Phase-Based Multi-Gait Learning for a Salamander-Like Robot

Xian Guo; Yang Liu; Yongchun Fang; Zhiang Liu

REVIEW 4 major objections 5 minor 29 references

Phase-Based Multi-Gait Learning for a Salamander-Like Robot

T0 review · 4 major / 5 minor · reviewed 2026-08-03 · deepseek-v4-flash

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Pith's one-line read The paper claims that a salamander-like robot can learn 22 distinct omnidirectional gaits from a single policy that outputs only five signed phase velocities, with no reference motions or hand-designed gait patterns.

desk verdict A genuinely useful hardware gait-learning result that is undermined by a circular ablation and a missing baseline comparison. read the letter →

arxiv 2511.08299 v2 pith:VVB6DTMR submitted 2025-11-11 cs.RO

Zhiang Liu , Yang Liu , Yongchun Fang , Xian Guo This is my paper · ORCID

classification cs.RO

keywords salamander-likerobotomnidirectionalgaitsphasevariablesreinforcementlearninggaitdiscoverymorphologicalsymmetrydataaugmentationquadrupedlocomotion

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 attempts to show that a salamander-like quadruped robot can learn a broad repertoire of omnidirectional gaits—22 distinct commands spanning forward, backward, lateral, diagonal, oblique, curved, and in-place rotational motions—without any reference trajectories, demonstrations, or manually predefined gait patterns. The authors argue that a single policy, which only modulates the rate and sign of phase variables controlling each leg and the spine, is expressive enough to produce all of these behaviors when trained with a phase coverage reward and symmetry-based data augmentation. If true, this would mean that bio-inspired robots with sprawling postures do not need motion capture data or hand-designed central pattern generators to achieve diverse, flexible locomotion; a general-purpose learning recipe could suffice. The claim is supported by simulated trajectories and physical robot experiments showing trot-like walking, lateral walking, and in-place rotation with smooth gait transitions.

What carries the argument

The load-bearing mechanism is the phase variable φ_i for each of five body parts (four legs and the spine), updated by φ_i ← φ_i + ω_i Δt with signed phase velocity ω_i ∈ [-π/2, π/2] as the only action. The trajectory generator maps each phase to joint angles via fixed sinusoidal templates (e.g., θ = A cos φ, A sin φ + θ_offset), so the policy cannot alter amplitude or waveform; it can only speed up, slow down, or reverse each body part's rhythmic cycle. Two auxiliary mechanisms carry the argument: a phase coverage reward that encourages each leg to traverse its full 2π phase range, and a symmetry group C2×C2 whose transformations are applied to states and actions as data augmentation, makin

What would settle it

Train the same framework on a command at a very low speed (long stance phase) or on a terrain where optimal gaits require asymmetric leg cycles; then check whether the learned policy can produce the required stance/swing asymmetry. If the fixed sinusoidal template forces a symmetric duty cycle that fails on that command, the expressivity limitation is demonstrated. Alternatively, directly measure foot-contact phases on the physical robot and compare the stance/swing duty cycles against the sinusoidal template's prediction.

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Extended reading notes

Core claim

The central discovery is that a learning framework based on bidirectional phase variables—one per leg plus the spine—can yield a single controller that tracks 22 translational and rotational commands on a salamander-like quadruped without reference motions. Each body part's phase evolves by a signed phase velocity that the policy outputs, and a fixed sinusoidal trajectory generator converts phases to joint angles. A phase coverage reward (mean plus minimum leg coverage) prevents static or under-explored phase behavior, while a C2×C2 symmetry augmentation (front-back and left-right reflections applied to states and actions) biases the policy toward equivariant, symmetric behavior. The authors

Load-bearing premise

The framework's success rests on the premise that all needed gaits can be expressed by one signed phase velocity per body part feeding a fixed sinusoidal joint template with constant amplitude and offset; if a desired gait requires different joint waveforms, amplitudes, or asymmetric stance/swing timing, no policy in this action family could produce it.

Editorial extensions

If this is right

If correct, a single policy, not a library of per-gait policies, suffices for a sprawling quadruped to execute 22 distinct translational and rotational commands.
The phase-based action space abstracts away joint-angle trajectories, suggesting that the same framework could be transferred to other bio-inspired morphologies by redefining the per-body-part phase mapping.
Symmetry-based data augmentation acts as a physics-informed prior that improves sample efficiency and enforces motion-level and task-level symmetry without extra real-world interaction.
Smooth gait transitions emerge naturally from phase continuity and random phase initialization, even though the policy was never trained on sequential command switches.
The framework removes the need for motion capture datasets or expert-designed CPG parameters for bio-inspired robots, which is especially relevant for morphologies without recorded motion data.

