Gait-Net-augmented Implicit Kino-dynamic MPC for Dynamic Variable-frequency Humanoid Locomotion over Discrete Terrains

Junchao Ma; Junheng Li; Quan Nguyen; Ziwei Duan

REVIEW 4 major objections 5 minor 1 cited by

Gait-Net-augmented Implicit Kino-dynamic MPC for Dynamic Variable-frequency Humanoid Locomotion over Discrete Terrains

T0 review · 4 major / 5 minor · reviewed 2026-08-09 · deepseek-v4-flash

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Pith's one-line read The paper claims that a lightweight neural network, Gait-Net, predicts per-step duration for a centroidal-dynamics MPC so that humanoids co-optimize foot location, timing, and contact forces, enabling variable-frequency dynamic walking…

desk verdict Real humanoid hardware results, but the paper overclaims 'simultaneously optimize step duration': Gait-Net is trained on randomized step durations, so the preferred-duration claim is not established. read the letter →

arxiv 2502.02934 v3 pith:YDFQJERH submitted 2025-02-05 cs.RO cs.SYeess.SY

Junheng Li , Ziwei Duan , Junchao Ma , Quan Nguyen This is my paper · ORCID

classification cs.ROcs.SYeess.SY

keywords humanoidlocomotionmodelpredictivecontrolvariablegaitfrequencydiscreteterraincentroidaldynamicsneuralnetworkfootstepplanningsequentialconvexoptimization

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

Most humanoid controllers fix step duration in advance or solve for it as an extra nonlinear decision variable. This paper argues that a small trained network can pick the step duration for each upcoming footstep, converting a hard multi-linear optimization into a sequence of cheap quadratic programs. The resulting controller simultaneously adjusts where a foot lands, how long the swing lasts, and what ground-reaction forces are applied, and it demonstrates this on hardware: a small humanoid walks at 0.75 m/s, clears gaps up to 20 cm, steps over obstacles, recovers from pushes, and pushes a 35 kg cart, using only the next step's terrain information. The sympathetic reading is that learned timing prediction plus convex optimization can capture dynamic, variable-frequency gait without solving full nonlinear MPC online.

What carries the argument

Gait-Net is a lightweight neural network that maps the robot's current state feedback and desired foot location to a preferred MPC sampling time dt (interpreted as step duration), trained on 600 seconds of simulated whole-body MPC data with randomized step durations and impulses, with inputs reduced via PCA from 16 to 6 features. The sequential solving mechanism is a custom SQP-style algorithm: each iteration linearizes the bilinear centroidal dynamics around the previous solution's search directions, solves a convex QP for the increments of force, foot location, and CoM position, then uses Gait-Net to update dt and an analytical inverse-kinematics step to refresh the spatial-momentum and pose references; this loop repeats until the search direction is small, and the final dt is held fixed for the whole footstep.

What would settle it

Run the hardware controller over a terrain course whose gap sizes exceed 20 cm and with push magnitudes outside the training distribution, while disabling the fallback that reverts to nominal dt; if the Gait-Net's predicted step durations systematically fail to keep the feet inside the feasible terrain—measured by foot placement error or falls—the claim of robust variable-frequency co-optimization is falsified.

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Extended reading notes

Core claim

The core discovery is that variable-frequency bipedal locomotion can be cast as a centroidal-dynamics MPC in which the per-step sampling time dt—equivalently the step duration—is predicted by a lightweight Gait-Net trained on 600 seconds of whole-body-MPC walking data, rather than being an optimization variable. With dt fixed as a parameter, the multi-linear couplings among contact force, foot location, and timing disappear, and each sequential iteration becomes a QP over force, foot, and CoM search directions. To preserve kinematic feasibility without joint states in the optimization, the algorithm recomputes the spatial-momentum and centroidal-pose reference trajectories from the latest foot and CoM solutions using a fast analytical IK, embedding implicit kino-dynamics into the reference design. On a 24-DoF, 16-kg humanoid, the controller produces natural step-duration adjustments—quick small steps near terrain boundaries, longer steps over gaps—and keeps the robot stable under pushes, unknown loads, and terrain discontinuities.

