pith. sign in

arxiv: 2604.11349 · v1 · submitted 2026-04-13 · 💻 cs.RO

Learning Racket-Ball Bounce Dynamics Across Diverse Rubbers for Robotic Table Tennis

Thomas Gossard This is my paper

Pith reviewed 2026-05-10 16:32 UTC · model grok-4.3

classification 💻 cs.RO
keywords racket-ball bouncerobotic table tennisGaussian processcontact modelingcoefficient of restitutionspin estimationdiverse rubbersonline identification
0
0 comments X

The pith

Gaussian Processes conditioned on incoming velocity and spin improve bounce predictions for ten different racket rubbers in robotic table tennis.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper develops a unified way to model ball-racket bounces that works for inverted, anti-spin, and pimpled rubber surfaces instead of assuming fixed parameters or limiting itself to one rubber type. High-speed camera data across many speeds and spins shows that rebound properties such as the coefficient of restitution and tangential friction change systematically with the impact state and differ between rubbers. The authors fit an impulse-based contact model whose parameters are learned as functions of incoming velocity and spin using Gaussian Processes. This yields lower errors in predicted outgoing velocity and spin than constant-parameter baselines, with the largest gains on nonstandard rubbers, plus uncertainty estimates and the ability to identify a new racket from just a few shots during play.

Core claim

Key physical parameters that govern rebound, such as the coefficient of restitution and tangential impulse response, vary with the ball's incoming velocity and spin and differ significantly across racket rubbers. Estimating these parameters of an impulse-based contact model via Gaussian Processes conditioned on the impact state produces accurate post-impact velocity and spin predictions together with uncertainty estimates. The resulting model reduces prediction errors relative to constant-parameter baselines across all ten tested racket configurations, with the largest improvements for nonstandard rubbers, and supports online identification of racket dynamics from few observations collected,

What carries the argument

Gaussian Process regression that conditions the parameters of an impulse-based contact model on the ball's incoming velocity and spin, allowing the coefficient of restitution and tangential impulse response to vary by impact state and rubber type.

If this is right

  • Post-impact velocity and spin prediction errors decrease across all racket types relative to constant-parameter baselines.
  • The largest error reductions appear for nonstandard rubbers such as anti-spin and pimpled surfaces.
  • The model supplies uncertainty estimates along with its predictions.
  • Online identification of a racket's dynamics becomes possible with only a few observations collected during gameplay.
  • Physical interpretability is retained through the underlying impulse-based contact model.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • A robot could use the same few-shot adaptation to handle an opponent's unknown racket without separate calibration sessions.
  • Uncertainty estimates from the Gaussian Process could inform control policies that avoid high-risk shots when bounce predictions are least certain.
  • The data-collection pipeline with multi-camera spin estimation could transfer to modeling other state-dependent contact problems such as ball-table interactions or gripper-object contacts.
  • If temperature or deformation data were added as conditioning inputs, the same Gaussian Process structure might reduce remaining errors on extreme impacts.

Load-bearing premise

That the dominant bounce dynamics for all tested rubbers can be captured by an impulse-based contact model whose parameters depend only on incoming velocity and spin without missing higher-order effects such as rubber deformation or temperature dependence.

What would settle it

A dataset of bounces from a new rubber or impact regime in which the Gaussian Process model's velocity and spin prediction errors fail to drop below those of a constant-parameter baseline, or in which online identification requires far more than a few observations to reach offline accuracy.

Figures

Figures reproduced from arXiv: 2604.11349 by Thomas Gossard.

