ICROS-SICE International Joint Conference 2009
August 18-21, 2009, Fukuoka International Congress Center, Japan

Modeling of Rebound Phenomenon between Ball and Racket Rubberwith Spinning EffectAkira Nakashima1 , Yosuke Kobayashi1 , Yuki Ogawa1 and Yoshikazu Hayakawa1,21

Dept. of Mechanical Science and Engineering, Graduate School of Engineering, Nagoya University, Nagoya, Japan(Tel: +81-52-789-2746; E-mail: a nakashima@nuem.nagoya-u.ac.jp)
2
RIKEN-TRI Collaboration Center, RIKEN, 2271-103, Anagahora, Shimoshidami, Moriyama-ku, Nagoya, Japan

Abstract: We model the rebound phenomenon between a ping-pong ball and the table/racket rubber. In the model
between the ball and racket rubber, it is assumed that the kinetic energy of the contact velocity is stored as the potential
energy due to the elasticity of the rubber. This assumption leads to that the impulse in the horizontal direction is proportional to the contact velocity. The vertical and horizontal COR in the models of the table and rubber are identiﬁed withthe measured velocity and spin of the ball by high-speed vision sensors. The models are veriﬁed by other experimental
data.
Keywords: Rebound phenomenon, Racket rubber, Vision sensors

1. INTRODUCTION
We aim to realize a robot to play table tennis with ahuman as a typical example of robots in uncertain environment since playing table tennis is a dexterous task forhumans. For simplicity, consider the situation of Fig. 1,
where a robot tries to hit a ﬂying ball. In this situation, thestrategy of the robot playing table tennis can be decomposed as the following subtasks: 1) To detect the states of
the ﬂying ball with vision sensors; 2) To predict the ball
trajectory; 3) To determine the trajectory of the racket
attached to the robot for the hit ball to achieve desired
trajectory. The number 1) means the image processing
algorithm to obtain the position, the translational and rotational velocities of the ball. The number 2) means theprediction of the position and translational/rotational velocities of the ball for the next task of the determination.
The number 3) means the determination of the trajectories of the position and orientation of the racket attachedto the robot for the ball to follow desired trajectory. In
the subtasks 2) and 3), the ball rebounds from the table
and the racket rubber. It is therefore necessary to model
the rebound phenomenon between the ball and the table/rubber in order to achieve the subtasks 2) and 3).
The rebound phenomenon between a ball and a surface has been studied by many researchers. Garwin [1]proposed the coefﬁcient of restitution (COR) in the di-rection parallel to the surface. He determined from experimental data which the contact is the rolling, sliding
or both together during the impact and whether the kinetic energy is stored as elastic one or not. Brody [2]considered the effect of friction of the surface with theassumption of no compression of the ball. He derived
the condition of the rolling contact which is related tothe translational and rotational velocities just before therebound. Shibukawa [3] derived the condition which determines whether the rolling contact happens or not during the impact. As speciﬁed situations of the rebound,
R. Cross [4] considered the effect of friction between theball and strings in tennis. He also considered other situations, e.g., the horizontal COR between balls of various
properties and a solid surface [5]. Furthermore, he investigated the bounce of a spinning ball being incident near
the normal to the surface as an example of situations ofhitting a ball by a racket because this situation was notinvestigated [6]. For the normal COR, R. Cross [7] investigated the relation between the normal COR and the
dynamics hysteresis of the impact force. Kawazoe [8]modeled the normal COR of a table tennis rubber as alinear function of the velocity just before the rebound,which shows the dissipation of the elastic energy. In thesestudies, the elastic properties of the ball and surface were
not considered analytically or evaluated as the values of
coefﬁcients on restitution experimentally.

Fig. 1 A robot try to hit a ﬂying ball.

We model the rebound phenomenon between a pingpong ball and the table/racket rubber. The model between
the ball and table is the generalized one of Shibukawa’s
model [3]. In the model between the ball and racket rubber, it is assumed that the kinetic energy of the contact velocity is stored as the potential energy due to the elasticityof the rubber. This assumption leads to that the impulsein the horizontal direction is proportional to the contact
velocity. The vertical and horizontal COR in the modelsof the table and rubber are identiﬁed with the measured
velocity and spin of the ball by high-speed vision sensors.
The models are veriﬁed by other experimental data.

