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Bouncing model for the table tennis trajectory prediction and the strategy of hitting the ball

Authors:

Abstract

The bouncing model plays an important role in the trajectory prediction process when the humanoid robots play table tennis. A nonlinear bouncing model based on Momentum Theorem and Moment of Momentum Theorem considering spin of the ball is established to describe the collision between the ball and the table. With the constructed bouncing model, the hitting strategy is discussed to hit the ball the way we expect. The experiments show that with the existing aerodynamic model, we can predict the table tennis trajectory within the error range we expect.
Bouncing Model for the Table Tennis Trajectory
Prediction and the Strategy of Hitting the Ball
Han Bao, Xiaopeng Chen ZhanTao Wang Min Pan, Fei Meng
Key Laboratory of Robots and Systems
Ministry of Education, P.R. China
Intelligent Robotics Institute, School of
Mechatronic Engineering, Beijing Institute of
Technology
Beijing, 100081, P.R. China
bao_han327@126.com,xpchen@bit.edu.cn
Measurement Technology
Beijing Aerospace Institute
for Metrology and
Measurement
Technology
Beijing, 100081, P.R. China
zhantao_wzt@163.com
Key Laboratory of Robots and Systems
Ministry of Education, P.R. China
Intelligent Robotics Institute, School of
Mechatronic Engineering, Beijing Institute of
Technology
Beijing, 100081, P.R. China
sonyejin@bit.edu.cn, mfly@yahoo.com.cn
Abstract – The bouncing model plays an important role in the
trajectory prediction process when the humanoid robots play table
tennis. A nonlinear bouncing model based on Momentum Theorem
and Moment of Momentum Theorem considering spin of the ball is
established to describe the collision between the ball and the table.
With the constructed bouncing model, the hitting strategy is
discussed to hit the ball the way we expect. The experiments show
that with the existing aerodynamic model, we can predict the table
tennis trajectory within the error range we expect.
Index Terms - error of prediction; bouncing model; the
strategy of hitting the ball
I. INTRODUCTON
The research on the Ping-Pong robot began when John
Billiney suggested having a robot table tennis game in 1983
[1]. In the recent years the Ping-Pong robot has become a
hotspot region of the robotic technology. To make a humanoid
robot playing Ping-Pong like a professional player is exciting
and challenging task for its complexity and it is attracting
more and more people now to investigate.
On the basis of trajectory prediction, setting the paddle’s
velocity and orientation, and calculating the joint parameter
with inverse kinematics, Ping-Pong robot can hit the ball in
the desired direction. In the whole procedure, the flying
trajectory prediction is the key to hit the table tennis the way
we expect. The trajectory of the Ping-Pong could divide into
two segments, flying in the sky and bouncing from the table.
Lots of research has been done to construct the aerodynamic
model of the ball when it flies in the sky and gets the position
and velocity information [2] [3] [4] [5]. But few of people
study of the collision between the ball and the table. One
approach is adopting the concept of restitution coefficient to
establish the linear model [2] [6], and another is constructing
the nonlinear model using Momentum Theorem and Moment
of Momentum Theorem [7] [8].
This work was supported by the Research Fund for the Doctoral Program
of Higher Education of China (Grant No. 20101101120006), the National
High-Technology Research and Development Program (Grant No
2011AA040201), the 111 Project (Grant No. B08043), National Science
Foundation for Distinguished Young Scholar (Grant No. 60925014). The
corresponding author is Xiaopeng Chen.
The advantage of the linear model is with few of
computing and easy to solve, but with the disadvantage of
neglecting the spin of ball, the model is not with the enough
accuracy. The nonlinear model takes into account the effect of
the spin of the ball and it fits the actual motion of the table
tennis. The hitting model of the paddle is similar to the model
the ball collides with the table, but few researches have been
done to it. In this paper the hitting model will be established
and the strategy of hitting the ball will be discussed.
