Computer Simulation of the Relationship between Flat. Service and Service Return in Tennis Singles

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1 Inenaional Jounal of Spo and Execise Science, (): Compue Simulaion of he Relaionship beween Fla Sevice and Sevice Reun in ennis Singles Ching-Hua Chiu * Gaduae Insiue of Spos & Healh Managemen, Naional Chung Hsing Univesiy, aichung, aiwan, 4, ROC Received 5 p 1; cceped Jun 1 bsac In ennis singles maches, he seving playes y hi an ace by placing he ball o hei advanage. In conas, he eceiving playes y o posiion hemselves fo he bes defence o eun he ball in ime and avoid losing poins. his sudy is an aemp o esablish a mahemaical model on how o aac by opimally placing a fla sevice and o eun he sevice wih he opimal defence posiion. Fou membes of a college ennis eam (heigh: 175.5±.1cm; weigh: 7.3±5.4 g; age: ±3.1 yeas) wee he subjecs of his sudy. he expeimen was conduced in hee sages. Fis, a fligh algoihm based on he physics of he fla sevice was esablished. Nex, spline funcions wee used o consuc an algoihm fo he 3D space available fo defence duing sevice eun. Finally, daa on he ime of sevice eun wee colleced hough a high-speed video camea and hen eyed ino a compue pogam o es and simulae he fligh and defence algoihms. When he seving playe inceased he impac speed, he eceiving playe was unable o hi i bac. lso, when he placemen was close o he eceiving playe s laeal sides, he seving playe also inceased his chances of winning a poin. Howeve, if he sevice bounced on he gound igh in fon of he eceiving playe, i was moe liely o be euned. he placemen findings wee defined as he U effec in his sudy. his expeimen also defined he eceiving playe s oiginal defence space and helped idenify posiions vulneable o aac, which enabled successful sevice euns. In conclusion, he simulaion esuls of his expeimen coespond o eal condiions; heefoe, he model esablished in his sudy is qualified o be a aining ool fo ennis playes in seving and sevice euning. Keywods: lgoihm, U effec, Defensive space Inoducion ennis playes who excel a seving have an edge ove ohes, while hose who do no, have difficuly in becoming elie [1]. lhough diffeen body heighs, am lenghs and muscula sengh lead o diffeen seving posues, playes ae supposed o follow he basic pinciples of a high conac poin, eadiness o swing he ace, and an appopiae shif of he cene of gaviy. In conas, he eceiving playe should focus on pevening he seving playe fom obaining any advanages [1]. Seele e al. conduced a wind unnel expeimen on ennis ball sufaces []. Fou inds of balls wee esed in ha expeimen: unused new balls and hose used ove hee diffeen ime peiods. hey found ha he balls had diffeen dag coefficiens. he dag coefficien fo he unused new ball was.7.9. Wang found diffeen coefficiens of esiuion (CR) fo diffeen ennis cous. ed clay cou had a CR of.69 [3]. Wong found ha he speed of he ennis ball when i ouched he gound did no emendously ale he CR [4]. Haae e al. conduced an expeimen on he impac of he *Coesponding auho: Ching-Hua Chiu el: Fax: hangcaso@yahoo.com.w ball on he ace and found ha CR was educed wih inceasing impac speed [5]. his elaionship may sem fom he vibaion damping poduced by he ace handle and sings. Seveal sudies invesigaed he velociy of he ennis ball. uagawa and Kojima found ha he oaion