- 2295 -PR0002/09/0000-2295 ¥400 © 2009 SICE

Fig. 4 Detection of the translation and rotation velocities.
Fig. 2 Experimental system.

they are the same as the other one. The left of Fig. 3
shows an example of the image data of the camera. Let
us consider the detection of the center of the ball. Since
the ball is white and the background is much darker thanthe ball, all the area of the ball can be easily detectedby the simple scanning shown in the right of Fig. 3. In
the scanning of (a), the arrows are scanned in turn fromthe left and top until the brightness of the scanned image is larger than a speciﬁed threshold. The scanningof (b)–(d) are similar to the one of (a). The positionsof the detected data with the threshold are recorded as
ξi := (ui , vi ) ∈ R2 (i = 1, · · · , 4). Consider the mediators of the lines between ξi+1 and ξi (i = 1, · · · , 4),
ξ5 := ξ1 as shown in the left of Fig. 4. Deﬁne ξc as thecenter of the ball. The center ξc is obtained by the leastsquares method with the following performance function:
�2
4 ��
�
(ξi+1 − ξi )T ξc − 12 (�ξi+1 �2 − �ξi �2 )�
,Vc (ξ) :=�ξi+1 − ξi �2
i=1

2. EXPERIMENTAL SYSTEM AND BALL
DETECTION
Fig. 2 shows the experimental system for detecting theball states. The table tennis is an international standard
one whose size is 1525(W)×760(H)×2740(D) [m]. Theball is shot out from the automatic ball catapult ROBOPONG 2040 (SAN-EI Co.) which is set at the end of thetable. The parameters of the ball are m = 2.7 × 10−3 [kg]and r = 2.0 × 10−2 [m]. The ﬂying ball is measured
by the high-speed vision sensors with 1000 [fps] (Hamamatsu Photonics K.K.). The array and pixel sizes are
232×232 and 2.9 × 10−5 [pixel/m]. The sampled data
are quantized as 2D image data with the monochromebrightness of 8bit (0–255). The focal length of the lens is
3.5 × 10−2 [m]. ΣB is the reference frame. The racket
is an usual one (Butterﬂy K.K.) and is ﬁxed at the table
whose surface is normal to the x-axis of ΣB . The cameras are calibrated with respect to ΣB . The 3D positionin ΣB can be therefore calculated with the corresponding
2D image positions of the two cameras [9].
By this experimental system, we get the translationaland rotational velocities of the ball for the modeling of
the rebound phenomenon. It is assumed that there is noeffect of the air resistance. By this assumption, the translational and rotational velocities of the ball are constants
when the ball is ﬂying freely. Here, the image processing and detection of the 2D image data of the two cameras are described with respect to the only one because(1)which is the sum of the squares of the distances of ξ fromthe mediators. Deﬁne pb ∈ R3 as the corresponding 3D
position vector of ξc ∈ R2 . The translational velocitiesof the ball just before and after the rebound are obtainedby the mean of all the velocities between the (i+1)th andith position as
�N −2
vb [i]
pb [i + 1] − pb [i], vb [i] :=, (2)vb = i=0
N −1
Δtwhere Δt = 1[ms] is the sampling time and N is thenumber of the frames before or after the rebound. Notethat [i] denotes the (i + 1)th frame.
Next, consider the detection of the rotational velocity of the ball. The right of Fig. 4 shows the ball in theith frame, where ω ∈ R3 is the rotation axis, pblj :=plj − pb and plj ∈ R3 is the feature point on the ball(j = 1, · · · , Nl ). For the jthe feature point, the following rotational relationship is hold [10]:� Δt,pblj [i + 1] = eω� Δt pblj [i], eω� Δt ≈ I3 + ω

(3)

3×3

� ∈ R
where ωis the skew-symmetric matrix whichis corresponding the cross product of ω and eω� Δt is linearized because of the small Δt. (3) is rewritten asp�blj [i]ω = −vblj [i], vblj [i] :=Fig. 3 Image data and Detection of the center of the ball.