II. THE SYSTEM OF THE PING-PONG ROBOT
We use two humanoid robots and four 200fps high speed
cameras and two low speed cameras to construct the
experiment system. The construction of system is like the
Fig1. The high speed cameras fix on the ceiling of the robot
side to collect initial kinematic information of the ball, and the
low speed cameras fixed behind the robots to get the robots’
location. The table tennis we use is the official ball suggested
for game by the ITTF, with the diameter 0.04m, the radius
0.02m, the mass 0.0027kg, the inertia I along the diameter
is 2
2/3mr . a
ρ
is the density of the air whose value is 1.29
kg/m3 in Beijing, and g is the acceleration of gravity with the
value 2
9.8015m/s . The word coordinate frame of the system
is defined as the Fig.1 shows, with the X axis along the
parallel to the longer edge of the table, the Z axis vertical to
the table flat, and the Y axis defined according to the right
hand rule.
III. THE BOUCING MODEL BETEEWM THE BALL AND
THE TABLE AND THE STRATEGY OF HITTING THE
BALL
The velocity and angular velocity are both changed after
the ball bounces off the table as in Fig2 shows. From amounts
of experiments, we find that the inaccuracy of the bouncing
model is the important factor which causes the error of the
trajectory prediction. So we must have an accurate bouncing
model that can describe the process the ball collides with the
table.
2002
978-1-4673-1278-3/12/$31.00 ©2012 IEEE
Proceedings of 2012 IEEE
International Conference on Mechatronics and Automation
August 5 - 8, Chengdu, China
For the modeling, we present some assumptions.
Assumption 1: In the Z direction, the emergence velocity is
proportional to the incidence velocity, with the equation:
ez v iz
vkv=− , (1)
where the v
k stands for the restitution coefficient vertical to
the table, and its value is a table dependent constant.
Fig. 1 The Ping-Pong Robot System.
Fig.2 Ball velocity before and after bouncing
Assumption 2: The angular velocity along the Z direction
z
ω
is not changed after the bouncing.
Assumption 3: During the collision process, the ball
either slides or rolls, without the situation that the ball both
slides and rolls.
Assumption 4: The direction of the velocity of the point
where the ball contacts with the ball is not changed.
Now, we go to the bouncing model. The incidence
velocity of the contact point is:
(,,0)
bi ix iy iy ix
vvrvr
ωω
=− + , (2)
The direction is r
v
:
bi
r
bi
v
vv
=. (3)
The impulse S along r
v
is that:
SN
μ
=−
³. (4)
In the Z direction, according to the Momentum Theorem, The
impulse
z
Sis defined as:
()
z
ez iz
SNmvv
μ
==−
³, (5)
where
μ
stands for the dynamical coefficient of friction of the
table. According to the Momentum Theorem and Moment of
Momentum Theorem, we can find that:
()
x
ex ix
Smvv=−
, (6)
()
yeyiy
Smvv=−
, (7)
()
xiyey
Sr I
ωω
=−
, (8)
()
yexix
Sr I
ωω
=−
, (9)
where
x
Sand y
Sis the component of S, with the equation:
22
()
()()
ix iy
x
ix iy iy ix
vr
SS
vrvr
ω
ωω
=
−++
, (10)
22
()
()()
iy ix
y
ix iy iy ix
vr
SS
vrvr
ω
ωω
+
=
−++
. (11)
From (4), (5), (6), (7), (10) and (11), we can find that:
22
22
(1 ) | | ( )
()()
(1 ) | | ( )
()()
vizix iy
ex ix
ix iy iy ix
viziy iy
ey iy
ix iy iy ix
kvv r
vv
vrvr
kvv r
vv
vrvr
μω
ωω
μω
ωω
+−
=−
°−++
°
®++
°=−
°−++
¯
, (12)
and from (4), (5), (8), (9), (10) and (11), we can find:
22
22
3(1 )| |( )
2( ) ( )
3(1 )| |( )
2( ) ( )
viziy ix
ex ix
ix iy iy ix
vizix iy
ey iy
ix iy iy ix
kvv r
rv r v r
kvv r
rv r v r
μω
ωω
ωω
μω
ωω
ωω
++
=−
°−++
°
®+−
°=−
°−++
¯
. (13)
According to (12) and (13), the emergence velocity of the
contact point is:
22
22
2.5 (1 ) | |
(1 )
()()
2.5 (1 ) | |
(1 )
()()
viz
bex bix
ix iy iy ix
viz
bey biy
ix iy iy ix
kv
vv
vrvr
kv
vv
vrvr
μ
ωω
μ
ωω
+
=−
°−++
°
®+
°=−
°−++
¯
, (14)
with the assumption 4, if the condition:
22
(1 ) | | 0.4
()()
viz
ix iy iy ix
kv
vrvr
ωω
+
−++
, (15)
satisfies, the emergence velocity of the contact point is zero.