Load-bearing premise

The entire variable-frequency behavior rests on the trained Gait-Net predicting a step duration that is accurate for states, terrains, and disturbances it never saw during its 600 seconds of simulated training data; if this prediction is wrong, the claimed co-optimization of timing and placement collapses to a fixed-duration fallback.

Editorial extensions

If this is right

The controller requires only one-step terrain preview, so it can be coupled with onboard perception that sees the next step's boundaries.
Variable step durations emerge automatically: short quick steps near terrain boundaries and longer steps to clear gaps, matching stride length and state feedback.
Computational cost stays at QP level: normalized solve time is 0.76 versus 7.28–12.05 for the compared alternatives, and real-time deployment runs on an Intel NUC i5.
The controller rejects disturbances, handling push recovery, unknown loads, a 35-kg cart push, and uneven terrain by adapting step duration on hardware.
MPC weights have a broad stable region, from 10% to 300% of the hand-tuned values, so the approach is not brittle to parameter tuning.

Reading between the lines

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

The Gait-Net is essentially a learned prior for an optimization-difficult variable, and the same pattern could be applied to other legged robots or manipulation tasks where contact timing and location are coupled, such as quadruped gait transitions.
The one-step preview limitation is a direct consequence of encoding terrain constraints as functions of CoM position; extending to longer horizons would require either predicting future CoM or adding terrain constraints as functions of the planned CoM trajectory, possibly using Gait-Net itself.
Because the Gait-Net predicts dt at the start of each footstep and the MPC then holds that dt throughout the step, the framework effectively performs event-triggered timing updates, suggesting a family of controllers where a learned policy sets the time grid and a convex optimizer fills in the spatial variables.

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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

I read the paper. The genuinely new piece is the architecture: a small network inside a sequential convex MPC predicts the MPC sampling time dt, and the kinematic reference trajectory is updated each iteration from the latest foot and CoM solutions. The hardware evidence is real: 0.75 m/s walking over a 20 cm gap on a 16 kg humanoid, push recovery, unknown load, and pushing a 35 kg cart. That is a meaningful engineering result. The authors are also unusually candid: Remark 2 says the linearization is standard SQP, the limitations section admits only one-step preview is used, and Section V-E4 reports a 2.8% fallback rate to nominal dt. I take the engineering contribution seriously.

The soft spot is the central claim. The abstract says the controller "simultaneously optimizes step location, step duration, and contact forces." It does not. In Algorithm 1, dt is produced by Gait-Net after each QP solve and then fixed; it is not a decision variable in the QP. That would be fine if described as learning-based step-duration selection, but "simultaneously optimize" is inaccurate.

The deeper problem is the training signal for Gait-Net. Section IV-A1 says the step duration at the start of each stride was drawn uniformly from [150, 400] ms. So the labels are random draws, not optimal or preferred durations. A network cannot learn a "preferred" duration from independent random labels. The reported RMSE (6.8 ms on dt) is less than half the standard deviation of a uniform [30, 80] ms distribution (~14.4 ms), which strongly suggests the network is echoing the dt currently or previously in effect, not predicting a preferred future dt. At deployment, the query at the start of a stride contains states from the previous stride, so the learned mapping is on shaky ground. The hardware works, but that does not establish the co-optimization claim; the network may just be a reasonable heuristic mapping state features to dt.

I would not call this fatal. The authors are honest about the fallback, and the system demonstrates useful behavior. But the paper should be revised to (1) stop calling dt optimization, (2) explain what Gait-Net actually learns given randomized labels, and (3) provide direct evidence that predicted dt responds to the intended physical quantities, ideally with ablations and statistical repetitions. Code or training data would help considerably.