momen poduced by he un and hip joins had a diec influence on he speed of single-handed and wo-handed bachand ennis soes [6]. anabe and Io used 3D phoogaphy o analyze he velociy of a ennis sevice [7]. hey found ha he impac speed of he ennis sevice was deemined by he angle and angula velociy of he shoulde join. In his eseach on ennis seves, asi oo fou aiwanese college ennis playes as paicipans [1]. He found ha he maximum seving speeds anged fom 5 o 65 m/s. Fo compue simulaions, Chiu used a numeical algoihm o simulae he fligh disance fo badminon [8], ennis, volleyball, and socce. He found ha using a.1s ime ineval, he eo in he esuls yielded by he algoihm was negligible. When holding sevice, a ennis playe can seve wice a mos. Fo he fis sevice, he fla sevice is ofen used because i is fase and allows he eceiving playe less ime o eac, which educes he eun ae. Howeve, he focus of he second sevice is on he success pecenage and placemen. Sevices wih opspin, placed on he eceiving playe s laeal

2 6 Inenaional Jounal of Spo and Execise Science, Vol.. No. 1 sides, ae adoped o peven successful eun soes [1]. In ohe wods, accuae pedicion of he impac speed and ball placemen is cucial o he seving playe. Howeve, few pevious sudies have exploed he elaionship beween he offensive and defensive saegies fo sevice and sevice eun, especively. his sudy is an aemp o esablish a mahemaical model fo sevice and sevice eun o incease he acics and saegies available o ennis playes. In he fuue, he poposed algoihm can be developed ino compue sofwae, which can be used by ennis coaches and playes o acquie pacical saegies. Mehods Subjecs he subjecs wee fou male membes of he ennis eam a Naional aiwan Chung Hsing Univesiy (heigh: 175.5±.1 cm; weigh: 7.3±5.4 g; age: ±3.1 y). In sanding posiion, hei maximal heigh fo he ball conac poin was 75.±.4 cm. hey wee all igh-handed, and hey had paicipaed in maches fo moe han hee yeas. hey had no hisoy of injuy fo he las six monhs, and hey showed no sign of soeness o pain duing he expeimen. Mahemaical Modelling he cene of he ennis cou was aen o be he Caesian coodinae sysem OXYZ (Figue 1a). he seving playe was on he lef side of he cou, while he eceiving playe was on he igh side. ( =, 1,..., n) epesened a ceain ime poin duing consecuive fligh imes. he sevice was assumed o be eleased a ime poin. If he eceiving playe could eun he seving befoe ime poin n, which is when he ball ouches he gound a second ime, his mean ha he sevice was euned successfully. If no, he seving playe gained one poin. Figue 1. (a) Illusion of sevice and sevice eun. (b) Momen of impac. (c) Coodinae sysem fo he momen of impac. lgoihm fo he fligh of he ennis ball o bee undesand he eys o a successful sevice and sevice eun, wo algoihms wee consuced fo simulaions. One of he algoihms deal wih he fligh of he ennis ball. Caesian coodinae sysem OXYZ was se in he cene of he cou (Figue 1c). Nex, a anslaion coodinae sysem OX Y Z was se up, and is coodinae oigin was defined as he ennis ball s cene of mass (COM) p ; his was also he iniial posiion veco. α efeed o he azimuh angle of he sevice, β was he angle of elevaion, and V was he iniial velociy veco elaive o ai (Figue 1c). he fligh pocess began wih he seve and ends when he ball ouches he gound a second ime. he iniial posiion veco of he ennis was assumed o be p = [ X, Y, Z ]. he posiion veco of COM a was assumed o be p K = [ X, Y, Z ]. he hoizonal fligh disance fom P o P was d ( 1 / d = [( X X ) + ( Y Y ) ] ). In P, he elemens X, Y epesened he posiion poins of plane XY in he Caesian coodinae sysem, and Z was he heigh of he conac poin. Fo he fligh pocess, wo neighbouing ime poins and + 1 wee andomly seleced. he ineval beween hem was assumed o be = + 1. When and + 1 appoximae each ohe, he movemen of COM fom o