In that case, the ball is rolling, without the sliding friction.
Substituting the equal condition of (15) into (12) and (13), the
emergence velocity and angular velocity is:
0.6 0.4
0.6 0.4
ex ix iy
ey iy ix
vv r
vv r
ω
ω
=+
®=−
¯, (16)
2003
0.4 0.6 /
0.4 0.6 /
ex ix iy
ey iy ix
vr
vr
ωω
ωω
=−
®=+
¯. (17)
The sliding situation and rolling situation is illustrated in Fig3.
Using the condition (15), we can get the emergence
velocity and emergence angular velocity respectively like:
ei i
vAvB
ω
=+, (18)
ei i
Cv D
ωω
=+ , (19)
while in the sliding situation:
100
01 0
00 v
A
k
α
α
ªº
«»
=−
«»
«»
¬¼
00
00
000
r
Br
α
α
ªº
«»
=−
«»
«»
¬¼
3
00
2
300
2
000
r
Cr
α
α
ªº
«»
«»
«»
=«»
«»
«»
«»
¬¼
3
100
2
3
01 0
2
001
D
α
α
ªº
«»
«»
«»
=−
«»
«»
«»
«»
¬¼
,
where
α
is equal to 22
(1 ) | |
()()
viz
ix iy iy ix
kv
vrvr
μ
ωω
+
−++
, while in
the rolling situation:
0.6 0 0
0 0.6 0
00 v
A
k
ªº
«»
=«»
«»
¬¼
00
00
000
r
Br
α
α
ªº
«»
=−
«»
«»
¬¼
00.6/0
0.6 / 0 0
000
r
Cr
ªº
«»
=«»
«»
¬¼
0.4 0 0
0 0.4 0
001
D
ªº
«»
=«»
«»
¬¼
.
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6OLGLQJ
:KHQ%DOO
5ROOLQJ
%HIRUH
5HERXQG
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5HERXQG
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5HERXQG
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Fig.3 Sliding and Rolling When Impact
IV. THE STRATEGY OF HITTING THE BALL
The situation the ball bounces off the paddle is like that
the ball bounces off the table in Fig4. If we want to hit the ball
the way we expect, we must calculate the paddle’s velocity
and orientation with the known incidence velocity and angular
velocity and desired emergence velocity and angular velocity.
The multilayered rubber paddle can cause the analysis of the
model complex with the function of reserving the energy, so
to have a model we can solve, we must simplify the problem
that we have a paddle without rubber. In our Ping-Pong robot
system we also do like this.
Fig.4 Ball velocity before and after hitting
Let’s simplify the model further that we hit the ball along
the paddle’s normal vector, and the ball always rolls never
slides. With the conclusion we get in the part III, we can have
equations like:
0.6( ) 0.4
0.6( ) 0.4
()
p
ex px pix px piy
p
ey py piy py pix
pez pz pv piz pz
vv vv rw
vv vv rw
vv kvv
−= − +
°−= − −
®
°−=− −
¯
. (20)
Solving (20) we can get the paddle velocity
p
v
, and with it we
can calculate the final angular velocity with:
0.4 0.6( ) /
0.4 0.6( ) /
pex pix piy py
pey piy pix px
pez piz
vvr
vvr
ωω
ωω
ωω
=−
°=+
®
°=
¯
. (21)
Now we can talk about the hitting strategy. If we want to
hit the ball with more spin and with a higher speed, we can
calculate the paddle velocity to meet our needs. The only pity
is we can’t control the spin independently unless we relax the
constraint that we hit the ball along the paddle’s normal vector
and allow having another velocity component t
v, which
perpendicular the
p
v.