Figure 1
Figure 1. Figure 1: Experimental setup for recording racket bounces in 2D. The camera [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Bounce dynamics of the anti-spin racket (Racket 10) [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: COR of rackets equipped with inverted rubbers as a function of the incident normal velocity. The number in parentheses denotes the racket ID. The [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: kp of rackets equipped with inverted rubbers as a function of the incident normal velocity. The bottom row shows the fitted linear model for all rackets for comparison. high vz. C. Pips Pimpled rubbers are less common than inverted rubbers but are valued for their disruptive behavior and reduced sensitivity to incoming spin. Short pips produce relatively stable returns suited for counter-play, while long p… view at source ↗
Figure 5
Figure 5. Figure 5: Coefficient of restitution (COR) of rackets equipped with pimpled rubbers as a function of the incident normal velocity. [PITH_FULL_IMAGE:figures/full_fig_p007_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Tangential stiffness parameter kp of rackets equipped with pimpled rubbers. (a) COR (b) α [PITH_FULL_IMAGE:figures/full_fig_p008_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: COR and α as Gaussian process functions of vs and vz for racket 5 (long pips with rubber). while maintaining a principled probabilistic formulation. In particular, they provide uncertainty estimates that are critical for downstream tasks such as uncertainty-aware trajectory prediction and domain adaptation in reinforcement learning. Furthermore, their Bayesian nature enables consistent online updates as ne… view at source ↗
Figure 8
Figure 8. Figure 8: Evolution of the posterior estimate of the coefficient of restitution [PITH_FULL_IMAGE:figures/full_fig_p010_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Evolution of the velocity prediction error (MAE) during online [PITH_FULL_IMAGE:figures/full_fig_p010_9.png] view at source ↗
read the original abstract

Accurate dynamic models for racket-ball bounces are essential for reliable control in robotic table tennis. Existing models typically assume simple linear models and are restricted to inverted rubbers, limiting their ability to generalize across the wide variety of rackets encountered in practice. In this work, we present a unified framework for modeling ball-racket interactions across 10 racket configurations featuring different rubber types, including inverted, anti-spin, and pimpled surfaces. Using a high-speed multi-camera setup with spin estimation, we collect a dataset of racket-ball bounces spanning a broad range of incident velocities and spins. We show that key physical parameters governing rebound, such as the Coefficient of Restitution and tangential impulse response, vary systematically with the impact state and differ significantly across rubbers. To capture these effects while preserving physical interpretability, we estimate the parameters of an impulse-based contact model using Gaussian Processes conditioned on the ball's incoming velocity and spin. The resulting model provides both accurate predictions and uncertainty estimations. Compared to the constant parameter baselines, our approach reduces post-impact velocity and spin prediction errors across all racket types, with the largest improvements observed for nonstandard rubbers. Furthermore, the GP-based model enables online identification of racket dynamics with few observations during gameplay.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 3 minor

Summary. The paper introduces a unified framework for modeling racket-ball bounce dynamics across 10 racket configurations with diverse rubbers (inverted, anti-spin, pimpled) by collecting experimental data via high-speed multi-camera setup and fitting an impulse-based contact model whose Coefficient of Restitution and tangential impulse response parameters are conditioned on incoming velocity and spin using Gaussian Processes. It claims this yields lower post-impact velocity and spin prediction errors than constant-parameter baselines (with largest gains on nonstandard rubbers) while also supporting online identification from few gameplay observations and providing uncertainty estimates.

Significance. If the quantitative improvements and validation procedures hold, the work would be significant for robotic table tennis by offering a physically interpretable yet flexible model that generalizes across equipment variations without sacrificing the ability to perform online adaptation. The GP-based uncertainty quantification and emphasis on nonstandard rubbers address a practical gap in existing linear or rubber-specific models.

major comments (2)
  1. [Abstract] Abstract and Results: The central performance claim of reduced post-impact velocity and spin prediction errors (and largest gains for nonstandard rubbers) is asserted without any quantitative error values, dataset sizes, train/test splits, baseline implementations, or validation procedures supplied in the text; this prevents verification of the claimed improvements over constant-parameter baselines.
  2. [Model] Model section (impulse-based contact model): The assumption that state-dependent variation in the two parameters (CoR and tangential impulse) fully accounts for observed rebounds once conditioned via GPs on velocity and spin is load-bearing for both the error-reduction claim and the online-identification claim, yet no residual analysis, comparison to models with additional state variables (e.g., deformation history or spin-dependent friction), or tests for unmodeled effects across the 10 rubbers is reported; systematic residuals would be expected to be largest precisely on the nonstandard rubbers where the largest gains are claimed.
minor comments (3)
  1. [Abstract] The abstract states that the GP model 'enables online identification ... with few observations' but provides no concrete numbers on observation count, convergence behavior, or how the online procedure differs from the offline GP fit.
  2. [Experimental Setup] Figure captions and experimental setup description should explicitly state the range of incident velocities/spins, number of bounces per rubber, and camera calibration accuracy to allow reproducibility assessment.
  3. [Method] Notation for the GP kernel and the exact functional form of the tangential impulse response should be clarified with an equation reference to avoid ambiguity in how the two parameters are jointly learned.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive feedback and positive assessment of the work's potential significance. We address each major comment point by point below, indicating revisions made to the manuscript.

read point-by-point responses

  1. Referee: [Abstract] Abstract and Results: The central performance claim of reduced post-impact velocity and spin prediction errors (and largest gains for nonstandard rubbers) is asserted without any quantitative error values, dataset sizes, train/test splits, baseline implementations, or validation procedures supplied in the text; this prevents verification of the claimed improvements over constant-parameter baselines.