- 2296 -pblj [i + 1] − pblj [i]
. (4)
Δt

rolling, by using the contact velocity given by⎡⎤
vbx − rωby
vbT := [vbx vby 0]T + ω × r = ⎣vby + rωbx ⎦ , (6)0where r := [0 0 − r]T ∈ R3 is the contact point of theball from its center and r ∈ R+ is the ball radius. For themodeling, we make the following assumptions:
Assumption 1: During the impact of the rebound, thetype of the contact between the ball and table is a pointof contact. This means that any moment does not effecton the ball during the impact.
Assumption 2: The differences between the translational and angular momentums before and after the rebound equal the impulses at the rebound. Therefore, theimpulse of the rotation is given by r × P , where P ∈ R3
is the impulse in the translational direction.
Assumption 3: The following simple bounce relationship in the z direction holds:

Fig. 5 Ball velocities just before and after the rebound.
If the direction of ω is given, (4) can be rewritten asθω =�vblj [i]�,�pblj [i] × nω �

(5)where ω := θω nω , �nω � = 1. The rotation axis ω is obtained by the least squares method with the performance
function given by combining the relations of (4) or (5)with respect to the frame i and feature point j.

3. MODELING OF REBOUND�= −et vbz
vbz

3.1 Rebound between the ball and table
Figure 5 shows the rebound of the ball from the table,
where (vb , ωb ) and (vb� , ωb� ) are the translational and rotational velocities of the ball just before and after the rebound and ΣB is the reference frame with the z-axis normal to the table. The simplest model of the rebound phe�
= −et vbz ,nomenon is described by the equations of vbz��vbx = vbx and vby = vby , where et is the coefﬁcient ofrestitution of the table in the z direction. In this model, it
is assumed that there does not exist friction on the table.
Therefore, the components x and y of the velocity vb do
not change. However, since there actually exists the friction, the velocity vb� just after the rebound changes dueto the friction and the rotational velocity ωb as shown in
Fig. 6. The red and blue circles represent the cases of
the top and back spins respectively and the ball velocities
just before the rebound are the same. It is easily con�in the case of the back spin is
ﬁrmed that the velocity vbz
greater than in the case of the top spin. In addition, therotational velocity ωb� also changes due to the friction. In
order to predict the ball trajectory after the rebound fromthe table, it is necessary to consider the friction.
For integrating the effect of the friction in the reboundmodel, it is very important to consider the type of the contact during the impact, i.e., the sliding and rolling contact. It can be determined which the contact is sliding or(7)

Assumption 4: The impulse in the x and y directions
Pxy := [Px Py 0]T ∈ R3 is given by
Pxy = −λvbT
, 0 ≤ λ ≤ μ|Pz |,�vbT �

(8)

where μ is the dynamical coefﬁcient of friction between
the ball and table.
Assumption 5: The contact velocities vb and vb� justbefore and after the rebound are in the same direction.
That is, the following relation holds:�= νvbT , ν ≥ 0.
vbT

(9)

If ν �= 0, λ = μ|Pz |.
Assumptions 4 and 5 are necessary to express the sliding and rolling contacts during the impact. Assumption 4means that the impulse in the x and y directions is related
to the one in the z direction as shown in Fig. 7 (a), where

Fig. 6 Defference of vb� when ωb is the top or back spin.

Fig. 7 Sliding and rolling during the impact.