V. EXPREMENT
Establishing the aerodynamic model for the process the
table tennis flying in the air together with the established
bouncing model, we can have the whole trajectory predicted.
The Ping-Pong ball flying in the air is given four forces as the
Fig5 shows.
g
F
stands for the gravity, b
F
stands for the buoyancy force,
2004
d
F
stands for the air resistance, and m
F
stands for the
Magnus forces caused by the spin. As b
mm, according to
[7], neglecting the buoyancy force, resultant force F has
following form:
b
F
g
F
d
F
m
F
Fig. 5 The four forces given to ball in the air
23
11
88
da ma
F
mg C D v v C D w v
ρπ ρπ
→→ → →
=− + ×
, (22)
where d
Cis the drag coefficient with the value 0.5 in Beijing,
m
Cis the Magnus coefficient which can be received form
offline training, b
mis the mass of the air with the same
volume as the ball. We can have the initial kinematics
parameter for the ball as [6] trained for the trajectory
prediction using the method of maximum error minimization
as:
min max , ,
xi xi yi yi zi zi
pp pp pp
∧∧
½
ªº
§·§·§·
−−−
®¾
¨¸¨¸¨¸
«»
©¹©¹©¹
¬¼
¯¿
(23)
where the i
p
stands for the value collected by the high speed
vision system, and i
p
stands for the value we predict.
With the parameter trained, we can predict the whole
trajectory with our system to hit the ball. Fig6 is the three
dimension ball trajectory and prediction trajectory. Fig7~9 are
vision data versus prediction data in each axis respectively.
From the figure bellow, we can find the model we established
is exactly accurate.
Table.1 list the error in position, velocity, and hitting
time between vision data and prediction data, as well as the
angle between the actual velocity and prediction velocity.
With the Table.1 we can find that the error is in our tolerance
to hit the ball the way we expected.
VI. CONCLUSION
In this paper, the bouncing model is established when
the table tennis collides with the table, and extended to the
case when the Ping-Pong robot hit the ball. With the model,
we can have a good prediction of the trajectory and hit the ball
the way we desired. Experiments show that the bouncing
model we constructed is suited and helpful to let the robot
hitting the ball.
0
1
2
3
-1
0
1
0
0.5
1
X / m
3D trajec t predic tion
Y /m
Z /m
original dat a
predict data
Fig.6 3D ball trajectory versus prediction trajectory
00.05 0.1 0. 15 0.2 0. 25 0.3 0.35 0.4 0. 45 0.5
-0.5
0
0.5
1
1.5
2
2.5
time /sec
X /m
original dat a
predict data
error
Fig.7 Vision data versus prediction data in X axis
00.05 0.1 0. 15 0.2 0. 25 0.3 0.35 0.4 0. 45 0.5
-0.05
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
time /sec
Y /m
original dat a
predict data
error
Fig.8 Vision data versus prediction data in Y axis
2005
00.05 0.1 0. 15 0.2 0. 25 0.3 0.35 0.4 0. 45 0.5
-0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
time /sec
Z /m
original dat a
predict data
error
Fig.9 Vision data versus prediction data in X axis
Table.1 The error of position, velocity and hitting time of prediction
Position error (mm) Velocity error (m/s) Angle
(e)
Error of
Time
(ms)
X Y Z X Y Z
31.2 27.1 25.1 0.09 0.03 0.31 2.69 13.8
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[3] Russell L. Andersson, “A Robot Ping-Pong Player Experiment in Real-
Time Intelligent Control,” The MIT Press
[4] Z. Zhang, D. Xu, “High-Speed Vision System Based on Smart Camera
and Its Target Tracking
Algorithm
,” Robots, vol. 31, pp. 229-234, 2009.
