American Control Conference Marriott Waterfront, Baltimore, MD, USA June -Jul, WeB8. Modeling of Reound Phenomenon of a Rigid Ball with Friction and Elastic Effects Akira Nakashima, Yuki Ogawa, Yosuke Koaashi and Yoshikau Haakawa Astract We derive analtical models of the reound phenomenon etween a ping-pong all and the tale/racket ruer. In the reound model of the tale, the determination of the tpe of the contact during the impact is derived as a generalied condition in the -dimensional space. In the model etween the all and racket ruer, it is assumed that the kinetic energ of the tangent velocit is stored as the potential energ due to the elasticit of the ruer. This assumption leads to that the impulse in the horiontal direction is proportional to the tangent velocit. The models are verified eperimental data. I. INTRODUCTION It is a challenging task for a root to pla tale tennis with a human ecause plaing tale tennis is a deterous task for humans. For simplicit, consider the situation of Fig., where a root tries to hit a fling all. The deterous can e implied the strateg of the root plaing tale tennis, which can e decomposed as the following sutasks: ) To detect the states of the fling all with vision sensors; ) To predict the all trajector; ) To determine the trajector of the racket attached to the root for hitting the all to achieve desired trajector. The numer ) means the image processing algorithm to otain the position, the translational and rotational velocities of the all. The numer ) means the prediction of the position and translational/rotational velocities of the all for the net task of the determination. The numer ) means the determination of the trajectories of the position and orientation of the racket attached to the root for the all to follow desired trajector. In the sutasks ) and ), the all reounds from the tale and the racket ruer. It is therefore necessar to model the reound phenomenon etween the all and the tale/ruer in order to achieve the sutasks ) and ). Fig.. A root tries to hit a fling all. Authors are with Mechanical Science and Engineering, Graduate School of Engineering, Nagoa Universit, Furo-cho, Chikusa-ku, Nagoa, Japan akira@haa.nuem.nagoa-u.ac.jp, and Y. Haakawa is also with RIKEN-TRI Collaoration Center, RIKEN, 7-, Anagahora, Shimoshidami, Moriama-ku, Nagoa, Japan. The reound phenomenon etween a all and a surface has een studied man researchers. Garwin [] proposed the coefficient of restitution (COR) in the direction parallel to the surface. He determined from eperimental data which the contact is the rolling, sliding or oth together during the impact and whether the kinetic energ is stored as elastic one or not. Brod [] considered the effect of friction of the surface with the assumption of no compression of the all. He derived the condition of the rolling contact which is related to the translational and rotational velocities just efore the reound. Shiukawa [] derived the condition which determines whether the rolling contact happens or not during the impact. As specified situations of the reound, Cross [4] considered the effect of friction etween the all and strings of a racket in tennis. He also considered other situations, e.g., the horiontal COR etween alls of various properties and a solid surface []. Furthermore, he investigated the ounce of a spinning all eing incident near the normal to the surface as an eample of situations of hitting a all a racket [6]. For the normal COR, Cross [7] investigated the relation etween the normal COR and the dnamics hsteresis of the impact force. From the reviewed studies, the friction and elastic effects are ver important facts in the reound phenomenon. It is here assumed that a tale tennis all does not deform when reounding. The friction effect is dominative in the case of the reound on the tale and determines the tpe of the contact (sliding/rolling) during the impact, while the elastic effect is dominative in the case of the reound on the racket. As mentioned in the aove reviews, these effects are not considered analticall in the reound. The friction effect was considered in onl the -dimensional case in [] and the elastic effect was considered onl eperimentall in the some previous mentioned studies. We