MAKAROV, N.V.

SHOLUKHA, V.A.

ZINKOVSKY, A.V.

English

ISBS (International Society of Biomechanics in Sports): Conference Proceedings Archive

SOFTWARE FOR MATHEMATICAL HODEUING AND COMPUTER SIMULATION OF SPORTS MOVEMENTS AND TRAINING APPARATUS m ZINXOVSKY, A.V; HAKAROV, N.V.; SHOLUKHA, V.A. ' Politechnical institut Leningrad w USSR TO mke the analysis of interaction of a sportsw with an apparatus we ust develop tbe matheratical mdel and tben w mist pbow the realizatim of this rtbd using a computer. Then the mt hportanta sped of this problem is tbo degree of detailization of tbo mdel and, M a result, i t ' s  adequacy t o  the types of mverent mder sMy. T!M developrent of softvare for the Pdcl includes several guestions, vhich are closely connected with each otber, i. e. : selection of the mathematical d l ,  rqistration of tbe experhntal data and i t s  loading into the computer, analysis of tbe parametus of the d l  with regard to the c x p e r h t a l  data, variation of the parameters and functions with the air of obtaining mvemenb vith the given properties (characteristics), presenting of tbe results of calculation in a f a n  of diagram, tables and kinematic s c b .  Ibse  very a& deternine the structure of software, &tined for selection and analysis of the adequate rodels, detellinirq the interaction of the sprtsman and apparatus. At present it seems that the simple apprCdch is the appr0dch based on tbe use of d y n d a  for tbe systea of solids (Wittenburg J. , 1971). Let us suppose that the segments of tbe sprtsllan's body and tbe levers of the sh la to rs  (apparatus) are a l l  solids interacting via generalised forces, which represent tbe forces ard m a t s ,  deperding on parameters and tine. Fence, our prcble~ is limited to the analysis of tbe closed kinuatic din with tbe non-stationary innr  connections and external (probably non-stationari) interactions. Let us use the traditiowl approach based on the introduction of the reaction ties, and let ~lr .cite dom the equations of movetent for the ramified chain vith n-segments (in the Neirtsn-Euler fon)  . The radius-vectsr r of the fulcru is directed froa the inertioml basis to the conditional fulcra of k e  sjstem ( i t  my be a real fu lcm,  e.g. ankle joint in the phase with on? fulcm!. Ceonetrical and mwinertial characteristics (GHIC) of the system i n  i-s mbile basis mi / I i / ,  P c i ,  <c;:g~ss, lenqb psition of the centre of lnasses and the central tensor of inet?ia, respctivhly) are qlven or are calalated for the s w t s  either usinq apcropriate formulas or directly ifor the appdratus or siaulator). Note Lbt the ;roblea of recalculation of GXIC i s  qulte useful in softdare in th case of changlg ties (onlof:), a:d a t  t?e saw t i= one mst stare inf~nat ion about the system vith the mximua number of l ink as the basic in!:=:isn. me value of pdrajleter n is essentially non-linear and deternines the difficulty of tk a l g o r i * ~  of calculation (integrationof the equation of mvemts). This is why, in the phase of prxessinq o! ttie exprimntal data, as a m!e, w try to elininate 'mecessary' d q e e s  of freedon. b a value for the reciprocal mbi1it.j of the links ve recommend the use of the linear regressiowl estimation: . c : . ( , + , ~ !  - j ,  + pc. , -- , < = I , ? ?  , (1) 2- ( P- S vhere 4 ,  Q. represents the man value durinq the period of observation 1 of the value of the relative shift and speed (velocity). After calculation: ve get the measure of the relative mobility of the links. me sequence of Ute connection of seqnents in tbe mcdel ve shall determine in the structural matrix? with the dknsiom (n x n )  consisting of zeros and ones aMf bavirq a triangular form, if the nueration of tbe segments i s  arranged i n  tle direction of increasing froa a e  conditiondl fulcm. The interlinks pivot VIII Symposium ISBS -315 - Prague 1990 d i o a s  represent th highest interest tm ttm point of vieu of pa ra r t r i a t i an ,  h v e r  for dad- of the equations of m v e m t  l e t  us CO~ide]: tht in each pivot point tbcre exists an hbr-sagmt l~ ra t  l(i, th i-s body of tbe absolute m m t  gi, is detenined f r m  tbe fomla:  - - ,.T y = /y .I -C ( 3 )  T - r &re Lj -[ , h-ln.]: T i s  for transpsition. Let ri vedon connecthg tbe