Maximum Impulse Approach to Soccer Kicking for Humanoid Robots

Grzegorz Ficht; Sven Behnke

REVIEW 3 major objections 6 minor 11 references

Maximum Impulse Approach to Soccer Kicking for Humanoid Robots

T0 review · 3 major / 6 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read An analytic maximum-impulse plan accelerates the leg to peak velocity at impact, kicking the ball 42% farther in simulation than the waveform-based baseline.

desk verdict Sensible engineering contribution: a simple four-phase kick planner beats their own waveform baseline by 42% in simulation and works on hardware, but the 'maximum impulse' mechanism is asserted rather than directly verified. read the letter →

arxiv 2412.01480 v1 pith:O3PUOYZY submitted 2024-12-02 cs.RO

Grzegorz Ficht , Sven Behnke This is my paper · ORCID

classification cs.RO

keywords humanoidsoccerkickin-walkmaximumimpulseconstant-accelerationtrajectoryZMPgaitcentroidalmodelNimbRo-OP2Xlegswingplanning

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 claims that a powerful soccer kick for a walking humanoid can be generated analytically from a maximum-impulse principle: accelerate the leg as fast as hip torque and leg inertia allow, hit the ball at peak foot velocity, and fold the resulting motion into the existing gait as an offset. The kick is split into four phases (prepare, swing, continue, and return) with constant-acceleration trajectories, and the gait's step frequency is set to the total kick time so the robot stays balanced. In simulation the maximum-impulse kick sent the ball 7.53 m on average, 42% farther than the waveform-based kick it replaces, and on the physical NimbRo-OP2X the ball crossed the full 5.5 m field. The sympathetic reader would take this as evidence that simple dynamics-aware planning, not motion capture or optimization, is enough to get near-optimal kick power while remaining in-walk.

What carries the argument

The carrying object is a four-phase leg-swing trajectory built from constant-acceleration equations of motion. In the swing phase the leg accelerates to the target kick velocity $\omega_k$ with constant angular acceleration $\alpha_k = \tau_h/I_l$, where $\tau_h$ is hip actuator torque and $I_l$ is leg inertia from a five-mass centroidal model; the prepare and return phases are symmetric accelerate-decelerate segments, and the continue phase extends the motion past contact by $\theta_{ext}$ to desensitize impact timing. These offsets are added to the ZMP gait, and the gait frequency $f_g=1/t_k$ is matched to the total kick time, so the maximum-impulse motion is executed while the walking controller keeps the robot balanced.

What would settle it

Measure foot velocity at impact with motion capture across a range of kick angles and compare it with the predicted $\omega_k$ from $\alpha_k = \tau_h/I_l$: if the measured velocity is systematically below prediction when the knee is not synchronized, the single-rigid-body leg model is the weak point.

Watch

Extended reading notes

Core claim

The central claim is that kick power is an impulse problem, and impulse is maximized by maximizing foot velocity at impact; under a sagittal-plane assumption this reduces to reaching the hip's maximum angular velocity at the contact angle. The paper derives the leg swing angle $\theta_k = \mathrm{atan2}(z_h - r_b, x_b - r_b)$ from ball and hip geometry, chooses the shortest swing time $t_{sw}=\omega_k/\alpha_k$ using maximum hip acceleration $\alpha_k=\tau_h/I_l$, and pads this with prepare, continue, and return phases so the swing starts and ends at zero offset from the gait. The kick duration $t_k$ sets the next step frequency $f_g=1/t_k$, which is what makes the powerful motion executable inside a ZMP-based walk. The result is a parameterizable, constraint-aware kick whose simulated mean distance is 7.53 m with standard deviation 0.42 m, compared with 5.28 m with standard deviation 0.31 m for the waveform baseline, and whose hardware trial crossed the full 5.5 m field.

Load-bearing premise

The planner treats the whole leg as one rigid body with a fixed inertia and full hip torque throughout the swing, so the claimed peak foot velocity depends on that simple model being true at every joint angle.

Editorial extensions

If this is right

Simulated kicks average 7.53 m (SD 0.42) versus 5.28 m (SD 0.31) for the waveform-based kick, a 42% increase in distance.
The robot never fell in ten trials despite the step frequency dropping from 2.4 Hz to 0.7 Hz, so the kick survives the gait disturbance it introduces.
On hardware the NimbRo-OP2X kicked the ball the full 5.5 m field length, with residual momentum at the goal.
The same derivation applies in the lateral plane, which the paper states as the path to omnidirectional kicks.
The approach is parameterizable: desired kicking velocity and ball position enter directly, so the same planner can be retargeted without re-tuning.

