Advances in Judo Biomechanics research

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Table of Contents
Aknowledgments pag 8
Foreword
By Dr Matteo Pellicone
President of the Italian Judo Federation FIJLKAM pag 9
Foreword
By Mr Sergey Soloveychik
President of the European JudoUnion EJU pag 10
Foreword
By Mr Marius L. Vizer
President of the International Judo Federation IJF pag 11
Introduction to the English Edition 2009
“Very strong roots for a big tree”. pag 12
Introduction to the Italian Edition 1988
Zen, Physics and Judo. pag13
Part One
Biomechanics for modern Sport
Chapter 1 Biomechanics
1.1 Classical Biomechanics pag 17
1.1.1 Biomechanical Athlete pag 18
1.1.2 Warm up and Biomechanics of Muscular System pag 19
1.1.3 Deformation of Biomechanical Athlete, Energy, and Fatigue
1.1.4 Biomechanics of Motor Actions
1.1.5 Human Body Equilibrium and Stability
1.1.6 Human Body Centre of Mass
1.1.7 On Site Body’s Rotation (Tai Sabaki)
1.1.8 Locomotion (Ayumi Ashi, Tsugi Ashi )
1.1.9 Reflexes and Motor Control
1.1.10 Biomechanical Classification of Sports
1.1.11 How to find a good Judoka? With a multiregression equation!
1.2 Advances in Biomechanics
1.2.1 Astonishing Information on Muscular Contraction
1.2.2 Fractals in Heart Rate
1.2.3 Fractals in Breathing Patterns
1.2.4 Multifractals in Human Gait
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Part Two
Judo
Chapter 2 Three basic Judo Principles
2.1 Straight line unbalances (Happo No Kuzushi)
2.2 Abdominal Energy Exploitation (Hara gei)
2.3 Placement and basic grips (Shizen on Tai; Kihon Kumi Kata)
Chapter 3 Three advanced Judo Principles
3.1 Static and Dynamic Rotational Unbalances (Tai Sabaki)
3.2 Initiative as exploitation of kinetic energy and angular momentum
(Sen [ Renzoku and Renraku Waza])
(Go no Sen [Bogyo Waza])
(Sen no Sen [Kaeshi Waza])
The Russian approach to Initiative
3.3 Relative Range, Grips, and Timing (Mai Ai, Kumi Kata, Kobo Ichi)
The Japanese way
The Russian way
Chapter 4 Biomechanical principles of Judo Training
4.1 Conditioning: Classical linear approximation
Some special advanced methods
4.1.1 Advances in linear conditioning: software help
4.1.2 Advances in linear conditioning: Special Biomechanical Instruments
4.2 Conditioning: New Trends non linear approximation
4.3 Technical training
4.3.1 Technical teaching methods historical Analysis (Go Kyo and Others)
4.3.1.1 Cognitive classifications,
4.3.1.2 Cognitive motor lernings arrangments
4.3.1.2.1 Children area
4.3.1.2.2 Adult area
4.3.2 Technical teaching methods for elite athletes’ modern biomechanical approach
Chapter 5 Biomechanics of Throwing Techniques. (Tachi waza)
5.1 Biomechanics of Falls Control (Ukemi)
5.1.1 Non Orthodox Falling Techniques (Agonistic Ukemi)
5.1.2.1 Turnouts: In search of new way
5.2 Biomechanical classification of judo throwing Techniques
5.2.1 Biomechanical Analysis of some selected researches on Throws
a. Kinematic and Kinetic Parameters
b. Energy Cost
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Chapter 6 Biomechanics of Groundwork Techniques. (Ne Waza)
6.1 Holds Physical principle and Classification (Osae Waza)
6.2 Choking Physical Principle and Classification (Shime Waza)
6.2.1 Physiological effects.
6.3 Joint Locks Physical Principle and Classification (Kansetsu Waza)
6.3.1 Physiological effects.
Chapter 7 Competition (Shiai)
7.1 Competition Classical Approach for Coaches and Athletes
Competition: classical evaluation for Coaches
a. Energy Consumption
b. Athletes motion pattern
c. Grips (for Interaction)
Competition: classical Athletes approach
a. Study of the starting phase
b. Connecting grips with throwing
c. Tricks and New techniques
d. Shortening attack time
e. Skill techniques evolution
f. Connection Tachi Waza-Ne Waza
7.2 Initiative as Psychological tool, Strategy and Tactics
7.3 Scientific Studies on Competition a Survey
a. Competition statistical study
b. Male Athletes
c. Female vs. Male Athletes
d. Female Athletes
e. Studies on Dynamics of competition (Kalina Method)
f. Conclusion about competition scientific studies
7.4 Competition at light of Advanced Biomechanics.
a. Basic Biomechanical parameters able to obtain the most effective performance
a.1 Shifting Velocity
a.2 Attack Speed against Reaction Capability
a.3 Bodies’ Relative Positioning Management
b. Biomechanics of competition: some classical remarks.
c. Acting External forces on the Couple of Athlete System
d. Couple of Athlete System internal forces, motion analysis
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e. Interaction (Throws) Connection on Shifting Velocity
f. Athletes’ Interaction
Chapter 8 Advanced Coaching -Match Analysis in Judo
8.1 First Step in Match Analysis: Physiological data for training
8.2 Second level in Match Analysis: Technical biomechanical Improvement,
Action Invariants and Competition Invariants.
8.3 Third Step in Match Analysis: Local and Whole Fight Strategies
8.4 Main Use of Judo Match Analysis: Adversary’s scouting.
8.5 JUDO:The Next Generation (virtual training)
Bibliography
Appendices
Judo beyond Legend
Man and Judo Scientific Complexities
Appendix I
A Physical Complex System
Biomechanical theory of judo competition
Appendix II
Non Linearity in Human Body
Movement and Man at the end of the random walk
Appendix III
How Athletes shift during fights
Competitions’ Judo Patterns in Computational Biomechanics and fighting strategy indication




In fact at every shifting pace linked, in competition, to the Couple System; it is possible to apply a specific technique (cfr.Gleeson 5; Sacripanti 8.3).
It is interesting to remember that, during fight, two very different facets are connected without solution of continuity: Psychological facet and technical facet.
The “Optimum” athlete’s fighting skill (cfr. Biomeccanica del Judo ) could be defined  as a variable combination of previous facets, respect to the specific fighting situation.
Then it is possible to write the following formula:
S (p, T) = Fighting Skill;   S (p) = inner or/and outer Psychological facet; S (T) = Technical facet.
Fig.7.2.a  Fighting skill as function of psychology and technics
If we analyze the psychological facet during fight, it is necessary to review from the Technical –Psychological point of view, a very important aspect previously analyzed from the Technical Biomechanical point of view ( cfr. Biomeccanica del Judo) thi...
The Initiative: in that meaning: The Psychological-Technical dominance of one component of the “Couple System” over the other component.
In the fighting dynamics the Initiative is basic, if we must develop active attack plans.
It is also clear that its preservation should be connected to execution speed, muscular strength, technical skill and psychological pressure.
Sometimes it is possible to permit accordance in psychological facet to earn a gain in biomechanical initiative or vice versa.
The judo champion is a man which faces, solves and masters all the competition aspects.
The aspects of every competition can be grouped in three classes.
Pure psychological aspects.
Psycho-Technical aspects.
Pure technical aspects.
For example some aspects are presented in the following indicative list:
Psychological factors
Public, Referees, Coach, Opponent, Anxiety, Negative Mood, Overestimation, Underestimation, Poor Concentration.
Technical –Psychological factors
Combination techniques (chap), feints (chap), jamming grips (Kumi Kata) (chap), right exploitation of fighting time, right exploitation of penalties, active obstructionism in competition, competition right pace variation, tricking techniques (chap ), ...
Technical factors
Rotational Unbalances (Chap), Kinetic energy exploitation ( cfr), Angular Momentum exploitation (cfr), relative distance (cfr), grips ( Kumi Kata ) (cfr), shifting direction, attack direction (cfr), body weight balance, muscular conditioning, right ex...
Strategy and Tactics
Judo competition is a fight under referee regulations, between two athletes who aim a same goal: to grasp the victory.
Normally when the grips (kumi kata) are caught the fight start, with all possible attacks, defensive actions and counter blows which flow till to the final victory.
Generally the actions developed are grounded, both on the competition regulations and on the fight specific situations.
These opportunities come from the athletes’ personality and technical capability.
A thousand of competitions showed the experimental soundness of these concepts, while the biomechanical analysis shows us their scientific soundness.
Every fight has his specific way to the victory, grounded on athletes’ technical and psychical capabilities.
A good athlete understands that basic notions like: unbalance, combination techniques (Renzoku and Renraku Waza), or timeliness, are variable concepts that depend on a lot of parameters as: his own actions, opponent actions, shifting speed of the coup...
Competition Strategy is interested in coordinating the external and internal forces, or the strains balanced with the relative motions, till to catch the victory.
Strategy gives general concepts to the athletes to rule the fight till to the victory.
