SICE Annual Conference 21 August 18-21, 21, The Grand Hotel, Taipei, Taiwan Analsis of Effects of Reounds and Aerodnamics for Trajector of Tale Tennis Junko Nonomura Mechanical Science and Engineering, Graduate School of Engineering, Nagoa Universit, Furo-cho, Chikusa-ku, Akira Nakashima Mechanical Science and Engineering, Graduate School of Engineering, Nagoa Universit, Furo-cho, Chikusa-ku, Yoshikau Haakawa Mechanical Science and Engineering, Graduate School of Engineering, Nagoa Universit, Furo-cho, Chikusa-ku, and RIKEN-TRI Collaoration Center, RIKEN, 2271-13, Anagahora, Shimoshidami, Moriama-ku, Astract In this paper, we firstl anale the effects of the reounds and aerodnamics for trajector of a tale tennis all. We firstl anale the effect of the aerodnamics with a criterion of evaluation, where the half area of the tale is considered as 9 divided areas. Furthermore, the drag and lift coefficients are identified assuming that the rotational velocit is invalid during the all fling. With the identified coefficients, the modeling errors of the tale and racket are secondl verified the criterion mentioned previousl. Some conclusions are finall shown. Inde Terms trajector, Reound Phenomenon, aerodnamics, Tale tennis I. INTRODUCTION A human detect a lot of information of eternal world with his ees, i.e., the sense of vision. With the otained information, he etract some useful specified information which is necessar for tasks, e.g., catching, throwing and hitting a all, and so on. Therefore, it is ver important and useful for a root in uncertain environment to use vision sensors and have algorithms for ojective tasks with its otained vision information. We aim to realie a root to pla tale tennis with a human as a tpical eample of roots in uncertain environment since plaing tale tennis is a deterous task for humans. For simplicit, consider the situation shown in Fig. 1, where a root tr to hit a fling all. In this situation, the strateg of Fig. 1. A root tries to hit a fling all. Fig. 2. trajectories of the cases of the top or ack spin. the root can e decomposed as the following sutasks: 1) To detect the states of the fling all with vision sensors. 2) To predict the all trajector. 3) To determine the trajector of the racket attached to the root for the hitting to achieve desired all trajector. The numer 1) means the image processing algorithm to otain the position, the translational velocit and the rotational velocit of the all. This algorithm is needed to performed in real time for the net task of the prediction[2]. The numer 2) means the prediction of the position and translational/rotational velocities of the all for the net task of the determination. The numer 3) means the determination of the trajector of the position and orientation of the racket for the all to follow desired trajector. In the sutasks 2) and 3), the all reounds from the tale and the racket ruer. Furthermore, the fling all is affected aerodnamic forces. It is therefore necessar to model the reounds and the aerodnamics. Suppose in this paper that the translational and rotational velocities can e measured an appropriate method[2]. Therefore, we concern on the analsis of effects which should e considered in the prediction of the all - 1567 - PR1/1/-1567 4 21 SICE
d n u o e R t = 2. t = 2.5 Fig. 3. t = 1.5 t = 3. t = 1. t = 3.5 t =.5 t = 4. Reound of the all from the racket ruer. trajector. In Fig. 2, the red and lue circles represent the cases of the top. The ack spins respectivel and the all velocit just efore the reound is the same as the each other case. It is easil confirmed that the velocit normal to the tale in the case of the ack spin is greater than that in the case of the top spin. In addition, the rotational velocit after reound also changes due to the friction. In order to predict the all trajector after the reound from the tale, it is necessar to consider the friction of the tale[1]. Second, let us consider the reound phenomenon etween the all and the racket ruer. Fig. 3 shows an eample of the reound of the all from the racket ruer. The green circles represent the same point on the all. It is confirmed that the rotational velocit aout the ais normal to the images changes to the inverse direction after the reound. This can not e epressed considering onl the friction and is the specific phenomenon in the case of the ruer. In order