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T
(
k
)
Δ
β
Thus, we obtain
Δe
(
k
)
(
k
). From the tuning law obtained
by (11), the change of adjustable parameters can be rewritten as
≈−G
N
(
k
)
φ
−η
(
k
)
G
2
N
(
k
)
2
Δ
β
(
k
)=
||
φ
(
k
)
||
β
(
k
)+
η
(
k
)[
x
d
(
k
+1)
− F
N
(
k
)]
G
N
(
k
)
φ
(
k
)
.
(16)
By using (16), we have
Δe
(
k
)=
η
(
k
)
G
2
N
(
k
)
2
[
T
(
k
)
||
φ
(
k
)
||
−x
d
(
k
+1)+
F
N
(
k
)+
G
N
(
k
)
φ
β
(
k
)]
,
−η
(
k
)
G
2
N
(
k
)
2
[
e
(
k
+1)]
.
=
||
φ
(
k
)
||
(17)
We can rearrange (17) as
Δe
(
k
)=
−η
(
k
)
G
2
N
(
k
)
2
e
(
k
)
||
φ
(
k
)
||
2
.
(18)
1+
η
(
k
)
G
2
N
(
k
)
||
φ
(
k
)
||
Substitute (18) into (14), we obtain
1
2
.
ΔV
(
k
)=
−η
(
k
)
G
2
N
(
k
)
2
e
2
(
k
)
η
(
k
)
G
2
N
(
k
)
2
||
φ
(
k
)
||
||
φ
(
k
)
||
−
(19)
1+
η
(
k
)
G
2
N
(
k
)
||
φ
(
k
)
||
2
2+2
η
(
k
)
G
2
N
(
k
)
||
φ
(
k
)
||
With the learning rate give by (12), the change of Lyapunov function candidate
can be rearranged by
ΔV
(
k
)=
−γ
G
N
(
k
)
2
1+
γ
G
N
(
k
)
G
u
2
2(1 +
γ
G
N
(
k
)
G
u
γ
G
N
(
k
)
G
u
2
1
e
2
(
k
)
G
u
−
≤
0
.
(20)
2
)
Remark
: According to (20), sign or direction of
G
N
(
k
) is not needed.
The importance thing that we need to estimate is
G
u
. With the following
examples, the parameter
G
u
is clearly designed and discussed.
5
Experimental Setup and Results
In this experimental setup, the proposed control algorithm is implemented to
control a 7-DOF Mitsubishi PA-10 robotic arm system. The overall system con-
figuration is illustrated in Fig. 1. Seven FRENs are designed to control each of
joints independently. This robotic system is operated in the velocity mode con-
trol which means FRENs must generate the velocity commands to move very
joints to follow the desired trajectory.
To begin the controller design, the suitable IF-THEN rules are all needed to
be specified. Based on the knowledge of PA-10, those IF-THEN rules can be
given as the followings:
If
e
(
k
)isPLThen
ω
1
(
k
)=
β
PL
(
k
)
φ
1
(
k
), Large
e
in positive; Fast
ω
in positive,
If
e
(
k
)isPS Then
ω
2
(
k
)=
β
PS
(
k
)
φ
2
(
k
), Small
e
in positive; Slow
ω
in positive,
If
e
(
k
)isZ Then
ω
3
(
k
)=
β
Z
(
k
)
φ
3
(
k
), Reach desired position; Stop moving !!!,
If
e
(
k
)isNSThen
ω
4
(
k
)=
β
NS
(
k
)
φ
4
(
k
), Small
e
in negative;
Slow
ω
in negative,
If
e
(
k
)isNLThen
ω
5
(
k
)=
β
NL
(
k
)
φ
5
(
k
), Large
e
in negative;
Fast
ω
in negative,
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