- 2297 -the friction force in the x and y directions is proportionalto fz with μ and its direction is the opposite one of vbT .
λ in (8) is the magnitude of Pxy and is equal to or smaller
than μ|Pz | because the contact can be changed from sliding to rolling during the impact as shown in Fig. 7 (b).
Note that the static friction force does not work on the ballwhen the contact is rolling. Assumption 5 means that the�after the rebound does not becomecontact velocity vbT�= 0. The
negative because the rolling starts just after vbT
condition of ν �= 0 means that the contact is sliding during the impact.
From Assumptions 1, 2, the following equations hold:mvb� − mvb = PIωb� − Iωb = r × P ,Fig. 8 Rebound of the ball from the racket rubber.
Next consider the rolling case of (ii). From ν = 0 and(14), λ is given by

(10)
(11)

where m and I = 23 mr2 are the mass and moment ofinertia. Combining (7) and (10) yields the relation in thez direction:
Pz = −m(1 + et )vbz .

λ=

(12)

�
, has to satisfy
The contact velocity after the rebound, vbT(9) of Assumption 5. Substituting (13) into (9) yields
�
�
1
r2
λ
+
+ 1.
(14)
ν=−
�vbT � m
I

3.2 Rebound between the ball and racket rubber
Fig. 8 shows an example of the rebound of the ballfrom the racket rubber. The green circles represent thesame point on the ball. It is conﬁrmed that the rotational
velocity about the axis normal to the image plane changes
to the inverse direction after the rebound. This imply thatthe contact velocities between the ball and the rubber before and after the rebound are opposite each other. Thiscan not be expressed by considering only the friction andis the speciﬁc phenomenon in the case of the rubber due
to the storage of its elastic energy. In order to achieve desired ball trajectory after the rebound from the racket, itis necessary to consider this phenomenon.
The left of Fig. 9 shows the rebound of the ball fromthe racket rubber, where the meanings of the variables
are the same as in Section 3.1 with respect to the racket
frame ΣR attached to the racket as the z-axis normal to

It is sufﬁcient to check whether ν of (14) is positive or
not when the contact during the impact is supposed to besliding, that is, λ = μ|Pz |. Combining λ = μ|Pz |, (12),I = 23 mr2 and (14) leads to(15)

which is deﬁned as ν in the case of the sliding. It follows
that (i) if νs > 0, ν = νs and λ = μ|Pz | (the case of thesliding); (ii) if νs ≤ 0, ν = 0 (the case of the rolling).
Let us consider the sliding case of (i). Substituting (8)with λ = μ|Pz | and (12) into (10) and (11), we get thetranslational and rotational velocities after the rebound,vb� and ωb� , as the functions of those before the rebound:vb� = Av vb + Bv ωb
ωb� = Aω vb + Bω ωb ,
where

⎤
⎡
1−α
0
0
0
αr
1−α
0 ⎦ , Bv:=⎣−αr 0
Av := ⎣ 0
0
0
−et
0
0
⎡
⎤
⎡
3α
3α
0
0 − 2r 0
1− 2
3α⎦ , Bv:=⎣ 0
1−
Aω := ⎣ 3α
0
0
2r
2
0
0
0
0
0
⎡

α := μ(1 + et )|vbz |.
�vbT �

(19)

By the similar calculation of the case (i) with (19), we get
the coefﬁcient matrices of (16) and (17) as follows:
⎤
⎡3
⎡
⎤
2r
0
0
0
0
5
5
0 ⎦ , Bv:=⎣− 2r
0 0⎦Av:=⎣ 0 35
5
0 0 −et
0
0 0
⎡
⎤
⎡2
⎤
3
0 − 5r 00 0
5300⎦ , Bv:=⎣ 0 25 0⎦ . (20)Aω:=⎣ 5r
0
0
0
0 0 1

For deriving the relation in the x and y directions, thecontact velocity vb� after the rebound is calculated by thedeﬁnition of (6) with (10), (11) and Assumption 4:
�
�
r2
vbT
1
�
+
+ vbT .
vbT = −λ(13)m
I �vbT �

5|vbz |,
νs := 1 − μ(1 + et )2
�vbT �

2m�vbT �.
5

(16)
(17)
⎤
0
0⎦
0
⎤
0
0⎦
1

(18)

Fig. 9 The rebound between the ball and the rubber.