Tactics is based on the capability to rule and hold the transitory action.
Then we define Strategy: the plan or the flexible connection of more plans based on the coordination of physical efforts, harmonized with relative movement finalized to the fight victory instead we define Tactics: the capability to utilize the transit...
Tactics is connected to do technical actions naturally, as natural perception of the actual situation like the Zen mystic status called Satori  ( cfr Introduction of first edition).
Tactics, in our advice, can be connected not only to specific attack motion on a transitory fight situation, but also to Kaeshi Waza as solution of transitory  situation like improvise attack applied by the competitors. In the next figures we can see ...
Fig.7.2.f.g.h.i. -Tactic for throwing (Mae Mawari by Kokga) to overcome defense on Seoi (Finch)
Fig.7.2. j.k.m.n- Tactic for throwing (Mae Mawari by Ciano) to overcome defense on Tama Guruma (Zahonyi)
7.3 Scientific Studies on Competition:  a Survey
Conclusion about competition scientific studies
Naturally the historical data analysis about competition, times, techniques and kind of competitions statistically evaluated, could be source of very important information, starting from judo competition tendential evolution, until to the accurate kno...
If we only think at the statistic as a kind of photograph of the past, they will give us the past situation and nothing else.
But if the measurable output of a system is viewed as data that include a patterns and some error, a major consideration in forecasting is to identify and fit the most appropriate pattern.
The critical task in forecasting is to separate the pattern from the error (random) component so that the former can be used for forecasting.
The general procedure for estimating the pattern of a relationship is through fitting some functional form in such a way as to minimize the error component. One form of this estimation is the mean squared errors.
A major consideration in the selection of forecasting method for judo is the type of patterns in the data. These patterns may represent characteristics that repeat themselves with time, or they may represent turning points that are not periodic in nat...
In general a data series can be described as consisting of two elements the pattern and the randomness.
The objective of forecasting is to distinguish between those two elements using the forecasting method that can most appropriately do so.
It is also possible that pattern to be thought of as consisting of sub patterns, or components, each of which can be considered separately.
The components most frequently used in describing elements of pattern are normally described in scientific literature as: Trend; seasonality and cycle.
Knowledge of the type of sub-patterns included in the data can be very useful in selecting the most appropriate forecasting method.
As previously seen the data can be fractioned for team, weights, sex and more subtle fragmentations, doing so the use of data forecasting flows into scouting and spying the adversaries. (cfr Chapter 8.4)
7.3.1  Competition at light of Classical and Advanced Biomechanics.
The Study of the motion of the Couple of Athletes Systems on the mat, during competition, could be solved by the application of statistical mechanics.
The analysis of the Couple system singles out that this system is in stable equilibrium, if still it is isolated, when it goes the only force applied is ( by friction) the ground reaction force, but this force for the third principle of Newtonian mech...
Then if we analyze the system inside the force flows from the grips to the mat, and becomes attacking effective, flowing from the mat to the points of application (grips) to the adversary body.
The Couple system achieves "random" shifting by changing couple velocity direction in push/pull forces produced by Athletes themselves to generate specific "situation" in order to apply winning techniques.
In this case "random" means that statistically there is not a preferential shifting direction.
The motion can be accomplished by friction between soles and mat on the base of the III  principle of Dynamics; the general equation describing the situation is the II  Newton's Law F =ma.
In the generalized force F will appear both friction and push/pull contribution.
The friction component is proportional to the velocity Fa = -μν.
The changing in velocity and direction produced by push/pulls are created by resultant of force developed by the two Athletes themselves.
They are, with regard to the whole contest time, acting impulses in very short intervals of time.
Then the generalized force is F = Fa + F' and the general equation of the motion has the well-known structure of Langevin's Equation:
Because time average of  ; the “Couple of Athletes System” does move of Brownian Motion. (See Appendix I)
Fig.7.3.1.aa.bb.cc, Timing Japanese Application
Fig.7.3.1.dd.ee.ff.gg.hh.ii.Powerful Russian Application
Chapter 8   Advanced Coaching
Match Analysis in Judo.
In these years, with the rapid evolution of science and technology in the human life, with the growing of telematics and complementary technologies also the world of Sports undergoes slow but irreversible modifications into the way of more wide applic...
How to define Match Analysis?
Remembering the previous definition of Judo competition, we can answer: Match Analysis is the study of a clash of interests, based on the utility theory.
Match Analysis could be seen as the master key in all situation sports (dual or team) like Judo, to help in useful way the difficult task of coach or better of National or Olympic coaching equips.
The birth of Match Analysis may refer to the single Athlete performance study for simple cyclic Sports. It can be considered as extension from the “simple” before described case to two interacting athletes’ complex field, or again to more complex syst...
For many years Match Analysis in Wrestling or Judo or also in team Sports, was only some data sheets compiled by a technical observer, with special symbols, or specific information on the Athlete or on the adversaries. (see Fig…. )
Fig.8.a, Judo Notational Analysis data sheet from The Judo Textbook- Hayward Nishioka & James West; Edit. Black Belt Communication 1979 ISBN 08975500636-9780897500630.
For example into the book Modern Judo of Koblev, Rubanov and Nevzerov it is possible to find the very interesting Russian Stenographic system of fight registration.
This registration system was utilized by Russian coaches during the sixty and seventy years to collect informations on adversaries and athletes of the Soviet Uninon team.
Fig.8.b, The meanings of these symbols are the following: up to down from left to right respectively
Throws with grips to one leg, Throws with grips to two  legs, Overturn from the knees, Fore sweep, Side sweep, Internal heel sweep,  Fore trip, Back trip, Internal cross-buttock, Cross-buttock to two legs, External hook foot, Leg lever, Legs scissor, ...
Starting the computer era the first Match Analysis advanced utilizations was simply the same data sheets treated by more powerful statistical methods.
From year to year, in connection with the advance in technology: like very small and portable high speed digital camera recording, at first with the only slow motion, now equipped with specific analysis software, with the terrific increase in power an...
The most sophisticated software is based on the HHMM (Hidden Hierarchical Markov’s Models).
However normally until today, in judo world, match analysis considered as motion analysis was a powerful tool to gather data for scientific studies, mainly statistical or biomechanical ones.
A lot of studies results analyzed in this book come back from match analysis in the meaning of motion analysis.
Fig.8.c.d, Infrared camera for motion analysis and markers on the subject
From one other way, these systems are utilized to obtain very useful data for coach, like real time feedback of special action, or special data base on technical skill of his own athletes or about the adversaries; with a more defined capability to fin...
For Coaches in general it is possible to utilize the Match Analysis systems at three different useful levels:
First Level to obtain physiological data about the energy consumption in competition to specialize ever better conditioning methods for competition.
Second Level to obtain data for Technical Biomechanical improvement and Competition Invariants.
Third Level to obtain data about Local and Whole Fight Strategies.
In general performance is skill connected, but situation sports must be considered interactive processes between two opponents (dual or team).
The behaviour in dual situation Sports like Judo depends from athletes’ skill level, from the changing situations, and from random events that can happen during fight.
A correct Biomechanical model of Judo fight must be able to describe both the interaction between athletes (throwing or control techniques see Chapter 5 and 6) and the dynamic evolution on the mat (motion equation see Appendix I and III).
The two Physical-Biomechanical principles of Throwing Techniques (Chapter 5) were single out from the static position of the athletes but, because the motion on the mat is pseudo-uniform it is possible to apply the Galilean Relativity Principle to the...
Match Analysis is strictly based on video analysis both in real time and off line, in many team sports real time match analysis is performed with fruitful application, see Volley, Hockey, Football, some Soccer teams and so on, but in fighting sports, ...
Perhaps, in a next time will be possible to develop real time match analysis with the match analyst help, as support of the coach also in judo competition.
Team Sports match Analysis software utilize, for off line data base evaluation, many advanced mathematical instruments, from Data Mining algorithms to the Hierarchical Hidden Markov Models algorithms that are able to single out  automatically hidden s...
For the Judo fights these algorithms must be focalized on very specific particulars of the images, like grips, competitive invariants, direction of the techique application, or specific techniques linked together.
If we ask to the advanced mathematical methods Judo fights  must be included into the theory of adaptive complex systems, such systems are made by interactive agents ( the Athletes ) which continuously fit oneself  to the changing situation ( Strategy...
Normally dual situation sports must be analyzed, biomechanically speaking, studying both motion and interaction.
Match Analysis can be able to give sound information’s about these specific fight aspects.
The previous competition definition accounts us, which it is not possible to use the Newtonian physics to study such complex systems, like fight.
To do that, we need not only of more advanced technologies, but also of more advanced Biomechanical and Physical methodologies.
Among others we single out , Statistical Mechanics, Games Theory, Stochastic System Analysis, and Chaos Theory.
The advanced Biomechanical study of dual situation sports is able to give us a unified vision of such sports linked together by the motion class.
The Brownian class of motion (see Appendix I , II and III), in every application like classic, active, fractional and so on.
The differences must be found in the interaction phase which is different in each sport.
