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so as to keep the target within the field of view. The position of target
p
0
(
t
) is
computed using the center of gravity of the acquired image. Next, the future trajec-
tory of the target object is estimated as a polynomial function of the time
t
such as
p
0
n
(
t
). The estimated trajectory
p
0
n
(
t
) is fitted in with the past sequence of the tar-
get positions such as
p
0
(
t
n
)
p
0
(
t
1
) based on successive least-squares
estimation, where t
t
n
is the sampling time, and
n
is the number of sampling.
Step 2
. First, we set a virtual plane expressed as
,
p
0
(
t
n
−
1
)
,···,
ax
+
by
+
cz
=
d
,
(3.2)
where
a
,
b
,
c
,and
d
are constants. The plane is defined so that the ball hits the bat
in it. By solving the equation
abc
p
0
n
(
t
)=
d
(3.3)
we get the hitting time
t
=
t
b
. As a result, the desired position
p
d
(
t
b
)=
p
0
n
(
t
b
) of
the end-effector at the hitting time is computed. In general,
t
n
<
t
b
,and
n
should be
a large number to achieve accurate estimation. Then, the desired orientation
φ
d
(
t
b
)
at the hitting time is arbitrarily given in order to control the direction of the hit ball.
Step 3
. The joint angle vector at the hitting point is
q
b
∈
R
n
, and the boundary
condition is written as
q
b
=
q
d
(
t
b
)=
l
−
1
(
p
d
(
t
b
)
,
φ
d
(
t
b
))
(3.4)
where the function
l
−
1
() means the inverse kinematics.
In order to use the past sequence of the ball position with the current ball position,
we modify (3.1) to
q
d
=
f
(
q
b
,
t
)
.
(3.5)
We adopt a fifth order polynomial as the trajectory function
f
, in order to control the
position, the velocity, and the acceleration, continuously:
5
i
=0
k
i
(
q
b
)
t
i
q
d
(
t
)=
.
(3.6)
As a result, the trajectory of a manipulator is determined by the coefficients
k
i
.The
coefficients
k
i
are
k
0
=
q
d
(0)
(3.7)
k
1
=
q
d
(0)
(3.8)
1
2
q
d
(0)
k
2
=
(3.9)
20 [
q
b
−
3
q
d
(0)]
(3.10)
1
2
t
b
t
b
[8
c
v
+12
q
d
(0)]
x
+
t
b
[
c
a
−
k
3
=
q
d
(0)]
−
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