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Preprints of the 5th IFAC Symposium on Mechatronic Systems
Marriott Boston Cambridge
Cambridge, MA, USA, September 13-15, 2010

A Real-Time Measuring Method ofTranslational/Rotational Velocities of a
Flying Ball
Akira Nakashima ∗ Yoji Tuda ∗ Chunfang Liu ∗Yoshikazu Hayakawa ∗,∗∗∗
Mechanical Science and Engineering, Graduate School ofEngineering, Nagoya University, Furo-cho, Chikusa-ku, Nagoya, Japan(e-mail: a nakashima@nuem.nagoya-u.ac.jp)∗∗
RIKEN-TRI Collaboration Center, RIKEN, 2271-103, Anagahora,
Shimoshidami, Moriyama-ku, Nagoya, Japan.

Abstract: We propose a real-time measuring method of both the translational/rotational
velocities of a flying ball using two high speed cameras. The rotational velocity is estimatedby the kinematics which relates the velocity to the position variation of the centroid of a featurearea marked on the ball. The position variation is calculated by the correlation between two
images with respect to the translational displacement in the images. The corresponding 3D
position of the feature area is necessary for the estimation by the kinematics. It is calculated by
the centroids of all the feature areas on the ball in the left and right images with the assumption
where the centroids coincide with each other in the 3D space. The proposed method is verified
with respect to some configurations of the cameras and texture patterns of the feature areas bynumerical simulations. The effectiveness of the method is verified by experiments.
Keywords: speed measurement, computer vision, real-time, image matching
1. INTRODUCTION
It is a challenging task for a robot to play table tennis
with a human because playing table tennis is a dexterous
task for humans. For simplicity, consider the situation ofFig. 1, where a robot tries to hit a flying ball. The strategyof the robot playing table tennis can be decomposed as thefollowing subtasks: 1) To detect the states of the flying ballwith vision sensors; 2) To predict the ball trajectory; 3)
To determine the trajectory of the racket attached to therobot for hitting the ball to achieve desired ball trajectory.
The number 1) means the image processing algorithm toobtain the position, the translational and rotational velocities of the ball. The number 2) means the prediction of theposition and translational/rotational velocities of the ballat the time for the robot to hit the ball. The number 3)means the determination of the trajectories of the positionand orientation of the racket for the ball to follow desired
trajectory. The algorithm in the subtask 1) is necessary tobe performed in real time for the next task. In the subtasks

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

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2) and 3), the ball rebounds from the table and the racket
rubber. It is therefore necessary to model the reboundphenomena between the ball and the table/rubber in orderto achieve the subtasks 2) and 3). The rebound models
have been proposed by Nakashima et al. (2009, 2010). This
paper is concerned with the real-time measurement of therotational/translational velocities of the ball.
A vision sensor is effective with the measurement of thevelocities of the ball since it is a non-contact sensor andcan measure both the translational/rotational velocities
while an optical rotation meter can not measure boththem at the same time. It is necessary that some textures
are marked on the ball as feature areas in order todetect the movement of the ball by the vision sensors.
It is also necessary that the sampling frequency of avision sensor is high since the rotational velocity of theball is very large, e.g., 2000–3000 [rpm] in the case ofamateur players or 3000–5000 [rpm] in the case of pro
players (Tamaki et al. (2004)). Nakabo et al. (1996)
developed a high speed vision camera with the sampling
frequency 1000 [Hz] and the resolution 128 × 128 [pixel]
and applied it to high speed target tracking. Watanabe
et al. (2005) proposed a real-time measuring method of
the rotational velocity with the high speed camera in thecase where the position of the ball is fixed. They achieved
measuring the velocity under 1200 [rpm]. In their method,
the kinematics is used which relates the rotational velocity
to the position variation of the centroid of the featurearea. The position variation is calculated by tracking thefeature area in two images with the self-windowing of Ishii
et al. (1996), where it is assumed that a feature area in

