axis for Posture1 and Posture2, respectively.
In the simulations, the stability is considered by
calculating the loci of system eigenvalues when the
value of
K
b
is changed from 0.0 to 1.0. To obtain the
system eigenvalues, first the modelbased control
system shown in Fig. 4 is reexpressed using the state
variable approach, and then the eigenvalues of the
system matrix are calculated.
Fig. 6 shows the loci of system eigenvalues. In
this figure, eigenvalues of 11±47j and 16±73j are
related to the first natural frequency of the torsional
vibration for Posture1 and Posture2, respectively. Fig.
6 indicates that the control system is stable on the
condition of 0.9
≤
K
b
≤
0.0.
(a) Posture1 (N.F.=7Hz). (b) Posture2 (N.F.=12Hz).
Fig. 5. Correlation of natural frequency with the
posture.
Table 1. Simulation and experimental conditions.
Parameter
Value
Unit
J
m
1.362×10

5
J
g
2.048×10

5
J
l
2.852
Moment of inertia
Posture1
Posture2
J
l
1.205
kg·m
2
K
s
889.6
Torsional stiffness
K
g
6967.3
N·m/rad
C
s
0.0137
Damping coefficient
C
g
22.553
N·m·s/rad
Gear reducer
Reduction ratio
R
g
100
Velocity loop gain
K
v
0.15
A/(rad/s)
Integral time constant
T
i
1.0
s
Torque constant
K
t
0.316
N·m/A
Voltage constant
K
e
0.316
V/(rad/s)
Phase resistance
R
4.5
Ω
Phase inductance
L
0.0189
H
Current loop gain
K
c
118.84
V/A
Current feedback gain
K
cb
1.0
Feedback gain
K
b
0 or 0.7
Reducedorder model
Electrical part
ω
e
188.4
rad/s
Natural frequency
Damping ratio
ζ
e
1.0
Mechanical part
Posture1
ω
n
151.2
Natural
Freq.
Posture2
ω
n
163.3
rad/s
Damping ratio
γ
n
0.7
Posture1
R
n
8.363
Inertia
ratio
Posture2
R
n
3.256
International Journal of Control, Automation, and Systems Vol. 1, No. 3, September 2003
267
(a) In the case of Posture1 (N.F.=7Hz).
(b) In the case of Posture2 (N.F.=12Hz).
Fig. 6. Loci of system eigenvalues.
(a) Without modelbased control.
(b) With modelbased control.
Fig. 7. Simulation results of
ω
l
/
ω
cmd
in the case of Posture1.
(a) Without modelbased control.
(b) With modelbased control.
Fig. 8. Simulation results of
ω
l
/
ω
cmd
in the case of Posture2.
3.3. Simulation of frequency response
Fig. 7 and Fig. 8 show the Bode plots of the trans
fer function
ω
l
/
ω
cmd
. In these simulations,
K
b
is set to
0.7 after considering simulation results of the stabil
ity. These figures indicate that the proposed model
based control equivalently increases the cut off fre
quency of the system and the damping ratio between
the reducer’s input shaft and the driven machine part.
3.4. Simulation of time response
Then, the time response is calculated by the
RungeKutta method in order to verify the suppres
sion effect on the residual vibration. Fig. 9 and Fig.
10 show simulation results. In these figures,
Arm ac
cel
. represents the vibration acceleration in the direc
tion of rotation at the point of L1 or L2 in Fig. 5.
In these simulations, a trapezoidal velocity profile
is assigned. The constant acceleration in the start
phase is 1000 min
1
/28 ms. The cruise velocity is
2000 min
1
.
The constant deceleration in the arrival