IEEE Systems, Man and Cybernetics Magazine - July 2022 - 64

to the deformation of the artificial muscle when it is subjected
to a single input. In this case, each artificial muscle
deforms as represented by a sigmoid function for the input.
The parameters of the function, expressed as (9), are
determined by fitting with the experimental results. Pa is
the average of the three inputs, and max
f
P ,max
is the maximum
contraction ratio. In addition, k is the sigmoidal gain, and
Pth,
and k are parameters to be determined by fitting.
.
f =
fmax
1+exp
k P
a
PP
-
ath
max
k
i
=
(9)
Contraction Rate and Bending Angle
The actuator is bent by the contraction force (, ,)
Fi 12 3
generated in each artificial muscle. The torques produced
at the tip of the actuator by the contraction force are equal
to the torque required to bend the actuator at a certain
angle. Equation (10) shows the equilibrium of the torque.
3
|Fl = t - L .
EI
i
i=1
=
i
EIi
Each torque arm ( ,,)li 123i
1
=
c
lasincosb
lasin22
lasincosb
=- cc+^h (11)
,
is denoted by
,
(12)
3 =- cc-^h (13)
,
where a and b are constants based on the geometry of the
actuator. Each curved artificial muscle has the same bending
angle
i . In addition, the radius of curvature of each muscle
is varied by li from that of the actuator. When (1) and (2)
hold for the deformation of the actuator, the length of each
artificial muscle, Li
(, ,), can be expressed as (14).
i 12 3
=
LL^^ lhh=-0 ft i=- iii1
i
ff i=+
i
where ( ,,)i 123i
cial muscle.
f =
L ,
l
(14)
(15)
is the contraction rate of each artifi(10)
i
=
LK CC Pli
=1
0 ta^ -- @|
3
6 12fh
+ 2 at
Kk P
t =+ -
:1
a
exp
EI CK Pl
|
i
3PP -1
kD
-
=
ath
max
1
.
(18)
Control System Design
The proposed control system is illustrated in Figure 6. The
control system is a 2-DOF control system. It consists of
each operator designed to satisfy operator theory by using
the recommended model and a feedforward controller by
the model derived by using M-SVR [17]. C is a proportional
integral controller. For deviation ei, it is expressed as (19);
however, when output (x, y, z) is treated as a deviation, it is
difficult to use to determine the control input to each artificial
muscle. Therefore, the difference between length L
of each artificial muscle derived from the target value and
length rL
of each artificial muscle derived from the output
is used as deviation value e.
MSVR-Based
Inverse Model
Length
Converter
r = (x, y, z)
b
rL = (L1, L2, L3)
Length
Converter
Figure 6. Control systems.
64 IEEE SYSTEMS, MAN, & CYBERNETICS MAGAZINE July 2022
A
+
e
C
+
+
--
t
+
rc
r∗
e∗
B-1
u
D-1
ω
N
y = (x, y, z)
i
2
i
bi
i
The second moment of area I of the actuator is given by
(16). Ac and Io are the cross-sectional area and the second
moment of area when the artificial muscle and the silicon
cover are considered as a cylinder, respectively. The flexibility
of the actuator varies with the bending of the direction angle.
3
ID DA l
IA ab
=- +
=+
| r 44
=1 64 ^
od
^
idh
+
36 2
i
2
ci
.
oc cch
22 22
sincos
From (7), (10), and (15), i is given by (17).
.
3
i =
LC CPl
2
6 ^h @
|
ab i
i
12f
+
EI CPl2
1
-=
3
ai
i
i
However,
the input-output relationship expressed by
(17) is close to linear and does not fully represent the characteristics
of the actuator. Therefore, coefficient KP
derived from (9) is introduced and expressed as (18).
ta
^h
3
|
=1
i
(17)
(16)
IEEE Systems, Man and Cybernetics Magazine - July 2022

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