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one image laps over the corresponding area in the otherimage of the next frame. This technique is called as the
tracking-based method in this paper. It is also necessary
to calculate the corresponding 3D position of the featurearea in order to use the kinematics. They obtained it byassuming that the optic axis is through the center of theball. On the other hand, Tamaki et al. (2004) measured
the rotational velocity by an off-line processing with theimage registration (Szeliski (1994)) and the knowledgeof the ball shape. The image registration estimates thehomogeneous transformation matrix of a point on the ballform an image to the other by minimizing the sum of thesquares of the intensity residuals between the two images
with a nonlinear optimization. The rotational velocity is
calculated by the estimated rotation matrix.
In the tracking-based method, it is necessary to track afeature area in two images in order to calculate the corresponding position variation. However, the self-windowing
used by Watanabe et al. (2005) can not be applied because
the feature area in the two images does not lap over eachother in the case where the spin of the ball is larger than2000 [rpm]. Furthermore, the translational velocity can
not be measured because of the assumption of the fixed
center of the ball. The image registration can measure
the velocity 5000 [rpm] as in the results of Tamaki et al.
(2004) since it does not require tracking a feature area in
two images. However, it is difficult to apply it in real-time
because of the nonlinear optimization for obtaining thehomogeneous transformation.
Therefore, we propose a real-time measuring method ofboth the translational/rotational velocities of the ball using the two high speed cameras without tracking. The
rotational velocity is estimated by the kinematics and a
representative position variation of all the feature areas,
which is calculated by the correlation between the two
images with respect to the translational displacement in
the images. The cost of the calculation is small because
the calculation does not estimate the homogeneous transformation. The corresponding 3D position of the featurearea is estimated by the centroids of all the feature areas on
the ball in the left and right images with the assumption
where the centroids coincide with each other in the 3D
space.

Ball
y
frame

Σb
Table

z
Automaticball catapult

x

Ball

z
y

ΣB

x

Base
frame

u1
v1
Cameraframe(Left)

z
x

Σ C1
y

u2

High speed
camerasz

v2

Cameraframe(Right)

ΣC

x
2

y

Fig. 2. The experimental system and the frame configurations.
jth feature areaz

Centroid offeature area

pbl j

pl j

y

Σb
x

z

pb
y

ΣB

x

Fig. 3. The feature areas on the ball and the position of
the centroid.

2.1 System Configuration

parameters of a ping-pong ball are m = 2.7 × 10−3 [kg]and r = 2.0 × 10−2 [m]. The ball is shot out from theautomatic ball catapult of ROBO-PONG 2040 (SAN-EICo.) which is set at the end of the table. The flying ballis measured by the two high-speed cameras with 980 [fps](Hamamatsu Photonics K.K.). The array and pixel sizes
per meter are 232×232 and αu , αv = 2.0 × 10−5 [pixel/m].
The focal length of the lens is f = 3.5 × 10−2 [m]. The
sampled data are quantized as 2D image coordinates withthe monochrome brightness of 8bit (0–255). ΣB is thereference frame at the corner of the table. ΣCi (i = 1, 2)
are the camera frames with their z-axes coinciding with theoptical axes. ξi := [ui vi ]T ∈ R2 is the image coordinatesof the ith camera. The frames ΣCi are calibrated withrespect to ΣB . Figure 3 shows feature areas on the ball(j = 1, · · · , Nl ). Σb is the frame attached to the centerof the ball. pb ∈ R3 is the position of the center of theball with respect to ΣB . The feature position plj ∈ R3 isthe centroid of the jth feature area with respect to ΣB(j = 1, · · · , M ). The corresponding image coordinate ofplj is defined as ξlj ∈ R2 . The relative feature position isdefined as pblj := plj − pb .

Figure 2 shows the experimental system and the coordinate frames. The table is an international standard one
with the size of 1525(W)×760(H)×2740(D) [mm]. The

In latter, we assume that there is no effect of the air resistance. Therefore, the translational/rotational velocities ofthe ball are invariant during measuring the flying ball by
the cameras.

The target system and preliminary are shown in Section 2.
The tracking-based method is briefly described in Section
3. The real-time measuring method is proposed in Section
4. In Section 5, the proposed method is verified withrespect to some configurations of the cameras by numericalsimulations. An appropriate configuration for the method
is determined. Appropriate texture patterns of the featureareas are also verified here. In Section 6, the method
is verified by experiments. It is checked whether theprocessing time is shorter than usual rally time or not.
Conclusions are shown in Section 7.
2. SYSTEM CONFIGURATION AND PRELIMINARY

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rotation
axis

ξ1

(1)

ω

vbl [n]jξ2Featureposition

(a)
(2)

(1)

r

pb [n ]

(2)

(3)

Ball

C

ξc

ξ4

(b)

pbl j [n]

(3)

ξ3

(4)

Fig. 5. Detection of the translation and rotation velocities.