pivot points (or directed to the centre of rass for end seqmnt), 6 =$I? C is tbe wlum or absolutely angular velocities, tben the dyMnics equation for tk system of apparatus-sportsman i s  *re He= 5 m; - tbe to ta l  mss of the systea. R - i s  the aain vector of the external forces influencing upon tbe sfi&a, th? gravity fcrce i s  distinguished into the separate col~ponent in th? Lvton equations arid in the tom of (i igenters the equa%ors of hriler. Vectors pi represent the .in moents a c t i q  upon the i-s body of the systen, 1.e. the mrents of tbe concentrated external forces must be included into Ji, In equation ( 3 )  w use c-n designations for scdlar, vector and double products and tbe tensors are larked with tvo dots underneath, b i l e  the points above are for differentiation by tifie ( t ) .  Let us write don  the des ig~ t ions  used in (0: n E - i s  the s i q l e  tensor, * - mans conjugation, VIII Symposium ISBS Prague 1990 to evaluate the integral prolprties of the s y s b  let as write Qvn tbe upmsiora for tba total mechanical energy and for the -tic m w t  relatively t o  the inertial bash: here we use the dcterr inat io~ I = (1,. . . , I )  - id tbe mtrh - line with the length of n, consisting of ones; c - i s  the mtrix of -ion (n x n) ,  ~ w u i s t i q  of tenson and the expressions for m v m t  of the velocity and acceleration of the centre of lasses of the system are given below: vl (0 . R, = g:'. 2 +,zciicrc, = R~ = + ;kl , - - - - 1 - 1  (':I - - . % 2 .. n . (9 )  I?: = R c  = +z (i.,.rc,.-c' :5.,L.cL)-cLk: j , /p,; '  , - - - .  - [ ' f  - .  - - Let us mite dom tm mre correlations which ray be wd both for direct calculation of i t s  components and for control of result of nmerical calculations. M e  corrclationr represent tbe changes in +he muent of the inpulse and of the total energy o! the uhole system here Zit f q ; l T i s  matrix-colunn, containing relative arqular velocities. Pinishinq with t.e analysis of general equations and correlations let us note, that some segments i n  the systen may be considered as fictitious, and this fact ensures the possibility to model the arbitrary intersegmental comections. The numerical realizdtion of tk equations of m v e ~ n t s  ( 3 )  and t k i r  accogpanying correlations makes i t  possib:e to consider a number af problems concerninq processing of the result of registration of mvementj, mielling new mveiwnts and the qiven properties of the apparatus (simulator). On the basis of this probler is tle assqt ion about the adequacy of the resulting miel sportssan-appratus md of the real properties of the systel. For creation of the adequate sodel w sugest to consider the a d q c y  of the d e l  to tbe apprabus and t l e  sportsun separately. In both cases the degree of proximity of the mathematical model and the real object may be considered based on tie result3 of the mst s i v l e  ewpericents vhich are registered, for instanse wit3 . a help of shooting a film at hiqh speed and of tensor and accelerometric iteasurements. The principle difference of the systen lies in the fact that the interval generalized forces, realizing the mvewnts of the apparatus (simulator), as a rule are of the stationary natme and they may be described with the system of parameters [eharacteristics of damping, and springs, cwfficients of friction, etc.), unlike the non-stationary mwnts, . created by muscular efforts of a sprtssan. ROB th point of view of calculations tbe problems are lidted to t.e calculating of the parameters of the intern1 links of the simulator (apparatus) and to finding out the characteristics of the generalized forces for a sportsman versus tine. kt us suppose that in the results of the ewperiwnt ve obtained the tine dependencies of the generalized mvewnts. I t  i s  obvious that the lost economical (in the general case) seem to be a variant of estimation of these dependencies with the help of the sinale prdmtrical cubical smootbinq splines (Reinsch C.B. 1967). I t  rakes it possible to  consider the rje2erilised valws, chanqinq i n  a comparatively wide range, detemined by the registered infomtion about the w e x n : .  