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Extending the kinematic theory of rapid movements with new primitives

Miguel A. Ferrer, Moises Diaz, Jose J. Quintana, Cristina Carmona-Duarte

TL;DR

The paper extends the Kinematic Theory of rapid movements by introducing the Kinematic Theory Transform (KTT), which generalizes both the trajectory between virtual targets and the velocity profile. It replaces the traditional arc-and-lognormal pairing with general parametric curves (notably clothoids) and six bell-shaped velocity models, enabling data-driven selection via a generalized framework integrated into iDeLog. Across human handwriting, animal movement, and robotic motion, the approach yields improved trajectory and velocity reconstructions, with clothoids providing notable temporal fidelity gains. This work delivers a flexible, interpretable toolkit for modeling spatiotemporal sequences in handwriting biometrics and robotics, and sets the stage for future 3D extensions and broader primitive families.

Abstract

The Kinematic Theory of rapid movements, and its associated Sigma-Lognormal, model 2D spatiotemporal trajectories. It is constructed mainly as a temporal overlap of curves between virtual target points. Specifically, it uses an arc and a lognormal as primitives for the representation of the trajectory and velocity, respectively. This paper proposes developing this model, in what we call the Kinematic Theory Transform, which establishes a mathematical framework that allows further primitives to be used. Mainly, we evaluate Euler curves to link virtual target points and Gaussian, Beta, Gamma, Double-bounded lognormal, and Generalized Extreme Value functions to model the bell-shaped velocity profile. Using these primitives, we report reconstruction results with spatiotemporal trajectories executed by human beings, animals, and anthropomorphic robots.

Extending the kinematic theory of rapid movements with new primitives

TL;DR

The paper extends the Kinematic Theory of rapid movements by introducing the Kinematic Theory Transform (KTT), which generalizes both the trajectory between virtual targets and the velocity profile. It replaces the traditional arc-and-lognormal pairing with general parametric curves (notably clothoids) and six bell-shaped velocity models, enabling data-driven selection via a generalized framework integrated into iDeLog. Across human handwriting, animal movement, and robotic motion, the approach yields improved trajectory and velocity reconstructions, with clothoids providing notable temporal fidelity gains. This work delivers a flexible, interpretable toolkit for modeling spatiotemporal sequences in handwriting biometrics and robotics, and sets the stage for future 3D extensions and broader primitive families.

Abstract

The Kinematic Theory of rapid movements, and its associated Sigma-Lognormal, model 2D spatiotemporal trajectories. It is constructed mainly as a temporal overlap of curves between virtual target points. Specifically, it uses an arc and a lognormal as primitives for the representation of the trajectory and velocity, respectively. This paper proposes developing this model, in what we call the Kinematic Theory Transform, which establishes a mathematical framework that allows further primitives to be used. Mainly, we evaluate Euler curves to link virtual target points and Gaussian, Beta, Gamma, Double-bounded lognormal, and Generalized Extreme Value functions to model the bell-shaped velocity profile. Using these primitives, we report reconstruction results with spatiotemporal trajectories executed by human beings, animals, and anthropomorphic robots.
Paper Structure (15 sections, 11 equations, 6 figures, 2 tables)

This paper contains 15 sections, 11 equations, 6 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Kinematic Theory Transform
  3. Generalizing the trajectory between virtual target points
  4. Clothoids as curves that link virtual target points
  5. Generalization of the velocity bell shaped curve
  6. Gaussian function
  7. Gamma function
  8. Beta function
  9. Lognormal function
  10. Double-bounded lognormal
  11. Generalized Extreme Value function (GEV)
  12. KTT Evaluation In Complex Movements
  13. Database collection of different movements
  14. Experimental performance and statistical study
  15. Conclusion and Discussion

Figures (6)

  • Figure 1: Example of a spatiotemporal sequence with an inflection point between $sp_0$ and $sp_1$ and just one stroke reconstructed with arcs.
  • Figure 2: Procedure to sample the generalized link between virtual target points of a primitive. Example particularized for the case of Clothoid and Lognormal.
  • Figure 3: Same example as Fig. \ref{['fig3']} but reconstructing the trajectory with clothoids
  • Figure 4: Comparison between the original velocity and the reconstructed velocity with Script Studio and iDeLog with the KTT.
  • Figure 5: P-values results of the non-parametric Mann-Whitney U-test. Comparison in terms of $SNR_t$ and $SNR_v$ between arc of circumference and clothoid across all the databases and bell-shaped functions.
  • ...and 1 more figures