- 2298 -Table 1 Veriﬁcation of the table rebound model

the surface. In order to express the effect of the elasticity
parallel to the surface, we model the tangent motion of
the racket rubber as the motion of the virtual mass mαwith the spring ks and the distance s(t) ∈ R3 as shownin the right of Fig. 9. Note that mα is the equivalent massconsisting of the mass of the ball and the deformed areaof the rubber.
For the modeling, we make the following assumptions:
Assumption 6: The rebound in the z direction does
not cause any effect on the ones in the x and y directions.

α
vbT .
mα

(−2.8, −3.3, 279)(−3.8, 3.0, 196)(−4.0, 3.1, 199)(−3.8, 2.0, 209)(−3.8, 2.0, 194)5
6(−3.6, −2.1, 216)(−4.1, −2.2, −286) (−3.1, 2.0, −224) (−3.0, 2.1, −206)(−3.8, −1.8, −216) (−3.0, 1.7, −161) (−2.9, 1.7, −150)(−2.7, −2.2, −170) (−1.9, 1.9, −100) (−1.7, 1.9, −90.6)(−2.6, −2.2, −165) (−1.7, 2.0, −93.7) (−1.5, 2.1, −83.9)

4. PARAMETER IDENTIFICATION AND
MODEL VERIFICATIONThe coefﬁcient of restitution of the table et can be calculated as
�
h2
,
(25)
et =
h1
where h1 and h2 are the ﬁrst and second heights of the
dropped ball. et is identiﬁed as et = 0.93 by meaning
25 data. The dynamical friction coefﬁcient μ is estimatedas μ = 0.25 by measuring the drastic changed value ofthe spring balance instrument at the sliding where theweighted ball on the table with the weight 2.0[kg] ispulled.
The experimental veriﬁcation of the rebound model ofthe table is shown in Table 1. The experimental data arelimitted to the cases of the pure top spin (ωby < 0) or pureback spin (ωby > 0) with vby � 0. Hence (vbx , vbz ) [m/s]and ωby [rad/s] are only shown in Table 1. The number 1
and 2 are the cases of the top spin and the number 3–6 are
the cases of the back spin. The errors of the velocity androtation are ev = 1.0–11.1 [%] and eω = 1.5–10.5 [%].
There is a little big error of the order of 10 [%] since
the changes between frames of the data may be hiddenin the quantization errors. These was happened in thecases where the spins are smaller than the ones of theother data. Except for these results, the errors are smaller
than 10 [%]. Therefore, this veriﬁcation can show thevalidation of the model.

(21)

(22)

By (22) and Assumption 9, the impulse Pxy ∈ R3 in thex and y directions is obtained as
Pxy =

Model

1

4

(21) is solved with respect to vbT as
smax =

� , v� , ω� )
(vbx
bz
by

3

Assumption 8: The elasticity of the rubber is uniform
in the x and y directions. This means that the stiffness ks
is same in any x and y directions.
Assumption 9: The time rate of s(t) is a constantduring the impact.
Assumptions 1–4 are also assumed to hold here.
From Assumptions 7 and 8, the following energy conservation holds:

�(vbx , vbz , ωby )2

Assumption 7: The kinetic energy of the virtual mass
is conserved as the elastic energy without the dissipation.
smax ∈ R3 is the maximum displacement of the virtual
mass with the same direction of the contact velocity vbT .

1
1
ma �vbT �2 = ks �smax �2 .
2
2

No.

� T
T�
−kα s(t)dt = −kp vbT , kp :=mα kα ,
2
0
(23)

where T ∈ R+ is the time interval of the impact. From
Assumption 1–3 and 6, the equations of (10), (11) and(12) also hold here. Combining (23), (10), (11) and (12),we get the translational and rotational velocities after therebound, (vb� , ωb� ), are given by the same functional relationships as (16) and (17) with the coefﬁcient matrices asfollows:
⎤
⎡
⎡
⎤
0
0
0 0 0
1 − kpv
0 ⎦, Bv:=kpv⎣0 0 r ⎦1 − kpv
Av:=⎣ 000−er
0 −r 0
⎡
⎤
⎡
⎤
0 0 0
1
0
0
⎦
0
Aω:=kpω⎣0 0 −r⎦, Bv:=⎣0 1−kpω r2001−kpω r2
0 r 0
kp
kp
,
(24)
kpv:= , kpω :=mI
�
Fig. 10 Veriﬁcation of vbz
of the rubber case.

where er is the coefﬁcient of restitution of the rubber.