derive analtical models of the reound phenomenon etween a ping-pong all and the tale/racket ruer. In the reound model of the tale, the determination of the tpe of the contact during the impact is derived as a generalied condition in the -dimensional space. In the model etween the all and racket ruer, it is assumed that the kinetic energ of the tangent velocit is stored as the potential energ due to the elasticit of the ruer. This assumption leads to that the impulse in the horiontal direction is proportional to the tangent velocit. The models are verified eperimental data. II. EXPERIMENTAL SYSTEM AND BALL DETECTION Fig. shows the eperimental sstem for detecting the 978--444-74-7//$6. AACC 4
Racket Ball Automatical all catapult rotation ω ais v l [i] j p l j [i] Tale (a) p [i] C () r Vision cameras Fig. 4. Image data and calculation of the rotational velocit. Fig.. Eperimental sstem. all states. The tale is an international standard one with the sie of (W) 76(H) 74(D) [mm]. The all is shot out from the automatical all catapult ROBO-PONG 4 (SAN-EI Co.) which is set at the end of the tale. The fling all is measured the high-speed vision sensors with 9 [fps] (Hamamatsu Photonics K.K.). The arra and piel sies per meter are and α u,α v =. [piel/m]. The sampled data are quantied as D image data with the monochrome rightness of 8it ( ). The focal length of the lens is f = [mm] and the chip sie is. 4. [mm]. is the reference frame. The racket is a standard one (Butterfl K.K.) which is fied with respect to and is mounted such that its contact surface is normal to the -ais. The cameras are calirated with respect to. The translational and rotational velocities are calculated as the constant assuming that there is no effect of the air resistance. Figure shows an eample of the image processing. The figure (a) is the image data where the there are the racket and the all with the lack feature points. The figure () is the inaried one with an appropriate threshold. The white circle represents the all. In the figure (c), the green closed line is the oundar of the all. The four ellow points on the oundar of the all are detected the simple scannings as in the figures (c-) (c-4). In the scanning of (c-), the arrows are scanned in turn from the left and top until the rightness of the scanned image is larger than a specified threshold. The scanning of (c-) (c-4) are similar to the scanning of (c-). The positions of the detected data are recorded as ξ i := (u i,v i ) R (i =,,4). Consider the Racket Ball Feature point (a) () (c) (d) () () () ξ4 () (c-) (c-) (c-) (c-4) Fig.. Image data and detection of the center of the all. () ξ ξc ξ (4) () () ξ mediators of the lines etween ξ i+ and ξ i (i =,,4), ξ := ξ as shown in Fig. (d). Define ξ c as the center of the all. The center ξ c is otained the least squares method with the performance function, which is the sum of the squares of the distances of ξ from the mediators. The center of the all p is calculated ξ c in the left and right images [8]. The translational velocities of the all the mean of all the velocities etween the (i + )th and ith position as v = N i= v [i] N, v [i] := p [i + ] p [i], () t where t = /9 [s] is the sampling time and N is the numer of the frames. Note that [i] denotes the ith frame. Net, consider the detection of the rotational velocit of the all. Figure 4 (a) shows the feature points which are marked on the all as the lack prints. This image can e otained the appropriate gra scale transformation, sharpening and inariation. Figure 4 () shows the all in the ith frame, where ω R is the rotation ais, p lj := p lj p and p lj R is the feature point on the all (j =,,N l ). For the jthe feature point, the following rotational relationship holds [9]: p lj [i + ] = e ω t p lj [i], e ω t I + ω t, () where ω R is the skew-smmetric matri which corresponds to the cross product of ω. Note that the terms of the higher order with respect to t are approimated to ecause of the small t. Sustituting the nd eq. into the st eq. of () leads to p lj [i + ] = p lj [i] + t ωp lj [i], where ωp lj [i] is calculated as ωp lj [i] = ω p lj [i] = p lj [i] ω = p lj [i] ω. Then, we get p lj [i]ω = v lj [i],v lj [i] := p l j [i + ] p lj [i]. () t III. MODELING OF REBOUND A. Reound etween the all and tale Figure 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 and is the reference frame with the -ais normal to the tale. The simplest model of the reound phenomenon is descried the equations of v = e tv, v = v 4