3 Inenaional Jounal of Spo and Execise Science, (): i + can be egaded as moion a a consan acceleaion 1 [8-11]. he posiion veco P + a was compued wih equaion (1): P 1 = P + V (1) In equaion (1), V = [ Vx, Vy, Vz ] and = [ x, y, z ] efeed o he velociy and acceleaion veco of COM, especively. V + 1 and could be wien as follows: V + 1 = V + () = F m (3) / In equaion (3), F epesened wo exenal foces, gaviy and dag foce, ha wee imposed on he ennis ball duing is fligh; m efeed o he mass. he iniial velociy veco was epesened byv, and V = zy ( α, β)[ Vs,, ] (Figue 1c). Vs epesened he elease speed, and zy ( α, β ) epesened he oaion coodinae ansfomaion maices: cosα sinα cosβ sinβ zy( α,β) = sinα cosα 1 (4) 1 sinβ cosβ he fligh of he ennis ball has hee foces acing on i: gaviy, dag and lif (Figue ). Since a fla sevice has no spin, he lif foce was in his sudy. Meanwhile, F in could be wien as follows: F 1 = ρ C d s V u mg (5) + In equaion (5), he densiy of ai ρ was 1.3 g/m 3 a sandad amosphee pessue [1]. s efeed o he dag aea of he ennis, he dag coefficien C d was.78 [], he gaviy acceleaion veco G = [,, g] wheeg= 9.81 m/s, u was he uni veco fo he dag in, and u = V / V. Z epesened he coodinae value along he Z-axis, and efeed o he adius of he ball; heefoe, i was assumed ha Z. ssuming ha he ball ouched he gound a, he velociy veco could be wien as V = [ Vx, Vy, evz ]. he diecion of he velociy componens fo V on he Z-axis was opposie o he oiginal diecion, so he velociy of V was equivalen o he oiginal velociy componens muliplied by he coefficien of esiuion e. a Figue. Model of defensive space. (a) he heighs of he fou planes, B, C and D wee h 1, h, h 3 and h 4 especively. (b) ime paamees fo he defensive space. Requiemens fo successful fligh ove he ne ccoding o Inenaional ennis Fedeaion ules [13], ha ball adius should be.33 m, he ne heigh H is ne.914 m, and he disance beween he side and cene lines is m. Consequenly, he equiemens fo successful fligh ove he ne in ime peiod ae: X.5m, Y 4.115m, and Z H ne ( X.5m : ange of eo b oleance). Requiemens fo valid placemens he posiion veco of he age placemen was assumed o be Q = [ X,Y, ]. he azimuh of he sevice could hen be compued fom he wo posiion vecos P and Q: Y Y 1 α = an ( ) (6) X X

4 8 Inenaional Jounal of Spo and Execise Science, Vol.. No. 1 he angle of elevaion was β, and β = βmin + m (m =, 1,, n; = ( β m ax βmin)/n ; n is an inegal; : angle of sep). he minimal angle of elevaion β was inpu ino min he equaion, and he angle was gadually inceased fom β o min β o calculae he hoizonal fligh disance and m ax locae β fo he age aea. Fo, which was when he gound is fis ouched, ds, he eo fo hoizonal disance beween P and Q, could be compued using equaion (7): ds 1 / = [( X X ) + ( Y Y ) ] (7) n eo oleance of.5 m was pesumed fo ds (Figue 1a). heefoe, he condiions fo when he ball fis ouches he gound ae Z i