Variatlane af t3e p r 3 x t e x  frc: Smthinr akes i: pssibie t~ ~Dta!? infinite sets of curies of Ce VIII synposium ISBS -317 - Prague 1990 ganeralized cdts frm tb intorpolation to strriqbt l h ,  dram accordirq to the method of least squares. )or the o p t b  cstiration of the motbed parmtur values we suggest a W z h q  the function of noncoafanity of the calculated and &itionally learued l ln ra t i c  and force values. In th case of wuent with sbqle support and in the pmsence of a forceplate for the support reaction, forces this function light be of the folloving fon: here to , t, - tbe s t a r t i q  and finishing wlents of time, - fragment intervals of consideration do not take into account the errors of m h t i o n  on the ends of the i n  terval, the tensor, uhose matrix i s  diaqcnal and contains the weight coerficients , deteniniq the deqraa of input of different criteria. Besides tbe parameters of smtbiq, a l l  other system wmters  enter into tbe function of (11)-type, namely QIIC (for a sportspan), par-ten of the intersegmental connections for an apparatus - simulator. b a rule, the resome of the computer is lhited due to this fact. We sqqest to range a l l  the parameters in accordance with the possible input into the J values durirq their variation, for exaqle in the range of the errors of wsurements. We think a great belp in such a case will be the procedure allowing one to receive the interaction bet'm, the problem of e~perilental data (choosing of the adequate !&el) and the problem of modelling of given systens uith their followinq integration vith high accuracy of the wvemts equations and 'input of noiseQof the results of inteqration (distortion of the phase picture, e.g. vith the help of an evenly distributed error of the given mplitude). Along vith this we can vdry  the distribution of GHIC, distort frequency of skills and oMe: parameters in the systen. In the result of such studies the procedure of sinidration (11) my be c a ~ i e d  out i n  several consecutive stages. For qiver, calculations, described klov, we used the procedure of the com3ined search for t!x linisup, and this prccedcre consists of the meUcd of pseudo-accidental search n t h  the use of th.e evenly distributed sequeme (LF search) and of the Relder-Xead's n?thcd (J.A. Welder, R.l(ead, 1965) - h e  nrhcd of t !  deforped polyhedron. The exanple of calculation described furt!!er is perforzed vith the electrcnis coaputer of I E I  KIAT using a prcqran p&qe in FORTRAN. me package permits one to realize research of analoqs containing up to 16 ?miti and is orqanized as several interacting problem vhich reflect the structure described a t  the keginning of t:e article. Figures 1-3 represent results of the calculation of test mvewnts of six-eleaents (jmp dobn to t3e strain gave platform which further repulses upvards). Track =thing parameters and inertial lass characteristics were selected under condition of mini%: function of tjp ( 9 )  taking into account wmplimentirq strain measurements. The graphs of behaviour of horizontal and vertical ( ) colaponents of support reactions by the results of optidsation as t the same figure the behavi~ur of the cocplete energy ( E )  c w e  is represent&. represented in the P i p e  1. Lf ' ' Figure 3 represents the curves of alteraction of interunit m m t s  in the ankle, knee-joint and hip joint ( a,, n,, a 1. Tbe b e a c t e r  of w e  khaviour and their values correspond to the executed Poveaent and p e d t  to appneciate pwer consuption and amplitude values of intermit w m t s .  VIII Symposium I W  Prague 1990 Figure 2: VIII Symposium ISBS Prague 1990 I 5!! , . , / .  I ?: I ,  I - .  " " 1  ---.. - 4 t '  1 I ---_-?/ I GEAR C.W. The autontic inteq:ition of ordinary differential equations. - Com.ACK, 1971, 14. WaCER J.A., XW) 8. A sinplex wthcd for functim uninization. - The Comp. Journal, 7 ,  1965, 308-311. RPINISCB C.E. Smthing by spline furtions. - W r .  Ma., 10, 1967, 177-183. UIlTJiBORC J .  Dynamic of systems of rigid bodies. - B.C.Tekner, Stuttqart, 1977. VIII Symposium ISBS Prague 1990 

SOFTWARE FOR MATHEMATICAL MODELLING AND COMPUTER SIMULATION OF SPORTS MOVEMENTS AND TRAINING APPARATUS

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SOFTWARE FOR MATHEMATICAL MODELLING AND COMPUTER SIMULATION OF SPORTS MOVEMENTS AND TRAINING APPARATUS

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