Reading between the lines

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

Inference: if the rigid-body inertia and torque assumptions transfer, the 42% advantage should reproduce on any humanoid whose hip torque and leg inertia are known; a natural test is to run the same planner on a different robot and compare impact speed, not just distance.
Inference: because the paper only optionally synchronizes the knee, the analysis suggests most of the kick's impulse comes from the hip; varying knee extension timing and measuring ball speed would map how much of the remaining margin is knee-driven.
Inference: the step-frequency coupling $f_g=1/t_k$ is a stability bet—it trades gait rhythm for single-support time—so an interesting extension would be to let the planner slow the step only as much as the ZMP controller allows, rather than setting frequency from kick time alone.
Inference: the continue phase's extension angle $\theta_{ext}$ is a deliberately tunable robustness knob; tuning it against ball-position noise in simulation could yield a principled trade-off between impact timing robustness and post-impact leg velocity.

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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

The paper is a straightforward, useful engineering contribution rather than a scientific breakthrough. What is actually new: a four-phase kick planner that computes constant-acceleration trajectories up to a torque-limited hip acceleration, sets the gait frequency to the kick duration, and embeds the whole thing in an existing ZMP-based gait controller. The combination appears to work. In simulation, the mean kick distance is 7.53 m vs 5.28 m for the waveform baseline—a 42% improvement over ten trials—and the robot did not fall. On hardware they got a full-field kick in the lab. That is a legitimate, practically relevant result for RoboCup-sized humanoids.

The math is simple but mostly correct for the model. They equate maximum ball impulse with maximum hip velocity, use α_k = τ_h / I_l with a fixed leg inertia from a five-mass centroidal model, and split the motion into prepare/swing/continue/return. The derivation is self-contained, and no parameters are fitted to the outcome. The baseline comparison is against their own previous method, which is the right comparison to make. The self-citations are the gait and robot model, not the kick result. That part is clean.

The soft spots are real but not fatal. The biggest one is Eq. (4): a real leg is a multi-link chain, the inertia at the hip changes with knee angle, and hip torque drops with velocity. The paper never measures foot velocity at impact, so the 'maximum impulse' mechanism is asserted rather than verified. A knee-synchronized or torque-speed-limited trajectory might do as well or better. Still, the simulation result is an end-to-end comparison of two planners, and it is positive even if the mechanism is not isolated. The hardware evidence is a single anecdote, and ω_k and θ_ext are not reported, which would matter for reproduction. No code or data shipped. These are worth fixing or acknowledging, but they do not undo the 42% result.

Who is this for? People working on humanoid in-walk kicks and reactive gait control; RoboCup teams will get more from it than a general robotics audience. It deserves a serious referee—it needs a sensitivity analysis on the inertia assumption and at least a few more hardware kicks, but the core comparison is sound enough to publish after revision.

Referee Report

3 major / 6 minor

Summary. The paper presents an analytic, four-phase kick planner for humanoid robots, consisting of Prepare, Swing, Continue, and Return phases. Trajectories are derived from constant-acceleration kinematics under a rigid-leg assumption with the hip accelerating at alpha_k = tau_h / I_l (Eq. 4). The kick is integrated into a ZMP-based walking gait by adjusting the gait frequency to the total kick time. The approach is evaluated in MuJoCo simulation on a NimbRo-OP2X model, reporting a 42% greater mean kick distance over a waveform-based baseline (7.53 m vs. 5.28 m, 10 trials), and in a single hardware demonstration where the ball traveled the full 5.5 m field.

Significance. If the claims hold, the approach offers a simple, parameterizable, and physically motivated kick planner that noticeably outperforms the prior waveform-based method in simulation while preserving gait stability. The kinematic derivations are self-contained, the simulation comparison uses an external baseline without fitting parameters to the outcome, and the hardware demonstration shows real-world feasibility. However, the central 'maximum impulse' mechanism is not directly validated: no impact-velocity measurement, no sensitivity analysis, and the quantitative advantage rests on simulation only. The strengths lie in the clean analytic form and the practical integration with an existing ZMP-based gait.

major comments (3)