Reading between the lines

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

The expressiveness of the discovered gaits is bounded by the sinusoidal template: with constant amplitude and offset, any behavior requiring swing/stance asymmetry, variable amplitude, or multi-frequency joint patterns is inexpressible; testing on slow walking or uneven terrain would reveal this ceiling.
The paper's symmetry claim is specifically about C2×C2 (front-back and left-right) reflections; gaits that are not symmetric under this group—such as turning with a preferred direction or asymmetric terrain adaptation—would likely require a different or additional inductive bias.
The coverage reward applies only to leg phases, not the spine, yet the spine still produces effective motion; understanding how the spine phases self-organize under the leg-driven reward could yield a simpler design principle for body-limb coordination.
The 22 commands are discrete samples of a continuous command space; the policy's ability to interpolate to arbitrary directions and turning rates is not measured, so the true omnidirectional capability is an open empirical question.

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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

Worth knowing: this paper shows a single learned controller producing 22 different gaits on a physical salamander-like quadruped, and the hardware phase data is convincing. The policy outputs signed phase velocities for five body parts, and the robot demonstrably trots, side-steps, walks diagonally, curves, and rotates in place, with smooth transitions between commands. Those are real physical behaviors, not just simulations.

What's new: signed phase velocity extends CPG-RL [22], and the coverage reward plus C2×C2 augmentation are applied to a sprawling morphology for the first time. Lateral, diagonal, and in-place rotation on a salamander-like robot are genuinely new results as far as this literature goes. Credit where earned: the authors built the hardware, got it working, and reported phase trajectories rather than only reward numbers.

The soft spots are in the evaluation, not the core idea. The most important one is the coverage-reward ablation (Table IV): it compares cumulative rewards, and the full method adds a non-negative reward term (r_coverage, Eq. 14) that the w/o Cov baseline deliberately does not have. Higher totals for 'Ours' are structurally guaranteed, so the table tells us nothing about whether coverage improves gait quality. The qualitative comparison in Fig. 7 (larger phase oscillations with coverage) is a step in the right direction, but the paper needs reward-independent metrics — stride length, phase range, task error, power — to actually support the claim. Second, there is no direct comparison against the existing CPG/CPG-RL controllers the authors criticize. A simple baseline would sharpen the contribution. Third, the action space is limited to fixed-sinusoid joint trajectories with a single phase-velocity parameter per body part; the policy cannot reshape waveforms. That is a real expressivity bound and the paper should say so explicitly instead of implying open-ended discovery. Fourth, no code or data is released, which makes replication difficult.

None of these are fatal. The central result — one controller, 22 commands, real hardware, plausible phase coordination — holds up. This deserves peer review, but the ablation needs to be redesigned and external baselines added. The 'first to demonstrate' claim also needs a closer look at the surrounding literature. Overall: send it to review, and let the authors strengthen the metrics before it goes to press.

Referee Report

4 major / 5 minor

Summary. The paper proposes a phase-based reinforcement learning framework for a salamander-like quadruped robot. Each of five body parts (four legs and spine) is assigned a phase variable, and the policy outputs only five phase velocities (Eq. 7). The reward includes task progress, turning, undesired-motion penalties, and a phase-coverage term (Eq. 14). Morphological symmetry is injected via C2×C2 data augmentation (Sec. III-D, Table II). The authors report that a single policy learns 22 omnidirectional gait commands, with simulations and real-robot experiments showing forward/backward, lateral, rotational, curved, diagonal, and oblique movements, plus smooth transitions between gaits. Ablations are presented for the coverage reward and symmetry augmentation.

Significance. If the results hold, the paper makes a useful contribution: it shows that a very low-dimensional phase-frequency action space, trained without reference motions, can produce a varied omnidirectional gait repertoire on a bio-inspired quadruped, and it demonstrates lateral and in-place rotational locomotion on physical hardware. The real-robot phase trajectories (Figs. 6–7) and the smooth command-switching behavior are valuable empirical evidence. The symmetry augmentation is a clean and principled component. However, the quantitative evaluation has a circularity problem in the coverage-reward ablation, and the claims of 'discovering' dynamic and symmetric gaits are partly built into the reward and augmentation. With more careful evaluation and more modest framing, the core result could still stand.

major comments (4)