(c)

(d)

Fig. 4. Image data and Detection of the center of the ball.
2.2 Basic Relations
A 3D position Ci p ∈ R3 expressed in ΣCi is projected tothe corresponding image coordinate ξi ∈ R2 as pxi · ¸fu
 pαu  ,(1)
ξi = i =
yi
vi
pzi
αv
Ci
T Ci
where p := [pxi pyi pzi ] . p is related to B p ∈ R3
expressed in ΣB by the homogeneous transformation:Ci
p = Ci pB + RCi B B p,(2)
where Ci pB ∈ R3 and RCi B are the position and rotation matrix of ΣB with respect to ΣCi . The pair of(Ci pB , RCi B ) is obtained by the calibration of ΣCi withrespect to ΣB .
3. TRACKING-BASED METHOD
The tracking-based measuring method is briefly described
here. The image processing and variables are describedwith respect to only one of the images of the two cameras
because they are the same as the other one. Then, thesubscript i is dropped. The calculation of ξlj from the jthfeature area is omitted for space limitation. It is assumed
that any feature points are tracked in every frame.
3.1 Estimation of the translational velocity

Vc (ξc ) :=

¯2
4 ¯¯
X
(ξi+1 − ξi )T ξc − 1 (kξi+1 k2 − kξi k2 )¯2

kξi+1 − ξi k2

i=1

,

(3)
which is the sum of the squares of the distances of ξc fromthe mediators. The center of the ball pb is calculated byξc in the left and right images. The translational velocityof the ball is obtained by the mean of all the velocities
between the (n + 1)th and nth position asPN
vb [n]
pb [n] − pb [n − 1]
vb = i=2, vb [n] :=
,
(4)
N −1
∆t
where ∆t = 1/980 [s] is the sampling time and N is thenumber of the frames used for the data processing. Notethat [n] denotes the nth frame.
3.2 Estimation of the rotational velocity
Next, consider the calculation of the rotational velocityof the ball. The right of Fig. 5 shows the ball in the nthframe, where ω ∈ R3 is the rotation axis. For the jthfeature point, the following rotational relationship is holdby Murray et al. (1994):
ω ∆t
ω ∆t
b ∆t,
pblj [n] = eb
pblj [n − 1], eb≈ I3 + ω(5)b ∈ R3×3 is the skew-symmetric matrix which
where ωω ∆tcorresponds to the cross product of ω. Note that eb
is linearized due to the small ∆t. (5) is rewritten as
pblj [n] − pblj [n − 1]
. (6)∆tThe rotational velocity ω is obtained by solving the linear
least squares method with the performance function whichis the sum of the square of the difference between boththe terms of (6). The analytical solution of the linear leastssquares method is given by
pbblj [n]ω = −vblj [n], vblj [n] :=The left of Fig. 4 shows an example of the image data of
the camera. Let us consider the detection of the centerof the ball. Since the ball is white and the background ismuch darker than the ball, all the area of the ball can beeasily detected by the simple scanning shown in the rightof Fig. 4. In the scanning of (a), the arrays of the image arescanned in turn from the left and top until the brightnessof the scanned image is larger than a specified thresholdfor detecting the while area of the ball. The scanning of(b)–(d) are similar to the one of (a). The positions of
the detected data as the parts of the white area with thethreshold 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. 5. Define
ξc as the center of the ball. The center ξc is estimated by
the least squares method with the following performance
function:734

ω = (AT A)−1 AT b,
where

(7)






pbbl1 [2]−vbl1 [2]
..
..








.
.




 pbbl1 [N ]  −vbl1 [N ] 




..
..
 , b := .
A := .
.

 pb [2]  −v [2]  blM 


blM




..
..




