InTech uses cookies to offer you the best online experience. By continuing to use our site, you agree to our Privacy Policy.

Robotics » Mobile Robotics » "Motion Control", book edited by Federico Casolo, ISBN 978-953-7619-55-8, Published: January 1, 2010 under CC BY-NC-SA 3.0 license. © The Author(s).

Chapter 20

Three Degrees-of-Freedom Hybrid Stage With Dual Actuators and Its Precision Motion Control

By Yonmook Park
DOI: 10.5772/6976

Article top

Overview

Schematic of the hybrid stage.
Figure 1. Schematic of the hybrid stage.
Configuration of the coarse stage.
Figure 2. Configuration of the coarse stage.
Configuration of the fine stage.
Figure 3. Configuration of the fine stage.
Configuration of the voice coil motor.
Figure 4. Configuration of the voice coil motor.
Block diagram of the hybrid stage control system.
Figure 5. Block diagram of the hybrid stage control system.
Block diagram of the coarse stage and fine stage controllers.
Figure 6. Block diagram of the coarse stage and fine stage controllers.
Step responses of the hybrid stage by a step input of 1000 nm magnitude: (a) X-axis step response, (b) Y-axis step response.
Figure 7. Step responses of the hybrid stage by a step input of 1000 nm magnitude: (a) X-axis step response, (b) Y-axis step response.
Euclidean norm histories of current inputs of the four VCMs.
Figure 8. Euclidean norm histories of current inputs of the four VCMs.
Position stabilities of the hybrid stage: (a) X-axis position stability, (b) Y-axis position stability.
Figure 9. Position stabilities of the hybrid stage: (a) X-axis position stability, (b) Y-axis position stability.
X-axis incremental step response and orientation angle response of the hybrid stage: (a) X-axis incremental step response, (b) Orientation angle response during the X-axis incremental step motion.
Figure 10. X-axis incremental step response and orientation angle response of the hybrid stage: (a) X-axis incremental step response, (b) Orientation angle response during the X-axis incremental step motion.
Y-axis incremental step response and orientation angle response of the hybrid stage: (a) Y-axis incremental step response, (b) Orientation angle response during the Y-axis incremental step motion.
Figure 11. Y-axis incremental step response and orientation angle response of the hybrid stage: (a) Y-axis incremental step response, (b) Orientation angle response during the Y-axis incremental step motion.
Bidirectional repeatabilities of the hybrid stage: (a) X-axis bidirectional repeatability, (b) Y-axis bidirectional repeatability.
Figure 12. Bidirectional repeatabilities of the hybrid stage: (a) X-axis bidirectional repeatability, (b) Y-axis bidirectional repeatability.

Three Degrees-of-Freedom Hybrid Stage With Dual Actuators and Its Precision Motion Control

Yonmook Park1

1. Introduction

Nanotechnology can be defined as the research, development and processing of materials, devices and systems in which structure on a dimension of less than several hundreds of nanometer is essential to obtain the required functional performance. There are currently two different approaches to nanotechnology. The first approach is called engineering nanotechnology. This approach involves using classical mechanical and electrical engineering principles to build structures with tolerances at levels approaching a nanometer. The other approach is called molecular nanotechnology. This approach is concerned with self-assembled machines. Due to the startling progress of nanotechnology, we can make high density memories and flat panel display panels with the help of nano-positioning systems. Furthermore, it allows us to manipulate a molecule with scanning probe microscopes.

One of the largest challenges in the field of nanotechnology is precision motion control of macroscopic stages. Especially, precision motion control on the nanometer level that delivers precision position stability with high bandwidth is a very important issue for industrial and scientific applications that especially include the lithography and inspection of integrated circuit patterns (Lee & Kim, 1997; Kwon et al., 2001; Pahk et al., 2001) and the fabrication and operation of high-density magnetic data storage devices (Chung et al., 2000; Kim & Lee, 2004; Lee & Kim, 2004; Du et al., 2005).

In general, traditional stages are designed to operate with respect to a number of kinematic constraints, which are assemblages of mechanical parts and need to be compounded in order to perform multiple degrees-of-freedom (DOF) motion (Shan et al., 2002). These kinematic constraints provide contact friction and error accumulation (Awabdy et al., 1998), and thus, it is very difficult to implement a high-performance motion control system with high bandwidth as well as a precision multiple DOF stage that is capable of large travel with nanometer position stability.

For stages only using coarse actuators such as linear motor or hydraulic actuator, there are nonlinear friction in low-speed motion and resonance mode in high-frequency motion. Thus, it is very difficult to achieve precision motion control performance with only coarse actuators although they provide large travel. One of the methods to overcome their limitation is adopting fine actuators such as piezoelectric actuator or voice coil motor (VCM). But, the travel of fine actuators is about several hundreds of micron meters to several millimeters, which is the limitation of them. Thus, if we design a hybrid actuation system with coarse and fine actuators, we can utilise their advantages and mutually compensate their drawbacks. So far, there have been several studies for the design of hybrid actuation systems. For instance, ultraprecision dual-servo systems have been proposed by Lee & Kim, 1997, Kwon et al., 2001, and Pahk et al., 2001 for lithography steppers. Dual-stage actuation systems consisting of a VCM and a microactuator have been developed by Fan et al., 1995 and Li & Horowitz, 2001 to obtain high servo bandwidth and perform the disturbance rejection. A novel control design that aims to achieve a low-hump sensitivity function for a dual-stage system in hard disk drives has been studied by Du et al., 2005. Note that, according to Du et al., 2005, a servo control system with low-hump sensitivity function is able to reduce the contribution from disturbance to a system. Fundamental control designs of dual-stage hard disk drive systems have been presented by Chung et al., 2000, Kim & Lee, 2004 and Lee & Kim, 2004, and performance enhancement methods of dual-stage servo systems have been proposed by Wu et al., 2002 and Li et al., 2003.