This chapter will deepen the most important aspect of match analysis for coaches and the present main use worldwide applied the adversaries scouting.
First level in Match Analysis:  Physiological data for Conditioning
As already said, the first goal of the match analysis is to obtain from the motion analysis data to evaluate the relative energy system contribution during judo fight.
Through this approach, more fitted conditioning programs will be able to be individualized for each athlete, prescribing ideal duration and intensity of conditioning judo activity.
The second utilization of these data is connected to the right nutrition diet because to perform a good high level performance, it must be able to activate different energy sources, (rightly restored by a correct diet) in the Athlete’s body.
Obviously if we would obtain a high level performance, we must condition the energy sources more activated during the competition.
The goal is to warrant the right energy contribution during performance.
In general for judo the most important energy source is the anaerobic system, this means to be able to mobilize many times, the sufficient energy for strong and fast contractions, in short times.
These kind of energetic sources are well known ATP and CP.
In addiction during the fight, for the athlete is also necessary to realize very strong contractions based on other sources, glycogen depletion with simultaneous increasing in lactate.
The aerobic metabolism affects different aspect of the performance.
It affects better economy of muscular contraction in fight, the fast catabolises and the fast rebuilding of performance capability both during fight and rest.
It is interesting to consider that more work for conditioning purpose is today performed by weightlifting programs, this is a more speedy conditioning system, but very often because these programs, taken by other non complex cyclic Sports, fall in the...
Perhaps, at high level, it will be more time consuming but more effective for nonlinear conditioning point of view, to use varied and right judo activity like old Japanese Dojo, with modern specific ergometers.
I agree with the very beautiful book of Mr Pulkkinen that physiological data and energy cost taken by free wrestling style fights, could be similar to judo, but only in first approximation and those data can be used only in lack of judo data, because ...
Normally it is a very hard problem, to take useful measurement of physiological parameters during competition; because it is not possible to take blood samples during official fight, but only before and after the competition.
The first difficulty arises from the high variety of very different physiological indicators that are connected to the Human body complex system.
Other difficulties arise from the non linearity of the human body responses, it should be considered as property of this complexity, the large number of internal body structures and processes which interact non-linearly each others, with the possible ...
The third difficulty is the basic linear approach that, as underlined assumption, is connected till now to the physiological indicators.
In one word it is very difficult to connect, unambiguously the physiological linear parameters to the “real state” of nonlinear human body.
Probably the best approach, for Elite Athletes, could be to have for each athlete a data base with relative energy consumption both on personal throwing and on specific ground work taken in similar fight condition. These data could be used as referrin...
For a better evaluation it must be added and considered: the mean displacement speed of couple of athletes (motion), each single attack velocity (interaction), and the stops during the fight.
All that could be evaluated by digital data obtained by the fight movie, (for the evaluation of physical parameters like speed in term of oxygen it is possible referring to the appendix I; and for the attack cost to the Franchini approximation see Ene...
Tab.8.1.a defences applied in Nage Waza ( Matsumoto & c.)
in term of Oxygen consumption few data can be collected on this argument , peraphs the most indicative one are  the old data from Matsumoto  ( see Table 5.2.1 energy cost)
In formula it is possible to write:
Normally the metabolic data for situation Sports like Judo are taken or from literature or from indirect evaluation.  However advanced technologies could be able to take direct data on mean metabolic consumption in competition. Following up, we show s...
Thermograms of a judo technique: Koshi Guruma.
The equation utilized.
The results obtained in term of Energy evolution in time diagrams.
Fig.8.1.a, Thermograms of a judo technique: Koshi Guruma
(Sacripanti equation)
S= Athlete’s body surface
Ts   = mean skin surface temperature
Ta = mean environment temperature
σ = Stefan-Boltzmann constant
ε = skin emissivity
k = thermal conductivity of air
Sp = lungs effective surface
Re = Reynolds number.
Pr = Prandtl number
l = thoracic dimension of athlete
h=athlete height
P= Athlete weight
Sc= Sherwood number
D=molecular diffusity in air
λ=latent heat of vaporization
es= skin water vapour partial pressure
ea= environment water vapour partial pressure
Ms,a=water vapour mass
R= gas constant
Tva,vs= virtual temperature of skin                            and   environment
Fig.8.1.b Energy consumption recovered by thermal emission ( Sacripanti)
8.2   Second level in Match Analysis: Technical biomechanical Improvement,    Action Invariants and Competition Invariants.
Other important data could be singled out from tape, first about biomechanical quality of athlete’s techniques and second statistics and frequency that can be useful for coach to evaluate the performance capability with some “ad hoc” indexes.
Other special indexes are able to find into the fight’s structure some special situations more frequently happened, which we call for convention “Fight Invariants” and “Competitive Invariants”.
These special situations must be studied and repeated during the specific post fight technical training. These repetitions should be utilized by the athletes for learning the better way of governing such situations ( interaction) that are the most imp...
Biomechanical Improvement and Action Invariants
With the slow motion of the fight can be understood the state of athlete’s technical preparation and to take data for his technical improvement.This aspect of Judo Match Analysis is worldwide part of the today technical training and improvement. The b...
Tobi Komi  (jumping in )
Mawarikomi (spinning in)
Hikidashi   (pulling out)
Oikomi     (dashing in)
Daki  (to hug holding)
Debana  (Thwarting the opponent)
Nidan Biki  ( two stage pull)
Ashimoki (leg grab)
Sutemi ( body drop)
All these biomechanical actions are suitable to improve the Kuzushi -Tsukuri phase in real competitive positioning. But today with the grow of scientific studies in the world, these old (already useful qualitative approach), are ousted from very updat...
Fig.8.2.a Biogesta –Saga 3D system
Fig.8.2.b Advanced biomechanical researches onSuwari Seoi by Saga system ( Poitiers Fr.)
Diag.8.2.a, Suwari Seoi “Action Invariants” identified by mechanical analysis
from: Blaise & Trilles  Comparative mechanical analysis of the same judo  throwing : Seoi Nage, realized by five experts of the Judo French Federation -  Science & Motricité n  51 —49-68 -2004/1
It is interesting that in advanced biomechanics these Actions Invariants should be traced back to the Hamilton –Lagrange Equation and to the Hamilton Action principle.
With S = the Action; and L= the Lagrangian of the system.
In fact if we consider in a first approximation constant the external energy of the system (gravitational field) it is possible to write:
According to Hamilton's principle, the true evolution of S(q,t) is an  evolution for which the action is stationary (a minimum, maximum, or a saddle point).
Normally in Judo Action it is asked for the minimum; then this is the called the principle of the minimum action.
In Judoistic terms the Action Invariants, should be recognized as the minimum path, in time, of body’s shift to acquire the best Tsukuri-Kuzhshi position for every Judo Throws.
This is possible in a conservative field, if we consider a non conservative field, it is necessary to consider for the global balance the heat Q emitted, and in this situation it is not possible to find a minimum of the action.
Then in the case where it is possible to find a minimum, the two following biomechanical principles are true:
Best is the Judo Technique, minimum is the Athletes’ energy consumption.
Best is the Judo Technique, minimum is the Athletes’ motion path.
The modern second level of match analysis today tends on deeper the study of the most important competitive aspect of throwing:  the Tsukuri-Kuzushi phase.
A lot of studies have been developed in the world to understand the best way to achieve the better relative position inside the couple of athletes systems; there are to remember for example:
Competitive Invariants
The basis of the Match Analysis is the statistical approach to competition data
The second important aspect single out by Weer and others coaches was the grip skill study.
In classical way it is sufficient to see hands position and power applied on the adversary’ body
But there is one other more subtle aspect of the fight, which was singled out in the first edition of this book; but practically this thing has been passed over in silence.
If we think deeper,  the motion on the mat , it is the results of many pushes and pulls applied by the grips, but it is not possible to apply push or pull without the contact to the mat by the feet.
Thinks for example to apply the same push-pull forces in the couple system, wearing roller-skates, then obviously it will be impossible to apply anything!
Now after that, it is understandable the meaning of the so called “Biomechanics Grips Paradox”.
What is the most important aspect of the Grips (Kumi Kata)?
Feet Position is the Grips most important aspect!
Arms position is essential only in defining the forces’ directions to throw the adversary, but without a strong support base the arms position is unimportant.
This is a new kind of fight vision, not to see simply in short way: or the arms position, or the bodies’ relative position, or the power applied to the adversary.
But the advanced way is to approach the system as whole seeing at the Couple of Athletes and not at the single athlete.
This new vision is the right biomechanical vision, or in other words the advanced modern vision of the Judo competition.
This other aspect, very important, from the strategic point of view, as seen, came from the study not of the single athlete, that coach normally perform during competition, but from the analysis of the whole system  “couple of athletes closed”.
If the competition is approached in this way many interesting aspects of the adversary, both from the strategic and technical (Throwing) point of view get out from the system observation.