ω v ω Assumption : The following simple ounce relationship in the direction holds: v v = e t v () Assumption 4: The impulse in the and directions P := [P P ] T R is given Fig.. Ball velocities just efore and after the reound. and v = v, where e t is the coefficient of restitution of the tale in the direction. In this model, it is assumed that there is no friction on the tale. Therefore, the components and of the velocit v do not change. However, since there actuall eists the friction, the velocit v just after the reound changes due to the friction and the rotational velocit ω as shown in Fig. 6. The red and lue circles represent the cases of the top and ack spins respectivel and the all velocities just efore the reound are the same. It is easil confirmed that the velocit v in the case of the ack spin is greater than in the case of the top spin. In addition, the rotational velocit ω 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. For integrating the effect of the friction in the reound model, 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, (4) where r := [ r] T R 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 : 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 : 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 is the impulse in the translational direction. P = λ v T v T, λ µ P, (6) where µ is the dnamical coefficient of friction etween the all and tale. Assumption : 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. (7) If ν, λ = µ P. Assumptions 4 and are necessar to epress the sliding and rolling contacts during the impact. Assumption 4 means that the impulse in the and directions is related to the one in the direction as shown in Fig. 7 (a), where the friction force in the and directions is proportional to f with µ and its direction is the opposite of v T. λ in (6) is the magnitude of P and is equal to or smaller than µ P ecause the contact can e changed from sliding to rolling during the impact as shown in Fig. 7 (). Note that the static friction force does not work on the all when the contact is rolling. Assumption means that the tangent velocit v T after the reound does not ecome negative ecause the rolling starts just after v T =. The condition of ν means that the contact is sliding during the impact. From Assumptions,, the following equations hold: Before Reound µf ω f mv mv = P (8) Iω Iω = r P, (9) v v T After Reound µf ω (a) Case where the contact is sliding during the impact f v T v Before Reound f After Reound ω f v ω µf v v T v T = () Case where the contact is changed from sliding to rolling during the impact Fig. 6. Deference of v when! is the top or ack spin. Fig. 7. Sliding and rolling during the impact. 4
where m and I = mr are the mass and moment of inertia. Comining () and (8) ields the relation in the direction: P = m( + e t )v. () For deriving the relation in the and directions, the tangent velocit v after the reound is calculated the definition of (4) with (8), (9) and Assumption 4: ( ) v T = λ m + r vt I v T + v T. () The tangent velocit after the reound, v T, has to satisf (7) of Assumption. Sustituting () into (7) ields ν = λ v T ( m + r I ) +. () It is sufficient to check whether ν of () is positive or not when the contact during the impact is supposed to e sliding, that is, λ = µ P. Comining λ = µ P, (), I = mr and () leads to ν s := µ( + e t) v v T, () which is defined as ν in the case of the sliding. It follows that (i) if ν s >, ν = ν s and λ = µ P (the case of the sliding); (ii) if ν s, ν = (the case of the rolling). Let us consider the sliding case of (i). Sustituting (6) with λ = µ P and () into (8) and (9), we get the translational and rotational velocities after the reound, v and ω, as the functions of those efore the reound: v = A v v + B v ω (4) ω = A ω v + B ω ω, () where α αr A v := α,b v := αr e t A ω := α r α r,b ω := α α α := µ( + e t ) v v T. (6) Net consider the rolling case of (ii). From ν = and (), λ is given λ = m v T. (7) B the similar calculation of the