[Section II-A, Eq. (4)] The claim that the swing phase produces the maximum possible impulse rests on modeling the leg as a single rigid body with constant inertia I_l and constant available torque tau_h. The manuscript does not report numerical values for I_l, tau_h, alpha_k, omega_k, or theta_ext, and it does not verify that the planned hip velocity is actually reached at impact. Because the 42% distance advantage is attributed to this maximum-impulse mechanism, the authors should add a sensitivity analysis over leg configuration or knee angle, report planned versus achieved foot velocity in simulation, or explicitly state the validity range of Eq. (4). Without such evidence, the distance improvement cannot be directly attributed to the maximum-impulse principle.
[Section II-A and Section III] The paper equates maximizing ball impulse with maximizing hip and knee velocities, but only the hip is optimized; the knee is only 'optionally synchronized.' For a two-link leg, foot velocity at impact includes a shank contribution from knee velocity, so maximizing hip velocity alone does not in general maximize foot velocity. The manuscript does not analyze this contribution or report whether the knee was synchronized in the experiments. The title and abstract claim 'maximum impulse,' but the approach is more accurately described as maximizing hip velocity under a rigid-leg assumption.
[Section III, hardware evaluation] The hardware result is a single anecdotal kick whose distance is capped by the 5.5 m field length, so it cannot substantiate the quantitative 42% improvement claimed from simulation. The text states that 'the actual distance of the kick would have been greater' without measurement support. Please report the number of hardware trials, measured distances with a note that the field length is an upper bound, and ideally the planned versus achieved impact velocity or at least the achieved ball speed.

minor comments (6)

[Abstract and Section III] There are typos with missing spaces: '42 %further' and '135 cmtall'; these should be corrected.
[Figure 3] The tilde notation for measured trajectories is not explained in the caption; add axis labels, units, and a legend for the measured curves.
[Section II-A, Eq. (4)] The source of tau_h from 'hip actuator specifications' should be clarified: is it the peak stall torque, the continuous torque, or the torque available at the commanded speed? This affects the maximum achievable acceleration.
[Section III] The comparison would be strengthened by reporting confidence intervals or a statistical test for the 42% improvement, since only 10 trials are performed.
[Section II-E] The phrase 'on the spot in-walk kicks' is unclear; clarify whether the robot is walking in place or executing a step in place, and how the gait frequency reduction affects the ZMP stability margin.
[Section IV] The statement that 'the derivations also hold for the lateral plane' is not demonstrated; either provide the analogous equations or soften the claim.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the kick derivation is self-contained constant-acceleration kinematics with physical parameter inputs, and the experimental claim is measured against an external baseline without fitting.

full rationale

The paper's derivation chain is: impulse is maximized by maximizing foot velocity at impact (Section II), target kick angle from ball and hip geometry (Eq. 1), swing time from desired velocity and acceleration (Eq. 2), and leg acceleration from hip torque and five-mass centroidal leg inertia (Eq. 4). None of these steps defines the output in terms of the predicted distance; the acceleration alpha_k is computed from actuator specifications and a separately developed centroidal model [9], not from the measured kick distance. The four-phase trajectories are then integrated into a ZMP gait by setting step frequency to the required kick duration (Eq. 9). The only self-citations are the gait/centroidal model [7]-[9] and the robot description [10], which are independent inputs, and the baseline [4] is used as a comparison method rather than as a justificatory premise. The experimental result (7.53 m vs 5.28 m in simulation, hardware kick across 5.5 m) is an empirical measurement of a planned trajectory, with no parameter fitted to those distances. The reviewer's concern that Eq. (4) assumes a rigid leg and constant inertia is a correctness/transferability risk, not circularity: it does not make the prediction equivalent to an input. No circular step can be exhibited from the text.

Assumptions & free parameters 2 free parameters · 5 assumptions · 0 invented entities

The approach rests on standard kinematics plus several domain simplifications: planar sagittal motion, a rigid leg with constant inertia and torque, maximum foot velocity equating to maximum ball impulse, and stability of the underlying ZMP gait. The only hand-chosen parameters are the desired kicking velocity and extension angle, which are not reported. No new entities are introduced.

free parameters (2)

desired_kicking_velocity (ωk) = not reported; user-set
The method requires a target hip angular velocity at impact; the maximum-impulse claim depends on setting it as high as feasible. The paper does not report the value used in experiments or a procedure for choosing the maximum feasible value.
extension_angle (θext) = not reported
The Continue phase duration and follow-through depend on θext (Eq. 6); the value used in experiments is not reported.

