LAMES, M.

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

MATHEMATICAL MODELLING OF PERFORMANCE AND UNDERLYING ABILITIES IN SPRINTING LAMES, M. r University of Wainz m i n z  FRG 4 1. Problem There are tvo traditional sethods of analysinq behaviour in sprinting competition: apart from measuring stride-frequency and stride-lerqth, the mthod of speed curves is comn. The latter one is usually tbe result of t im measurements taken on certain intervals on the course. In case of a loondash usually every 10 aetrrs are seasured. In order to get t!x speed w e  the difference b e Y m  t.~, neighbouring measurements i s  taken and divided by the distaoce bet.wen them. This procedure produces the mean velocity for every internal. Fiyxe 1. Sped cxve dr'in as a pljqone and step-fwtion Sew critical :e~drkj have t s  be applied: - The comn represention of sped curves is a polygon which comects the average speed levels at  3.2 midpoints of the intervals. This suqqests a continuous function ard it is a simplification to a s s w  se?n sped in +be middle of an interval. k t u l l y  the procedure supplies a non-continuous step-function (Piq.1). a - Detailed analysis of the results i s  not satisfactory: if one simply asks for the location of m x i ~  speed the anser can only be the interval with maximum average speed. me cannot even h sure that the actua! oaximin is located in this interval, kcduse in unlucky cases it miqht as well h in a neighbourinq one. - Takinq differences - in cases of acceleration c w e s  differences of differences - makes the error for 3 velocity measurements systematically larger than the error w have already for a single time measuremt. These errors are amplified by the aITmgmnt in a aeasuremnt chain, where tbe random error of one interval becomes the systemtic one of the next.  Apart from these nor? methodological objections one should consider a general aia of analysing per:o.mce in  competition: t~ establish a linl bebeen behaviour (visible) and i ts  conditions /invisible/, betieen dexription and erpianation. This desideratlu applies to all bionechanical measurewnts. V I I I  Symposium ISBS - 291 - Prague 1990 In order to generate practically useable information tor train& it ia not always sufficient to stick to =re description of ubat is going on, but one bas to aim at  tbe Werlyirq conditions of this bebavioar. W r e  i s  broad consensus on tmderlyirq abilities in our exanple, the 100 n - dash: the basic abilities are mction time, ability of acceleration, sprintirq rpced and sprinting durance. The operational definition of these basic abilities appear - exept for reaction time - to be problematic though: - The ability of acceleratia includes the aspects of high acceleration and of acceleration over a long ti=. I t  i s  a complex ability vhich must not be aecessarily onedimensional. Operational definitions can only a h  at one aspect: the initial acceleration represents the larimu munt  of acceleration, the distarre with positive acceleration or the correspondinq t h e  used to reach maximum speed stand for i t s  duration. - The conventional mthod measures sprintiq speed as the maxim average speed in an in twa l .  So, apart fron errors due to the original time measurements, w have a systematic error: laxinun average speed underestbtes by definition maxirun speed. (This holds althoyh - for otber error-sources - we usually observe an overestination of m x i m  speed (Pig. 3 )).  - The operational definition of sprintinq endurance as a d i f f e r e ~ e  of differences (maxima speed ninus final speed ) increases the influence of errors from the original measurements. Cne reason for the problelr cited abave is tbat input data  (interaediate times) c m t  be transformed into a satisfactory description of behaviour in competition. 'be resulting step-function i s  non-continuous and supplies only average speed per interval. 2. nodel!ing of s~rint ing behaviour with analytic functions Pacing these problels the idea caw up to describe sprinting behaviour with analytic displacexnt-, speed- and acie1erat:oc- functions obtained by (non-linear) regression. The advantages of a regressio~l  appr0dch are obvious: - Erron l n  the o r i q i ~ l  Easurements are smthed by the regression function. This holdi k a c j e  one doesn't interpolate but minirites the Squared Sum of Errors (SSE). Cowred with t$e conventi:cd: met$& .w expect a daapirq of errors instead of an aqlification due to the use of d~fferences. - sped and acceleration are M longer dettrninated by using differences but by differentiation of the fitted function. - Using continuous functions we have speed and acceleration valws for any point on the course. Especially the deterxination of maxinun sped location results in a p i n t  and not in a lOm-interval. although Uis point i s  of course s t i l l  subjected to e m n  these are mt systematic any a r e .  