Model Analysis And Design Of Ellipse Based Segmented Varying Curved Foot For Biped Robot Walking

Boyang Chen; Chao Song; Jie Zhao; Xizhe Zang; Yue Zhang

REVIEW 3 major objections 4 minor 31 references

Model Analysis And Design Of Ellipse Based Segmented Varying Curved Foot For Biped Robot Walking

T0 review · 3 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read A biped foot shaped from three elliptical arcs cuts walking energy by up to 18.5%.

desk verdict Good experiments, but the core contact model has a likely ratio error that undermines the analytical claims as written. read the letter →

arxiv 2506.07283 v1 pith:QPECLCDN submitted 2025-06-08 cs.RO

Boyang Chen , Xizhe Zang , Chao Song , Yue Zhang , Jie Zhao This is my paper

classification cs.RO

keywords bipedrobotgaitenergyefficiencyvaryingcurvaturefootanalyticalcontactmodelellipticalarcnonlinearprogrammingdesignHLIPcontrollateralwalking

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

Biped robots burn battery power against a simple constraint: every step is a collision, and the foot's rollover shape decides how much of that collision is wasted. This paper argues that a foot whose frontal-plane profile is three elliptical arcs—a relatively flat mid arc flanked by more curved hind and fore arcs—reduces that waste while still admitting a closed-form contact model. The authors derive the model, design the hind and fore arcs with a nonlinear program, and verify the foot on the TT II biped in marking-time, straight-walking, and lateral-walking experiments. Across all tasks the ESVC feet consume less energy than line and flat feet, with the largest gain, 18.52% average-power reduction at 0.2 m/s lateral walking, achieved by the flattest variant, ESVC5.

What carries the argument

The central object is the ESVC (Ellipse-based Segmented Varying Curvature) foot: a biped foot whose frontal-plane rollover profile is three elliptical arcs, a relatively flat mid ellipse flanked by more curved hind and fore ellipses. The argument runs on the contact model: homogeneous transformation matrices between the foot frame, the contact frame, and the ground frame are derived using elementary functions, with an approximate elliptic arc-length formula and a sine-wave compensation for its error. A nonlinear program fixes the hind and fore ellipse parameters from a known mid ellipse by imposing tangent-slope consistency at the segment points and a perpendicular-bisector alignment assumption. This machinery carries the paper because it turns foot shape into quantities a model-based controller can use directly, without elliptic integrals or lookup tables.

What would settle it

Re-run the lateral-walking protocol at 0.2 m/s with mass- and inertia-matched feet and randomized repeated trials; if the ordering ESVC5 < ESVC3 < ESVC1 < Line Foot disappears once construction differences are controlled, the central energy-efficiency claim fails. A first check is simply weighing the five feet, since a mass difference correlated with the ordering would already cast doubt.

Watch

Extended reading notes

Core claim

The paper's central claim is that a segmented varying-curvature foot, built from three elliptical arcs, is both analytically tractable and energetically superior to conventional line and flat feet for bipedal walking. It establishes this in three steps: an elementary-function contact model (the foot frame, contact-point frame, and ground frame are related by homogeneous matrices, with the elliptic arc length approximated and then error-compensated), a nonlinear program that fixes the hind and fore ellipses once a mid ellipse is chosen, and physical experiments on the TT II robot with five feet. The measured average power shows ESVC feet below line and flat feet in every condition; the flattest mid-curvature variant, ESVC5, gives the largest margins, including an 18.52% reduction over the line foot in lateral walking at 0.2 m/s. The authors' interpretation is that the ESVC5 rollover geometry most closely resembles a biological foot, letting the contact point roll further and dissipating less impact energy.

Load-bearing premise

The load-bearing premise in Sec. 4.2 is that the five compared feet differ only in rollover shape: the paper reports no foot mass, inertia, material, or trial-to-trial variance, so the measured energy differences could come from physical differences between the feet rather than from the geometry.