to achieve desired all trajector after the reound from the racket, it is necessar to consider this phenomenon[1]. Finall, we should consider the effect of the aerodnamic forces. Fig. 4 shows the aerodnamic forces for a fling all, i.e., lift and drag. The lift is proportional to the spinning of the all. The drag disturs the all motion and is proportional to the translational velocit. The trajector of the all changes easil these two effects since the weight of the all is small and the rotation velocit is high, e.g., 3[rpm]. In order to achieve desired all trajector after the reound, it is necessar to consider these phenomena. In this paper, we anale the effects of the reounds and aerodnamics for trajector of a tale tennis all. We firstl anale the effect of the aerodnamics with a criterion of evaluation, where the half area of the tale is considered as 9 divided areas. Furthermore, the drag and lift coefficients are identified assuming that the rotational velocit is invalid during the all fling. With the identified coefficients, the modeling errors of the tale and racket are secondl verified the criterion mentioned previousl. Some conclusions are finall shown. the reference frame with the -ais normal to the tale. Since there eists the friction etween the all and the tale, the velocities v changes to v. The model etween tale and the all is given (5) and (6)[1]. It is ver important to consider the tpe of the contact during the impact, i.e., the sliding and rolling contact. This can e determined using the tangent velocit given v T := [v v ] T + ω r = v rω v + rω, (1) where r := [ r] T R 3 is the contact point of the all from its center and r R + is the all radius. For the modeling, we make the following assumptions: Assumption 1: During the impact of the reound, the tpe of the contact etween the all and tale is a point contact. This means that an moment does not effect on the all during the impact. Assumption 2: The differences etween the translational and angular momentums efore and after the reound equal the impulses at the reound. Therefore, the impulse of the rotation is given r P, where P R 3 is the impulse in the translational direction. Assumption 3: The following simple ounce relationship in the direction holds: v = e t v (2) Assumption 4: The impulse in the and directions P := [P P ] T R 3 is given P = λ v T v T, λ μ P, (3) where μ is the dnamical coefficient of friction etween the all and tale. Assumption 5: The contact velocities v and v just efore and after the reound are in the same direction. That is, the following relation holds: v T = νv T, ν. (4) If ν, λ = μ P. From Assumptions, the following reound model is derived[1]. v = A v v + B v ω (5) ω = A ω v + B ω ω. (6) where 1 α αr A v := 1 1 α, B v := αr e t Airflow Lift Drag II. REBOUND MODELS OF TABLE AND RACKET RUBBER A. Reound Model of Tale Fig. 5 shows the reound of the all from the tale, where (v,ω ) and (v, ω )are the translational and rotational velocities of the all just efore and after the reound. Σ B is Fig. 4. ω Lift and Drag of aerodnamics. - 1568 -
ω v ω Vision cameras (15fps) 2) Vision cameras (9fps) Automatical all catapult 1) Fig. 5. Σ B v velocities just efore and after the reound 3α 2r 1 3α 2 A ω := 3α 2r, B ω := 1 3α 2 1 { μ(1 + et ) v v α = T (ν s > ) 2 5 (ν s ) ν s =1 2 5 μ(1 + e t) v v T. ν s > means the case of the sliding contact and ν s means the case of the rolling contance. B. Reound Model of Racket Ruer Fig. 6 shows the reound of the all from the racket ruer, where the meanings of the variales are the same as in SusectionII-A with respect to the racket frame Σ R attached to the racket as the -ais normal to the surface. In order to epress the effect of the elasticit parallel to the surface, For the model, we make the following assumptions: Assumption 6: The reound in the direction does not cause an effect in the and directions. Assumption 7: The impulse in the and directions P R 3 is related to the tangent velocit v T P = k p v T. (7) From Assumption 1-3, 6 and 7, the reound model of the racket is derived as the same form as (5) and (6). The coefficient matrices are as follows: v ω Σ B Visual field of vision cameras (15fps) 2)Visual field of vision cameras (15fps) Σ B B Tale Position of arrival Fig. 7. Tale Fling distance 1)Visual field of vision cameras (9fps) Default position Automatical all catapult Eperimental sstem and fling distance. 