and ds. 5 m. a b Figue 3. (a) Expeimen seing. (b) Diecion ligh plae. (c) Pepaed posiion of he eceiving playe. c Defence algoihm he defensive space efeed o in his sudy is he maximal space wihin which he sevice can be euned befoe i ouches he gound a second ime. he cene of he ennis cou was he oigin of he Caesian coodinae sysem OXYZ (Figue 1). he seving playe was on he lef side of he cou; he hi he ball fom he oigin of he anslaed coodinae sysem OX ' Y' Z'. he eceiving playe was on he igh side of he cou; he pepaed a he oigin of he cylindical coodinae sysem OXsYsZs o eun he sevice (Figues 1 and 3). he calculaion seps fo he enie pocess ae as follows: Sep 1: he fligh equaion was adoped o calculae he posiion veco P elaive o he Caesian coodinae sysem OXYZ a (Figues 1a and 1c). P is hen coodinaely ansfomed ino R (=[,, h ] ), he posiion veco of he cylindical coodinae sysem OXsYsZs ( : adius; : azimuh; h : heigh). Sep : he defensive space fo he sevice eun was esablished (Figues a and b). he pepaed posiion of he eceiving playe was assumed o be he oigin of he cylindical coodinae sysemoxsyszs. he defensive space consised of fou planes (, B, C and D) which paalleled he OXsYs plane. he heighs of each plane in he cylindical coodinae sysem OXsYsZs wee indicaed by h 1, h, h 3 and h 4. Each plane had 5 conac poins (Figues and 3a), so ogehe he fou planes had 1 conac poins. Sep 3: Plane was used as an example hee (Figue a). hee wee 5 conac poins on plane (Figue ). One of he conac poins was he oigin. he ohe 4 conac poins wee disibued evenly in eigh diecion axes: eas (E), wes (W), souh (S), noh (N), nohwes (NW), souhwes (SW), souheas (SE) and noheas (NE). Each diecion axis was compised of hee conac poins and he oigin. he 4

5 Inenaional Jounal of Spo and Execise Science, (): conac poins fomed he hee concenic cicles S 1, S S 3, whose adii wee 1, and 3, especively. Sep 4: he movemen of he eceiving playe along he Xs axis of plane was elaboaed in Figue. he ime he eceiving playe spen moving fom he oigin of he cylindical coodinae sysem OXsYsZs o defence poins, 1, and 3 (1) (1) (1) (1) ( R = [, 1,, 3 ] ) was designaed as,, 1 3 (1) (1) (1) (1) (1) ( = [, ] ). Spline inepolaion was 1 3 (1) applied o he colleced daa a and o consuc a se of spline funcions. his way, he ime spen by he eceiving playe in moving fom he oigin o along he Xs axis of plane could be calculaed wih equaion (8): 1 (1) ( ) = Spline R,, ) (8) ( Sep 5: Following sep 4, he ime he eceiving playe spen in moving fom he oigin of cylindical coodinae sysem OXsYsZs (Figue b) o on any of he eigh diecion axes was assumed o be (1) ( ) ( ) ( ) ( 5 ) ( ) ( ) ( ) [ =,,, ]. he azimuh angle fo any of he eigh diecions is = [,π/ 4,π/,3π/ 4,π,5π/ 4,3π/,7π/ 4,π] (Figue ). Spline inepolaion is hen applied o he above wo ses of colleced daa o calculae he ime he eceiving playe spen moving a he azimuh angle fom he oigin of OXsYsZs o : = Spline (,, ) (9) Sep 6: he ime he eceiving playe spen in jumping o planes, B, C o D and moving fom he oigin of he cylindical coodinae sysem OXsYsZs a he azimuh angle B C D o was assumed o be,. he heigh of he fou planes was epesened by H ( H = [ h1,h,h3, h4 ] ), and he ime spen in moving fom he oigin a he angle o defence poin was epesened as B C D = [, ]. Spline inepolaion was applied o he above wo ses of colleced daa o