The Competitive Invariants singled out by the author are the so called “Guard Position” which are the hold or grip positions that the couple of athletes closed system got during the fights. ( see Biomechanical analysis of competition)
These positions could be classified on the basis of two relative ranges: distance between the heads, and distance between feet in two main groups connected to the shifting velocity, each group could be divided in three subclasses related form left to ...
Fig.8.2.c, Six Classes of Guard Position ( Competition Invariants) related to the couple shifting velocity. (Sacripanti)
These positions are strongly connected to the preferred fighting motion pace of each athlete, which reveals a lot of technical information about the fighting preference and the special class of Throwing utilized (Tokui Waza). In particular the author ...
Fig.8.2.d.e.f.g.h.i.j,, The Six Classes of Guard Position related to the couple shifting velocity in real competition. (Finch, Zahonyi )
Then if the coach sees at the Guard Position (Competitive Invariant) and understand the pace motion of the adversary he can preview the biomechanical class of his preferred Tokui Waza.
Remembering that the biomechanical classes are connected to the shifting speed, (Chapter 3 ) or that is easier to apply Techniques of couple of forces at high shifting speed, than techniques of the physical lever, it will be easier, for example, to re...
Today with the fight evolution elite athletes are able to change guard position during fight, by the way the connection speed throws is always valid and useful to contrast the pace motion changed.
Obviously in such situation to attempt the victory means to connect all these information in a whole fight approach.
Actions Invariants, Competitive Invariants, Renraku Waza, Renzoku Waza, Kaeshi Waza, and various stances, should be carried out under the most different situations every time more fight similar, during training like Yaku Soku Geiko, or Sute Geiko, wer...
Through the constant practice afforded by specialized Randori on the previous arguments, it is possible to build a creditable approach to real competition with a complete skill that will function favourably in contest application.
8.3 Third level in Match Analysis: Local and Whole Fight Strategies.
The third level in Match analysis is pointed to the Strategic Teaching associated to judo fight.
This is the most advanced and complex coaching level, because the strategy is the last goal of teaching it could be useful to define clearly this concept.
All people knows about strategy utility the classic definition of Sun Zu, in his book the Art of War: "People should not be unfamiliar with strategy, Those who understand it will survive, Those who do not understand it will perish", but less people kn...
Then as Von Noimann teaches; Strategy could be definite as if a player begins to play with a complete plan: a plan which specifies what choices he will make in every possible situation, for every possible actual information which he may possess at tha...
Starting from this scientific definition which enables us to understand some factual aspect of strategy, we will give a more “sportive” definition of the strategy connecting also the definition of tactics more often misunderstood.
Then we define Strategy the plan or the flexible connection of more plans based on the coordination of physical efforts, harmonized with relative movement finalized to the fight victory instead we define Tactics as the capability to utilizein right w...
On the basis of these definitions it is possible to understand the deep difference between these two activities.
A strategic plan can be studied and coached in advance, and then it is possible to connect it to the rational analysis of the fight.
While tactical capability is essentially founded on instant intuition of technical action, then it is not possible to teach it, in any way (it is a special skill gift of a Champion).
Now if we consider the contribution of off line supports, they where able to contribute at study and prepare strategies at two levels of difficulties: Local and Global.
A) “Local Strategies”.
Local Strategies are founded on the study of special situations that happen in small zones of the mat surface.  Also these strategies can be divided in three sub classes: Renzoku waza, Renraku waza, and Standard Strategies. The first two are surface i...
The last one is surface dependent and technically less dependent.
The first two are already been extensively analyzed in the chapter 3, but only for resuming we speaks a bit again about renzoku and renraku waza.
These technical tactical complexes are studied from the old time in Japanese Martial Arts.
Referring to judo Renzoku and Renraku Waza are applied in the study of classical initiative (Sen) (see chapter 3).   Both these studies are a very advanced proprioceptive training, built by the old Japanese experience, with the study of Renzoku and Re...
In the next figure we can see Adams applying his special Renzoku Waza techniques: Ko Uchi Gari-Kuchiki Taoshi .
Fig.8.3.a.b.c.d.e , Renzoku Waza applied by Adams against  Doherty
Today in Elite competitions such Olympics, World Championships, and Continental Championships, the fighting level is so improved that every athlete is able to carry out an all out attack, but also the Uke’s physical and athletically capability are inc...
Fig.8.3.f.g.h.i.j.k , Renzoku applied by Inoue
However they are more difficult to apply in real competition, one last form of Renzoku as continous attack, still today, is often applied during also high level fights. In the next figure we can see the special Ken Ken Uchi Mata  form preferred by  Ya...
Fig.8.3.m.n.o.p.q.r.s.t, Renzoku applied by Yamashita
This Renzoku like movement is common legacy in Japanese fighting style, in the next figures it is possible to see a variation on the theme ( the rotation is applied without ken ken to help the sided throwing movement )
Fig.8.3.u.v.x.w, Renzoku like movement Japanese’s legacy (Zahonyi)
The other form of Renzoku is today more often applied in real contest, where the Athletes carry “all out” attack, with tenacity, in fact in real competition when the attack is started, there is no “in between” but the action will be a rather complete ...
In this optics Athletes prepare Renzoku combination, which for almost every technique can culminate with the strong help of Tori bodyweight falling down, or with a variant of the same technique, or with a special prepared technique on the same directi...
Fig.8.3. y.x.z.aa.bb.cc, Renzoku applied by Tanabe
Other combinations in front - back -side direction are possible by owaza-owaza. In the next figure we can see a front back combination (O Waza-O Waza) applied by Nomura on Yekutiel.
Fig.8.3.dd.ee.,ff.gg.hh.ii, O waza-O waza applied by Nomura
If the couple stops his motion Tori can apply until four different techniques mixed among couple of forces and physical lever in the next figure we can see four different attack performed by Angelo Parisi , all attack applied to the same fixed leg.
Fig.8.3..jj.kk.ll,mm.nn.oo.pp. Four (!) different attacks applied by Parisi
The last type of local strategies born from the analysis both: of fighting surface and competition rules. Then it is possible to deduce as example the following two local strategies:
Study of the fight in the corner.
Study of the fight along the fighting area limit.
B) “Global Strategies”
The off line analysis of the fight video could be a very useful source for coaching about Athletes, both: light and heavy weight, fighting style.
For example:
Elite light weight Athletes generally like renraku waza with different kind of attack.
Elite light weight Athletes generally like lever tekniques like kata guruma or suwari seoi
Elite medium weight Athletes generally like renraku waza with different direction of attack.
Elite heavy weight Athletes generally like to attack with one or two direct techniques.
Elite heavy weight Athletes generally like to attack with couple techniques like o uchi, ko uchi.
Elite heavy weight Athletes generally use to attack in makikomi variation.
With the Athletes’ growing fitness and technical preparedness every more techniques are applied with lateralized direction for throwing (biomechanically speaking in the side directions the human body structure is less able to defend himself than in th...
One simple example of general fight strategy is the time evolution of attacks and penalties in series of specific fight. In the next figures there is one example from Calmet “Apport des TICE dans l’observation des gestes sportifs”. 8th  JORRESCAMP  (J...
Fig. 8.3. qq.rr, Example of general competition strategy  ( Calmet)
From these and other analysis it is possible for coach to build up some global strategies like:
Study of kumi kata changing during fight.
Study of fight with different relative ranges.
Study of special tachi waza-ne waza connection.
In equal points condition, study of a right strategy founded on attack changing speed.
In leading condition, study of a right defence fighting strategy.
In losing condition, study of a right attack strategy founded on technical-psicologial pressure.
Fig.8.3.ss.tt, Evolution to dynamics O soto gari
Fig.8.3 all. Technical evolution from classic to modern
8.4 Main Use of Judo Match Analysis: Adversary’s scouting.
The match analysis can give to coaches and athletes, a lot of useful information but normally his use is only partial and underused.
The main use of the match analysis systems, at present, is only restricted to scouting opponent’s capabilities and technical way to fight.
The video playback for scouting allows understanding what technique the opponent is using and what weakness he might have.
With the analysis of a videotape of an opponent it is possible to see attack pattern and cues, such  as personal grips or stance that the adversary can give off just in the time he is about to enter into a throwing technique. The knowledge of the adve...
In Rotterdam 2009 the author presented a paper on match analysis and knowledge obtainable by shifting patterns
But this very important and complete paper was forgotten and became practically unknown.
The following figures, taken by Matsumoto and co workers, show the analysis’ results of the use of the Tatami area, performed by the Japanese experts.
Fig8.4.a  Use of the “Tatami” area in judo fight  ( taken by Matsumoto and co-workers).
It is possible to write in mathematical form:
The first term is a fractional derivative, the second is connected to the initial condition of the process, and the third is always the random ( push/pull) force acting on the COM.
In this case is important to know the mean square displacement of the point:
From this expression it is possible to understand that we are in presence of different diffusion processes, identified by the Hurst parameter H.
In particular this parameter is time independent and it describes the fractional Brownian motion with anti-correlated samples   for 0 = 0
2) The mean over time and directions of two push/pull forces product give us information about force variation in time (8):
Checking these conditions allows us to see easily that the motion of the Athletes can be described in terms of statistical mechanics as Brownian motion over an unlimited surface.