[5] Lei Sun, Jingtai Liu, Yingshi Wang, Lu Zhou, Qi Yang, Shan He, “Ball’s
Flight Trajectory Prediction for Table-tennis Game by Humanoid Robot,”
International Conference on Robotics and Biomimetics, Guilin,
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[6] G. Meng, Y. Chen, “Mechanical Analysis on Table Tennis Sport,”
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[8] R. K. Adair, P. J. Brancazio, “The physics of baseball,” American
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In this paper, a novel ball trajectory prediction method is proposed for a robot ping-pong player. The method is based on the dynamic model in the sky and the bouncing dynamic model between the ball and the table. First, the ambiguity of the dynamic model in the sky is removed. Then, the bouncing model between the ball and the table is proposed to get the relationship of the emergence speed and the incidence speed. The corresponding coefficients of models are trained according to the 3D Cartesian coordinates of the ball calculated from a high speed stereo vision system. With known dynamic models, the trajectory can be predicted if several initial trajectory points of the ball from the vision system are given. Experiments show that the trajectory is able to be predicted accurately in several milliseconds. The result of our method is better than that by LWR based approach.
Article
A high-speed vision system based on smart camera and distributed parallel processing is designed and applied to tracking the fast-moving ping pong ball. At the same time, a set of quick and efficient target recognition and tracking algorithm based on grey image is proposed. This algorithm is applied to distinguishing the moving ping pong ball from the changing background with less computation and good robustness. In order to improve the real-time performance further, dynamic window is used. The experimental results demonstrate that the system can distinguish and track a moving ping pong ball quickly. It takes only 6 ms to capture and process a grey frame with a size of 640×480 pixels.
Article
The illumination of the ordinary—of why the sky is blue or why the stars shine—is not the least important role of physics and physicists. Then can't we add to the list of deeper queries some of the questions that seemed so important to me in my youth: How can Babe Ruth hit so many home runs? What makes Carl Hubble's curveball and screwball swerve in their trips to the plate? And if baseball plays no known role in the fundamental structure of the universe (see The Iowa Baseball Confederacy by W. P. Kinsella for a contrary position), it is not of trivial importance in the perception and appreciation of that universe by some of its inhabitants. Although not quite so important now, in the period between the Civil War and World War II baseball was a significant part of what defined the United States. Forty years ago, Jacques Barzun, a preeminent student of American culture and a native of France, said, “Whoever wants to know… America had better learn baseball.” But, even as the game itself is subtle and complex, I have found subtleties and complexities in my attempts to know the physical bases of this American game. For almost a century and a half, baseball has played a significant role in defining the United States; in defining the physics of baseball we confront the ill‐defined physics of the world in which we live.
Conference Paper
To improve and validate the performance of humanoid robot, the research on a table-tennis game by both humanoid robots is supported by 863, the national hi-tech program. Due to the limitation of the visual feedback system and the motion ability of the humanoid robot, a precise model of the table-tennis game is necessary to predict the ball's trajectory. This paper details the dynamic model of the ball's flight trajectory as well as its parameters calibration by photron fastcam, a high speed camera with 2 kfps to capture the trajectory of ball. With the model's characteristics, ball's flight trajectory prediction based on a few observation points is employed as well as its noise sources are discussed. According to the requirement of table-tennis game by humanoid robot, a novel nonlinear observer is developed to improve the prediction accuracy. Finally, Experiments results with binocular vision show that the model and trajectory prediction proposed by the paper outperform the requirement of the game.
An Approach to Hit Point Prediction for Ping Pong Robot
  • Bo Peng
  • Yongchao Hong
  • Sensen Du
Bo peng, Yongchao Hong, Sensen Du, et al., "An Approach to Hit Point Prediction for Ping Pong Robot," Journal of Jiangnan University (Natural Science Edition), vol. 6, pp. 433-437, 2007.
Mechanical Analysis on Table Tennis Sport
  • G Meng
  • Y Chen
G. Meng, Y. Chen, "Mechanical Analysis on Table Tennis Sport," Journal of Northeast Heavy Machinery Institute, vol. 20, pp. 80-83,1996.
Dynamic Model based Ball Trajectory Prediction for a Robot Player
  • Xiaopeng Chen
  • Ye Tian
  • Qiang Huang
  • Weimin Zhang
  • Zhangguo Yu
Xiaopeng Chen, Ye Tian, Qiang Huang, Weimin Zhang, Zhangguo Yu, "Dynamic Model based Ball Trajectory Prediction for a Robot Player," International Conference on Robotics and Biomimetics, Tianjin, China,2010.