case (i) with (7), we get the coefficient matrices of (4) and () as follows: A v :=,B v := r r e t A ω := r r,b ω :=. (8) Reound Fig. 8. t =. t =. t =. t =. t =. t =. t =. t = 4. Reound of the all from the racket ruer. B. Reound etween the all and racket ruer Figure 8 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 image plane changes to the inverse direction after the reound. This implies that the contact velocities etween the all and the ruer efore and after the reound are opposite to each other. This can not e epressed considering onl the friction and is the specific phenomenon in the case of the ruer due to the storage of its elastic energ. In order to achieve desired all trajector after the reound from the racket, it is necessar to consider this phenomenon. Figure 9 (a) shows the reound of the all from the racket ruer, where the meanings of the variales are the same as in Section. with respect to the racket frame attached to the racket as the -ais normal to the surface. In order to epress the effect of the elasticit parallel to the surface, we model the tangent motion of the racket ruer as the motion of the virtual mass m α with the spring k s and the distance s(t) R as shown in Fig. 9 (). Note that m α is the equivalent mass consisting of the mass of the all and the deformed area of the ruer. 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 is related to the tangent velocit v T v v ω ω (a) Reound etween the all and the ruer Fig. 9. P = k p v T. (9) Racket vt Ball m Racket m α vt () Virtual spring model parallel to the surface The reound etween the all and the ruer. s s 4
Assumptions 4 are also assumed to hold here. From Assumption and 6, the equations of (8), (9) and () also hold here. Comining (9), (8), (9) and (), we get the translational and rotational velocities after the reound, (v,ω ), are given the same functional relationships as (4) and () with the coefficient matrices as follows: A v := k pv k pv,b v :=k pv r r e r A ω :=k pω r r,b ω := k pωr k pω r k pv := k p m, k pω := k p I, () where e r is the coefficient of restitution of the ruer. IV. MODEL VERIFICATION The parameters of the all are m =.7 [kg] and r =. [m]. The coefficient of restitution of the tale e t can e calculated as h e t =, () h where h and h are the first and second heights of the dropped all. e t is identified as e t =.9 averaging data points. The dnamical friction coefficient µ is estimated as µ =. measuring the drastic change in the value of the spring alance instrument at the sliding where the weighted all on the tale with the weight.[kg] was pulled. A. The case of the reound on the tale The reound model on the tale is verified the 4 cases of (a) Top spin, () Back spin, (c) Side-top spin and (d) Side-ack spin as illustrated in Fig.. Figure shows the verifications of the cases of the top and ack spins, where the circles and squares represent the top and ack spins. Since the eperimental data of (a) and () is the cases of the pure top spin (ω < ) and pure ack spin (ω > ) with v, ω, ω, (v,v ) [m/s] and Fig.. (a) Top spin () Back spin (c) Side-top spin () Side-ack spin Situations of the verification of the tale. - -. -4-4. v - - -4 - - -. -.4 -.6 -.8 Ep. Value [m/s] Fig.. -4-4 -. - - -4 v Ep. Value [m/s] ω -4 - Ep. Value [rad/s] Fig.. - - ω - Ep. Value [rad/s] Verification of the top and side spins. - v - - Ep. Value [m/s] ω - - - Ep. Value [rad/s] - - 4 4 v 4 Ep. Value [m/s] v 4 Ep. Value [m/s] ω - Ep. Value [rad/s] Verification of the side-top and side-ack spins. ω [rad/s] are onl shown. The horiontal and vertical aes are the eperimental values and the calculated values from the model of the translational and rotational velocities after the reound. The solid lines represent the cases where these values are the same. The general situations are represented Fig., which shows the cases with the multi rotational aes where the circles and squares represent the side-top and side-ack spins. Note that the side spin is ω >, the top spin is ω < and the ack spin is ω >. It is confirmed that the almost data in Figs. and are close to the solid lines. It ma e caused the quantiation errors of the image data that a few data is a little far from the solid lines. The image coordinates (u,v) is related to the cartesian ones ( c, c, c ) as u = f c α u c and v = f c α v c, where the parameters f, (α u,α v ) are shown in Section. c =.