The motion equation can be solved ( with constant variation methods ) The solution states that the system velocity is directly proportional to push/pull impulse u and inversely proportional to the total mass m, then the biggest Athletes move themselve...
In these cases it is correct to evaluate only mean values of the quantities, for example the correlation function  gives us the delay time of measurable velocity variation (8)
The solution is    if we think of steady state
(t+ t’)>> (t-t’) the result is  and the delay time is   directly proportional to the Athletes' mass (8). If we put zero as the starting speed then it is possible to evaluate the kinetic energy mean value of more contests (8):  for t→ ∞. This expressio...
The C constant can be evaluated by a modified Einstein's method for the classic Brownian motion. Therefore if we think that the Athlete bio-system shows one of the lowest working efficiency or  , then (8) it is possible to write  or
From this equation it is possible to get (8) the square momentum is directly proportional to friction and to overall oxygen consumption  , while the correlation velocity function takes (8) the value   from which the speedy fluctuation is inversely pro...
From that it is possible to state that friction acts on forces and energy, but not on the velocities.
It is also possible to see the evolution of momentum correlation function, in formula:
This relation shows us very interesting information, the fluctuation of momentum and velocities has a time memory as large as retard time t* = m/μ.
This is the interval time after which the couple system heavily changes the momentum or the velocity from the previous ones.  However as it is possible to see the retard time changes with the Athletes mass, for the heavy weights, for example, it will ...
But the retard time is also inversely proportional to friction coefficient, from that if we consider a mat with zero friction the retard time becomes infinite, this means that the heavy weights can’t be able to change their position at all. This is a ...
It is important to remember that the correlation function  singles out  only mean u(t) , then it is possible to find momentum functions u(t)  very different in each detail but with same momentum correlation at different time.  The figure 2 shows us tw...
a) light weight athletes that have very few, but very strong, interactions during fight, with retard times proportional to:   .
b) Heavy weight athletes with continuous push pull interactions with retard time    .
In these two relations   is the so called collision time, the mean distance between two push pull statistically independent, which is different from the single push pull time δt.
From the experimental point of view it is easier to find the couple of athletes’ position on the mat than its momentum. After momentum correlation function integration, between 0 and t, remembering that  u=mv, it is possible to obtain a very interesti...
As it is easier to see that the mean square shift in time of couple centre of mass, is always function of energy but in one case it depends from the square time, in the other one it is linear function of time but in this case it is also friction inver...
If we study the bidimensional shifting paths on the mat (Tatami) of couple centre of mass , because the motion as we see to belongs to the class of Brownian motion, it will be valid the following relation:      it is possible to understand  the biomec...
it is the contact frequency or attack frequency which is possible to write as oxygen function by the relation:  .
Krylov developed the modelling of two hard spheres improved by Sinai to the system of two colliding discs, both this systems have been demonstrated that are chaotic.
In these demonstration was introduced a mixing parameters K=λ/r>1 that in our approximation takes the meaning:  λ= free mean path between two interaction;   r= Athletes inter range.
Then if this relation is fulfilled the motion of the Athletes’ couple centre of mass becomes chaotic.
If the Couple of Athletes system moves by Brownian motion, it will be possible to find, if not the real path, the trajectory most probable; in fact, if  f(q,t)dq  is the probability to find the couple of Athletes in the position q in the interval dq a...
Where K= - μq is the push –pull coefficient; and D is the diffusion coefficient.
Now remembering the Einstein relationship, the diffusion coefficient D can be correlated with the time evolution of centre of mass of Athletes couple.
Both in the limit of very short time interval or very long time interval, as regards to retard time, or to the square mean shift on the mat which is connected to the energy; remembering the previous relations it is possible to write: for short times  ...
The function f(q,t)  gives us the most probable shifting path in time, which is singled out by the  maximum probability points of the same function during its time evolution.
Tacking in account the well known work of Smoluchovski  on the Brownian Motion , the “ Physical   that produce the random evolution of the contest allows us to obtain the basic  probability of this Markovian  process.
Then for dual sports it is possible to obtain from the transition probability Q the solutions of conditional Probability , which give at infinite time limit the probability to find an athlete between x and x + dx at time t , in mathematical form we ca...
That gives us the solution
The experimental proof of this model can be founded in some Japanese works, on the world championship of the 1971.
VIII EXPERIMENTAL CHECK (VERIFICATION- VALIDATION)
The results about shifting trajectories study for Couple of Athletes system during contest, analysed by statistical mechanics techniques concerning "random" situations not repeatable in time, with a definite probability frequency, could appear only a ...
Fig 2  (1, 2, 7, 12)  judo contest motion patterns in 1971 Japan championships.
IX PHYSICAL PRINCIPLES AND INTERACTION TRAJECTORIES
"Couple of Athlete" closed system.
The interaction second face was solved by the author in the years 1985-1987 and led to the corollaries, about the use of forces in space, (static conditions) , with the analysis of flight paths and symmetries ( dynamic conditions ) and with the identi...
Using Galileo's principle of relativity it will be possible to extend the validity of the known results from static to dynamic conditions of contest.
1) Couple of forces techniques
For this class of techniques, we will apply the principle of simultaneous actions; then the problem of body motion in space can be simplified in the summation of a flat motion in sagittal or frontal plane, plus an eventually simplified motion in space.
So the first motion, produced by the application of the principal couple of forces, is a rotary flat motion independent of gravity force.
In terms of variational analysis the first motion flight path is obtained by the “extremals” of general function.   For r=-1 with solutions x=a- b sinθ e y = c – b cosθ in this case the "extremal" is the circle arc of radius b and centre B (a, c).
The first motion will be described by the general equation (with (z=s )
or
The second trajectory (applied in another group of the same class of techniques) is the summation of motions produced by gravity force plus a secondary couple of forces, acting in a perpendicular plane to the gravitational field.
The second flight path is, with good approximation, the parabola arc with vertex V coincident with the rotation centre B of principal couple of forces  Fig 3,4,5
Fig 3,4,5  Technique of couple of forces –Uchi Mata
2) Techniques of physical lever.
Considering the biomechanical athlete as a rigid cylinder, applying the stopping point ( fulcrum), then the starting impulse must be regarded as necessary and sufficient to perform the unbalance, that is to shift the barycentre perpendicular out from ...
So the athlete can be assimilated to a symmetric heavy top, falling down in the gravitational field.
Because the starting impulse is acting during a short time lapse, the trajectory, in a force field, is given by the solution of the variational principle :
In our case the external field is conservative, then it is possible to apply the principle of minimum action, that is:
The body will go along the path of the least transit time.
The Jacobi form of the principle of minimum action gives us other information:
The ρ parameter measures the length of the path and makes sure the body will go along a geodetic of a special symmetry. In this case it is possible to show that it is going along a spiral arc, geodetic of a cylindrical symmetry.  Fig 6,7
Fig 6,7 Technique of lever  Tai Otoshi
In fact the kinetic and potential energy will be:
( =     V=Mgl         W=0
( = t    V= Mgl cos(      W=
( =       V=0           W=
Where it is possible to write       e
The other hand ,the angular functions will be:
and
If      it is possible to write
if the angular momentum is , L=     the equations of motion will be:
If we remember that both energy and angular velocity ( are constat of motion then it will be :
With  s=cos( and remembering the values of     e     the body’s flight psth will be given by the function :
which derive from the equation:
With     and
The Initial conditions  ( =0   and      give us the solution E’= Mgl = cost.
Then the flight path between     e       is a geodetic arc of a cylindrical symmetry.
From the classical mechanics, we remember that for a helicoidal motion simultaneously both this solutions are true:
Then the Helix pitch is:
If we go to the limit, it is possible to understand whay the increase of the rotational component is equivalent to the Athletes lowering, because shortenig of the helix pitch, means to reduce the flight path of the adversary, reducing in the same time...
X PROBABILISTIC ANALYSIS OF INTERACTION
"Couple of Athlete" closed system.
In the study of the specific interaction for the Couple of Athletes closed system, the attack mechanics guarantees that there are only two solutions: to be successful or not, faced to a defence which is effective or not.
Then comes the question of what probability of success have defence and attack, and how can they be connected whit each other by the probabilistic analysis.
In the first approximation very strong direct an attack, that is without "shams" or "combinations", are independent. That means we have a series of Bernoulli tries.
By applying the binomial distribution to the attack, the probability of having 2 successes whatever results excluded Ippons, every 10 tries over the 8 possibilities of attacks in 5 directions is:
While the defence probability to have 2 successes (excluded Ippon), every 10 tries over the 6 kind of defence around himself (2π ) is:
The mathematical probability of 2 successes every 10 strong direct attacks is 2.8 % while 2 every 10 defence is 2.5 %.
So this case the probabilistic analysis shows us that in Couple of Athletes closed system, result for strong direct attack would be just a bit easier than defence.
