Reactions in the phases of impact-friction interaction of the foot with the supporting surface

Results of the analysis of experimental data obtained during the study of the tribological (impact-elastic-friction) interaction of the "foot–bearing surface" pair on a standard dynamometric platform and using inertial sensors are presented. It is proposed to model shock pulses in the basis of generalized Hermite functions. One of the advantages of these functions is the locality property, which consists in the fact that their behavior and construction on the shock segment does not depend on the behavior and construction on other segments. Another advantage is to define an even weight function in the form of a Gaussian, i.e. in the form of a certain normal distribution, the parameters of which are used in analyzing the dynamics of bipedal walking, taking into account the longitudinal shock pulses and the moments of transverse shock pulses. The transition to dimensionless variables allows to move to canonical distributions and build models for longitudinal momentum and transverse momentum moment. It is noted that in the canonical version, the Hermite functions satisfy a certain ordinary differential equation, which provides the basis for its use in modeling the dynamics of bipedal walking, taking into account the forces of friction. The results of the study can be used in the design of support elements for ankle orthoses and bipedal robots.

1. Donatelli R.A. J. Orthop. Sports Phys. Ther., 1985, no. 3(7), pp. 91–95, DOI: 10.2519/jospt.1985.7.3.91

2. Daminov V.D., Zimina E.Yu., Rybalko N.V., Kuznetsov A.N. Robotizirovannyye tekhnologii vosstanovleniya funktsii khod'by v neyroreabilitatsii (Robotic Technologies for Restoring Walking Function in Neurorehabilitation), Moscow, 2010, 128 р. (in Russ.)

3. Xupeng Xie, Ruoyu Geng, Jun Ma, Boyu Zhou. AKF-LIO: LiDAR-Inertial Odometry with Gaussian Map by Adaptive Kalman Filter, 2025, https://doi.org/10.48550/arXiv.2503.06891.

4. Rao B., Kazemi E., Ding Y., Shila D.M., Tucker F.M., Wang L. CTIN: Robust Contextual Transformer Network for Inertial Navigation, 2021, https://doi.org/10.48550/arXiv.2112.02143.

5. Musalimov V.M., Perepelkina S.Yu., Erofeev M.A. Journal of Instrument Engineering, 2024, no. 10(67), pp. 831–836, DOI: 10.17586/0021-3454-2024-67-10-831-836. (in Russ.)

6. Musalimov V.M., Paasuke M., Gapeeva E., Ereline Ya., Erofeev M.A. Scientific and Technical Journal of Information Technologies, Mechanics and Optics, 2017, no. 6(17), pp. 1159–1166. (in Russ.)

7. Wilson S., Eberle H., Hayashi Y., Madgwick S.O.H., McGregor A., Jing X., Vaidyanathan R. Mechanical Systems and Signal Processing, 2019, vol. 130, pр. 183–200.

8. Jang S., Park S., Moon S., Kim J. and Lee S. Proc. of the 7th Intern. Conf. on Sensor Networks (SENSORNETS 2018), 2018, pр. 161–167, DOI: 10.5220/0006716601610167.

9. Beletsky V.V. Dvunogaya khod'ba: model'nyye zadachi dinamiki i upravleniya (Bipedal Walking: Model Problems of Dynamics and Control), Moscow, 1984, 288 р. (in Russ.)

10. Vukobratovich M., Frank A., Djurich D. Mechanics, 1972, no. 1(131). (in Russ.)

11. Musalimov V.M., Erofeev M.A., Monakhov Yu.S., Malov M.S. Journal of Instrument Engineering, 2023, no. 8(66), pp. 652—659, DOI: 10.17586/0021-3454-2023-66-8-652-659. (in Russ.)

12. Musalimov V.M., Erofeev M.A. Journal of Instrument Engineering, 2025, no. 8(68), pp. 689–695, DOI:10.17586/0021-3454-2025-68-8-689-695. (in Russ.)

13. Lurye A.I. Analiticheskaya mekhanika (Analytical Mechanics), Moscow, 1961. (in Russ.)

14. Suetin P.K. Klassicheskiye ortogonal'nyye mnogochleny (Classical Orthogonal Polynomials), Moscow, 1976, 328 р. (in Russ.)

15. Jahnke E., Emde F., Lösch F. Tafel noherer funktionen, Stuttgart, 1960.