calculae he ime he eceiving playe spen in moving fom he oigin o R(,, h ) (Figue b): = Spline H,, h ) (1) h ( Sep 7: s shown in Sep 6, he eceiving playe moved fom he oigin o R (,, h ) o eun sevice; he ime spen is epesened wih. h could be calculaed using seps 1 h o 6. If was longe han h, o when he ball ouched he gound fo he second ime, his mean ha he eceiving playe did no inecep he ball and he seving playe gained one poin. Possibiliy of acing he cou on he eceiving playe s side was divided ino 1 ecangles of he same size. he cene of each ecangle was defined as he age placemen fo he seve. One hunded placemens wee simulaed o deemine he possibiliy of acing. Daa Collecion Pocedue he colleced daa came fom aing picues of wo subjecs euning sevice wih wo high-speed video cameas ( Hz). he oigin of he coodinae sysem OXsYsZs seved as he cene of cicle ( = m) fo he hee concenic cicles S 1, S and S 3, whose adii wee 1, and 3. ( 1 = 3 m, = 8 m, 3 = 18 m). diecion ligh plae was placed 5 m noh of. wo video cameas wee placed on eihe side of 5 m away. he cameas could shoo he subjecs movemen and he signals of he diecion ligh plae. he subjec sood on he oigin in a pepaed posiion (Figues 3a and 3b) wih his heels sepaaed by.5 m. he heels wee.5 m away fom he Ys-axis, and an isosceles iangle was fomed by he oigin of OXsYsZs and he wo heels (Figue 3b). On he diecion ligh plae, each of he eigh diecions had an indicaion ligh (Figue 3c). he expeimene andomly uned on one of he diecion lighs and ecoded he ime fo he sevice eun. Fo cicle S 3, each of he eigh diecion axes had a plumb line (Figue 3) o which fou ennis balls (a, b, c and d) wee aached a diffeen heighs (h 1 =.15 m, h =.9 m, h 3 = 1.7 m, h 4 =.75 m). One of he eigh diecion lighs was hen andomly pessed (each diecion ligh could only be pessed once). Seeing he ligh, he subjec had o move fom he pepaed posiion o he diecion he ligh indicaed o hi ball a. Ball a was hi in one of he eigh diecions, and each movemen ime was ecoded. Subsequenly, he movemen ime in hiing balls b, c and d in each diecion was ecoded. he above pocedue was applied o cicles S and S 1 o ecod he movemen ime spen in hiing balls a d in he eigh diecions. he ime consumed in hiing he fou balls a d on S was also ecoded. In oal, 1 balls wee hi. Each ball was hi wice, and his which oo less ime wee chosen as he es value. he subjecs hi he balls a inevals of 9 1 seconds. Daa nalysis he compue pogamming language Boland C ++ was used o simulae he fligh of he ennis ball and o es he fligh and defence algoihms. Fou physics paamees had o be confimed befoe simulaion and esing. One was he dag coefficien. ccoding o he gaph developed fo he ennis ball s dag coefficien by Seele e al. [], he coefficien C d was.78. nohe paamee was he coefficien of esiuion. ccoding o Wong [4], on a ed clay cou, he coefficien of esiuion fo a fla sevice wih no spin was.69. he nex paamee was he ime sep fo simulaing he fligh of he ennis ball. he ime sep was assumed o be. 1s in his sudy because he hoizonal disance of he fligh emained he same (able 1). he elaive eo was below.%(. 1s ). he final paamee was angle sep of β fo locaing he placemen of he ennis ball. es esuls showed ha =. 5 (able ) povided he bes esuls since he poduced eo fo adoping =. 5 o pedic he placemen was 1.68% ( ds.5 m ), which was wihin he accepable ange ds.