assumptions (5)

standard math Constant-acceleration kinematics describe the leg swing (Eqs. 1-8).
These are standard kinematic identities; they are correct for the idealized model but not for real position-dependent acceleration.
domain assumption The kick is planar in the sagittal plane, so a single hip angle θl defines the swing.
Section II states the motion is mostly in the sagittal plane; lateral and knee degrees of freedom are neglected except for optional modifiers.
domain assumption Maximum foot velocity at impact maximizes ball impulse.
Section II opening states this; it assumes a rigid impact and neglects effective mass, spin, and foot-ball collision details.
domain assumption The hip actuator delivers constant torque τh and the leg has constant inertia Il (Eq. 4).
Both are treated as constants over the swing; real torque and inertia vary with configuration, and the knee is only optionally synchronized.
domain assumption The ZMP-based gait with built-in controllers remains stable during the kick and the step-frequency change from 2.4 Hz to 0.7 Hz.
Section III reports the robot did not fall in 10 trials, but no stability analysis is given; this is an empirical assumption.

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Cite this review

Pith. "Pith review of Maximum Impulse Approach to Soccer Kicking for Humanoid Robots." pith.science (2026). https://pith.science/paper/O3PUOYZY

@misc{pith2026241201480,
  author       = {Pith},
  title        = {Pith review of: Maximum Impulse Approach to Soccer Kicking for Humanoid Robots},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/O3PUOYZY}},
  note         = {Machine review of arXiv:2412.01480}
}

read the original abstract

We introduce an analytic method for generating a parametric and constraint-aware kick for humanoid robots. The kick is split into four phases with trajectories stemming from equations of motion with constant acceleration. To make the motion execution physically feasible, the kick duration alters the step frequency. The generated kicks seamlessly integrate within a ZMP-based gait, benefitting from the stability provided by the built-in controls. The whole approach has been evaluated in simulation and on a real NimbRo-OP2X humanoid robot.

Figures

Figures reproduced from arXiv: 2412.01480 by the authors.

**Figure 3.** Generated and measured (˜) leg motion trajectories during an [PITH_FULL_IMAGE:figures/full_fig_p002_3.png] view at source ↗

**Figure 2.** NimbRo-OP2X kicking with the presented approach. From a walking state, the calculated swing motion builds up the leg velocity. At impact, the [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗

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Reference graph

Works this paper leans on

11 extracted references · 10 canonical work pages

[1]

11em plus .33em minus .07em @technote 4000 4000 100 4000 4000 500 `\.=1000 = #1 #1 #1 0pt [0pt][0pt] #1 * \| ** #1 \@IEEEauthorblockNstyle \@IEEEauthorblockAstyle \@IEEEauthordefaulttextstyle \@IEEEauthorblockconfadjspace -0.25em \@IEEEauthorblockNtopspace 0.0ex \@IEEEauthorblockAtopspace 0.0ex \@IEEEauthorblockNinterlinespace 2.6ex \@IEEEauthorblockAinte...

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S.-J. Yi, S. McGill, and D. D. Lee, ``Improved online kick generation method for humanoid soccer robots,'' in The 8th Workshop on Humanoid Soccer Robots, IEEE-RAS International Conference on Humanoid Robots (Humanoids), 2013

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Bormann, T

F. Bormann, T. Engelke, and F.-T. Sell, ``Developing a reactive and dynamic kicking engine for humanoid robots,'' Technical report, Universit \"a t Hamburg, Tech. Rep., 2019

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Pavlichenko, G

D. Pavlichenko, G. Ficht, A. Amini, M. Hosseini, R. Memmesheimer, A. Villar-Corrales, S. M. Schulz, M. Missura, M. Bennewitz, and S. Behnke, `` RoboCup 2022 AdultSize winner NimbRo: Upgraded Perception, Capture Steps Gait and Phase-Based In-Walk Kicks ,'' in Robot World Cup XXV. 1em plus 0.5em minus 0.4em LNCS (LNAI), vol. 13561 , Springer, 2022, pp. 240--252

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Pavlichenko, G

D. Pavlichenko, G. Ficht, A. Villar-Corrales, L. Denninger, J. Brocker, T. Sinen, M. Schreiber, and S. Behnke, `` RoboCup 2023 Humanoid AdultSize Winner NimbRo: NimbRoNet3 Visual Perception and Responsive Gait with Waveform In-Walk Kicks ,'' in Robot World Cup XXVI. 1em plus 0.5em minus 0.4em LNCS (LNAI), vol. 14140 , Springer, 2023, pp. 337--349

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S. Behnke and J. St \"u ckler, ``Hierarchical reactive control for humanoid soccer robots,'' International Journal of Humanoid Robotics (IJHR), vol. 5, no. 03, pp. 375--396, 2008