Who is this for? Legged-locomotion researchers working on variable-frequency MPC. It deserves a serious referee: the engineering is compelling, and the flaw is in the paper's framing rather than in the controller's demonstrated usefulness. I would engage with it, but I would not cite the co-optimization claim as-is.

Referee Report

4 major / 5 minor

Summary. The paper proposes a Gait-Net-augmented sequential convex MPC framework for humanoid locomotion, in which a lightweight neural network predicts the MPC sampling time dt (interpreted as step duration) and feeds it to a centroidal-dynamics-based MPC that also optimizes foot placement and contact wrenches via iterative QPs. The authors claim simultaneous optimization of step location, step duration, and contact forces for variable-frequency locomotion over discrete terrains, and they validate the method in 2-D and 3-D simulation and on a small humanoid robot, including push recovery, unknown loads, and gap/obstacle traversal.

Significance. If the central claim holds, the work would be a practical contribution to real-time humanoid locomotion, combining a lightweight learned timing policy with a convex MPC to achieve variable-frequency stepping with one-step terrain preview. The sequential QP linearization of the bilinear centroidal dynamics is standard and the hardware demonstrations, including pushing a cart of 219% robot mass and traversing a 20-cm gap, are notable experimental assets. However, as detailed below, the training signal for Gait-Net does not support the 'preferred step duration' or 'simultaneous optimization' claims, which undermines the core contribution.

major comments (4)

[Sec. IV-A1, Table I] The training labels for Gait-Net are randomized step durations drawn uniformly from [150, 400] ms at the beginning of each stride, as stated in Sec. IV-A1. Such labels contain no optimality or 'preference' signal, so the network cannot learn a preferred step duration. The reported prediction RMSE of 6.79e-3 s on dt is inconsistent with regression to an independent future draw from a uniform [0.03, 0.08] s distribution, where a constant predictor already achieves an RMSE of about 1.44e-2 s. This strongly suggests data leakage or echo of the currently/previous active dt rather than prediction of a future preferred duration. Please specify the exact input-label pairing used in training and provide evidence that the network's output at deployment (state at stride start plus desired foot placement) is not merely retrieving the previous dt. Without this, the central claim that Gait-Net 'determines the preferred step duration' and that step duration is simultaneously optimized is unestablished.
[Abstract, Sec. I-E, Algorithm 1, Remark 4] The abstract and contribution list state that the MPC 'simultaneously optimizes step location, step duration, and contact forces,' but in the actual algorithm dt is not an optimization variable: Algorithm 1 line 8 sets dt_{j+1} via Gait-Net, and Remark 4 states that the MPC runs with this same dt for the entire footstep. Thus step duration is a learned predicted parameter, not a co-optimized decision variable. The phrasing 'simplifying step duration optimization to the parameter level' is acceptable only if the network's prediction itself is the solution of an optimization (e.g., trained to imitate an optimal dt), which is not the case given the random labels. Please revise the claims to describe a learned timing heuristic that adapts dt, or provide evidence that the network's output satisfies an optimality condition.
[Sec. V-E, Figs. 9-11] The hardware experiments are presented as single-run demonstrations: Figs. 9 and 11 show step durations and velocity tracking for individual runs, and Fig. 10c mentions 'over 10 forward-pushing trials' without reporting success rates, means, or variance. Given the paper's robustness claims (push recovery, unknown loads, uneven terrain), statistical repetitions with error bars or at least quantitative success/failure counts are needed to support those claims. Please add aggregate statistics over repeated runs or soften the robustness statements accordingly.
[Table III] Table III is difficult to interpret because the 'Proposed' column appears to contain multiple variable-size entries and multiple solve-time values without clear column separation. The rows for 'Optimization Variable Size' and 'Normalized Solve Time' seem to list several numbers (e.g., 'R34h-22 R27h-15 R21h-9' and '1.00 2.58 0.76') without indicating which configuration (CD vs implicit kino-dynamic, or different solver variants) each number refers to. Please restructure the table so each method or variant is a separate column with a single value per cell, or add a clarifying caption.

