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Optimal Trajectory Planning in a Vertically Undulating Snake Locomotion using Contact-implicit Optimization

T0 review · 4 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read A reduced-order, contact-implicit model based on Moreau's stepping-forward approach can predict the vertical undulation gait of the COBRA snake robot, matching trajectory shape, head velocity, and joint torques across simulations and…

desk verdict Useful contact-implicit ROM for vertical snake undulation, but accuracy claims rest on qualitative comparison and hand-set contacts; title overreaches. read the letter →

arxiv 2508.02953 v1 pith:7M53DZAT submitted 2025-08-04 cs.RO

Adarsh Salagame , Eric Sihite , Alireza Ramezani This is my paper · ORCID
classification cs.RO
keywords snakerobotverticalundulationcontact-implicitoptimizationMoreaustepping-forwardsecond-orderconeprogramdifferentialinclusionreduced-ordermodeltrajectoryplanning
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 tries to establish that a deliberately simple, reduced-order model of the COBRA snake robot can replace hand-designed shape functions for planning vertical undulation on flat ground. The model treats each body module as a point mass inside a virtual sphere, resolves ground contacts through Moreau's stepping-forward differential-inclusion scheme, and solves the resulting contact-implicit optimization as a second-order cone program. The authors argue that this formulation captures the essential contact and control allocation problem of snake locomotion, and they support the claim by comparing predicted trajectories, head velocities, and joint torques against two higher-fidelity simulations and the physical robot. If correct, the model offers a computationally cheap basis for optimal gait and trajectory planning without pre-specifying when or where contacts occur.

What carries the argument

Moreau–Jean time-stepping with contact constraints expressed as a differential inclusion, solved as a second-order cone program. At each step, an unconstrained velocity $\tilde{v}$ is predicted, then corrected by contact impulses $f_{c,i}$ subject to complementarity conditions $g_i \ge 0$, $f^n_{c,i} \ge 0$, $g_i f^n_{c,i} = 0$, a friction cone $|f^t_{c,i}| \le \mu_i |f^n_{c,i}|$, and a semi-implicit position update $q_{n+1} = q_n + \Delta t v_{n+1}$. The optimization over the next state and contact forces replaces any hand-scheduled contact pattern; contact forces and gaits emerge from the constraints.

What would settle it

Run the same vertical undulation gait on the physical COBRA robot over two surfaces with clearly different friction properties (for instance, polished vinyl and rubber mat) while keeping joint commands fixed; if the SOCP model with a single $\mu$ does not reproduce the change in head velocity, travel distance, or slip direction seen on hardware, the point-contact Coulomb assumption is falsified for this gait. A more direct check would place a force plate under the robot and compare measured ground reaction forces with the model's predicted short-duration force spikes.

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Extended reading notes

Core claim

The central claim is that the SOCP-based reduced-order model (Eqs. 13-19) reproduces the vertical undulation of COBRA: over a ten-second gait the model and hardware both travel roughly one meter forward, with matching movement direction, trajectory shape, head-module velocity, and actuation torque profiles. The model's predictions are not identical to hardware; the paper reports that joint backlash attenuates the physical robot's joint motion and shortens its travel, while the analytical model shows larger oscillations, shorter travel, and sharp torque spikes from numerical integration instabilities. The authors interpret the comparison as validation of the model's ability to capture the overall dynamics, with discrepancies attributed to ground reaction force estimation and tunable contact hyperparameters rather than to a wrong model structure.

Load-bearing premise

The model's predictions rest on representing the ground as flat with thirteen point contacts under Coulomb friction with a fixed friction coefficient ($\mu = 0.5$) and on hand-set spring-damper parameters in the comparison simulations; if real ground contact is distributed, compliant, or direction-dependent, the agreement shown may not carry to other surfaces.

Editorial extensions

If this is right

  • The same SOCP can be used directly as a planner: joint torques, contact forces, and base motions are decision variables, so a vertical undulation trajectory needs no predefined contact schedule.
  • Because the model is a small convex program with a dozen states, it can be evaluated quickly enough for iterative gait design on the physical robot, where a 500 Hz controller already runs in the head module.
  • If the model generalizes to other gaits, the contact-implicit formulation transfers to three-dimensional contact patterns and loco-manipulation, as the conclusion sketches.
  • The torque profiles match hardware up to actuation limits, so the model can be used to check feasibility of planned motions within the robot's 10 Nm saturation.