6 3 Inenaional Jounal of Spo and Execise Science, Vol.. No. 1 able 1. he numeical ime sep (s) (acual:.1) fo he fligh algoihm of he ennis ball f (s) Disance(m) Relaive eo (%) Noe: f epesened he fligh ime; Vs =45m/s, α=, β =, Z =.75m. Relaive eo = 1% (hoizonal disance )/ able. he eo fo adoping o pedic he placemen (deg) β (deg) ds (m) Relaive eo (%) (acual:.1) Noe: =.1 s; Vs=45m/s; α= ; conac posiion (m, m,.75m); placemen (1m, m, m). Relaive eo = 1% ( ds /.5). Resuls and Discussion Defensive space he defensive space efes o he maximal ange whee he eceiving playe can eun sevice befoe he ball ouches he gound a second ime. he success of he sevice eun is deemined by he eceiving playe s eacion and movemen imes [1]. he measuemen of he defensive space depends on he aea of he eceiving playe s physical aciviy [14]. Figue 4 showed ha he volume of each subjec s defensive space became lage as he defence ime inceased (able 3). In addiion, he fou subjecs had defensive spaces wih diffeen syles. ppaen diffeences could be found in he fou spaces a a defence ime of.6 s. When he defence ime inceased o 1.8 s, he defensive space was shaped lie a pie cha. Subjec D had he lages defensive space; his volume eached m 3 when he defence ime was 1.8 s. his esul shows ha subjec D was fase han he ohe subjecs in euning a sevice. Being speedy in movemen mean ha he eceiving playe had moe ime, which enhanced he possibiliy of a successful sevice eun. able 3. Defensive space of fou subjecs subjecs s=.6s s=.9s s=1.s s=1.8s Vol (m 3 ) Vol (m 3 ) Vol (m 3 ) Vol (m 3 ) 11.4 (1) 6.8 () 15.( 3) 44.5 (4) B 16.9 (B1) 75.9 (B) (B3) 419. (B4) C 15.3 (C1) 69.5 (C) (C3) (C4) D 13.8 (D1) 74.7 (D) 18.7 (D3) (D4) Noe: s epesened he movemen ime; Vol epesened he volume of defensive space (See figue 4). able 4. ime subjec B spen o eceive he soe Case Successful eceiving Vs(ms) Conac poin Placemen X(m) Y(m) Z(m) X(m) Y(m) Z(m) ime(s) f(s) a Yes b Yes c No d Yes e Yes f No Noe: Defence posiion: (m, 8.5m, m). f: suplus o deficien ime beween eceiving and he ball ouching he gound a second ime(see figue 5).

7 Inenaional Jounal of Spo and Execise Science, (): Figue 4. Subjecs ae epesened by, B, C and D. he seial numbes 1,, 3 and 4 epesen he defensive spaces fo defence imes of.6,.9, 1. and 1.8 s, especively. Peliminay es of fligh and defence algoihms he eceiving soes of he fou subjecs wee simulaed. Subjec B was used hee as an example (Figue 5; able 4). he playe sood in posiion (m, -8.m,.m), eleased hee balls a a speed of 5 m/s, and placed he balls a (m, 6m, m), (1.5m, 6m, m) and (3m, 6m, m) (able 4). he algoihm was used o simulae he eceiving soe of subjec B and o judge if he was able o inecep he ball (Figue 5; able 4). he fis ball was placed igh in fon of subjec B (Figue 5a), so subjec B was able o successfully eun i wih he defence ime of.788 s. Subjec B was able o inecep he ball.353s ahead of ouching he gound a second ime (able 4). he second ball was placed a lile o he lef side of subjec B. his ime, subjec B hi he ball.14 s ahead of i ouching he gound a second ime (Figue 5b; able 4). he hid ball was placed fa o he lef side of subjec B. his ime, subjec B failed o inecep he ball. he defence ime was.7s behind he ball ouching he gound a second ime (Figue 5c; able 4). Subjec B was hen simulaed o hi hee ohe balls ha wee eleased fom he posiion (m, -1.m,.5m) a hee diffeen speeds, 3 and 4m/s and had he same placemen (1.5m, 6m, m). When he elease speed was o 3m/s, subjec B could hi he ball (Figue 5d and 5e; able 4). When he elease speed was 4 m/s, subjec B was unable o inecep i, and he defence ime was.134 s behind he ball ouching he gound a second ime (Figue 5f; able 4). he simulaion esuls showed ha he close he placemen was o he eceiving playe s iniial pepaed posiion, he moe easily he could eceive he soe. On he ohe hand, he moe disan he ball was placed fom he eceiving playe, he moe