.
.
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4. REAL-TIME MEASURING METHOD

g i [n ]G i [n ]limin

The estimation of the translational velocity is same as theone in Section 3. The estimation of the rotational velocity
with the correlation between the images in the two frames
is proposed here.

g i [n ]g i [ n + 1]k

4.1 Kinematics between the image coordinates and therotational velocity
Substituting (2) into (1) yields
B

B

B

Ci

f r11 pxi + r12 pyi + r13 pzi + pBxi
αu r31 B pxi + r32 B pyi + r33 B pzi + Ci pBzi,
f r21 B pxi + r22 B pyi + r23 B pzi + Ci pBxivi =αu r31 B pxi + r32 B pyi + r33 B pzi + Ci pBzi

ui =

(8)

where rij is the component of RCi B . Differentiating (8) of
i = 1, 2 and combining the resultant equations lead toξ̇ = J (B p)B ṗ,:= [ξ1Tξ2T ]T4(9)
4×3

where ξ∈ R and J ∈ R
is the image
jacobian. Replacing (ξ, B p) to the feature point (ξlj , plj )leads toξ̇blj + ξ̇c = J (plj )(ṗblj + ṗb ),(10)
where ξblj := ξlj − ξc is the relative feature position vector
in the 2D image planes which corresonds to pblj . Notetherefore that these 2D and 3D feature potition variables
ξc and pb are related byξ̇c = J (plj )ṗb .
(11)
Substituting (11) into (10) and transforming the resultant
equation to the discrete formulation result in∆ξblj [n] = J (plj [n])∆pblj [n],
(12)
where
∆ξblj [n] := ξblj [n] − ξblj [n − 1]
∆pblj [n] := pblj [n] − pblj [n − 1].
Substituting (6) into (12) results in∆ξblj [n] = −J (plj [n])pbblj [n]ω∆t.
(13)Solving (13) with respect to ω, we get the following
kinematic relationship³´+ ∆ξ [n]blj
ω = − J (plj [n])pbblj [n]
.
(14)
∆t
A matrix A+ stands for the pseudo inverse matrix of A.
Note that plj [n] can be calculated by pblj [n] and pb [n] withplj [n] = pblj [n] + pb [n]. However, as mentioned in Section
1, pblj [n] and ξblj [n] can not be obtained and ξblj [n] can
not be corresponded to the previous one because it is
impossible to track the feature areas. In the following, theestimation methods of ∆ξlj [n] and plj [n].

miniFig. 6. The segmented 2D image data and the correlation
between the two images.
and (n + 1)th images is given by minimizing the following
performance function:
1×
Vi (ki , li ) :=(Ngi − ki )(Ngi − li )Ngi −ki Ngi −li ³´2
X X
gi [n + 1](s, t) − gi [n](s + li , t + ki )
(15)
s=1

t=1

This operation is illustrated in the right of Fig. 6, where(kimin , limin ) is the solution of the correlation. Then, we
estimate the image displacements ∆ξblj of all the featurepoints by (kimin , limin ) of the correlation as¤T£∆ξbl [n] = k1min [n] l1min [n] k2min [n] l2min [n] . (16)The searching ranges of (ki , li ) are Kmin ≤ ki ≤ Kmax
and Lmin ≤ li ≤ Lmax , which are modified in each frame
for reducing the processing time. In the first and second
frames, the search ranges are set to Kmin = Lmin = −Ng
and Kmax = Lmax = Ng . From the third frame, it
is assumed that the next (kimin , limin ) exists nearby theprevious ones. Therefore, the ranges are set toKmin [n] = kimin [n − 1] − K 0 , Kmax [n] = kimin [n − 1] + K 0(17)
Lmin [n] = limin [n − 1] − L0 , Lmax [n] = limin [n − 1] + L0 ,(18)
00where K and L are chosen as small ones.
4.3 Representative position corresponding to the estimatedimage displacement
In the previous subsection, we estimated the image displacement by the representative displacement ∆ξbl of theimage correlation. It is necessary to determine the representative 3D position corresponding to ∆ξbl in order to
use the kinematics (14) with ∆ξbl .
Firstly, as in Fig. 7, the centroids of the feature areas arecalculated byP
Ts,t∈D G(s, t)[s t]Pξbfi =− ξ ci ,(19)s,t∈D G(s, t)
where

¸
·
¸¾
¯ ·
Ngi
Ngi
¯
≤ s, t ≤ ξci +(s, t) ¯ ξci −2
2and [·] stands for the integer as an argument rounded
off. Secondly, the representative 3D position pbl ∈ R3 is
calculated by the centroids ξbf1 and ξbf2 . However, this is
not a position on the ball because the image centroids do
not coincide with each other in the 3D space. This causes
the reduction of the accuracy of the rotational velocity byusing (14). Therefore, thirdly, pbl is modified to a position½