1 K v r A v := 1 K v, B v := K v r e r r 1 K ω r 2 A ω := K ω r, B ω := 1 K ω r 2 1 K v := k p m, k ω := k p I where e r is the coefficient of restitution of the ruer. III. BALL MOTION WITH AERODYNAMICS The all motion is epressed the free-fall equation if there is no aerodnamics. However in the real world, the fling all is affected the aerodnamics. Fig. 7 shows the eperimental sstem to detect the fling distance. The Σ B is set the edge of the tale. s are shot out from the automatic all catapult and detected 2 pairs of vision sensors. The sampling frequenc of the right cameras is 9[fps]. These cameras are called high speed cameras. The initial position and translational/rotational velocities are detected the high speed cameras. The sampling frequenc of the right cameras is 15[fps]. These cameras are called middle speed cameras. The position of arrival is detected these cameras. The lue square is the visual field of the Σ R ω v Σ B (a) Reound etween the all and the ruer Root s court ΣΒ Opponent s court.51[m].46[m] Fig. 6. Racket Coordinate. Fig. 8. Criterion of evaluation - 1569 -
automatic all catapult all The effects of the aerodnamic forces c) Σ 115cm ) 69cm a) 23cm (9fps) Fig. 1. Eperiment sstem to confirm rotational velocit Fig. 9. The fling distance etween the measured value and the calculated value without aerodnamic forces. high speed cameras and the red square shows that of the middle speed cameras. The fling distance is measured this eperimental sstem. For comparison, the fling distance in the case of the free all motion is calculated using the measured initial position and velocit. We use each 24 case of top spin all and ack spin all. The initial translational velocit of the top/ack spins are - 6.2/-6.5[m/s] and the initial rotational velocit of the top/ack spins are aout -34/34[rpm] around the ais of Σ B. Fig. 8 shows the tale area divided into 9 areas. The sie of one area is.46.51[m 2 ]. The influence of the aerodnamic forces are evaluated the 9 divided areas since plaers ma devise their strategies choosing the one of these areas. Fig. 9 shows the relation etween the fling distances of the all with and without aerodnamics. The horiontal/vertical aes are the calculated/measured fling distances. The dotted line means that the calculated values equal to the measured values. The circles and crosses represents the cases of the top and ack. The differences etween the dotted line and circles/crosses show the influence of the aerodnamic forces. The lue/red arrows show the influences in the case of top/ack spin. The difference of the influence in the case of ack spin is smaller than the case of top spin. However it is aout.2[m] which is close to the half sie of the sides of the divided areas. Therefore, in order to hit into the desired area, it is necessar to consider the aerodnamic forces. IV. IDENTIFICATION OF AERODYNAMICS As the model including Aerodnamics, (8) is introduced [3]. m p = mg 1 2 C DρA ṗ ṗ + 4 3 C M πρr 3 ω ṗ (8) where m: mass, g: acceleration of gravit, ρ: air densit, A: projected area, r: radius, C D : drag coefficient, C M : lift coefficient, p: position of the all, ω: rotational velocit. The values are given m=2.7 1 3 [kg], g=[,, 9.8] T [m/s 2 ], ρ=1.184 [kg/m 3 ](25 C), A=12.5 1 4 [m 2 ] and r=2. 1 3 [m]. In order to identif C D and C M, we assume that ω is constant value. Fig. 1 shows the eperimental sstem to confirm the assumption. Blue frames a), ) and c) are the visual fields of the high speed cameras. The frame a) is placed 23[cm] awa from the center line and the frame ) and c) are awa from 69[cm] and 115[cm], respectivel. s are shot out from the automatic all catapult parallel. From Fig. 11 through Fig. 13 show the rotational velocit. The horiontal ais is the speed scale markings of the automatic all catapult and the vertical ais is the rotational velocit detected the vision sensors. The closed circles are the average of rotational velocit just after the time when the all is shot the automatic all catapult. The lack dotted line represents the standard variation of the rotational velocit. The lue line is the lineariation of the closed circles. The pink cross shows the average of rotational velocit in the case of a) - c). The pink dotted line represents the standard variation of the rotational velocit in the each case ±3σ. All the pink crosses are contained