[§IV-D, Table IV, Eq. (14)] The comparison between 'Ours' and 'w/o Cov' uses the cumulative reward as the evaluation metric, but the full method includes the non-negative phase coverage reward r_coverage = mean(c) + min(c) in its total return (Eq. 14) with weight w6 = 0.01 (Table III). The baseline omits this term. Therefore the higher cumulative reward for 'Ours' is structurally inflated by the very reward being ablated; Table IV does not demonstrate any behavioral benefit. The qualitative hardware phase data (Fig. 7) are more informative, but the quantitative claim that the coverage reward improves gaits is unsupported. Please re-evaluate with reward-independent metrics (e.g., achieved distance, turning error, phase coverage values, success rate), and report repeated training seeds rather than only variance across commands.
[§III-A, Eqs. (1)–(7), Table III] The action space is limited to five phase velocities; the leg joint trajectories are fixed to A·cos(φj), A·sin(φj)+θ_offset, A·cos(φj), and the spine to A·cos(φS), with constant A=π/12 and θ_offset=31π/180. Thus the policy cannot modulate amplitude, joint offset, or waveform shape. The claim that the framework 'discovers' novel gaits such as lateral walking and in-place rotation is therefore conditional on those behaviors being expressible in this one-parameter-per-body-part family. This is a load-bearing scope limitation for the 'diverse repertoire' claim. Please either test expressiveness directly (e.g., by comparing with a variable-amplitude or direct-joint-action baseline) or explicitly state that the discovered gaits are confined to this sinusoidal template.
[§III-D, Table II, §IV-B, Fig. 5] The symmetry of the learned behaviors is presented as a success of the method, but it is largely enforced by construction: the C2×C2 data augmentation explicitly transforms both states and actions (Table II), and the reward is G-invariant. Consequently, the central-symmetric trajectories in Fig. 5 and the paired phase patterns in Fig. 6 are expected consequences of the augmentation, not emergent discoveries. Please separate what is induced by the inductive bias from what is genuinely learned, and avoid wording that implies symmetry emerges independently. The useful empirical question is how much augmentation helps sample efficiency and robustness, not whether the policy becomes symmetric.
[§IV-B and §IV-C] The claim that this is 'the first to demonstrate that a salamander-like robot can achieve lateral, in-place rotational, and diagonal movements' is difficult to verify from the cited literature and may be too strong, especially since the action space and trajectory generator are specifically designed to allow such phase patterns. The authors should either provide a more detailed comparison with prior salamander-like robot controllers or temper the claim.

minor comments (5)

[General] The term 'dynamic' is used throughout (abstract, Sec. I, Sec. IV-D) but never formally defined. It appears to be identified with phase coverage and with periodic leg oscillation. Please give a precise criterion.
[§III-B] The pose stack and phase stack assume a fixed period T* = 5 s, but the policy can change phase velocities, so actual gait periods may vary. Please clarify how the stack length is chosen when the executed gait is not strictly T*-periodic.
[Table IV] The table reports variance within each gait category, but no variance over independent training runs. Given that PPO is stochastic, a single training run cannot support claims of 'consistently outperforms' or 'significantly lower variances'. Please add multi-seed statistics or state the number of seeds.
[Eq. (11)] The sign convention for r_turn = δ ω* Δψ should be stated explicitly: for ω* = -1 (clockwise), a positive Δψ would be penalized, which may be intended depending on the yaw convention. Please make the convention clear.
[References] The reference ordering appears unusual (e.g., [7] appears before [2] in the bibliography list). Please ensure the bibliography is in a consistent, journal-compliant order.

Circularity Check

1 steps flagged · score 6.0 of 10

Coverage-reward ablation is evaluated on a metric that includes the ablated reward term, making the quantitative 'improvement' partly by construction; the central multi-gait repertoire is otherwise supported by independent hardware evidence.

self definitional [Sec. III-C (Eq. 14) and Sec. IV-D (Table IV)]
"r_coverage = mean(c)+min(c) ... The total reward is computed as a weighted sum of the above terms, with corresponding weights w_i for i=1,...,6 ... For evaluation, we compute the total undiscounted cumulative reward ... The quantitative results in Table. IV show that our approach consistently outperforms the no-coverage baseline across all gait types. This improvement stems from the inclusion of the phase coverage reward."