minor comments (5)

[Sec. V-E1, Fig. 10c] The statement that 'foot positions are not the primary factors in predicting the MPC step duration' appears to contradict the observation in Fig. 10c that longer step durations favor near foot locations. Clarify whether the two statements refer to different regimes (e.g., PCA loadings vs. observed correlation in the pushing task) or whether one of them is imprecise.
[Sec. IV-B3, Eq. (20)] In the CoM position update equation, the superscript and subscript indexing (e.g., p_{c,k+1}^{j-1} and δpc,k+1) is not fully explained; please spell out the iteration index convention used for states and search directions.
[Sec. II, Sec. IV-B] The term 'implicit kino-dynamic' is used to describe a method that does not optimize joint states but updates reference trajectories via IK; define this term clearly at first use and consider whether a more conventional descriptor (e.g., 'kinematically updated centroidal MPC') would avoid confusion.
[Sec. V-E4] The sentence 'integrating a neural network into a sequential solver does not guarantee convergence..' contains a doubled period; also, the fallback mechanism (reverting to nominal dt when non-negative slope appears) could be discussed more explicitly in terms of its effect on the claimed co-optimization, even if it occurs only 2.8% of the time.
[Table I] The RMSE values in Table I are implicitly in seconds; state the units explicitly (e.g., 's' or 'ms') to avoid ambiguity.

Circularity Check

1 steps flagged · score 6.0 of 10

Gait-Net's 'preferred step duration' is fitted to randomized labels, so the central co-optimization claim reduces to reproducing random draws rather than optimizing step timing.

fitted input called prediction [Sec. IV-A and IV-A1; Algorithm 1 line 8; Table I.]
"we propose a lightweight Gait-frequency Network that maps current state feedback and desired foot placement to the preferred step duration for the upcoming stride ... At the beginning of each stride, we generate a randomized step duration between [150, 400] ms ."

The regression target for Gait-Net is an independent uniform random draw made at the start of each stride, not the minimizer of any objective. The network is therefore fitted to a random label, and the 'preferred step duration' is by construction that same random draw; there is no optimality signal for the network to learn. With h'=5 MPC steps per stride, a uniform step-duration draw of [150,400] ms makes dt uniform over [30,80] ms, for which a constant predictor already achieves about 14 ms RMSE. Table I reports 6.79e-3 s (~6.8 ms) RMSE, which is only achievable by regressing the already-active or previously executed dt embedded in the state (leakage), not by predicting an independent future preferred dt.

full rationale

The only load-bearing circular step is the Gait-Net timing signal. Section IV-A1 states that the label for each stride is a randomized step duration drawn uniformly, while Section IV-A says the network outputs the 'preferred step duration.' Since the label is an independent random draw carrying no optimality information, the 'preference' is definitional: the network is fitted to the random command, not to an optimized quantity. The reported RMSE being below the constant-predictor floor for a uniform target indicates that the fitted target is the active or previous dt, not a future preferred dt, so the claimed co-optimization of step duration is not supported by the derivation. The remaining framework—the sequential convex MPC, foot-location and friction constraints, and the analytical IK reference updates—is self-contained and externally validated on hardware and simulation, and there is no load-bearing self-citation. The circularity is partial: the timing prediction reduces by construction, while the foot placement and force optimization still have independent content.