D :=4.2 Estimation of the image displacement
Define Gi [n](s, t) ∈ N232×232 as the 2D image data of
the nth image of the ith camera. Also define gi [n](s, t) ∈NNgi ×Ngi as the segmented 2D image data of the inscribed
quadrangle of the ball as shown in Fig. 6. The values of
Gi [n] and gi [n] are assumed to be binarized to (0, 1) with
an appropriate threshold. The correlation between the nth735

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difference between the centroids of the feature areas on theball in both the images in the 3D space. The value of α ischanged from 0 to 90 [deg] with the increment of 30 [deg].
The translational speed is changed from 5 [m/s] to 15 [m/s]with the increment of 5 [m/s]. The translational direction
is set to [−1 0 0]T . The rotational speed is changed from1500 to 4500 [rpm] with the increment of 1500 [rpm]. The
axis of the rotation is the top spin, i.e., [0 − 1 0]T .

Representative
PositionCentroid of
Feature areas

Left image

The results are shown in Figs. 10–12. Fig.10 displays themeasured and real values of the translational speed in thex-axis with all the elevation angles. The results in the yand z-axes are omitted because of its trivia. The solidline represents the points that measured and real values
coincide with each other. It can be confirmed from Fig.10
that the accuracy of the measured value of translational
speed is high regardless of the elevation angles of theRight image

Fig. 7. The representative position of the feature areas.
e op 2e op 1

Representative
position
Modified
position

16

Right camera

Fig. 8. The modification of the representative position.
on the ball as in Fig. 8. The direction of the projection of
pbl on the ball is defined as1
epro := − √ (eop1 + eop2 ),
(20)
2
where
eopi := RBCi [0 0 1]T , i = 1, 2
are the normalized vectors in the optical axes of thecameras. Then, the modified position p0bl is given by(21)
p0bl = pbl + kepro ,where the value of k is obtained by solving the following
constraint:
kp0bl k = r.
(22)
r is the radius of the ball.
5. SIMULATION

14

12

10

8

6

4

4

6

8

10

12

14

16

true value of translational velocity[m/sec]

Fig. 10. The translational velocity in the x-axis.
1.5

inner product of measurement axis and true axis

Left camera

measurement value of translational velocity[m/sec]

e pro

1

0.5

0

-0.5

-1

-1.5

0

1000

2000

3000

4000

5000

rotational velocity[rpm]

Table
Ball
Ball

α

θ

y
z

∑

Fig. 11. The inner products between the true and measured
rotational axes.
7000

6000
measurement rotational velocity[rpm]The proposed method is simulated here. The measurement
accuracy of the method is examined under different camera
configurations. The parameters of the camera configuration are the distance between cameras d, the angle of the
optical axes of the cameras θ and the elevation angle ofthe cameras α as shown in Fig. 9. The parameters are set
to d = 0.30 [m], θ = 16 [deg] in order not to increase the

5000

4000

3000

2000

1000

x
B

0

d

0

1000

2000

3000

4000

5000

true rotational velocity[rpm]Fig. 9. The camera configuration.

Fig. 12. The rotational speed.

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camera. Figs. 11, 12 show the inner product between
the measured and true value of the rotational axis andmeasurements of the rotational speed respectively. The
black, green, blue and red markers represent 0, 30, 60 and
90 [deg] of the elevation angles. In Fig. 11, the solid line
and dashed line represent
the points that values of inner√product are 1 and 1/ 2 respectively. The measured values
on the solid line coincide the true values. The solid line
in Fig. 12 represents the points that measured and realvalues coincide with each other. In the case of the angle of
α = 90 [deg], the accuracy of the measurement is best; theangle between the measured and true the rotation axes
is smaller than 45 [deg] and the difference between themeasured and true speeds is smaller than 600 [rpm]. This isbecause the optical axes are perpendicular to the rotation
axis. However, the angel α = 90 [deg] is not appropriate
because the ranges of the images can not deeply over thetable in the optical axes. Therefore, the elevation angle isset to α = 75 [deg].
Figure 13 shows the comparison of 3 types of texture
patterns: the first consists of the big circles with the
low density distribution; the second consists of the small
circles with the high density distribution; the third consistsof the middle circles with the high density distribution.
The performance function of the correlation of (15) withrespect the (ki , li ) are illustrated under the figures of
the texture patterns. It is confirmed that there is only
one minimum extremum in the case of the first texture
while there are some minimum extrema in the other cases.
Therefore, the first texture pattern is used in the next
experiments.

