# fighting and team sports advanced bio mechanics

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8/14/2019 Fighting and Team Sports Advanced Bio Mechanics

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Keywords: sport biomechanics, stochastic modelling, Brownian motion, complexmany-particle systems, dual sports, self organization

About seventy years after the bizarre observations of the botanist Robert Brown in

1827, scientists such as Bachelier, Einstein, Perrin, Langevin, Kolmogorov,

Chapman, Wiener, Fokker and Planck approached the Diffusion Problem as aBrownian Motion of one or many particles, both from a deterministic and a

probabilistic point of view. Contemporary, at the end of the eighteen century, the

Chaos theory is born with some great initial ideas, concepts and results, of an

astonishing French mathematician Henry Poincar (a thesis supervisor of Louis

Bachelier), with the developments of Julia, Ljupanov, Hausdrorff, Serpinsky and

many other mathematicians. A century later scientists as Mandelbrot, introduced theconcept of Fractal geometry and Lorenz, the concept of chaos connected to the

sensitivity of solution equations. Later, Feigenbaum approached one of the most

beautiful and important themes in these arguments, the route from order into chaos,

orFeingbaumm universality. Fifty years ago studying scaling factor possibility and

self similarity, Harnold Edwin Hurst introduced the exponent H (0.5 for ordinary

Brownian Motion, 0.2 or 0.8 for Fractional Brownian Motion) connecting for the

first time Fractals and Brownian (Fractional) Motion. In modern times Elbing and

Schweitzer in Germany studied the so called Active Brownian motion, analyzed by

the Russian school of Klimontovich, while in Hungary, Helbing, Farkas and Vicksek

analyzed the people panic escape. In the 1990s the field of Sport Biomechanics the

first study of competition of judo as dual situation sport was approached, while in

1997 a ultimate study of the contest for all the dual fighting sport was done.

This paper presents the extension of a general model for all the situation sportsboth team and dual. The interesting thing is that the motion of the centre of mass for

the couple of athletes in fighting sports competition is very well modelled as a

classical Brownian motion, while the motion in team sports competition is modelled

by a more general class of Brownian motions such as active Brownian motions. It is

possible to classify the team sport competitions as continuous in time or as a cyclic

Markov system. The objective of the present study is to present a computational

Computational Biomechanics, Stochastic Motion

and Team Sports

E. Grimpampi1, A. Pasculli2 and A. Sacripanti3,4

1,3 Facolt di Medicina e Chirurgia,

University of Rome Tor Vergata, Italy2Facolt di Scienze MM.FF.NN.,

University G. DAnnunzio, Chieti.Pescara, Italy4Dipartimento Tecnologie della Fisica e Nuovi Materiali (FIM),

ENEA- Italy

8/14/2019 Fighting and Team Sports Advanced Bio Mechanics

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model of the motion of a single athlete in a team and to compare the resulting

trajectory with experimental data obtained in the field during competitions by match

analysis software. To this purpose, some results related to a paths ensemble of a

single player are discussed. It is assumed that the player has many interactions dueto tackles, strategy changing, adversary contact and so on. Between each interaction

it is assumed that he follows a straight line and his motion is characterized by

viscous, pushing and pedestrian like force. A random force (Langevin force) is

supposed to influence only the trajectory direction after each interaction.

Furthermore it is assumed that the time step between each interaction is a random

variable belonging to a Gaussian distribution. Consequently an average direction

along which the player moves is selected and other reasonable assumptions are made

in order to build an objective function. The main criteria is a selection of a

function correlated to the strategy of the player, around which, in a necessarily

randomly way, a tactic function should be added. The strategy depends on the

players role: for the numerical simulations in this paper, a forward player was

selected, with the average target to score. So it is straightforward to assume that

the line direction joining the player position and a point related to the goal, would be

the main strategy objective function around which a random angle rand,

expression of the tactic objective function, influencing the direction selected bythe player until a next interaction could be introduced. The random variable rand is