XI CONCLUSIONS
The biomechanical analysis of contest sports competitions has given us some very important results:
a) the motion of Couple of Athletes system is a Brownian motion;
b) interaction between athletes can be subdivided into two steps: the first is common to all contest sports (shortening of mutual distance ); the second step is peculiar to each sport.
Ex. long distance sports: direct blows to sensible or conventional body points ; very short distance sport: tools for throwing the adversaries by two physical principles ;
c) the kinetic energy of the athlete depends on oxygen consumption and the athlete's efficiency;
d) The capability of changing velocity does not depend on friction; it is inversely proportional to the mass and directly dependent on oxygen consumption and efficiency;
e) the variation time of velocity is dependent on the mass and inversely proportional to friction;
f) the attack speed at contact is given by square root of double of oxygen consumption multiplied by efficiency divided by body or limb mass;
g) the measure of variation of push/pull force is related to the measure of the friction in direct proportion to oxygen consumption;
h) the flight path of thrown athlete is a geodetic of three specified symmetry or their linear composition;
i) the couples of forces techniques are independent of friction; it is possible to use them whatever the shifting velocity is;
j)the physical lever techniques depend on friction ; that means it is possible to use them only for stopping the adversary;
k)the techniques of couple of forces are energetically the best;
l) among the techniques of physical lever, the maximum arm are energetically  the best;
m) attack frequency is directly proportional to impact velocity and indirectly to mutual distance;
n) attack frequency is directly proportional to kinetic energy for time and inversely to mass and square of distance;
o) the direct attack blow has a success probability of 66% compared to defence .
XII REFERENCES
(1) Y. Matsumoto &Y. Takeuchi & R.Nakamura-Analytical studies on the contest performed at the all Japan judo championship tournament KdK Report V Tokyo Japan 1978
(2) A.S. Mikhailov & A.Yu. Loskutov- Fundation of Synergetics II ( complex patterns) Springer -Verlag Berlin 1991
(3)A. Pasculli & A.Sacripanti -Teoria biomeccanica della competizione di karate ENEA /RT in print 1996
(4) A.Pedotti - Motor coordination and neuromuscular activities in human locomotion - Biomechanics of motion Ed. Springer Verlag N.Y. USA 1980.
(5) R. Petrov- Lutte libre et lutte Greco-Romaine Ed. Fila 1984.
(6) A.Sacripanti - Biomechanical classification of judo throwing techniques ( Nage Waza). Int. Simp. of Biomechanics in Sport Athens Greece 1987.
(7) A.Sacripanti - Biomechanical classification of wrestling standing techniques. Int. Simp. of Biomechanics in Sport Bozeman USA 1988.
(8) A.Sacripanti - Valutazione del costo energetico degli sport di combattimento in<> Report 3 Teoria biomeccanica della competizione ENEA/RT/INN/ 1990/07
(9) A.Sacripanti - Biomecanica do judo competiqao aos metodos de treino nas diversas estrategia de competiqao
I simp. de ciencias de disporto aplicadas ao judo Lisbona Portougal l991
(10) A.Sacripanti - Fondamenti di biomeccanica Ed. Filpjk Roma Italia l995
(11) A.Sacripanti - Biomeccanica degli sport di combattimento Ed. Filpjk Roma Italia l996
Appendix II
Non Linearity in Human Body
Movement and Man at the end of the random walk
Movement and man at the end of the Random Walks
1 Introduction –fractals in Human body Physiologhy
2 Inside the Body
2.1 Fractal dimension, Self -Similarity, Self- Affinity.
2.2 Fractals as Geometrical Self Organization.
2.3 Gauss and Pareto Inverse Power Law.
2.4 Random Walk and its limits.
2.5 Continuum limit of Fractional Random Walks.
2.6 Time series : some example of internal body answers.
2.7 Myosin II, Brownian Ratchet and Muscular Contraction.
2.8 On the boundary
3 Outside the body
3.1 Fluctuation of Surface Body Temperature
3.2 Human Balance : Centre Of Pressure  (COP)  vs  Centre of Mass (CM)
3.3 Multifractals in Human Gait Normal and diseases.
4 From Usual Movement to Sport Movement
4.1 Multifractals in Running Training
4.2 Situation Sports
4.3 Dual Sport with Contact
4.4 Active Brownian motion
4.5 Team Sports.
5 Conclusions.
6 Bibliography
Movement and Man at the end of Random Walks
1 Introduction
Fractals in Human Body Physiology
2 Inside the Body
2.1 Fractal Dimension : Self Similarity  and Self affinity
Since the fractals occupy an intermediate position  between standard geometric subject with integer dimensions, they can be conveniently characterized by their fractal dimension
Then an important parameter describing fractal geometrical structures is the fractal dimension which generalizes the usual integer valued topological dimension., for each structure it was empirically found that the total length L  varies as a power of...
(1)
The parameter D is called fractal dimension of the curve, another way to write the equation (1) is
(2)
Where N(l)  is the minimum number of boxes with side l needed to cover the fractal curve.
There are several generalization of the  fractal dimension but it is possible find them in every fractal book.  However data are usually a sequence of real numbers ( time series ) and in this case we may have not information about detail of the proble...
Assuming that the time series is determined by an unknown deterministic dynamics it os still possible under general conditions, to reconstruct its phase space and analyze the system “Kinematics” .This is the foundation of the non linear time series an...
Fractal dimensions, as seen,  can be introduced in various distinct way, each emphasizing a different geometric aspect of the pattern.
An important property of a fractal is its self-similar nature. In other words if we magnify some fragment of such pattern ( both static geometrical  or kinematics temporal ) we would see precisely the same  structure  reproduced on a new scale. Moreov...
2.2 Fractal as Geometrical Self Organization
Gauss and Pareto Inverse Power Law
Chance in physics is connected to the concept of probability, many process in real life are random and their dynamic evolution is very difficult to understand from a deterministic point of view.
Probability theory born  to explain the outcome of games, one of the simplest games is tossing a coin where one can find head or tail.
Normally the number of proof are determined and finite ( binomial distribution) but more known is the limiting case to the infinite proof number , this analysis was performed  by Gauss .
In definite form we, today, can define the world of Gauss “ simple”  as scientific world view, in this ( linear) theory the output  is proportional to the input, the algebra is additive , the presence of simple rules yield  simple result of the proble...
In mathematical form the system evolution is defined by the following equation:
(7)
In this  Langevin like equation we can see that the fluctuation is simply additive  and the  bell  curve that define the Gauss distribution is  the well known simple inverse curve of the normal distribution:
(8)
But a more complex scientific vision is connected to a not well known great Italian scientist, except  outside the social and economic world, Pareto.
By his law , it is possible to describe a more complex scientific vision of the world, this world is non linear, in which small changes  may produce divergence in the solutions.
It is a multiplicative world , in which  simple rules yield complex results, the processes are unstable the predictability is limited  and the description of the phenomena  is both  qualitative and quantitative, the  Pareto distribution it is also an ...
In mathematical form the system evolution is defined by the following equation:
(9)
In this more complex world  fluctuation is multiplicative and the Pareto inverse power law distribution satisfies the following  form:
(10)
Random Walk and its limits
If we consider a stochastic process going on time, for example the motion of a particle which is randomly hopping backward and forward; this example is known in the scientific world as “Random Walk” .
If we would known  the probability that after n hopping the particle will be in a position m , it could be common to connect  this probability in mathematical discrete form and to write:
(11)
This discrete probability equation has two very important limits in his continuum form.
The limit of the random walk with infinitesimal, independent steps  is called Brownian Motion.
The first form is the limit that explains more carefully the dynamic of a single particle in time  and  its mathematical form is known as Langevin Equation from the French Scientist  that proposes it in 1905.
(12)
In which the first term after the equal is the dissipation suffered  by the particle and the second is the stochastic fluctuation applied on it and, if Gaussian type,  this dissipation will be zero in mean over time.
The second form, is the limit that take in account more the point of view of the global probabilistic aspect of the random process analyzed in phase space , and its mathematical form is well known as Fokker Planck Equation from the two German  Scienti...
(13)
In the next figure (showing in it also the self similarity property of this random process) we can see one example of two dimensional Random Walk or Brownian Motion of a particle, that as before described, it is view in two different conceptual way,  ...
Continuum limit of fractional random walks
An interesting way to approach complex systems derives also from a special view of the Random walks, it is based to incorporate this complexity in it introducing  memory  in the random walks through fractional differences, this generalization has the ...
If we see more carefully at the dynamic aspect of the process it is possible to write a generalized fractional Langevin equation and to introduce the Fractional Brownian Motion.
In mathematical form it is possible to write:
(14)
In which the first term is a fractional derivative, the second is connected to the initial condition of the process ,  and the third is always the random force acting  on the particle.
In this case is important to know the mean square displacement of the particle :
(15)
From this expression it is possible to understand, that we are in presence of an anomalous diffusion  processes, identified by the H parameter usually called Hurst  parameter , in particular this parameter is time independent , and it describes the fr...
This important generalization come from certain situation occurring  either in the field of turbulence ( Frisch 1999) or from Biomechanics ( Collins and De Luca 1994) where there are the needs of a more flexible model necessary both: to control locall...