Assumptions & free parameters 6 free parameters · 5 assumptions · 0 invented entities

The central method relies on a learned network (Gait-Net) that is a fitted function, plus a set of hand-tuned MPC weights and bounds. No new physical entities are introduced. The assumptions are standard for centroidal MPC but the generalization of Gait-Net is the most consequential untested assumption.

free parameters (6)

Gait-Net network weights = not disclosed
The network is trained on whole-body MPC data to predict step duration dt, which is central to the variable-frequency behavior.
MPC control weights L1, L2 = Tables IX/X (hardware and simulation)
Hand-tuned state and input tracking weights used in the QP objective; the paper provides a sensitivity analysis showing a broad stable region.
MPC sampling time bounds = [0.045, 0.07] s hardware, [0.03, 0.06] s simulation
Ad hoc clipping limits on dt to ensure hardware feasibility.
PCA-selected feature set = 6 features chosen from 16 via PCA loadings
The reduced feature space is chosen to maximize variance while cutting inference time; the specific features are determined by PCA on training data.
Fallback threshold for non-negative slope = not quantified
The algorithm reverts to nominal dt when the search-direction slope is non-positive, but the exact trigger condition is not specified.
Nominal dt0 = 0.05 s
Initial MPC sampling time used at startup (Algorithm 1).

assumptions (5)

domain assumption Assumption 1: Kinematic assurance can be implicitly embedded in reference trajectory design if actual spatial momentum and pose align with designed trajectories.
Introduced in Sec. III-B to justify omitting joint states from the MPC optimization; the validity depends on the reference update loop staying close to true kinematics.
standard math Assumption 2: The bilinear product of search directions is negligible (δa·δb ≈ 0) after warm-starting.
Used in Sec. IV-B3 to linearize the cross terms in the angular momentum dynamics; the authors note it is mathematically identical to first-order SQP linearization (Remark 2).
domain assumption Periodic contact sequence with fixed h and h' (swing phase length).
The whole method presupposes a known, periodic foot contact schedule; variable-frequency is achieved only by scaling the MPC sampling time, not by changing the contact order.
domain assumption Gait-Net, trained on whole-body MPC simulation data, is a sufficient proxy for optimal step duration decisions.
The network's dt output is trusted for the entire footstep; if the training data does not cover the deployment distribution, the prediction quality degrades, as acknowledged in Sec. IV-A1.
domain assumption One-step preview of terrain is sufficient; constraints for later steps are deactivated.
Explicitly stated in the Limitation section (Sec. VI): only the next step location is constrained by terrain data, which limits look-ahead feasibility.

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Cite this review

Pith. "Pith review of Gait-Net-augmented Implicit Kino-dynamic MPC for Dynamic Variable-frequency Humanoid Locomotion over Discrete Terrains." pith.science (2026). https://pith.science/paper/YDFQJERH

@misc{pith2026250202934,
  author       = {Pith},
  title        = {Pith review of: Gait-Net-augmented Implicit Kino-dynamic MPC for Dynamic Variable-frequency Humanoid Locomotion over Discrete Terrains},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/YDFQJERH}},
  note         = {Machine review of arXiv:2502.02934}
}

read the original abstract

Reduced-order-model-based optimal control techniques for humanoid locomotion struggle to adapt step duration and placement simultaneously in dynamic walking gaits due to their reliance on fixed-time discretization, which limits responsiveness to various disturbances and results in suboptimal performance in challenging conditions. In this work, we propose a Gait-Net-augmented implicit kino-dynamic model-predictive control (MPC) to simultaneously optimize step location, step duration, and contact forces for natural variable-frequency locomotion. The proposed method incorporates a Gait-Net-augmented Sequential Convex MPC algorithm to solve multi-linearly constrained variables by iterative quadratic programs. At its core, a lightweight Gait-frequency Network (Gait-Net) determines the preferred step duration in terms of variable MPC sampling times, simplifying step duration optimization to the parameter level. Additionally, it enhances and updates the spatial reference trajectory within each sequential iteration by incorporating local solutions, allowing the projection of kinematic constraints to the design of reference trajectories. We validate the proposed algorithm in high-fidelity simulations and on small-size humanoid hardware, demonstrating its capability for variable-frequency and 3-D discrete terrain locomotion with only a one-step preview of terrain data.

Figures

Figures reproduced from arXiv: 2502.02934 by the authors.