- 2299 -Table 2 Veriﬁcation of the contact velocity
No. Before the rebound After the rebound Calculated by Model
1
2
3
4
5
6

−3.79

2.35

2.21

−3.87

2.12

2.26

−3.77

2.04

2.20

−3.92

2.03

2.29

−3.68

2.02

2.15

−3.74

2.37

2.18

5. CONCLUSIONS
We have modeled the rebound phenomenon between aping-pong ball and the table/racket rubber. In the model
between the ball and racket rubber, it was assumed thatthe kinetic energy of the contact velocity was stored as thepotential energy due to the elasticity of the rubber. This
assumption led to that the impulse in the horizontal direction was proportional to the contact velocity. The verticaland horizontal COR in the models of the table and rubber
were identiﬁed with the measured velocity and spin of
the ball by high-speed vision sensors. The models wereveriﬁed by other experimental data.

�Fig. 11 Veriﬁcation of vbx
of the rubber case.

REFERENCES[1] R. Garwin, “Kinematics of an ultraelastic roughball,” Am. J. Phys., Vol.37, pp.88–92 1969.
[2] H. Brody, “That’s how the ball bounces,” Phys.
Teach., Vol.22 pp.494–497, 1984.
[3] Kanji Shibukawa, Dynamics of Physical Exercise,
Daishukanshoten, 1969 (in japanease).
[4] R. Cross, “Effects of friction between the ball andstrings in tennis,” Sports Eng., Vol.3, pp.85–97,
2000.
[5] R. Cross, “Measurements of the horizontal coefﬁcient of restitution for a superball and a tennis ball,”Am. J. Phys., Vol.70, No.5, pp.482–489, 2001.
[6] R. Cross, “Bounce of a spinning ball near normal incidence,” Am. J. Phys., Vol.73, No.10, pp.914–920,
2005.
[7] R. Cross, “The bounce of a ball,” Am. J. Phys.,Vol.67, No.3, pp.222–227, 1999.
[8] Y. Kawazoe and D. Suzuki “Prediction of Racket
Performance Based on the Table Tennis Impact
Analysis,” Porc. Dynamics and Design Conf., 218,
2002 (in japanease).
[9] B. K. Ghosh, N. Xi and T. J. Tarn, Control in
Robotics and Automation: Sensor-Based Integration, Academic Press, 1999.
[10] R. M. Murray, Z. Li and S. S. Sastry, A Mathematical Introduction to ROBOTIC MANIPULATION,
CRC Press, 1994.

�
Fig. 12 Veriﬁcation of ωbyof the rubber case.

The experimental data in the case of the rubber are
also only the pure top or back spin. The parameters
are indentiﬁed as er = 0.503, kpv = 0.68 and kpω =2.25 × 103 [1/m2 ]. Figures 10–12 show the veriﬁcations
�
�
�
, vbx
and ωby, which are the translational and roof vbz
tational velocities after the rebound normal and parallelto the rubber. The red data are used for the identiﬁcationof the parameters and the blue data are used for the veriﬁcation. The solid line represents the set of the points
where the values of the experiment and model equal eachother. It is conﬁrmed that the parameters are identiﬁed
well because the red data are close to the solid line. It isalso conﬁrmed that the blue data are close to the solid line
and are distributed around the one with the almost same
variance as the red data. This can show the relevance ofthe model of the rubber.
Table 2 shows the experimental data of the contact velocities before and after the rebound and the ones after
the rebound which are calculated by the model. It is conﬁrmed that the calculated velocities are in the opposite directions of the velocities before the rebound and close to
the velocities after the rebound. Therefore, the proposedmodel can represent the speciﬁed effect of the rubber.

- 2300 -