8 3 Inenaional Jounal of Spo and Execise Science, Vol.. No. 1 difficul i was o hi he ball. he lowe he elease speed of he soe, he easie i was o hi he ball. his confimed he eason why mos ennis playes used he fla sevice as he fis seve [1], as i is fase and allows he eceiving playe less ime o eac and move. Since he findings of he peliminay es on he fligh and defence algoihms coesponded o he esuls of ohe sudies, he mahemaical model in his sudy can be adoped o simulae sevice and sevice eun. 5m/s 5m/s 5m/s m/s 3m/s 4m/s Figue 5. he elease speed Vs of he hee balls was 5 m/s (able 4), and hei placemens wee ( m, 6 m, m), (1.5 m, 6m, m), and (3m, 6m, m) (able 4). Subjec B was simulaed o hi he hee balls and obained he placemens shown in (a), (b) and (c). Nex, subjec B was simulaed o hi hee ohe balls eleased a hee diffeen speeds, 3 and 4 m/s wih he same placemen (1.5m, 6m, m). he simulaion esuls ae shown in (d), (e) and (f). 19% 6% 1% Figue 6. Illusion of how subjec B euned he sevice (Vs = 55m/s); conac poin of sevice (CP), posiion of sevice conac (-.1m,-1m,.75m). able 5. Conac poins of sevice and eceiving playe B s iniial pepaed posiion Conac poin eceive s iniial posiion Case Vs(m/s) X(m) Y(m) Z(m) X(m) Y(m) Z(m) ce(%) a b c Noe: Simulaion esuls of subjec B s sevice eun ae pesened hee(see figue 6). nalysis of ace asi found ha he speed of he fis sevice of fou ennis champions in caegoy was beween 5 and 65m/s [1]. heefoe, when conducing simulaions in his sudy, a elease speed of 55 m/s was adoped; hee diffeen posiions of sevice conac wee assumed a -.1, -1 and.75 m. he cou on he eceiving playe s side was assumed o have 1 placemens. When eceiving playe B pepaed in he middle posiion (m, 13m, m) fo he sevice eun (Figue 6a), he aea he was scoed on was found o accoun fo 19% of he whole aea (able 5). When he pepaed posiion of eceiving playe B was moved o (3m, 13m, m), he scoing aea was

9 Inenaional Jounal of Spo and Execise Science, (): educed o 6% (Figue 6b). If he pepaed posiion was moved o he sideline (6m, 13m, m), hen he undefendable aea inceased o 1%. he above analysis demonsaed ha he seving playe would have poblems acing when he eceiving playe pepaed in he middle posiion, while he side placemen would lead o acing. he simulaion esuls coesponded o acual ennis mach siuaions [1]; hus, he mahemaical model in his sudy was eaffimed o be eliable in simulaing sevice and sevice eun. ea(%) 16 1 U effec C B D Conclusions In a ennis mach, he eceiving playe ends o mae judgmens based on his pevious expeience. his is neihe objecive no effecive. his sudy is an endeavou o povide bee saegies by analyzing he elease speed, placemen and defensive space. he U effec is found hough expeimens and simulaion wih he fligh and defence algoihm developed in his sudy. he U effec demonsaes ha i is easie fo he eceiving playe o eun a sevice placed in fon of him and ha i is easie fo he seving playe o ace if ball is placed o he eceiving playe s laeal side. he simulaion esuls coespond o acual ennis mach siuaions, poving ha he algoihms of his sudy ae eliable and pacical. he simulaion esuls also show evidence ha he mahemaical model in his sudy can help he eceiving playe compue his vulneable aeas and he defensive space in