2.1. Developement of an appropriate del-function Since reaction tine is an additive parawter i t  is excluied fron the followinq considerations. Wel-building vith regression functions can be performed in t w  fundamentally distinct ways. Inductive model-buildiq condenses data into a function: 'Yhicb function do I know that looks almost the way may data do?' Deductive &el-building tries to generate regression functions fron asssunptions on the underlyinq process:' Which function describes the internal functioning' of tbe modelled system?' Indvctive wdel-buildinq has severe dravbacks. One just can't haw tk saw confidence in an inductive model as in a deductive one, although sometimes mnplexity of system or lackinq hvledge pernit only inductive models ( see PJCBSmm 1989). hying the deductive w c h  ve assume that the speed curve can be understood as a superposition of tm gro'rtb processes: acceleration vA and fatigue vp. With an additive superposition ve arrive at  the vIII Symposium IS85 - 292 - Prague 1990 v( t )  = vA(t) + vp(t) = A (l-e-R ) + P p e l t )  A,k,1>0, Ds. a v i s  a function of time vith 4 parameters and, typical for deductive mdelinq , these parameters have got interpretations in tbe original systea: V A = absolute speed-1st achieved by infinitely long acceleration vithout fatique, k = steepness of acceleration process, P = onset of fatigue and 1 = steepness of fatigue-iapad. P Note that one i s  not dealing with a rechanical wdel but with a systmriented one. In 1951 8WRY and TIUPWN have used a mdel vhich is identical to the acceleration coqne?t of the introduced one. Their model perford very uell i n  predicting speed-curves of 6by-dashes. Nso they found that the parameten A and k vere independent. Practical calculations vith our model forced a modification. Haviq only 11 data points but I prameters results in unstable estimates for the parameters. In addition to this, the tdo parameters of the fatigue-process are only lwsely determined by data. The tio reasons are that only the last msurexnts shov one significant impact of fatigue and that i t s  overall influence sprinting speed is ssall compared uith the influence of acceleration. These inductive considerations M, lacking quality of data lead to the elinination of praEter  1, because the steepness of the fatigue-inpct seem even less determinable than its onset. Elimination of a regression pa:aneter m n s  that an appropriate constant value for i t  is chosen instead of obtaininq an estimate by - a regression algorithm. The &el-building Frccess is resumed i n  Figure 2. Several kinematic aspects are involved: - data consist of intermediate times, - the &el-function is a sped-curve over tine, - usual representations are speed and acceleration curves over the course and - regression i s  based on displacement over tine (This has the advantage of making lise of the ram-tata without transformation). The regression function is obtained by integrating the miel function v: Acceleration is obtained by differentiation: * A critical r w r k  bas t? be made: for a startin? m e r  (t-0) holds s='m=3, vhi!e the &el ass=$ maxim1 acceleration at t=O (see Figure 2 ) .  As a consequence one has to admit that the model is not able ts  desc:ibe precisely what i s  goin? on on the first few meters. This is not surprising because one can't reasonably e v c t  , a description of the building-up of acceleration on the first me:ers by a mdel vhich has just the tire for 0 and 10 meters as relevant. input. In order to dsscribe this phase more precisely different methods hd to be applied. VIII Symposium ISBS Prague 1990 Figure 2: dnsa t ic  aspects of sprinting perfomme. 