Reading between the lines

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

  • A direct test of the load-bearing assumption would vary the ground surface under COBRA (e.g., low-friction vinyl vs. high-friction rubber) while keeping the gait fixed; the point-contact Coulomb model with a single $\mu$ predicts specific changes in slip and travel that could be checked against hardware.
  • The same contact-implicit program could be extended to uneven terrain by substituting measured depth maps into the gap function $g_i$, which the model already updates from $n_i^T J_{c,i} v$, turning the planner into a closed-loop reactive controller.
  • One could embed the SOCP in a receding-horizon model predictive controller by treating the measured joint state as the initial condition and using the predicted contact forces as feedforward; the paper does not claim this, but the structure of Eqs. (13)-(19) is compatible with it.
  • The observed torque spikes from numerical instabilities suggest a regularization or smoothing modification to the objective $\Phi(v_{n+1})$ that would make the planner's outputs directly executable without filtering; this is an engineering refinement, not a change to the model's core claim.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 5 minor

Summary. The paper introduces a reduced-order dynamic model of a 12-link snake robot moving on flat ground, in which each module is represented by a virtual convex sphere, ground contacts are governed by Coulomb friction, and the state trajectory is resolved through a Moreau-Jean time-stepping scheme formulated as an optimization problem (Eqs. 13-19). The model is evaluated on a prescribed vertical undulation gait by comparing its predictions against a Simscape reduced-order model, a CAD-based high-fidelity Simulink model, and hardware experiments on the COBRA robot. The central claim is that the analytical model accurately captures the head-module trajectory shape, movement direction, and velocity, with a close match in actuation torques. The manuscript frames this work as enabling contact-implicit optimal trajectory and acyclic contact planning for snake locomotion.

Significance. If the model-accuracy claim is substantiated, the work is a useful step toward applying differential-inclusion/Moreau contact-implicit formulations to a multi-contact snake platform: it offers a planar reduced-order model that is computationally efficient, it is validated against independent hardware data rather than only against itself, and the authors are transparent about observed discrepancies such as joint backlash, torque spikes, and shorter travel distance. These strengths make the central approach plausible and worth pursuing. However, the validation is currently qualitative: no error metrics, no sensitivity analysis, and no repeated-trial statistics are provided, and the most load-bearing claim of 'accurate' trajectory and velocity prediction is accompanied by the paper's own admission of larger oscillations and shorter travel. The contribution would meet the journal bar only after the accuracy claim is quantified and the optimization formulation is clarified.