**Figure 1.** Gait-Net-augmented Kino-dynamic MPC. Hardware Experiment Snapshots. (a). Push-recovery in locomotion (b). Locomotion while carrying an unknown 0.75 kg object; (c). Locomotion over unknown uneven terrain; (d). Dynamic walking over terrain with a 20 cm terrain gap at 0.75 m/s; (e). Dynamic walking over terrain gap and obstacle. Full experiment video: https://youtu.be/UqLDYHGL5EA timing within a unified framework by ad… view at source ↗

**Figure 2.** Control System Architecture optimization problem. In our framework, we proposed a GaitNet-augmented sequential CMPC algorithm that translates the original nonlinear problem into convex sequential subproblems. With the additional assistance of Gait-Net, we reduce the optimization variable and mimic a natural step duration decision in each iteration. The control framework converts user commands and contact sequence … view at source ↗

**Figure 3.** Illustration of the Gait-frequency Network Hence the optimization variables of the explicit kinodynamics MPC can be chosen as, Xkd = {q, q˙ , λk} h k=0, (7) The finite horizon optimization problem with a prediction of h steps can expressed as, min Xkd h X−1 k=0 [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗

Figures from the paper (10 more)

**Figure 4.** Figure 4: Feature Projection Bar Graphs along 6 Principle Axes. The feature with the highest projection in each axis (red bar) is selected to be part of the new feature space. Note that along principle axes 1 and 2, both left and right legs are equally weighted with opposite sig…

**Figure 5.** Figure 5: Bilinear Envelope Approximation by Neglecting Search Direction Product δa · δb. corresponding whole-body joint space trajectory, which is a significant part of ensuring the kinematics feasibility of the optimization results. h ref k = AG(q ref k )q˙ ref k , (11) Href k…

**Figure 6.** Figure 6: Comparison of Discrete Terrain Locomotion Performance in 2D Simulation. dynamic MPC with sequential CMPC subproblems and GaitNet. In the initialization stage (1-4), fj2m describes the mapping from joint-space general-coordinate states to spatial momenta. fref construc…

**Figure 7.** Figure 7: Locomotion over 3-D Stepping-stone Terrain. Simulation snapshots (left) and plot of measured foot locations (right). In the plot, only foot locations that are on the ground are visualized. The green dashed-line bounding box represents the CoM position threshold that tr…

**Figure 8.** Figure 8: Spatial Momenta Measurement vs. MPC Prediction along lG,x, kG,y, and kG,z of 3-D Stepping-stone Simulation Results. reference trajectories {x ref , p ref f } are continuously updated to match closely to the real spatial momentum and pose trajectories based on the lates…

**Figure 9.** Figure 9: Variable Step Durations under Unknown Disturbances. Hardware experiment of (1) Push-recovery, [PITH_FULL_IMAGE:figures/full_fig_p009_9.png]

**Figure 11.** Figure 11: Velocity X Tracking Plot. In discrete terrain locomotion hardware experiment with a 20-cm gap. predictions to be within the bound of [0.045, 0.07] s to ensure hardware feasibility. On relatively flat terrain with minimal perturbations, the MPC optimizes dt to the nomi…

**Figure 10.** Figure 10: Pushing Carts with Unknown Payload. The time instances of the robot hand’s contact with the cart are synchronized in the comparison plot to enhance visualization. Associated plots of spatial momenta are also provided in [PITH_FULL_IMAGE:figures/full_fig_p010_10.png]

**Figure 12.** Figure 12: MPC weight parameter sensitivity: robot performance with 10 [PITH_FULL_IMAGE:figures/full_fig_p011_12.png]

**Figure 13.** Figure 13: Kinematic Definition of 5-DoF Leg with Line-foot. [PITH_FULL_IMAGE:figures/full_fig_p014_13.png]

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Pith tools

Reviewed August 9, 2026 · model on record in the stance chip above.

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