assumed to be given by a Gaussian distribution as well. For all the numerical

simulations discussed in this paper the intrinsic generation of random numbers by

the Fortran 97 Compilator is considered. To introduce an equivalent force, due

essentially to tactic player reasoning, it is assumed that at each point of the field a

different variance (x,y), of the rand is associated. This means that a player tacticaction, specific to the area in which he is located, is considered as a random

perturbation, superimposed to the player strategic reasoning. Thus an average angle

m (x,y), function of the position as well, could be interpreted as the strategic

objective, while a variable term is the tactics action associated to each point. The

comparison between both single and multiple experimental paths and figures

obtained by the numerical methodology proposed in this paper are very interesting,

showing common Brownian path structures.

References[1] A. Sacripanti, Breve dissertazione su di un sistema fisico complesso

ENEA-RT-INN/24, 1997.

[2] D. Selmeczi et al., Brownian motion after Einstein and Smoluchowski:

Some new applications and new experiments., Acta Physica Polonica B, 38

(8), 24072431, 2007.

[3] G.E.P. Box, M.E. Muller, A note on generation of random normal deviates,

Ann. Math. Statist., 29, 610-611, 1958.

W. Ebeling, Active Brownian motion of pairs and swarms of particles, Acta

Physica Polonica B, 38 (5), 165716

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Abstract

In this paper we put the basis for a mathematical theory of competition in situation

sports, such as dual sports and team sports. It is shown that in dual contest sports,

the motion of the centre of mass of a couple of athletes is well described, with a

good approximation, by Classical Brownian Motion. In contrast, the problem of the

motion in team sports, like soccer, football, basketball, water polo, and so on, seems

more complex and it would be better to be modelled by a special class of Brownian

Motion, the well known Active Brownian Motion, with internal energy depot. In this

paper a special equation is proposed for the first time, describing the athletes motion

in team sports game and a numerical simulation of the trajectories. The motionpaths, obtained from the computational approach, are validated using experimental

data of actual games, obtained from motion analysis systems.

Keywords: sport biomechanics, stochastic modelling, Brownian motion, complex

many-particle systems, dual sports, self organization

1 Introduction

It is well known that the evolution of the self organizing complex organic systems is

described by their non linear evolution in time.

If we observe at microscopic and mesoscopic scale, with specific attention to thehuman body, we can find that all the inside physiological self organized complex

structures such as the DNA, the coronary artery tree, the Purkinjie cells in

cerebellum, the small intestine and others, exhibit the property of self-affinity, that is

the natural form of the well known geometrical property of self similarity.

Self similarity is a well known property of fractals structures, and we can find it

in the whole human body, in their static, kinematics and dynamics forms [1].

The connection among these different aspects of the human body as a complex

system is the generalized Brownian Motion in its every known formulation: classic,

fractional, active and so on.

Computational Biomechanics, Stochastic Motion

and Team Sports

E. Grimpampi1, A. Pasculli2 and A. Sacripanti3,41,3 Facolt di Medicina e Chirurgia,

University of Rome Tor Vergata, Italy2Facolt di Scienze MM.FF.NN.,

University G. DAnnunzio, Chieti.Pescara, Italy4Dipartimento Tecnologie della Fisica e Nuovi Materiali (FIM),

ENEA- Italy

8/14/2019 Fighting and Team Sports Advanced Bio Mechanics

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It can be shown that starting from fractals and finishing with multifractal aspects

of the human physiological complex systems or response, Brownian Dynamics is

one of the basic modelling of biological systems.

But if we study in deep the evolution of the macroscopic complex systems in timeconnected to the human body, starting from the motion of centre of mass in standing

still position to gait, the Brownian Motion shows its ubiquitous presence in the

description of these phenomena, as in the case of the Fractional Langevin equation

which describes the variability of the stride interval during walking [2, 3].

More surprising, if we study the time evolution of macroscopic self organizing

complex systems consisting in more than one human bodies, once more Brownian

Dynamics are present.

The above mentioned applications of Brownian Motion are well known in the

case of