major comments (4)
  1. [Section IV, Figure 7] The central accuracy claim, stated as 'the analytical model accurately captures the overall trajectory shape and movement direction, with a similar agreement observed in the predicted velocity of the head module,' is supported only by qualitative visual comparison. No error metric (e.g., RMSE, normalized trajectory error, or travel-distance error), no repeated experimental trials, and no confidence intervals are reported, and the same paragraph admits that the analytical model 'predicts larger oscillations with shorter travel distances.' Because the friction coefficient (mu = 0.5) and the Simulink spring-damper parameters (10^4 and 10^3) are hand-set, the observed agreement could be an artifact of parameter tuning. Please add quantitative trajectory and velocity errors, a friction-coefficient sensitivity study, and uncertainty bounds over experimental trials.
  2. [Section III, Eq. (16)] The complementarity constraint g_i f_i = 0 is bilinear and nonconvex, so the optimization problem in Eqs. (13)-(19) is not a second-order cone program as claimed, unless some convex relaxation or regularization is applied. The manuscript does not state how this constraint is handled in practice (e.g., complementarity smoothing, penalty relaxation, active-set iteration, or an MPCC solver) nor which numerical solver is used. Since the paper's stated computational benefit rests on the optimization being a tractable SOCP, this is a load-bearing algorithmic detail that must be clarified.
  3. [Section IV, first paragraph] The experiments validate a prescribed joint-position trajectory, not an optimized trajectory: the text states that 'this gait was defined by a prescribed joint trajectory, from which joint torques, contact forces, and base accelerations were computed.' The title and introduction claim optimal trajectory planning, but the optimization in Eqs. (13)-(19) includes only a regularization cost u^T u, and no planning scenario, objective, or optimized gait is presented. Please specify how the prescribed joint trajectory is enforced as a constraint or as an input to the optimization, and, if trajectory planning is a claimed contribution, include a demonstration of it.
  4. [Section IV, Figure 8] The contact-force comparison reveals qualitatively different behavior between the analytical/ROM point-contact models and the high-fidelity model: the point-contact models produce shorter, higher-magnitude impulses, while the high-fidelity model distributes contact over longer durations. Because contact modeling is the core of the proposed approach, the paper needs to quantify this discrepancy (e.g., contact duration, peak force, impulse) and discuss whether the point-contact model remains adequate for the intended planning use despite this qualitative mismatch. Without such a quantitative contact-level assessment, the central modeling claim is only partially verified.
minor comments (5)
  1. [Section III] There is a typo in 'optimiztion' before Eq. (13), and the Moreau time-stepping derivation is repeated in Eqs. (4)-(6) and (7)-(12); the duplication could be streamlined to improve readability.
  2. [Section III, Eqs. (2), (14)] The sign convention for the normal contact force is inconsistent: Eq. (2) states f_n <= 0 for compressive force, while Eq. (14) imposes f_n >= 0. Please clarify the relationship between these definitions and the contact normal n_i = [0,1].
  3. [Section III, Eq. (1)] The state dimension q in R^12 should be reconciled with the stated platform of 12 links and 11 actuated joints, and the dimensions of J_{c,i} and the contact normal n_i should be defined explicitly for the 13 contacts.
  4. [Section IV] Figures 7-9 would benefit from axis labels, units, and legend details; in particular, Figure 9 should state whether the hardware torque data are from joint current sensing and over how many trials, and Figure 8 should specify which representative link and which time window are shown.
  5. [Section III] For reproducibility, please report the time step used in the Moreau scheme, the convergence tolerance, the solver used for the optimization, and the runtime of the 10-second simulation window.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the contact-implicit SOCP model is validated against independent hardware and Simulink data, and self-citations concern the hardware platform rather than the claimed derivation.

full rationale

The derivation chain is self-contained: the claimed prediction is the SOCP contact-implicit model (Eqs. 13-19) built on Moreau's stepping-forward approach (Eqs. 4-12), which is introduced from external nonsmooth mechanics literature rather than from the authors' prior results. No predicted quantity (head trajectory, velocity, torques, contact forces) is defined in terms of the experimental data it is compared against: the prescribed joint trajectory is the same input to all models and the hardware, and the Section IV comparison is an external validation against independent measurements. The friction coefficient mu = 0.5 and the Simulink spring-damper coefficients (1e4 and 1e3) are stated model parameters, not fitted outputs, and the paper's own concession that the analytical model 'predicts larger oscillations with shorter travel distances, likely due to differences in ground reaction force estimation stemming from optimization constraints and tunable hyperparameters' is a limitation on accuracy, not a circular reduction. The COBRA self-citations concern the robot hardware platform and prior locomotion studies; they do not supply the mathematical result being claimed, and no uniqueness theorem or ansatz is imported from the authors' own work. No self-definitional, fitted-input-as-prediction, or imported-uniqueness circularity is present.

Assumptions & free parameters 5 free parameters · 4 assumptions · 1 invented entities

The central claim rests on hand-set ground contact parameters, a spherical-contact model abstraction, and the assumption that Coulomb complementarity on flat ground is sufficient. These choices are not independently measured, so the ledger includes five free parameters, four assumptions, and one invented modeling abstraction.