2.2. Deriving indicators for basic abilities With a ~ l y t i c  spaed- and acceleration functions i t  is possible to  ovucow sow of tba troubles with operational &!initions of basic sprinting abilities quoted above. - The k s t  indicator of reaction time is of course reaction time itself. - The abilitv of acceleration is described in its tvo aspects: amunt and duration. The indicator for the amount is aoa(0), that is the initial acceleration. Usual indicators of duration for acceleration are the ti= used to reach mxiaua speed (\) and tbe point on the course for this event s j s (  ). hlysing empirical speed curves reveals that speed is alwst constant kl betwen 40 and 130 r t e r s  thogh. , fixing the location of speed maxim is a sort of gaPbling. Por this reason as indicator tbe tir teps I '9 chosen. At that time acceleration bas not yet dropped to  zero but to a very wall value eps, i.e. eps=0.1 m/s . teps and seps =s(teFs) are by definition smaller than and s, and we expect then to be uch mre precise. - Scrintim sPnd is indicated as usual by w i w r  speed: vm=v(t,). - The qwtie!!t qv Of final speed by nxinum sped is taken as indicator of scrintina endunrce: qv = 103 v ; : ~ , ~ )  1 v(t,) It]. 3. Results of a pilot study Earlier the sensitivity of the conventional differences-etbd to errors i n  measurements was criticized and a higher roblrrtncss of tbe regression-metbod w postulated. A chance for testing tbese assqtions are the remarkable differences betwen htud t h s  reported for tbe 1OOcfinal at  Rome 1987. tETZELTER (1989) pointed out that interval t i r s  npoltrd hd ia t e ly  after the event deviated from those published by the official biowhanical comirrion sou ant& later. Obviously Me first measurement suffers n r h  mre fron errors than VIII symposium ISBS - 294 - Prague 1990 the last one vhich used high-frequeacy techniques. 1 Z ~ T  Figure I: S p e d  nwes from interval times reported imediately after the final at Row 1987 (dashed) and reported by the bioeechanical comission (solid). The curved lines are the results of the regression method for each set of data. In Fiqure 3 i n  addition to rhe step-functions the rqression curves are dram. Cne sees tbat they are not only relrarkably smtber but in particular that the tw curves lead almost to identical results. This is a convincinq indication of robustness because w know that one set of raw data suffers a lot from errors. Even the largest deviation between the t'x, curves at the end of Ben Johnson's dash is s ~ l l e r  than O.U/s. I 3 .2  'esdltj or. sprinting atilities The introduced mthcd supplies estimates for the parameters mentioned abave, which are only  a selection of possible variables. With this data as input, ideally based on a large n&r of cases, very sophisticated analyses of sprinting bebaviour and wderlyinq abilities are possible. S K ~  analyses are inappropriate Clcuqh ts t$e data base of this pilot study: the 16 100n-finalists of Row 1987. It 's ain is a methodolyical one. Bct even uid 2x8 cases descriptive and correlative results seem to be very interestinq. T.AELE 1 Descrip:ive statistics of 100m-finals at Row 198: *at-. 1 ./c 1 .an I st. dw.( .in I ui r?C11 tl.. 1 Q.103 1 9.#> 10.14 1 0.. 10.*O 11.19 / <."lth J C  r c c .  c 0  0.l.i.l i -I-'-- --- , - , I - F . o c ~ . ~ ~  ......nc. far  i :? u::::: ':: 1 :::ti II:::~ 1- - L _ _ -  VIII Symposium IS3S - 295 - Prague 1990 The desaiptive statistics ot 9 variables an given in  Table 1. Ron the lethodoloqical pbt of viev it is iqortant to note tbat GP and re are obviously batter estimates tlw C id 8, Por ren s, bas a m q  of 40 wters vhile the range ot re !! less than 10 rters. Tbe rape tendency can beeobserved for vow, but, as all vow have a decrew in sprEtiiq speed, their rawinum speed can be Pore precisely deteniwd, vbereaP pn do not wticeahly redm their sp6ed (lin qp911;41t). T A W  2 I n t u c o r n h t i a u  of vartabhr descrihiq p r f o M n r  cm 10Om (ulrper half: men, lower half: wm: l m l s  * for significaoa: 0.71 (9) and 0.83 ( I t ) ) .  * Table 2 shows tbe intercorrelations of variables for men and woen. Tvo aspects are of particular interest: the deterai~t ion of tbe corpler criterion of performance ( t  and the intercorrelations of basic abilities. lux iv l  speed accounts a l a t  rhp la ry  for the to~!) lOOrtk.  