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Marew, N

D. Marew, N. Perera, S. Yu, S. Roelker, and D. Kim, ``A biomechanics-inspired approach to soccer kicking for humanoid robots,'' arXiv preprint arXiv:2407.14612, 2024

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Ficht and S

G. Ficht and S. Behnke, `` D irect C entroidal C ontrol for B alanced H umanoid L ocomotion,'' in Climbing and Walking Robots Conference (CLAWAR). 1em plus 0.5em minus 0.4em Springer, 2023, pp. 242--255

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[9]

Ficht and S

G. Ficht and S. Behnke, ``Centroidal state estimation and control for hardware-constrained humanoid robots,'' in IEEE-RAS International Conference on Humanoid Robots (Humanoids), 2023

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G. Ficht and S. Behnke, ``Fast whole-body motion control of humanoid robots with inertia constraints,'' in IEEE International Conference on Robotics and Automation (ICRA), 2020, pp. 6597--6603

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Ficht, H

G. Ficht, H. Farazi, A. Brandenburger, D. Rodriguez, D. Pavlichenko, P. Allgeuer, M. Hosseini, and S. Behnke, ``Nimb R o- OP2X : Adult-sized open-source 3D printed humanoid robot,'' in IEEE-RAS International Conference on Humanoid Robots (Humanoids), 2018

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Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

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11em plus .33em minus .07em @technote 4000 4000 100 4000 4000 500 `\.=1000 = #1 #1 #1 0pt [0pt][0pt] #1 * \| ** #1 \@IEEEauthorblockNstyle \@IEEEauthorblockAstyle \@IEEEauthordefaulttextstyle \@IEEEauthorblockconfadjspace -0.25em \@IEEEauthorblockNtopspace 0.0ex \@IEEEauthorblockAtopspace 0.0ex \@IEEEauthorblockNinterlinespace 2.6ex \@IEEEauthorblockAinte...

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S.-J. Yi, S. McGill, and D. D. Lee, ``Improved online kick generation method for humanoid soccer robots,'' in The 8th Workshop on Humanoid Soccer Robots, IEEE-RAS International Conference on Humanoid Robots (Humanoids), 2013

work page 2013

[3] [3]

Bormann, T

F. Bormann, T. Engelke, and F.-T. Sell, ``Developing a reactive and dynamic kicking engine for humanoid robots,'' Technical report, Universit \"a t Hamburg, Tech. Rep., 2019

work page 2019

[4] [4]

Pavlichenko, G

D. Pavlichenko, G. Ficht, A. Amini, M. Hosseini, R. Memmesheimer, A. Villar-Corrales, S. M. Schulz, M. Missura, M. Bennewitz, and S. Behnke, `` RoboCup 2022 AdultSize winner NimbRo: Upgraded Perception, Capture Steps Gait and Phase-Based In-Walk Kicks ,'' in Robot World Cup XXV. 1em plus 0.5em minus 0.4em LNCS (LNAI), vol. 13561 , Springer, 2022, pp. 240--252

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Pavlichenko, G

D. Pavlichenko, G. Ficht, A. Villar-Corrales, L. Denninger, J. Brocker, T. Sinen, M. Schreiber, and S. Behnke, `` RoboCup 2023 Humanoid AdultSize Winner NimbRo: NimbRoNet3 Visual Perception and Responsive Gait with Waveform In-Walk Kicks ,'' in Robot World Cup XXVI. 1em plus 0.5em minus 0.4em LNCS (LNAI), vol. 14140 , Springer, 2023, pp. 337--349

work page 2023

[6] [6]

Behnke and J

S. Behnke and J. St \"u ckler, ``Hierarchical reactive control for humanoid soccer robots,'' International Journal of Humanoid Robotics (IJHR), vol. 5, no. 03, pp. 375--396, 2008

work page 2008

[7] [7]

Marew, N

D. Marew, N. Perera, S. Yu, S. Roelker, and D. Kim, ``A biomechanics-inspired approach to soccer kicking for humanoid robots,'' arXiv preprint arXiv:2407.14612, 2024

arXiv 2024

[8] [8]

Ficht and S

G. Ficht and S. Behnke, `` D irect C entroidal C ontrol for B alanced H umanoid L ocomotion,'' in Climbing and Walking Robots Conference (CLAWAR). 1em plus 0.5em minus 0.4em Springer, 2023, pp. 242--255

work page 2023