[m] is in the optic ais and the distance etween the all and the camera. The igger errors of the translational and rotational velocities are aout.[m/s] and [rad/s]. The corresponding distances in ( c, c ) of the errors in a frame are calculated as e v =. t and e ω = r t. From the relationships u = f c α u c and v = f c α v c, the errors in (u,v) are calculated as e v uv =.8 and e ω uv =.64 [piel]. Since e v uv and e ω uv are smaller than [piel], the errors of. [m/s] and [rad/s] ma e caused the quantiation errors. Because all the errors in Figs. and are smaller than these errors, the eperimental data shows the validation 44
v 4 v v (a) Top spin -. - -. - - - Ep. Value [m/s] 4 Ep. Value [m/s].8.6.4.. Ep. Value [m/s] () Back spin (c) Side-top spin ( > ) (d) Side-top spin ( < ) v v 6 ω 8 ω 8 ω Fig.. Situations of the verification of the racket. of the reound model on the tale. B. The case of the reound on the racket The reound model on the racket is verified the 4 cases of (a) Top spin, () Back spin, (c) Side-top spin (v > ) and (d) Side-ack spin (v < ) as illustrated in Fig.. The parameters in the reound model are indentified as e r =.8, k p =.9 [/kg]. Figures 4 and show the case of the pure top and ack spins and the case of the sidetop spin of v > and v <. The red data is used for the identification of the parameters and the lue data is used for the verification. The circles and squares represent the top and ack spin in Fig. 4 and the side-top spin of v > and the one of v < in Fig.. It is confirmed that the red and lue data are close to the solid lines with the errors due to the quantiation errors of the image data. Tale I shows some of the eperimental data of the contact velocities efore and after the reound and the ones after the reound which are calculated the model. It is confirmed that the calculated velocities are in the opposite directions of the velocities efore the reound and close to the velocities after the reound. Therefore, the proposed model can represent the specified effect of the ruer. V. CONCLUSIONS We derived analtical models of the reound phenomenon etween a ping-pong all and the tale/racket ruer. In the reound model of the tale, the determination of the tpe of the contact during the impact was derived as a generalied condition in the -dimensional space. In the model etween the all and racket ruer, it was assumed - - - v - Ep. Value [m/s] Fig. 4. 4 - -4-6 ω - Ep. Value [rad/s].. v Ep. Value [m/s] Verification of the top spin and ack spin. 4 - -4 - Ep. Value [rad/s] 6 4 Ep. Value [rad/s] 6 4 Ep. Value [rad/s] Fig.. Verification of the side-top spins of v > and v <. TABLE I VERIFICATION OF THE TANGENT VELOCITY No. Before the reound After the reound Calculated Model.79...87..6.77.4. 4.9..9.68.. 6.74.7.8 that the kinetic energ of the tangent velocit was stored as the potential energ due to the elasticit of the ruer. This assumption led to that the impulse in the horiontal direction is proportional to the tangent velocit. The models were verified eperimental data. As future work, it is necessar to reduce the quantiation errors and verif the model in additional reound situations. Furthermore, we would tr to make a method to detect the all state in real-time. REFERENCES [] R. Garwin, Kinematics of an ultraelastic rough all, Am. J. Phs., Vol.7, pp.88 9 969. [] H. Brod, That s how the all ounces, Phs. Teach., Vol. pp.494 497, 984. [] Kanji Shiukawa, Dnamics of Phsical Eercise, Daishukanshoten, 969 (in japanease). [4] R. Cross, Effects of friction etween the all and strings in tennis, Sports Eng., Vol., pp.8 97,. [] R. Cross, Measurements of the horiontal coefficient of restitution for a superall and a tennis all, Am. J. Phs., Vol.7, No., pp.48 489,. [6] R. Cross, Bounce of a spinning all near normal incidence, Am. J. Phs., Vol.7, No., pp.94 9,. [7] R. Cross, The ounce of a all, Am. J. Phs., Vol.67, No., pp. 7, 999. [8] B. K. Ghosh, N. Xi and T. J. Tarn, Control in Rootics and Automation: Sensor-Based Integration, Academic Press, 999. [9] R. M. Murra, Z. Li and S. S. Sastr, A Mathematical Introduction to ROBOTIC MANIPULATION, CRC Press, 994. 4