undulatory serpentine locomotion utilizing body dynamics of a ﬂuidic soft robot." Bioinspiration & biomimetics 8, no. 2 (2013): 026003

work page 2013

[2] [2]

Dynamics for vari- able length multisection continuum arms

Godage, Isuru S., Gustavo A. Medrano-Cerda, David T. Branson, Emanuele Guglielmino, and Darwin G. Caldwell. "Dynamics for vari- able length multisection continuum arms." The International Journal of Robotics Research 35, no. 6 (2016): 695-722

work page 2016

[3] [3]

Modal kinematics for multisection continuum arms

Godage, Isuru S., Gustavo A. Medrano-Cerda, David T. Branson, Emanuele Guglielmino, and Darwin G. Caldwell. "Modal kinematics for multisection continuum arms." Bioinspiration & biomimetics 10, no. 3 (2015): 035002

work page 2015

[4] [4]

Simulation of contact using a nonlinear damping model

Marhefka, Duane W., and David E. Orin. "Simulation of contact using a nonlinear damping model." In Proceedings of IEEE international conference on robotics and automation, vol. 2, pp. 1662-1668. IEEE, 1996

work page 1996

[5] [5]

Shape function-based kinematics and dynamics for variable length continuum robotic arms

Godage, Isuru S., David T. Branson, Emanuele Guglielmino, Gustavo A. Medrano-Cerda, and Darwin G. Caldwell. "Shape function-based kinematics and dynamics for variable length continuum robotic arms." In 2011 IEEE International Conference on Robotics and Automation, pp. 452-457. IEEE, 2011

work page 2011

[6] [6]

Pneumatic muscle actuated continuum arms: Modelling and experimental assessment

Godage, Isuru S., David T. Branson, Emanuele Guglielmino, and Darwin G. Caldwell. "Pneumatic muscle actuated continuum arms: Modelling and experimental assessment." In 2012 IEEE International Conference on Robotics and Automation, pp. 4980-4985. IEEE, 2012

work page 2012

[7] [7]

A snake-like swimming robot using IPMC actuator/sensor

Kamamichi, Norihiro, Masaki Yamakita, Kinji Asaka, and Zhi-Wei Luo. "A snake-like swimming robot using IPMC actuator/sensor." In Proceed- ings 2006 IEEE International Conference on Robotics and Automation,

work page 2006

[8] [8]

Theoretical modeling and experimental analysis of a pressure-operated soft robotic snake

Luo, Ming, Mahdi Agheli, and Cagdas D. Onal. "Theoretical modeling and experimental analysis of a pressure-operated soft robotic snake." Soft Robotics 1, no. 2 (2014): 136-146

work page 2014