Fig. 13. The comparison of 3 types of the texture patterns.

n=1

n=2

n=3

n=4

n=5

n=6

Fig. 14. An example of the experimental image data.
measurement. The upper and lower figures represent theleft and right image data respectively. The translationaland rotational velocities of the ball are estimated by thereal-time measurement in on-line.
Fig.15 shows the experimental result of the translational
velocity of the ball in the x-axis. It is confirmed thatthe measured values coincide with the true values well.
Figs. 16, 17 show the results the axis and speed of therotational velocity. In Fig. 16, the angles between almost
the measured axes and the true axis are smaller than
45 [deg]. In Fig. 17, there is the significant deviation of
about 700 [rpm] between the measured and true values of
the rotational speed. The processing time averaged with
10 frame is 15–30 [ms], which is very shorter than the rally
time of 200–500 [ms]. Therefore, the remaining time is longenough to predict the ball trajectory and plane the racket
trajectory.

6. EXPERIMENTS

7. CONCLUSIONSThe effectiveness of the proposed method is verified byexperiments. The operating system of the computer is
Windows XP sp2. The CPU is the Intel(R) Xeon(R)E5430, 2.66GHz. The physical memory is 2.00GB RAM.
The autmatical ball catapult is set to the configurations of
the cases of the top and back spins. The lines, markers andaxes of figures in latter are same as those in the simulation
results. The image data is also saved in HDD during
the real-time measurement of the ball velocity. Then, thetrue values in the latter are manually estimated by the
tracking-based method of Section 3 in off-line. Figure 14
shows an example of some parts (n = 1, · · · , 6) of the
saved image data of the expperiment of the real-time

We proposed the real-time measuring method of both thetranslational/rotational velocities of a flying ball using
the two high speed cameras. The rotational velocity wasestimated by the kinematics which relates the velocity tothe position variation of the centroid of a feature area
marked on the ball. The position variation was calculatedby the correlation between two images with respect to thetranslational displacement in the images. The corresponding 3D position of the feature area was calculated by thecentroids of all the feature areas on the ball in the left
and right images with the assumption where the centroids
coincide with each other in the 3D space. The proposed
method was verified with respect to the configurations

737

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Mechatronics'10
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7

6

measurement velocity[m/sec]

5

4

3

2

1

0

0

1

2

3

4

5

6

7

true velocity[m/sec]

Fig. 15. The experimental result of the translational velocity in the x-axis.

inner product of measurement axis and true axis

1.5

1

0.5

0

CRC Press.
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ball and racket rubber with spinning effect. In Proc.
ICCAS-SICE 2009, 2295–2300.
Nakashima, A., Ogawa, Y., Kobayashi, Y., and Hayakawa,Y. (2010). Modeling of rebound phenomenon of a rigidball with friction and elastic effects. In Proc. IEEE
Amer. Cont. Conf. (To be published).
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Tamaki, T., Sugino, T., and Yamamoto, M. (2004). Measuring ball spin by image registration. In Proc. 10th
Frontiers of Computer Vision, 269–274.
Watanabe, Y., Komuro, T., Kagami, S., and Ishikawa, M.
(2005). Multi-target tracking using a vision chip and its
applications to real-time visual measurement. Journalof Robotics and Mechatronics, 17(2), 121–129.

-0.5

-1

-1.5

0

1000

2000

3000

4000

5000

rotational velocity[rpm]Fig. 16. The experimental result of the rotational axis.
5000
4500

measurement rotational velocity[rpm]

4000
3500
3000
2500
2000
1500
1000
500
0

0

1000

2000

3000

4000

5000

true rotational velocity[rpm]Fig. 17. The experimental result of the rotational speed.
of the cameras and the texture patterns of the featureareas by the numerical simulations. The effectiveness ofthe method was verified by the experiments.
The processing time is short enough for a robot to playtable tennis. However, it is necessary to improve theaccuracy of the measured rotational velocity because it is
not high enough. An idea is to consider both the kinematic
relation and the image registration in the estimation.
REFERENCES
Ishii, I., Nakabo, Y., and Ishikawa, M. (1996). Target
tracking algorithm for 1ms visual feedback system using
massively parallel processing. In Proc. IEEE Int. Conf.
Robot. Automat., 2309–2314.
Murray, R.M., Li, Z., and Sastry, S.S. (1994). A Mathematical Introduction to ROBOTIC MANIPULATION.

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