Pure Brownian motion: next step is uncorrelated with previous step  H=0.5
Anti-Persistent Fractional Brownian motion: each step is negatively correlated with previous step H<0.5
Persistent Fractional Brownian motion: each step is positively correlated with previous step
H> 0.5
Time series : some example of internal body answers.
Many organs inside the human body could be controlled or analyzed by instrumentation which grip electric time series signals as answers of the specific organ.
For example Brain imaging data may generally show fractal characteristics - self-similarity, 1/f-like spectral properties ( like Pareto  inverse power law ).
Normally Self-similar or scale-invariant time series  like EEG, ECG, have 1/f-like power spectrums with these accepted classification
if a = 0, noise is white
if a = 2, noise is brown   (random walk )     (16)
if a = 3, noise is black   (Nile floods)
if 0 < a < 2, noise is pink  (J. S. Bach)
Fractional Brownian Motion has covariance parameterised by Hurst exponent 0 < H < 1
The Hurst exponent, the spectral exponent a, and the fractal (Hausdorf) dimension FD, are simply related:  2H+1 = a   or   2-H = FD  For example  classical Brownian motion has a = 2, H = 0.5 and FD = 1.5  as easily it is seen in the next figure
Wavelets are the natural basis for analysis and synthesis of fractal processes in the human body, they are used in the brain image analysis, but the same method could be applied to the heart analysis.
In the first two figures we can see, as example of diseases in white blood cells and circulatory dynamics. In the next four, respectively, it is possible to see:
One example of Severe Congestive Heart Failure, on signal of a Healthy Heart, again a Severe Congestive Heart Failure et the last example is the signal of a Cardiac Arrhythmia, Atrial Fibrillation
Another interesting properties of these signals is that  Fractal Complexity  degrades with Disease.
In the next example it is possible to see a degradation of a signal from complexity to a simplified signal index of disease.
Another hypothesis is connected to the time irreversibility of the signal; time irreversibility is greatest for healthy physiologic dynamics, which have the highest adaptability. Time irreversibility decreases with aging and disease in the first heart...
2.7  Myosin (II)  Brownian Ratchet and the Muscular Contraction
Now it experimentally well known that the model of Brownian Ratchet , give us a good explanation of the non processive motor , the Myosin II , using the thermal fluctuation and also the energy stored in the ATP  structure, can  move long the Actin  fi...
Till now two models are in competitions the Huxley and Simmons  power stroke ( lever arm) model and the Brownian ratchet.
The problem is to compute the real Myosin translation  to evaluate the correctness of the model.
The translation caused by the pivoting of the lever arm would be about 5 nm. New technologies for manipulating a single actin filaments allow to test the lever arm model  but the displacement varied considerably some report have shown myosin displacem...
However others have shown that if  myosin  is oriented correctly relative to actin filament  axis as in muscle the value increases to 10-15 nm out of model predictions.
For the ratchet  model one problem is that the scale of the motion is smaller than the Brownian motion of microneedles , because the average amplitude is between 30-40 nm .
Recently Yanagida et others  2000 , has shown , manipulating a single myosin head  and measuring  the displacements with a scanning probe. This assay allowed measurement of individual displacements of single myosin head with high resolution. The data ...
Similar substeps observed  are constant  in size with the repeat actin monomers (5.5 nm) , independent of force, and because some subsets are backward, it is more likely that a Myosin head may step along the Actin monomer repeat by biased Brownian Mot...
In the next figure we can see for the Myosin II  and V , both the models:
2.8 On the Boundary
In this years a very interesting model as Random Walk Model of  the Human Skin Permeation was presented by  Frash in 2002  in this model the skin normally made by heterogeneous  different material in different layer, was presented first as homogeneous...
Normally Effective path length  can be defined as the  thickness of a homogeneous membrane having  identical permeation properties as the skin, and the Effective diffusivity is a  diffusivity of a homogeneous membrane having  identical permeation prop...
The next figure shows the interesting results of this random walk model of the human skin
Random walk simulations for mass penetration  through SC with  logKcor_lip=0
Dcor/Dlip = 1.0/0.01
Outside the Body
As there is shown the geometric fractals and temporal fractals are widely common inside the human body, but if we remember that, also at microscopic level, Brownian motion is ubiquitous like in DNA, molecular motors for the seven myosin family, axon t...
In the following paragraphs there are show  some interesting examples.
3.1 Fluctuation of surface body temperature.
If the surface body temperature fluctuation are analyzed for example by a thermocamera we can see that the equation describing this topological situation in steady state condition,  in which the  ζ (t) is the random fluctuation of the incident absorbe...
(17)    Langevin like equation
Fluctuation in energy
(19). Fluctuation in temperature.
...
The previous equation shows that the surface temperature fluctuation of the human body is Brownian.  But the topology of the surface temperature taken by thermocamera easy shows that the superficial temperature  is a function  not only of time but als...
In the next frame the last consideration is very clear
If the body makes a motion or works in not controlled condition, then the equation considered as a whole could take this form in his free evolution captured by thermocamera:
(20)
Really speaking  the situation is  more complex because the specific heat cv  is not constant, but it is a very complicate  function at least of time, space, food , fat, .and body dimensions; some works in this field to identify the real shape of this...
In fact it is a mistake or a first approximation, to identify a constant number as specific heat for the human body that is, in reality, a complex engine with continuous production and dispersion of energy ( the metabolic heat).
In this case the function cv  better called “body’s Thermal Inertia” is not constant but a complex function with shape as lennard-Jones potential.
Also h the thermal coefficient is not really constant but is dependence is very more complex because for work like normal activity or sport movement in not controlled thermal condition, it must satisfy the following experimental  Sacripanti’s  relatio...
In the next figures we can see some thermograms of a judo technique taken by a thermocamera in a very pioneering work of the author and co –workers (1989) in which the thermal emission was connected to the oxygen consumption by means of the previous S...
3.2 Human balance Centre Of Pressure ( COP )  vs. Centre of Mass ( CM)
In static equilibrium the CM (centre of Mass) projection and the COP (centre of pressure) would lie on the same plane, on the vertical line COP would coincide as proportional model with the projection of the CM on the ground.
Both motions are similar but the COP motion is always larger than the CM projection motion.
This can be illustrated in Biomechanics using a simple model, the inverted pendulum, Winter 1998, Pedotti 1987 .for the anterior posterior balance.
The pendulum rotates around the ankle joint which we take as origin of the Cartesian system if we denote  as F the force acting on foot by force plate at the point (-ζ, η) which is the COP. The the system is described in Newtonian approximation by the...
(22)
The component Fz  is the same force as is obtained from the readings of the force transducers. For a small deviation around the vertical z axis, we may replace cosα by y/L and in the first approximation we may also set Fz= mg then the last equation wi...
(23)
After some easy manipulation and putting the equation in term of the angle π/2- α= (  we obtain:
(24)
This equation of course describes an unstable situation, the inverted pendulum topples over. However this classical procedure do not explain the Random Walks characteristics of quiet standing  coordinates of the COP  they can be explained by  the equa...
(25)
Here dB is the uncorrelated noise with zero mean
Posturogram- Random Walk of the COP coordinates.
3.3 Multifractals in Human Gait
Walking is a very complex voluntary activity, the typical pattern shown by  the stride interval time series  suggest particular neuromuscular  mechanisms that can be mathematical modelled.
The fractal nature of the stride time series of human was incorporated into a dynamical model by Hausdorff  using a stochastic model that was later extended by  Askhenazi  et others  so as to describe the analysis of the gait dynamics during aging.
The model was  essentially a random walk  on a Markov or short range correlated chain, where each node is a neural  that fires an action potential with a particular intensity when  interested by the random walker.
This mechanism  generates a fractal process with a multifractals aspects, in which the Holder time dependent exponent depends  parametrically on the range of the random walker’s step size.
The multifractal gait analysis is also used to study the fractal dynamics of body motion for patients with special aging problems or diseases, like Parkinson or post-stroke hemiplegic.
In the next figures we can see.
The variation of the time dependent Holder exponent , with the walker step size for free pace and metronome pace at different speed.
The different time series produced by free pace and metronome pace at different speed.
The relative acceleration signals and the related fractal values in post stroke  and Parkinson patients.
4  From  Usual  Movement to Sport Movement
4.1 Multifractals in Running training
The fractal nature of the physiological signals: heart and respiratory frequencies and oxygen uptake in long distance runs was compared and analyzed  by Billat et others 2001-2002-2004.
Today middle and long distance running  are characterized by  speed variability.
The statistics show that if there are considered  the last three world record on a middle or long distance running, it can be observed that velocity  varies of 5%.
In races the variation of the velocity and the choice of the optimal speed , obviously involve a complex interplay between physiological and psychological factors .
Multifractal analysis is used in biomechanics of running for classifying signals which  exhibit a rough behaviour .
This behaviour is  quantified calculating the holder exponent using multifractal analysis.