accodance wih a ceain pepaed posiion. he above findings demonsae ha he mahemaical model based on he wo algoihms can seve as a aining ool fo ennis playes in sevice and sevice eun X -xis(m) Figue 7. Defence cuves fo he fou subjecs (Vs = 65 m/s). he fou subjecs all sood 1 m behind he baseline as he iniial posiion. hei laeal sides exended fom (m, 13m, m) o (6m,13m, m). posiion of sevice conac (-.1m,-1m,.75m). U effec he fou subjecs sevice eun (Figue 7) was hen simulaed fuhe in his sudy. he elease speed was assumed o be 65 m/s. hei iniial pepaed posiion was 1 m behind he baseline (m, 1.89m, m). he subjecs hen moved owad he sideline unil hey go o he posiion (5m, 1.89m, m). he fou subjecs opimal aea fo euning he sevice was found o be 3 o 4 m away fom he cene line. In ohe wods, when he eceiving playe sood in his defensive aea, he seving playe would have difficuly acing. Each cuve in Figue 7 was dened in he middle, which was emed he U effec in his sudy. he U effec demonsaed ha i was easie fo he eceiving playe o eun a sevice placed in fon of him and ha i was easie fo he seving playe o ace if he ball was placed o he eceiving playe s side. he esuls coespond o asi s suggesion ha he sevice should be placed on eihe side of he eceiving playe o peven a successful eun soe [1]. Refeences [1] C.L. asi. (1999).Dynamical analysis of uppe exemiy in ennis fla sevices, Ph.D. Disseaion, Naional aiwan Nomal Univesiy. [] C. Seele, R. Jones & P. Leaney. (8). Impoved ennis ball design: incopoaing mechanical and psychological influences, Jounal of Engineeing Design, 19: [3] J.C. Wang. (1989). he impac dynamics of a ennis ball siing a had suface, Ph.D. Disseaion, Oegon Sae Univesiy( US). [4].L. Wong. (1993). he impac compaison of ennis ball on clay cou and had cou, Nomal Univesiy. Ph.D. Disseaion, Naional aiwan [5] S. J. Haae, M.J. Cae & S.R. Goodwill. (3). he dynamic impac chaaceisics of ennis ball ennis aces, Jounal of Spos Sciences, 1: [6] C. uagawa &. Kojima (5). un oaion oques hough he hip joins duing he one-and wo handed bachand ennis soes, Jounal of Spos Sciences, 3: [7] S. anabe &. Io. (7). hee-dimensional analysis of he conibuions of uppe limb join movemens o he hoizonal ace head velociy a ball impac duing ennis seving. Spos Biomechanics, 6: [8] C.H. Chiu. (1). he effecs of giving a hoizonal Soe, diving a ball, and chopping a ball on he flying ajecoy of he ennis ball, Physical Educaion of R.O.C., 58: [9]. C. Soong. (198). Biomechanical analyses and applicaions of sho pu, discus and javelin hows, In D. N. Ghisa (Ed.), Human body dynamics: impac, occupaional, and ahleic aspecsoxfod: Claendon Pess, pp [1] C.H. Chiu. (9a). Esimaing he opimal elease condiions fo wold ecod holdes in discus, Inenaional Jounal of Spo and Execise Science, 1: [11] C.H. Chiu (9b). Discoveing opimal elease condiions fo he javelin wold ecod holdes by using compue simulaion,

10 34 Inenaional Jounal of Spo and Execise Science, Vol.. No. 1 Inenaional Jounal of Spo and Execise Science, 1: [1] B. R. Munson, D. F. Young &. H. Oiishi (1994). Fundamenals of Fluid Mechanics, nd ed.. Canada: Lehigh Pess, pp [13] FI (Inenaional ennis Fedeaion), Rules of ennis. (9). available online: hp: ules.asp. (ccessed 7 Novembe, 9). [14] C.H. Chiu (3). he simulaion sysem of offence and defence fo badminon singles, Jounal of Physical Educaion in Highe Educaion, 5: 1-1. UHORS BIOGRPHY Ching-Hua Chiu Employmen Pof., Gaduae Insiue of Spos and Healh Managemen, Naional Chung Hsing Univesiy, aiwan. Degee PhD Reseach ineess Biomechanics and opimal conol Chungoodman@yahoo.com.w