Editorial extensions

If this is right

Robot gaits can be made more efficient without changing the control law: the ESVC foot drops into an HLIP-based controller through the same contact-frame transformations, so the benefit comes from geometry alone.
The flatter the mid ellipse, the better the energy performance in these experiments, which gives designers a monotone design rule: among ESVC feet, smaller mid curvature pays off, especially when frontal-plane motion is large.
The nonlinear program turns foot shaping into a solvable design choice—given a desired mid arc, the hind and fore arcs are determined uniquely—so foot morphology can be optimized per robot rather than hand-tuned.
Lateral walking is where the design matters most: savings of 6.8% to 18.5% over the line foot, versus only a few percent in sagittal walking, so frontal-plane tasks are the natural target for curved-foot bipeds.

Reading between the lines

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

The paper's self-stated limitation—no comparison with alternative ESVC methods and no quantitative foot-geometry metric—means the strongest interpretation (shape alone drives the gain) remains untested; a mass-matched single-ellipse foot would make the comparison decisive.
Because the paper's own conclusion says no floating-support theory was derived, the claim to watch is about the foot geometry in an existing controller, not a general theory of curved feet.
The sine-wave error compensation in the roll-angle domain is generic; it could be exported to any smooth convex foot, making model-based foot optimization possible beyond elliptical arcs.
Since frontal-plane savings dominate, adding a roll degree of freedom to the ankle, which the TT II leg lacks, might amplify the ESVC benefit further; the paper's data imply but do not test this.

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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

Quick take: the paper is worth reading for the design idea and the experiments, but I think the central contact model has a geometric error that needs to be fixed before the modeling claims are credible.

What's new: an analytical contact model for segmented elliptical feet using elementary functions (Anakhaev approximation plus a fitted compensation), an NLP that breaks the coupled major/minor axis design of single-ellipse feet, and real-biped walking experiments on five foot types that consistently show lower energy consumption for the ellipse-based shapes. That last piece is genuinely useful data for the foot-design community.

Where it gets soft: I checked the geometry of the mid-ellipse derivation and the stress-test note holds up. Eq.1 gives tan φ_mc = (r_mb^2 / r_ma^2) tan θ_m, but the correct relation is the reciprocal: the tangent of the angle to the vertical scales as (r_ma / r_mb)^2 times tan θ_m. As a result, at θ_m=0 the paper's Eq.2 sets φ_m=π/2 and Eq.3 then gives d_m = r_ma, and Eq.4 gives y_C = r_ma instead of 0. The contact point at zero roll would be at the side of the foot, not at the bottom. This is not a small perturbation; for ESVC5 the error in tan is about a factor of 50. Because the same wrong relation is used in the NLP constraints and in the transformation matrices for the HLIP controller, the analytical contact model is not internally consistent as written. The energy-efficiency claim is measured, not derived from the model, so the experimental trend survives, but the paper's stated contribution, the analytical model, does not.

The experiments have their own soft spots: no repeated trials or error bars, no mass/inertia data for the feet, and the flat-foot baseline is dropped in lateral walking. The 18.52% improvement is a single measurement, so treat it as preliminary. Minor: there is a leftover 'Example Appendix Section' and several typos.

Bottom line: the design concept and experimental comparison are valuable, and the paper deserves a serious referee. But the referee should be sent in with the expectation that the geometry in Sec. 2.2 needs to be corrected and re-derived. If the authors correct Eq.1 and the subsequent equations, the paper would be a solid contribution. As it stands, I would not trust the analytical model.

Referee Report

3 major / 4 minor

Summary. The paper proposes an Ellipse-based Segmented Varying Curvature (ESVC) foot for bipedal robots, consisting of three elliptical arcs in the coronal plane. It derives an analytical contact model using elementary functions, formulates a nonlinear program to design the fore/hind ellipses from a given mid ellipse, introduces an error-compensated approximation for rollover arc length, integrates the foot with an HLIP-based controller, and validates the approach in simulation and on the TT II biped robot. Physical experiments across marking time, straight walking, and lateral walking report consistent energy reductions for ESVC feet, with up to 18.52% improvement in lateral walking compared with a line foot.