The following equations give the formula of the exponent:
( 26)
Free and constant  heart rate scaling law behaviour
Heart rate spectrum and scaling exponent
4.2 Situation Sport
In the field of Biomechanics of Sport it is interesting to classify the sport , in order to study the athletic  performance.
There are many potential classifications, for example in function of performance energy expenditure,  but  one of the more useful one’s is the biomechanical classification that is in function of the most basic movement performed during the performance.
This classification allows to single out  the most basic complex movement that must be measured by specific scientific discipline as whole or in step size.
In alternative this most basic movement could be the goal of an expert group like: Sport Physiologist, Neurologist, Biomechanics, engineering, trainers, technicians.
The aim of this classification is to allow  to find rightly what kind of specific observational approach must be applied  to solve the problem by mechanical or mathematical models both qualitative or quantitative one’s.
This classification allows us to group all the sports in four big families.
Cyclic Sports
There are all the sports in which the basic movement is  repeated continuously in time like: gait, running, marathon cycling, swimming,  etc.
Cyclic Sports
There are all the sports in which the basic movement is applied only once  in the performance, like : discus,  shot put, hammer throw,  pole vault, high jump, long jump, triple jump, ski jump,  javelin throw, etc.
Alternate-Cycling Sport
There are all the sports in which two basic movements are applied alternatively in time, like 110 hurts , 400 hurts, steepchase , golf.
Situation Sport
There are all the sports with the presence of the adversary.
These sports, can be divided in two classes (without and with contact) and each class in two sub classes dual sports and team sports.
The first dual ones are tennis and ping pong, and as team sports we can find: volleyball, beach volley.
The last dual ones, in which athletes can contact together, are: fighting sports: judo, boxing, wrestling, karate, etc. and as team sport,  soccer, basketball, football, water polo, hockey, etc.
The situation sports  are sport in which  it is not possible to find a repeatable motion pattern for each specific game- For each game the motion in it, is a random process, then there are not basic specific movement during the motion, but it is possi...
In fact the correct way to analyze such macro phenomena is to study them. in two steps:  motion and interaction with basic repeatable movements..
And we can find astonishing  that motion for each class of these sports could be associate at one of the previous Brownian Motion that we show.
In fact if we consider for each sport the motion basic pattern of a big number of games from the statistical point of view, like classical Gaussian approach, it appears easily  that the motion belongs to the classes of Brownian Motion.
4.3 Dual sport
We take in consideration the couple of athletes as a single system, then the motion of the centre of mass system is definite by  a push pull random forces. That in formulas can be  express as:
(27)
The system is isolated no external forces less the random push-pull forces then the motion equation will be a Langevin like equation, ( Sacripanti 1992 ) and it is possible to write:
(28)
Tacking in account the well known work of Smoluchovski  on the Brownian Motion , the “ Physical   that produce the random evolution of the contest allows us to obtain the basic  probability of this Markovian  process.
Then for dual sports it is possible to obtain from the transition probability Q the solutions of conditional Probability , which give at infinite time limit the probability to find an athlete between x and x + dx at time t , in mathematical form we ca...
(29)     that give us  the solution
(30)
The experimental proof of this model can be founded  in some Japanese works  , on the world championship of the 1971.
In the next figure we can see the summation of motion patterns of 1,2,7, and 12  contests of judo .it is easy to see that the random fluctuation not have  a  preferential direction over the time this means that          and the motion of the Centre of...
4.4 Active Brownian Motion
Motion in team sport is better modelized by the active Brownian motion proposed by Ebeling and Schweitzer tacking in account the oxygen uptake from the particles and its consumption from the environment.
Particle with mass m, position r, velocity v, self-propelling force  connected to energy storage depot e(t); velocity dependent friction γ(v),  external parabolic potential U(r) and noise F(t) could satisfy the following Langevin like equation that co...
This is the motion system equation for the active Brownian motion
(31)
From the original work of Ebeling et co-workers it is possible to take in account the energy depot: space-dependent take-up q(r), the internal dissipation c e(t), and also the conversion of internal energy into kinetic energy d2 e(t) v2  then the rela...
(32)
Energy depot analysis (for q(r) = q0) that are special constant conditions, gives the following result, after some calculation,  in term of friction non linear coefficient:
(33)
This data can be specialized for human people playing  in team games, as it is shown in the following pages.
4.5  Team sport
In the first Sacripanti’s model, relative to the dual situation sports, the motion of the centre mass of couple of athletes systems is a classical Brownian Motion, that means there is not a special direction in their motion patters.
In the case of team sport, the situation is completely different, there is, in mean, a special preferential direction in motion pattern and every single athlete is not in stable equilibrium as the couple of athlete’s system.
In this case, it is necessary to adopt a different model for the motion, like the active Brownian motion  proposed by Ebeling and Schweitzer., to take in account the oxygen uptake from the environment. In this special case it is possible to write for ...
If we take the hypothesis that the energy E(t) is slowly varying, the previous equation can be simplified on the basis of the following considerations:
Then it is possible to obtain the special value for the energy     namely :
the equation achieves a term  kE0v  as shown by Ebeling, the friction coefficient  in this case will be:
(34)
considering also the potential interaction  against the adversary , collision or avoidance we can present the second Sacripanti model.
On the basis of the Ebeling and Schweitzer model and the Helbing equation, the following Langevin type  equation proposed in the  Sacripanti’s second model accounts of:
motion, oxygen uptake, kinetic energy from uptake and potential mechanical interaction like collision and avoidance manoeuvres:
(35)
In compact form it is possible to write
(36)
The specific preferred direction in motion patterns of the team sports, could be modelized  by the solution model proposed by Erenfest, but with a special modification made by the author it is possible to modelize the basic probability  of this Markov...
In effect for the team sports it is possible to write the transition probability  Q  in function of the attack strategy α.
The α  parameter  can vary from 1 to 5 , with these meanings  :
1= lightning attack ; 2=  making deep passes; 3= manoeuvring ; 4= attack by horizontal passes;     5= melina
The solution of the conditional probability P are connected to the limit of mean value in time for finding the athlete between x and x + dx at time t, in formulas:
(37)
In the next four figures it is possible to see that, in spite of the preferential direction present in each motion pattern, from the statistical point of view ( summation of several motion patters from several games)  also in team games the random flu...
Conclusions
From this article it is understandable that all the self organizing complex systems, especially the biological ones, like the human body , are better described by non linear evolutions equations that show themselves in their static, kinematics and dyn...
The only connection  among these different  aspects is the generalized Brownian Motion in every known formulation : classic, fractional, active and so on.
Its  results starting from fractals till to multifractals aspects assure us, at light of our knowledge , that  Brownian dynamics is one of the basic modelling of the mathematical alphabets of  Life.
6 Bibliography
Neurophysiology linear approximation mistake
Motion of the Projection of Couple of Athlete’ loci
Bibliography
In mathematical form it is possible to write:
(2)
In which the first term is a fractional derivative, the second is connected to the initial condition of the process, and the third is always the random force acting on the COP In this case is important to know the mean square displacement of the point...
(3)
From this expression it is possible to understand, that we are in presence of an anomalous diffusion process, identified by the Hurst parameter, in particular this parameter is time independent and it describes the fractional Brownian motion with anti...
Neurophysiologist’s linear approximation mistake
Normally the subjects’ postural performance is described by many parameters linked to the COP displacements in the posturogram. The COP surface area (90% c onfidence ellipse, see Fig. 1) evaluates the subject’s performance, the smaller the surface, the...
The COP lengt h indicates the net muscular force variation which allows the evaluation of the postural control. The co-ordinates of the average position of the COP on the medio/l ateral axis or X-axis (X COP mean) and antero/posterior axis or Y -axis ...
But the mistake is connected to the surface simplified evaluation. In general the fBm surface could be a function of time; this generalization is called multifractional Brownian motion.
Then on the basis of Hurst parameter’s variation it is possible to have three types of Brownian motions. As we can seen in the Appendix II
Pure Brownian motion: next step is uncorrelated with previous step H=0.5. Brownian diffusion
Anti-Persistent Fractional Brownian motion: each step is negatively correlated with previous step H<0.5. Ipo Brownian diffusion
Persistent Fractional Brownian motion: each step is positively correlated with previous step H> 0.5 Iper Brownian diffusion
Then it is possible to have the same surface with three curve density different, inside the confidence ellipse.
Obviously the better way to identify the surface area of the COP curve is to evaluate the perimeter, but the confidence ellipse is a 2D surface while the boundary of a Brownian motion is a curve of   4/3 D.
Motion of the Projection of Couple of Athlete’ loci
In the case of a couple of athlete system after stabilized grips, the motion of the projection of the CM of couple on the mat belongs to the class of Brownian motions.
It is obviously a 2D fBm (See Appendix I) equivalent to the two fBm performed by the projections of CM of each athlete.
The fBm is a complex motion, fractal related, and shows the autosimilarity property.
(See Appendix II).
In that case, the fractional dimension D, along a single axis, is in fact related to the Hurst scaling exponent H , since D =1—H for the present case.