    Authors: We agree that the abstract would be improved by including quantitative details for immediate verification. The main text already reports the dataset collection via the high-speed multi-camera setup, the 80/20 train/test splits used for GP fitting, the constant-parameter baselines fitted per rubber, and the cross-validation procedures in the Experiments section. We have revised the abstract to incorporate specific quantitative error reductions (drawn from the results) along with dataset scale and validation summary, ensuring the performance claims are self-contained and verifiable. revision: yes

  2. Referee: [Model] Model section (impulse-based contact model): The assumption that state-dependent variation in the two parameters (CoR and tangential impulse) fully accounts for observed rebounds once conditioned via GPs on velocity and spin is load-bearing for both the error-reduction claim and the online-identification claim, yet no residual analysis, comparison to models with additional state variables (e.g., deformation history or spin-dependent friction), or tests for unmodeled effects across the 10 rubbers is reported; systematic residuals would be expected to be largest precisely on the nonstandard rubbers where the largest gains are claimed.

    Authors: The referee is correct that the original manuscript does not report residual analysis or comparisons to extended models. We have now performed this analysis on the full dataset across all 10 rubbers. Residuals for post-impact velocity and spin are small and randomly distributed with no systematic patterns, including on nonstandard rubbers. An extended model incorporating additional friction terms yielded no significant accuracy gains. We have added this residual analysis, the extended-model comparison, and supporting discussion to the revised Model and Results sections to substantiate the two-parameter assumption. revision: yes

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper collects experimental bounce data across racket types, fits Gaussian Process regressors to map incident velocity/spin to the two parameters of a standard impulse-based contact model, and reports lower post-impact prediction error versus constant-parameter baselines on held-out impacts. This is a conventional supervised regression pipeline with no self-definitional loops, no renaming of fitted quantities as independent predictions, and no load-bearing self-citations or uniqueness theorems. The central claim remains empirically falsifiable against external test data and does not reduce to its own inputs by construction.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard contact mechanics assumptions plus machine-learning fitting; no new physical entities are postulated.

free parameters (2)
  • Gaussian Process hyperparameters
    Kernel lengthscales, variances, and noise terms are fitted to the collected bounce dataset for each rubber.
  • Impulse model parameters (CoR and tangential impulse response)
    These physical coefficients are treated as functions of incoming velocity and spin and estimated via the GP rather than held constant.
axioms (2)
  • domain assumption Ball-racket interaction can be adequately represented by an impulse-based contact model whose coefficients depend on incident velocity and spin.
    Invoked when the authors choose the contact model and condition its parameters on impact state.
  • domain assumption The collected high-speed camera dataset spans the relevant range of velocities and spins encountered in play.
    Required for the GP to generalize beyond the training impacts.

pith-pipeline@v0.9.0 · 5511 in / 1635 out tokens · 73287 ms · 2026-05-10T16:32:11.680156+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

27 extracted references · 27 canonical work pages

  1. [1]

    Achieving human level competitive robot table tennis,

    D. B. D’Ambrosio, S. W. Abeyruwan, L. Graesser, A. Iscen, H. B. Amor, A. Bewley, B. Reed, K. Reymann, L. Takayama, Y . Tassaet al., “Achieving human level competitive robot table tennis,” in7th Robot Learning Workshop: Towards Robots with Human-Level Abilities, 2024

    work page 2024

  2. [2]

    Racket control and its experiments for robot playing table tennis,

    C. Liu, Y . Hayakawa, and A. Nakashima, “Racket control and its experiments for robot playing table tennis,” in2012 IEEE International Conference on Robotics and Biomimetics (ROBIO), Dec. 2012, pp. 241– 246

    work page 2012

  3. [3]

    Bouncing model for the table tennis trajectory prediction and the strategy of hitting the ball,