Significance. If correct, the paper would offer a useful contribution: a fully analytical contact model for segmented elliptical feet, a design methodology that breaks the coupling inherent in single-ellipse designs, and real-robot evidence that rollover geometry affects energy efficiency. The authors are to be credited for building and testing five physical foot types on a real biped and for providing simulation and experimental data. However, the central geometric derivation contains a load-bearing error, and the experimental comparison lacks the statistical grounding needed to support the headline efficiency claim. As presented, the modeling and design contributions are not established; the empirical energy-efficiency result is suggestive but not rigorously supported.

major comments (3)

[Sec. 2.2, Eq. (1) and Eqs. (3)-(4)] Eq. (1) is geometrically inconsistent. For an ellipse with semi-axes r_ma (lateral) and r_mb (vertical), the contact point under a roll angle θ_m must have a horizontal tangent in the world frame. Using the standard ellipse parameterization, the correct relation with the paper's angle conventions is tan φ_m = (r_ma^2 / r_mb^2) tan θ_m, and thus tan φ_mc = (r_mb^2 / r_ma^2) cot θ_m. Eq. (1) instead states tan φ_mc = (r_mb^2 / r_ma^2) tan θ_m. The error is immediate in the circular limit r_ma = r_mb: Eq. (1) gives φ_mc = θ_m, whereas the radius to the contact point of a circle remains vertical in the world frame, requiring φ_mc = π/2 − θ_m. This mistake propagates into d_m (Eq. (3)), y_C^m (Eq. (4)), the transformation matrices (Eqs. (11)-(17)), the segment-point constraints and NLP formulation (Sec. 2.3, Eq. (24)), and the HLIP integration (Eqs. (39)-(43)). The analytical contact model, which is the paper's first contribution, is therefore not correct as written.
[Sec. 4.2, Table 2] The experimental evidence for the headline energy-efficiency improvement is incomplete. Table 2 omits the Line Foot baseline row for all tasks, yet the lateral-walking improvements (7.28% to 18.52%) are computed relative to the Line Foot; without the baseline values these percentages cannot be verified. In addition, each experimental condition appears to be a single recorded trial: no repeated trials, standard deviations, confidence intervals, or statistical tests are reported. The five foot types are also not characterized by mass, inertia, or material, so the measured energy differences cannot be attributed to rollover geometry rather than to mechanical differences among the feet. Please provide repeated trials (n ≥ 3) with mean ± standard deviation, include the complete data table with the Line Foot baseline, and report or otherwise control for foot mass and inertia.
[Sec. 2.5, Eq. (38) and Table 1] The error-compensation method is not reproducible as reported. The compensation law depends on the parameters δ_max and K_e, but the paper does not state how these are selected or give their numerical values for EA1, EA2, and EA3. The accuracy analysis in Table 1 also validates only the arc-length approximation in isolation, not the full contact model that includes the flawed geometric relations from Sec. 2.2. Since Eq. (1) is the foundation of the segment-point constraints and the NLP, the error analysis does not address the dominant source of modeling error. Please report the parameter values and validate the complete contact model against the actual foot geometry or measured contact points.

minor comments (4)

[Sec. 2.5] The text states that 'EA3 is closer to a circle', but EA3 has the largest axis ratio (r_ma/r_mb = 2.66), making it the most elongated of the three arcs; EA1 (ratio 1.22) is closest to a circle. The subsequent sentences about the boundary parameter angle λ* are also confusingly worded and should be rewritten.
[Table 2] The caption of Table 2 is incorrect: it reads 'Accuracy Performance of Different Elliptical Arc', which is the caption for Table 1. The table should be titled with the average power values by foot type and task.
[Eq. (36)] In the transformation matrix for the fore-ellipse case, the rotation entries use θ_m, but this expression should be in terms of the fore-ellipse rollover angle θ_f; this appears to be a typo.
[Appendix A] The manuscript contains an unfinished placeholder appendix ('Example Appendix Section', 'Appendix text.') that should be removed before publication.