The correlation is hiqher tor rn (P-0.97) than for wpn (F-0.83) but the w's sample h s  a broader range (0.51s versus 0.29s for wren). Figure I shows the impressive correlation. I(arium sped itself i s  correlated with duration and lenqtb of acceleration and to a smaller degree with initial acceleration. Beaction tire and sprintirg erdurance seer to be of unor iqortance for p e r f o w e  in the two sarples. A very astonishing result i s  the nrked but neqative correlation between initial amount and duration of acceleration. Nthoqh these findings are consistent with a one-dimensional concept of the ability of acceleration, the two aspects seem to be antithetic: one can either bave a large initial acceleration or a l o q  acceleration. An explanation of this finding could be selective adaption of strerqht abilities to contact time on the grolmd vbich decreases considerably. t loO [s] 10.3 10 1 Figure 4: Scatter diagrao for l a r h  sped t, ani total tire tlOO for the finals a t  Rome 1987. VIII symposium 1- - 296 - Prague 1990 a) IIatbDd The reqrusion mtaod provad to be s~parior to tha differences method. TIM advantages are: * - -thing of raw data rarma amplification of mn, - continuous anw Pvour aDasoatiPPPm step-miont, - values for any pint on amna versa r a lm  only for intervals. * Cna redm for tba excellent f i t  is that 10Ordmb are nm vith rari.n accaleration and M tactical mipulation of sprinting rpwd azm. Faxptio~ could be tbe last retan of eliminating heats vhen qualification is sure. Eut even in this case anly variables quantifying sprinting enturanca vould be affected. %is objection i ~ ~ p l i u  thoqb that thc -1-function not suitable for events lmger than M O  reten, because in those events running speed is very ach dater&ied by tactical considerations. The applied mdel is  obtained by m i o n  anb describes the additive superpotition of an acceleration-process ard a fatigue-process. Practical calculations @OSE tw restrictions: one fatigue-paramter ust be held constant and thz mdel is not able to &mibe precisely tbe building-up of spxd and acceleration on the f i a t  leters. Tbe lain xiencetific advantage is that tbe ratbod allows for calculation of parameten which can be interpreted as precire irdicators of basic sprinting abilities. b) Practical results A first result is  that values for mxim sped reported by the differurcs wtbod bave to be doubted. Because of the ananqenent of masuremts in a chain i t  is very likely that at least one interval shows values that they are too high. l%e error-struck Pasuremt published imediately after Johnson's victory at Rome 1987 - reprted a minun speed of 12.05~/s, the mre precise biomechanieal colaission 11.761/s. Tk regression aetbad results in a maxim sped of 11.66a/s. The lerqtb and duration of pmitive acceleration is a question of practical interest. ' I t  can nw be answered by pointirq out a certain point on the course. A better indicator for this aspect of the ability of acceleration is the length and duration of positive acceleration greater than an alwst negliqible threskld !suggestion: eps=o.lm/s2). Concrete results at  me pilot study on the tvo finals at Row 1987 are: - vith the exceptions of reaction time and duration of acceleration len are significantly superior to :own in all variables, - extreme groups differ demnstrably i n  duration and lenqth of acceleration and especially in r a x h  speed, - n x i m  speed is  clearly the mst i w m t  ability accounting for overall performace and - the ability of acceleration seeps to show a conflict bet'- initial acceleration and its duration. The special iapct of the introduced mthod is that the gap between description and explanation, bet'nen performance in competition and underlying abilities is closed. The abilities can MV be tested under optha1 conditions: during coapetition. 8 I: further investigation confim its excellent suitability and technical progress raLes data =re available, the introduced pethcd could become a routine-procedure of future trainiq i n  sprint. PmK, P. and LANES, I.: llathematische Wellierung md Leistunqsprognosen i n  der kichtathletilr (Ihtbclatical nodeling and predictimg performance in track and field-events). In: Sportwissenscbaft 19(1989)4, 420-435. W(RY, P.M. and TRWTON, I.V.:The velocity m e  of sprint nmniq. In: The Research Quarterly (1951) 22. LETZELTER, I(.: wettkampfverhalten und Sprinttraining: Zur Diagnose des Rmpoverbaltenr mn Heltklassesprintern (Performance in competition and training of sprint: Diagnosis of speed behaviw of wrd-class athletes). In: Die Lehre der Leichtathletik 28(1989)1, 50-67. VIII Symposium ISBS - 297 - Prague 1990 

MATHEMATICAL MODELLING OF PERFORMANCE AND UNDERLYING ABILITIES IN SPRINGING

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MATHEMATICAL MODELLING OF PERFORMANCE AND UNDERLYING ABILITIES IN SPRINGING

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