The comparison metric is the same total cumulative reward that includes r_coverage (Eq. 14) with weight 0.01. The w/o Cov baseline is trained without that term, so when both policies are scored on the full reward, the full method holds a structural advantage in the very metric used to claim the reward's benefit. The paper uses this reward-based comparison as the quantitative evidence that the coverage reward 'improves' gaits, so that specific claim reduces to the definition of the reward rather than to an independent behavioral metric. The separate phase-trajectory evidence in Fig. 7 is more independent but is qualitative and itself measures the phase coverage that the reward explicitly promotes.

full rationale

The paper's central contribution—a phase-velocity policy that tracks 22 translational/rotational commands—is not itself circular: the commands in Table I are task specifications, not gait definitions, and the learned phase coordinations (trot, lateral, in-place rotation) are demonstrated on the physical robot via phase plots and trajectories. The sinusoidal phase template (Eqs. 1–7) is an explicit ansatz rather than a hidden circular input. Symmetry-based augmentation is openly described as an inductive bias ('enforcing both motion-level and task-level symmetry'), so observing symmetric behaviors is a sanity check of that design choice, not a derived prediction. No load-bearing self-citation chain was found; references to the authors' prior CPG design [29] are contextual. The one genuine circularity is the coverage-reward ablation: comparing cumulative rewards computed from a total reward that includes the ablated r_coverage term makes the quantitative superiority of the full method partly self-validating. Because the robot's omnidirectional repertoire is still supported by reward-independent observations, the overall circularity is partial rather than total.

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

No new physical entities (particles, forces, fields) are postulated. The bidirectional phase variables, phase stacks, and pose stacks are algorithmic/representational constructs; their behavior is fully specified by Eqs. 1–8. Nothing here needs a falsifiable outside-handle. The free parameters above are hand-tuned constants of the trajectory generator and reward; they do the work of making the 'dynamic' and 'diverse' behaviors achievable.

free parameters (6)

Joint amplitude A = π/12 rad
Hand-set constant in the trajectory generator (Eqs. 1–4, Table III). Determines step size and spine bend magnitude; gait feasibility depends on it; not learned or justified from the robot's morphology.
Joint offset θ_offset = 31π/180 rad
Fixed baseline posture of the second (knee) joint (Eq. 2). Affects ground clearance and stance geometry; chosen by hand.
Reward shaping parameters α, β, ζ, ε, δ = 1, 1.5, 0.5, 2.2, 0.75
Weights inside the directional/penalty/turn terms (Eqs. 9–11). They define what 'good locomotion' means; the directional terms deliberately amplify lateral commands via (α+β|v*y|) and perp penalties via (ζ+ε|v*x|).
Reward weights w_i = [0.1, 0.1, 0.1, 0.1, 0.1, 0.01]
Trade-off among forward, perp, turn, undesired-turn, undesired-move, and coverage terms (Sec. III-C). The coverage weight 0.01 is the lever that produces 'dynamic' gaits.
Desired gait period T* = 5 s
Window length for pose-change estimation and phase stacks (Sec. III-B). The design assumes learned gaits are T*-periodic; if gaits are slower than 5 s, the state/reward signal degrades.
Sim-to-real dynamics parameters = not reported
Ground friction, mass distribution, joint PID gains 'carefully tuned to match' the physical robot (Sec. IV-A). Values are not given; hardware transfer depends on them.

assumptions (6)

standard math In a G-symmetric MDP, the optimal policy is G-equivariant and the optimal value function G-invariant (Sec. II-C, citing [25]).
Borrowed from [25]; the augmentation's expected benefit rests on this theorem. Not re-derived; treated as external result.
domain assumption The robot's morphology is symmetric under front-back and left-right reflections, with the Table II relabelings as the correct group action.
The C2×C2 augmentation (Sec. III-D) assumes swapping LF↔LH/RF↔RH (with cos → −cos) and spine cos → −cos is a true dynamical symmetry. Front and hind girdles/legs may differ in mass and actuation on the physical robot; the paper does not measure how approximate this symmetry is.
domain assumption The MuJoCo model, after tuning friction/mass/PID, faithfully reproduces the physical robot's dynamics.
All RL training happens in MuJoCo (Sec. IV-A); policy quality on hardware depends on this fidelity.
ad hoc to paper Phase coverage ratio cov_i = (2π − d_max)/2π is a valid proxy for motion quality/dynamicity.
Eq. 14 rewards coverage of the leg phase circle; the claim that learned gaits are 'dynamic rather than static' is operationalized by this metric, chosen by the authors rather than derived from locomotion principles.
domain assumption Leg and spine joint commands of a single sinusoid per joint (Eqs. 1–4) can express all useful gaits for this robot.
The whole action space is this template; the 'discovery' claim is bounded by its expressiveness. This is the weakest_assumption of the report.
domain assumption Pose changes measured over T* windows are a sufficient feedback signal for command-following.
The state uses [Δx, Δy, Δψ] estimated by transforming recent poses (Sec. III-B); instantaneous velocities are deliberately discarded.