Circularity Check

0 steps flagged · score 2.0 of 10

No load-bearing circularity: the energy-efficiency result is an independent physical measurement, and the contact-model/design chain is not defined in terms of its own outputs.

full rationale

The paper's central claim—that ESVC feet reduce energy consumption by up to 18.52% in lateral walking—rests on cumulative power measurements on the physical TT II robot (Sec. 4.2, Table 2), not on a derivation from the fitted contact model. The modeling chain (ellipse contact geometry, Eqs. 1-17; fore/hind-foot NLP, Eq. 24; rollover-length approximation from [29], Eq. 7; compensation, Eq. 38; HLIP integration, Eqs. 39-48) is a forward construction: each equation transforms geometric inputs into contact-frame outputs, and no step is defined in terms of the energy outcome it is used to explain. The compensation step (Eq. 38) uses δmax, the maximum error of Eq. 7 against numerical ground truth on the same elliptical arcs, so Table 1 is an in-sample calibration report rather than an out-of-sample prediction; that is a minor self-referential validation issue, but it is not load-bearing for the energy claim. The only self-citation is [31], used to point to implementation details of the real-time control framework, and it does not carry any load-bearing premise. Possible algebraic inconsistencies flagged by a close read (e.g., the reciprocal ratio in Eq. 1 and the apparent Y-coordinate value in Eq. 4) are correctness concerns about internal consistency, not circularity of the argument. The paper also explicitly concedes that no comparison with alternative foot-design methods was performed, which is a limitation rather than a circular step.

Assumptions & free parameters 3 free parameters · 5 assumptions · 1 invented entities

The central design depends on the rigid no-slip contact assumptions F1-F2, the ad hoc segmentation constraints F3-F4, an approximate arc length formula from the literature, and hand-tuned compensation parameters. The energy-efficiency claim itself is empirical and does not depend on these model choices, which is why the circularity burden is low.

free parameters (3)

Ke (compensation gain) = not stated
Used in Eq. 38 to scale the sine-wave error correction; chosen per ellipse to reduce approximation error.
delta_max (maximum approximation error) = varies per EA1/EA2/EA3
Determined from the error curve of Eq. 7 for each ellipse; effectively fitted to the data being corrected.
NLP weights w1..w4 = not stated
Weights in Eq. 24 trade off tangent matching, segment sizes, and foot width; chosen by hand.

assumptions (5)

domain assumption F1: foot-ground contact is rigid and the foot does not deform
Stated in Sec 2.1.3; underlies all contact geometry.
domain assumption F2: no slipping during support phase
Stated in Sec 2.1.3; rollover length equals arc length only if no slip.
ad hoc to paper F3: tangent slopes of adjacent ellipses are as consistent as possible at segment points
Design constraint in Sec 2.3 that makes the NLP well-posed.
ad hoc to paper F4: perpendicular bisector of S2S3 coincides with the major axis of the fore ellipse
Design constraint in Sec 2.3 that fixes the fore ellipse orientation.
standard math Arc length approximation Eq. 7 from Anakhaev is valid for the ellipse range used
Cited from [29]; error is measured and compensated in Sec 2.5.

invented entities (1)

ESVC foot (three elliptical arcs: hind, mid, fore)
purpose: Improve gait energy efficiency while keeping contact model analytical
A new hardware design introduced by the paper. Its claimed benefit is demonstrated only in the paper's own simulations and experiments, not by external independent data.

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Cite this review

Pith. "Pith review of Model Analysis And Design Of Ellipse Based Segmented Varying Curved Foot For Biped Robot Walking." pith.science (2026). https://pith.science/paper/QPECLCDN

@misc{pith2026250607283,
  author       = {Pith},
  title        = {Pith review of: Model Analysis And Design Of Ellipse Based Segmented Varying Curved Foot For Biped Robot Walking},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/QPECLCDN}},
  note         = {Machine review of arXiv:2506.07283}
}