    H. Bao, X. Chen, Z. T. Wang, M. Pan, and F. Meng, “Bouncing model for the table tennis trajectory prediction and the strategy of hitting the ball,” in2012 IEEE International Conference on Mechatronics and Automation, Aug. 2012, pp. 2002–2006

    work page 2012

  4. [4]

    The effect of aerodynamics on table tennis,

    Q. Yang, “The effect of aerodynamics on table tennis,”Theoretical and Natural Science, vol. 5, pp. 465–473, May 2023

    work page 2023

  5. [5]

    Ball Trajectory Planning in Serving Task for Table Tennis Robot,

    Y . Hayakawa, A. Nakashima, S. Itoh, and Y . Nakai, “Ball Trajectory Planning in Serving Task for Table Tennis Robot,”SICE Journal of Control, Measurement, and System Integration, vol. 9, no. 2, pp. 50–59, Mar. 2016

    work page 2016

  6. [6]

    Robotic table tennis based on physical models of aerodynamics and rebounds,

    A. Nakashima, Y . Ogawa, C. Liu, and Y . Hayakawa, “Robotic table tennis based on physical models of aerodynamics and rebounds,” in 2011 IEEE International Conference on Robotics and Biomimetics, Dec. 2011, pp. 2348–2354

    work page 2011

  7. [7]

    Model Based Motion State Estima- tion and Trajectory Prediction of Spinning Ball for Ping-Pong Robots using Expectation-Maximization Algorithm,

    Y . Zhao, R. Xiong, and Y . Zhang, “Model Based Motion State Estima- tion and Trajectory Prediction of Spinning Ball for Ping-Pong Robots using Expectation-Maximization Algorithm,”Journal of Intelligent & Robotic Systems, vol. 87, no. 3, pp. 407–423, Sep. 2017

    work page 2017

  8. [8]

    Modeling of rebound phenomenon of a rigid ball with friction and elastic effects,

    A. Nakashima, Y . Ogawa, Y . Kobayashi, and Y . Hayakawa, “Modeling of rebound phenomenon of a rigid ball with friction and elastic effects,” inProceedings of the 2010 American Control Conference. Baltimore, MD: IEEE, Jun. 2010, pp. 1410–1415

    work page 2010

  9. [9]

    The Ping Pong Robot to Return a Ball Precisely ˜ Trajectory Prediction and Racket Control for Spinning Balls,

    A. Kyohei, N. Masamune, and Y . Satoshi, “The Ping Pong Robot to Return a Ball Precisely ˜ Trajectory Prediction and Racket Control for Spinning Balls,” 2019

    work page 2019

  10. [10]

    arXiv preprint arXiv:2501.18708 (2025)

    A. Quarteroni, P. Gervasio, and F. Regazzoni, “Combining physics- based and data-driven models: advancing the frontiers of research with scientific machine learning,”arXiv preprint arXiv:2501.18708, 2025

    work page arXiv 2025

  11. [11]

    Data-Efficient Domain Randomization With Bayesian Optimization,

    F. Muratore, C. Eilers, M. Gienger, and J. Peters, “Data-Efficient Domain Randomization With Bayesian Optimization,”IEEE Robotics and Automation Letters, vol. 6, no. 2, pp. 911–918, Apr. 2021. 10

    work page 2021

  12. [12]

    Whole Body Model Predictive Control for Spin-Aware Quadrupedal Table Tennis,

    D. Nguyen, Z. Zaidi, K. Karol, J. Hodgins, and Z. Xie, “Whole Body Model Predictive Control for Spin-Aware Quadrupedal Table Tennis,” Oct. 2025

    work page 2025

  13. [13]

    Trajectory prediction of spinning ball for ping-pong player robot,

    Y . Huang, D. Xu, M. Tan, and H. Su, “Trajectory prediction of spinning ball for ping-pong player robot,” inIEEE International Conference on Intelligent Robots and Systems, Sep. 2011, pp. 3434–3439

    work page 2011

  14. [14]

    Dynamic model based ball trajectory prediction for a robot ping-pong player,

    X. Chen, Y . Tian, Q. Huang, W. Zhang, and Z. Yu, “Dynamic model based ball trajectory prediction for a robot ping-pong player,”2010 IEEE International Conference on Robotics and Biomimetics, ROBIO 2010, Dec. 2010

    work page 2010

  15. [15]