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Cite this review

Pith. "Pith review of Phase-Based Multi-Gait Learning for a Salamander-Like Robot." pith.science (2026). https://pith.science/paper/VVB6DTMR

@misc{pith2026251108299,
  author       = {Pith},
  title        = {Pith review of: Phase-Based Multi-Gait Learning for a Salamander-Like Robot},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VVB6DTMR}},
  note         = {Machine review of arXiv:2511.08299}
}

read the original abstract

Salamander-like robots are designed inspired by the skeletal structure of their biological counterparts. However, existing controllers cannot fully exploit these morphological features and largely rely on predefined patterns or joint trajectories, which prevents the generation of diverse and flexible gaits and limits their applicability in real-world scenarios. In this paper, we propose a phase-based learning framework that enables the robot to acquire a diverse repertoire of gaits without using reference motions. Each body part is controlled by a phase variable capable of forward and backward evolution, with a phase coverage reward to promote the exploration of the leg phase space. Additionally, morphological symmetry of the robot is incorporated via data augmentation, improving sample efficiency and enforcing both motion-level and task-level symmetry in learned behaviors. Extensive experiments show that the robot successfully acquires 22 representative gaits exhibiting both dynamic and symmetric movements, demonstrating the effectiveness of the proposed learning framework.

Figures

Figures reproduced from arXiv: 2511.08299 by the authors.

**Figure 3.** Diagram of the morphological symmetries of the salamander-like [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗

**Figure 4.** Schematic of the experimental platform. locomotion trajectories generated by the robot, followed by the analysis of phase evolution under multi-gait command switching. We also conduct ablation studies on phase coverage reward and symmetry-based data augmentation to assess their contributions, with results evaluated in terms of both phase behavior and cumulative reward. Visualizations of all hardware experiments are … view at source ↗

Figures from the paper (2 more)

**Figure 5.** Figure 5: Simulated robot trajectories under seven gait categories (see Table. I). [PITH_FULL_IMAGE:figures/full_fig_p006_5.png]

**Figure 8.** Figure 8: Robot trajectories observed in simulation for four symmetric gait [PITH_FULL_IMAGE:figures/full_fig_p007_8.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

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A novel design methodology of cpg model for a salamander-like robot,

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2024

Pith tools

Reviewed August 3, 2026 · model on record in the stance chip above.

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Cpg-rl: Learning central pattern genera- tors for quadruped locomotion,

G. Bellegarda and A. Ijspeert, “Cpg-rl: Learning central pattern genera- tors for quadruped locomotion,”IEEE Robot. Autom. Lett., vol. 7, no. 4, pp. 12 547–12 554, 2022

2022

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From cineradiography to biorobots: an approach for designing robots to emulate and study animal locomotion,

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2016

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Inverse kinematics and reflex based controller for body- limb coordination of a salamander-like robot walking on uneven terrain,

T. Horvat, K. Karakasiliotis, K. Melo, L. Fleury, R. Thandiackal, and A. J. Ijspeert, “Inverse kinematics and reflex based controller for body- limb coordination of a salamander-like robot walking on uneven terrain,” inProc. IEEE/RSJ Int. Conf. Intell. Robots Syst., 2015, pp. 195–201

2015

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Diffusion model-based path follower for a salamander-like robot,

Z. Liu, Y . Liu, and Y . Fang, “Diffusion model-based path follower for a salamander-like robot,”IEEE Trans. Neural Netw. and Learn. Syst., vol. 36, no. 8, pp. 14 399–14 413, 2025

2025

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Autonomous visual navigation with head stabilization control for a salamander-like robot,

Z. Liu, Y . Liu, Y . Fang, and X. Guo, “Autonomous visual navigation with head stabilization control for a salamander-like robot,”IEEE/ASME Trans. Mechatronics, 2025

2025

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Fast and efficient locomotion via learned gait transitions,

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2022

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Minimizing energy con- sumption leads to the emergence of gaits in legged robots,

Z. Fu, A. Kumar, J. Malik, and D. Pathak, “Minimizing energy con- sumption leads to the emergence of gaits in legged robots,” inProc. Conf. Robot Learn., 2022, pp. 928–937

2022

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Spine controller for a sprawling posture robot,

T. Horvat, K. Melo, and A. J. Ijspeert, “Spine controller for a sprawling posture robot,”IEEE Robot. Autom. Lett., vol. 2, no. 2, pp. 1195–1202, 2017

2017