read the original abstract

This paper presents the modeling, design, and experimental validation of an Ellipse-based Segmented Varying Curvature (ESVC) foot for bipedal robots. Inspired by the segmented curvature rollover shape of human feet, the ESVC foot aims to enhance gait energy efficiency while maintaining analytical tractability for foot location based controller. First, we derive a complete analytical contact model for the ESVC foot by formulating spatial transformations of elliptical segments only using elementary functions. Then a nonlinear programming approach is engaged to determine optimal elliptical parameters of hind foot and fore foot based on a known mid-foot. An error compensation method is introduced to address approximation inaccuracies in rollover length calculation. The proposed ESVC foot is then integrated with a Hybrid Linear Inverted Pendulum model-based walking controller and validated through both simulation and physical experiments on the TT II biped robot. Experimental results across marking time, sagittal, and lateral walking tasks show that the ESVC foot consistently reduces energy consumption compared to line, and flat feet, with up to 18.52\% improvement in lateral walking. These findings demonstrate that the ESVC foot provides a practical and energy-efficient alternative for real-world bipedal locomotion. The proposed design methodology also lays a foundation for data-driven foot shape optimization in future research.

Figures

Figures reproduced from arXiv: 2506.07283 by the authors.

**Figure 2.** Fig2.d represents the case just before the foot lifts off the ground which can [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗

**Figure 2.** Rollover Process of ESVC Foot Consider the case when the ESVC foot tilts from left to right, with the rollover range within the mid ellipse, as shown in Fig2.b. In this case, the minor axis of the ellipse coincides with the vertical axis of the foot surface. θ represents roll angle of the foot which can be obtained from inclinometer. The roll angle of mid ellipse is expressed as θm = |θ|. Then the rollover angle of … view at source ↗

Figures from the paper (5 more)

**Figure 3.** Figure 3: Accuracy Of Rollover Length For Different Ellitical Arcs [PITH_FULL_IMAGE:figures/full_fig_p019_3.png]

**Figure 4.** Figure 4: Snapshots of the simulation at walking speeds of 0.1m/s and 0.2m/s. [PITH_FULL_IMAGE:figures/full_fig_p023_4.png]

**Figure 5.** Figure 5: a.Snapshots of marking time of robot TT II; b.The Total Energy Input; [PITH_FULL_IMAGE:figures/full_fig_p025_5.png]

**Figure 6.** Figure 6: a.Snapshots of straight walking of robot TT II; b.The Total Energy Input Of [PITH_FULL_IMAGE:figures/full_fig_p026_6.png]

**Figure 7.** Figure 7: a.Snapshots of lateral walking of robot TT II; b.The Total Energy Input Of [PITH_FULL_IMAGE:figures/full_fig_p027_7.png]

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Reference graph

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Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [1]

Leng J, Fan S, Tang J, et al.M-A3C: a mean-asynchronous advantage actor-critic reinforcement learning method for real-time gait planning of biped robot, IEEE Access, 2022, 10: 76523-76536

work page 2022

[2] [2]

Ege M, Kucuk S.Energy minimization of new robotic-type above-knee prosthesis for higher battery lifetime, Applied Sciences, 2023, 13(6): 3868

work page 2023

[3] [3]

Mou H, Tang J, Liu J, et al.High Dynamic Bipedal Robot with Under- actuated Telescopic Straight Legs, Mathematics, 2024, 12(4): 600

work page 2024

[4] [4]

CurtzeC,HofAL,vanKeekenHG,etal.Comparative roll-over analysis of prosthetic feet, Journal of biomechanics, 2009, 42(11): 1746-1753

work page 2009

[5] [5]

Kwan M, Hubbard M.Optimal foot shape for a passive dynamic biped, Journal of theoretical biology, 2007, 248(2): 331-339. 28

work page 2007

[6] [6]

Mei Q, Kim H K, Xiang L, et al.Toward improved understanding of foot shape, foot posture, and foot biomechanics during running: A narrative reviews, Frontiers in Physiology, 2022, 13: 1062598

work page 2022

[7] [7]

Smyrli A, Ghiassi M, Kecskeméthy A, et al.On the effect of semiel- liptical foot shape on the energetic efficiency of passive bipedal gait, IEEE/RSJ International Conference on Intelligent Robots and Sys- tems(IROS), 2019, 6302-6307

work page 2019

[8] [8]

Ren L, Howard D, Ren L, et al.A generic analytical foot rollover model for predicting translational ankle kinematics in gait simulation studies, Journal of biomechanics, 2010, 43(2): 194-202

work page 2010