6.2 Dual-Servo Stage
6.2.2 Modeling and Mechanism Design
6.2.2.1 Coarse Motion Stage
the design and control of the fi ne servo mechanism composed of the fl exure hinge system with piezoelectric actuators. A few of them have made an effort to develop a new type of coarse motion actuator that can substitute the conventional DC or AC servo motor system. The ER clutch actuator is one of the most potential candidates for the coarse motion control in dual-servo system. Sakaguchi et al. [19] proposed a multi-cylindrical-type ER clutch for force display system. They analyzed its perfor- mance by comparing the proposed ER clutch with conventional powder clutch. Saito and Sugimoto [20] made a cylindrical-type ER clutch and applied it as a positioning actuator for a single-link rigid robot arm. Han et al. [21] presented position control of an X–Y stage mechanism driven by a pair of ER clutch actuators. From these works on the ER clutch as a positioning actuator, it has been proved that the ER clutch can be successfully adopted in various industrial fi elds to substitute conventional DC or AC servo motors.
This section presents a bidirectional ER clutch as a coarse motion actuator and a piezostack actuator associated with the displacement amplifi er as a fi ne motion actuator to construct a “smart” dual-servo system that wholly consists of smart materials [22]. After deriving the dynamic model for the coarse positioning stage, a sliding mode controller with the friction compensator is designed to achieve robust control performance. On the other hand, the Preisach model–based feed-forward compensator with PID feedback controller was designed to compensate the hyster- esis nonlinearity of the fi ne positioning system. These controllers are experimentally realized in a decentralized strategy, and the position control responses are evaluated in terms of accuracy in order to demonstrate the effectiveness of the “smart” dual- servo system.
Optical linear encorder Laser sensor
Fine motion table
Piezostack actuator
Driving motor
ER clutch
Coarse motion table Magnification device
(a)
Coarse positioning stage Piezoactuator
Fine positioning stage
ER clutch (b)
FIGURE 6.9 The dual-servo stage system. (a) Confi guration and (b) photograph.
Inner cylinder
Bidirectional rotary cylinder
Coupling Oil seal
Bearing
FIGURE 6.10 The bidirectional-type ER clutch.
cl cw ccw f
4 4
i c
2 3 3 3
i cl i c i cl f
ef v f
d d
4 4 ( )
2 ( ) sgn( )
3 E E 2
r A r A
r r
r l r r u u r l
h
β
τ = τ + τ − τ
⎡ ⎤
πη −
⎡ ⎤
= π⎢⎣ + π − ⎥⎦α ⋅ − ⎢⎣ + ⎥⎦θ − τ
= τ − τ − τ
∫ ∫
(6.22) where
τcw (or τccw, [Pa]) represents the shear stress of the ER fl uid when the electric fi eld is applied to the clockwise (or counterclockwise) rotating part of the ER clutch actuator
τf stands for the friction torque due to components of the ER clutch such as oil seal for leakage prevention
lcl, ri, and θ.
are the length, the radius, and the angular velocity of the inner cylin- der, respectively
In addition, h is the gap size, rc is the axis radius of the inner cylinder, and η is the dynamic viscosity of the ER fl uid. The parameters of α and β are the intrinsic values of the ER fl uid to be experimentally determined. In this test, a coquette-type elec- troviscometer has been used to obtain these parameters. From the measured yield shear stress data of the chemical starch/silicone oil–based ER fl uid, it was obtained as 135.46 for α and 1.59 for β [23]. The input, uE, can be defi ned by
, for clockwise torque transmission , for counterclockwise torque transmission
E
u E E
= ⎨⎧⎩− (6.23)
where E denotes the electric fi eld (kV/mm).
As presented in Equation 6.22, the torque transmission of the ER clutch actu- ator consists of three elements. Those are controllable fi eld-dependent torque τef, viscous friction torque τv, and friction torque τf. Figure 6.11 shows the measured torque responses of the ER clutch actuator. In order to exclude the effect of the fric- tion, a nonrotating-type (θ⋅ = 0) torque transducer has been employed to measure the transmitted torque. Therefore, the result shown in Figure 6.11 can be regarded as controllable fi eld-dependent torque, τef. It is observed from Figure 6.11b that con- trollable torque increases exponentially with respect to the input electric fi eld as expected from Equation 6.22. Moreover, one can fi nd that the measured torque is well accorded with the estimated result obtained by using the fi rst term of Equation 6.22. On the other hand, since the dynamic response of the fi eld-dependent torque, τef, reveals the behavior of the fi rst-order system, as shown in Figure 6.11a; it can be presented by introducing a time constant, λcl, as follows:
2 3 3
cl ef ef i cl i c
d 2
2 ( ) sgn( )
d r l 3 r r uE uE
t
⎡ ⎤ β
λ τ + τ = πα⎢⎣ + − ⎥⎦⋅ (6.24)
On the other hand, the governing equation of motion for the coarse positioning stage is given by
T ef f
J θ = τ − T (6.25)
where, JT is the total inertial load including the inertia of the ball screw, the coupling, and the inner cylinder of the ER clutch, and the equivalent inertia of the total moving mass. Tf is the total frictional torque including viscous and Coulomb friction given by
4 4
i c
3
f i cl c
v c
4 ( )
( ) sgn( ( )) 2
( ) sgn( ( )) r r
T r l t c t
h
c t c t
⎡ ⎤
πη −
= ⎢ + ⎥θ + θ
⎣ ⎦
= θ + θ
(6.26)
0.0 0.1 0.2 0.3
Time (s) 3.0 kV/mm
2.5 kV/mm 2.0 kV/mm 1.5 kV/mm 1.0 kV/mm
Torque (Nm)
(a)
1.8 1.9 2.0 2.1 2.2
–4 –3 –2 –1 0 1 2 3 4
–0.4 –0.3 –0.2 –0.1 0.0 0.1 0.2 0.3 0.4
Measured (positive) Measured (negative) Estimated
Transmitted torque (Nm)
Electric field (kV/mm) (b)
FIGURE 6.11 Field-dependent torque characteristics of the ER clutch. (a) Step torque response and (b) controllable torque. (From Han, S.S. and Choi, S.B., Proc. Inst. Mech. Eng.
Part C J. Mech. Eng. Sci., 218, 1435, 2004. With permission.)
where cv and cc are the coeffi cient of viscous and Coulomb friction, respectively. By substituting Equation 6.26 into Equation 6.25, one can obtain the following equation:
T ( ) v ( ) ef csgn( ( ))
J θ + θ = τ −t c t c θ t (6.27) Consequently, the governing equation of motion for the coarse positioning system considering the dynamic characteristics of the ER clutch actuator can be derived as follows:
{ }
2 3 3
T cl T v cl v i cl i c c c
( ) ( ) ( ) ( ) 2 2( ) ( ) sgn( ( ))
J λ θ +t J + λ θ + θ = παc t c t ⎡⎢⎣r l +3 r −r ⎤⎥⎦⋅ U t − f θ t (6.28) where
c( ) E( ) sgn( E( )) U t = u t β u t
⎡ ⎤
= πα ⎣⎢ + − ⎥⎦
c 2 3 3
c i cl i c
2( )
2 3
f c r l r r
Table 6.2 presents the dimensional specifi cations of the manufactured coarse posi- tioning stage.