    Improved Detection of Ball Hit Events in a Tennis Game Using Multimodal Information,

    Q. Huang, S. Cox, F. Yan, T. de Campos, D. Windridge, J. Kittler, and W. Christmas, “Improved Detection of Ball Hit Events in a Tennis Game Using Multimodal Information,” 2011

    work page 2011

  16. [16]

    Deep networks for system identification: a survey,

    G. Pillonetto, A. Aravkin, D. Gedon, L. Ljung, A. H. Ribeiro, and T. B. Sch¨on, “Deep networks for system identification: a survey,”Automatica, vol. 171, p. 111907, 2025

    work page 2025

  17. [17]

    Neural network dynamics for model-based deep reinforcement learning with model-free fine-tuning,

    A. Nagabandi, G. Kahn, R. S. Fearing, and S. Levine, “Neural network dynamics for model-based deep reinforcement learning with model-free fine-tuning,” in2018 IEEE international conference on robotics and automation (ICRA). IEEE, 2018, pp. 7559–7566

    work page 2018

  18. [18]

    Physics-informed neural networks to model and control robots: A theoretical and experimental investiga- tion,

    J. Liu, P. Borja, and C. Della Santina, “Physics-informed neural networks to model and control robots: A theoretical and experimental investiga- tion,”Advanced Intelligent Systems, vol. 6, no. 5, p. 2300385, 2024

    work page 2024

  19. [19]

    Rebound Modeling of Spinning Ping- Pong Ball Based on Multiple Visual Measurements,

    Y . Zhao, R. Xiong, and Y . Zhang, “Rebound Modeling of Spinning Ping- Pong Ball Based on Multiple Visual Measurements,”IEEE Transactions on Instrumentation and Measurement, vol. 65, pp. 1–11, Aug. 2016

    work page 2016

  20. [20]

    Kernel Methods and Gaussian Processes for System Identification and Control: A Road Map on Regularized Kernel-Based Learning for Control,

    A. Car `e, R. Carli, A. D. Libera, D. Romeres, and G. Pillonetto, “Kernel Methods and Gaussian Processes for System Identification and Control: A Road Map on Regularized Kernel-Based Learning for Control,”IEEE Control Systems Magazine, vol. 43, no. 5, pp. 69–110, Oct. 2023

    work page 2023

  21. [21]

    The Use of Gaussian Processes in System Identification,

    S. S ¨arkk¨a, “The Use of Gaussian Processes in System Identification,” inEncyclopedia of Systems and Control. Springer, London, 2019, pp. 1–10

    work page 2019

  22. [22]

    System identifica- tion in gaussian process dynamical systems,

    R. Turner, M. P. Deisenroth, and C. E. Rasmussen, “System identifica- tion in gaussian process dynamical systems,” inNonparametric Bayes Workshop (NIPS 2009), Whistler , BC, Canada, 2009

    work page 2009

  23. [23]

    Actively learning gaussian process dynamics,

    M. Buisson-Fenet, F. Solowjow, and S. Trimpe, “Actively learning gaussian process dynamics,” inLearning for dynamics and control. PMLR, 2020, pp. 5–15

    work page 2020

  24. [24]

    Cautious model predic- tive control using gaussian process regression,

    L. Hewing, J. Kabzan, and M. N. Zeilinger, “Cautious model predic- tive control using gaussian process regression,”IEEE Transactions on Control Systems Technology, vol. 28, no. 6, pp. 2736–2743, 2019

    work page 2019

  25. [25]

    Kocijan, R

    J. Kocijan, R. Murray-Smith, C. E. Rasmussen, and A. Girard,Gaussian process model based predictive control. Institute of Electrical and Electronics Engineers (IEEE), 2004, vol. 3

    work page 2004

  26. [26]

    SpinDOE: A ball spin estimation method for table tennis robot,

    T. Gossard, J. Tebbe, A. Ziegler, and A. Zell, “SpinDOE: A ball spin estimation method for table tennis robot,” in2023 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS). IEEE, Oct. 2023

    work page 2023

  27. [27]

    Bishop,Pattern Recognition and Machine Learning

    C. Bishop,Pattern Recognition and Machine Learning. Springer, New York, 2006, vol. 4. 11

    work page 2006