in the area etween the lack dotted lines in the case of a) and ). Then, the rotational velocit can e assumed not to during the all fling. The coefficients of C D and C M are identified minimiing the following cost function: V (C; p i ):= 1 2 p i(t) p(t; C) (9) where p i (t) R 3 is the measured all trajector (i=1,,n) and p(t; C) R 3,C:=[C D C M ] R 2 + is otained solving (8) numericall with the initial values of p and ṗ which are given p i () and ṗ i () pi(δt) pi() Δt, Δt = 1 15 [s]. The minimiation is dealt with for each data of P i (N=5). The identified C D and C M are.54±.74 and.69±.29. The result is verified another data not to e used in the identification. An eample is shown in Fig. 14 and Fig. 15. In Fig. 14 and Fig. 15, the lue, red and green lines represent the measured all trajector and the numericall simulated ones with and without the aerodnamics. It is confirmed that the red lines almost coincide with the lue lines. V. VERIFICATION OF BALL TRAJECTORY In older to use the model of the racket, we have to consider the arrival position of the all which is hit the racket. Fig. 16 shows the eperimental sstem for the verification of - 157 -
the racket reound model, the all is shot the automatic all catapult and the translational/rotational velocities just efore and after the reound from the racket are measured the high speed cameras. The predicted positions of arrival are calculated using the aerodnamics (8) with the identified C D and C M and the translational/rotational velocities just after the reound otained from the reound model. We verif the modeling error of the reound from the racket ased on the distance etween the calculated and measured positions of arrival with the criterion the 9 divided areas. Fig. 17, Fig. 18 and Fig. 19 show the trajector and the position of arrival. The lue line and dot represents initial translational/rotational velocit are given the high speed cameras. The red line and dot represents initial translational/rotational velocit are given the racket model. In the case of the top spin, the average error is.9[m] and the standard deviation is.3[m]. In the case of the ack spin, the average error is.4[m] and the standard deviation is.4[m]. Because it is confirmed that the positions of arrival with all of the errors are included in one area, the model of the racket can e used for the prediction of the all. VI. CONCLUSION In this paper, we analed the effects of the reounds and aerodnamics for trajector of a tale tennis all. We firstl analed the effect of the aerodnamics with a criterion of evaluation, where the half area of the tale is considered as 9 divided areas. Furthermore, the drag and lift coefficients were identified assuming that the rotational velocit is invalid during the all fling. With the identified coefficients, the modeling errors of the tale and racket secondl were verified the criterion mentioned previousl. Some conclusions are finall shown. As future work,it is necessar to consider the trajector of the racket for a root to hit the all to a desired position of arrival. REFERENCES [1] A. Nakashima, Y. Ogawa, Y. Koaashi and Y. Haakawa, Modeling of Reound Phenomenon of a Rigid with Friction and Elastic Effects, Proceeding of ACC21, Baltimore, Marland, USA,21 (To appear) Fig. 12. Rotational velocit around the all catapult and the frame ) ω ω Fig. 13. Rotational velocit around the all catapult and the frame c) [2] A. Nakashima, Y. Tsuda, Y. Koaashi and Y. Haakawa, A Real- Time Measuring Method of Translational/Rotational Velocities of a Fling, Proc of IFAC Smp. Mech. Ss., Camridge, Massachusetts, USA 21( To e accepted) [3] Naoki OKADA, Tomohiro YABUUCHI, Takua FUNATOMI, Koh KAKUSHO and Michihiko MINOH, Estimating Unoserved Motion of a Acquiring Its Phsical Model from Image Sequences, IEICE, D vol.j91-d No.12, pp.295-296, 28 (In Japanese) ω Fig. 11. Rotational velocit around the all catapult and the frame a) - 1571 -
Fig. 17. Trajector and arrival position in the direction of ais Fig. 14. Model identification and Trajector in the direction of ais Fig. 18. Trajector and arrival position in the direction of ais Fig. 15. Model identification and Trajector in the direction of ais Racket Automaticall all catapult Σ B Vision cameras (9fps) Tale Fig. 16. Eperimental sstem to verif the racket model Fig. 19. Trajector and arrival position in the direction of ais - 1572 -