free parameters (5)
  • Ground friction coefficient mu = 0.5
    Used in both Simulink models and friction cone constraints; hand-set, not measured, and cited as a tunable hyperparameter affecting ground reaction forces.
  • Ground spring stiffness (Simulink) = 1e4
    Chosen for spring-damper ground contact model; affects contact duration and force magnitude.
  • Ground damping coefficient (Simulink) = 1e3
    Chosen for spring-damper ground contact model; affects oscillation and energy dissipation.
  • Ground restitution coefficient = not reported
    Mentioned as a tunable hyperparameter influencing ground reaction force estimation, but no value is reported.
  • Virtual contact sphere radius = not reported
    The reduced-order model encapsulates each module in virtual spheres; the radius affects gap functions and contact timing but is not specified.
assumptions (4)
  • domain assumption The robot is a rigid-body system on flat ground with Coulomb friction and complementarity contact conditions.
    Invoked in Eqs. (1)-(3); if ground interaction is distributed or compliant, the model's contact representation is approximate.
  • ad hoc to paper Each module's contact can be represented by a convex sphere with a gap function.
    This is the core reduced-order modeling choice; no experimental measurement validates the spheres, only indirect trajectory comparison.
  • standard math Moreau-Jean time-stepping with a small regularization term yields correct contact resolution for this system.
    Borrowed from Moreau's 1988 differential-inclusion framework; assumed to hold for the 10-second vertical undulation scenario.
  • domain assumption The same prescribed joint trajectory is executed by all simulated models and the physical robot.
    Used throughout Section IV; hardware joint backlash is later acknowledged as a source of discrepancy.
invented entities (1)
  • Virtual convex contact spheres surrounding each robot module
    purpose: Provide gap functions that determine contact events in the reduced-order model
    A modeling abstraction introduced in Section III; no independent measurement supports the sphere representation, and its accuracy is only indirectly assessed through whole-body trajectory comparison.

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Cite this review

Pith. "Pith review of Optimal Trajectory Planning in a Vertically Undulating Snake Locomotion using Contact-implicit Optimization." pith.science (2026). https://pith.science/paper/7M53DZAT

@misc{pith2026250802953,
  author       = {Pith},
  title        = {Pith review of: Optimal Trajectory Planning in a Vertically Undulating Snake Locomotion using Contact-implicit Optimization},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7M53DZAT}},
  note         = {Machine review of arXiv:2508.02953}
}
read the original abstract

Contact-rich problems, such as snake robot locomotion, offer unexplored yet rich opportunities for optimization-based trajectory and acyclic contact planning. So far, a substantial body of control research has focused on emulating snake locomotion and replicating its distinctive movement patterns using shape functions that either ignore the complexity of interactions or focus on complex interactions with matter (e.g., burrowing movements). However, models and control frameworks that lie in between these two paradigms and are based on simple, fundamental rigid body dynamics, which alleviate the challenging contact and control allocation problems in snake locomotion, remain absent. This work makes meaningful contributions, substantiated by simulations and experiments, in the following directions: 1) introducing a reduced-order model based on Moreau's stepping-forward approach from differential inclusion mathematics, 2) verifying model accuracy, 3) experimental validation.

Figures

Figures reproduced from arXiv: 2508.02953 by the authors.

Figure 1
Figure 1. Shows COBRA performing vertical undulation for traveling through [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. COBRA system overview. conditions. We can then leverage the inherent redundancy of the snake body and reformulate snake locomotion as an acyclic self-manipulation problem. This work makes the following contributions: 1) intro￾ducing a reduced-order model based on Moreau’s stepping￾forward approach from differential inclusion mathematics, 2) verifying model accuracy, 3) experimental validation. The paper is organized… view at source ↗
Figure 3
Figure 3. The robot’s interactions with the environment are [PITH_FULL_IMAGE:figures/full_fig_p002_3.png] view at source ↗
Figures from the paper (5 more)
Figure 3
Figure 3. Figure 3: Illustration of our modeling and control approach based on proximal optimization for the constrained COBRA model with contact dynamics, [PITH_FULL_IMAGE:figures/full_fig_p003_3.png]
Figure 6
Figure 6. Figure 6: Shows the joint velocity for two joints during the vertical undulation [PITH_FULL_IMAGE:figures/full_fig_p004_6.png]
Figure 5
Figure 5. Figure 5: Shows the joint trajectory for two joints during the vertical [PITH_FULL_IMAGE:figures/full_fig_p004_5.png]
Figure 8
Figure 8. Figure 8: Comparison of ground contact forces between the ROM and high-fidelity model. The ROM exhibits intermittent point contacts, whereas the [PITH_FULL_IMAGE:figures/full_fig_p005_8.png]
Figure 9
Figure 9. Figure 9: Shows the actuation torque profiles for the ROM, high-fidelity [PITH_FULL_IMAGE:figures/full_fig_p005_9.png]

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