This chapter presents a three DOF precision hybrid stage that can move and align an object on it for the measurement of its three-dimensional image using the confocal scanning microscope (CSM). The CSM can observe a sub-micron meter-sized material due to its fine resolution and has a three-dimensional surface profiling capability. The hybrid stage consists of two individually operating x-y-θ stages, called the coarse stage and the fine stage. The coarse stage is driven by the three linear motors, and the fine stage is driven by the four VCMs. The coarse and fine stages are not mechanically interconnected and can be controlled independently.

For control of the hybrid stage, the author proposes a precision motion controller in this chapter. The precision motion controller consists of a position and velocity control loop, an anti-windup compensator to eliminate the windup problem that occurs in the controller, a generator of optimal force to optimally control the fine stage, a precision position determiner to determine the exact position of the fine stage and a perturbation observer that can observe the perturbation of the fine stage and compensate it. Note that, in this chapter, the exact position of the fine stage means the centre of the fine stage that is precisely determined by considering the orientation angle of the fine stage. The performances of the precision motion controller are evaluated by experiment.

The remainder of this chapter is organized as follows. In Section 2, the system overview of the hybrid stage is described. In Section 3, the hybrid stage control method is presented. In Section 4, the experimental results of the hybrid stage motion control are given. Finally, some concluding remarks are given in Section 5.

2. Hybrid stage with dual actuators

2.1. Overview

Fig. 1 shows the schematic of the hybrid stage presented in this paper. The objective of the hybrid stage is to move and align an object on it for the measurement of its three-dimensional image using the CSM. The CSM has a capability of the optical sectioning and can generate three-dimensional surface profile. The measurement principle of the CSM is based on the fact that only light reflected from the focal point of the objective lens contributes to the image, whereas all diffusely scattered light beams are filtered out by a pinhole. This creates a focused two-dimensional image of all object points that are located during the scanning process in the focal plane, similar to the contour lines of a map. Scanning the whole samples with an automatically varying focal plane results in a highly resolved and enlarged image of the corresponding surface section. The vertical and horizontal resolutions of the CSM are 30 and 140 nm, respectively.

media/image1.jpg

Figure 1.

Schematic of the hybrid stage.

The hybrid stage consists of two individually operating x-y-θ stages, called the coarse stage and the fine stage. The coarse stage produces the initial movement of an object, and the fine stage provides the final alignment of the object. Since the laser interferometer with 0.31 nm resolution is used as position sensor in the fine stage, the precision control for the final alignment of an object can be possible. The moving parts of the hybrid stage are sustained by air bearings so that they can float on the base plate without mechanical contact. The material of the base plate is granite, and the base plate is connected with an isolator that can suppress internal and external vibrations.

2.2. Coarse stage

The schematic of the coarse stage is shown in Fig. 2. The coarse stage is driven by the three linear motors, and uses the 14 air bearings as guide and the three linear encoders as position sensor. The linear motor can be moved by the following Lorentz force

FLM(t)=iLM(t)dlLM×BLM
(1)

where FLM(t) , iLM(t) , lLM(t) and BLM are force, current, coil length and flux density of the linear motor, respectively. A three-phase linear motor with the force of 233 N is used. The three linear motors are mechanically linked in an H-shaped rigid frame and can generate the x-y-θ motion of the coarse stage. Specifically, the stators of the two linear motors LM1 and LM2 are fixed to the base plate and parallel to each other. And the sliders of the two linear motors LM1 and LM2 are connected by the stator of the linear motor LM3 that floats on the base plate. Thus, the movements of the three linear motors LM1, LM2, and LM3 determine the x-y-θ motion of the coarse stage. The mass of each individual linear motor is 15 kg. The position sensor of the coarse stage is a linear encoder with the resolution of 5 nm. The coarse stage offers a large workspace of 500 500 mm2.

media/image7.jpg

Figure 2.

Configuration of the coarse stage.

Next, the kinematics of the coarse stage is derived. For the coarse stage shown in Fig. 2, let the vector pc(t) be given by

pc(t)=[xc(t)yc(t)θc(t)]T
(2)

where xc(t) and yc(t) are the Xc and Yc positions of the coarse stage, respectively, and θc(t) is the orientation angle of the coarse stage. Let the vector pLM(t) be given by

pLM(t)=[xLM3(t)yLM1(t)yLM2(t)]T
(3)

where xLM3(t) , yLM1(t) and yLM2(t) are the displacements of the three linear motors LM3, LM1 and LM2, respectively. In (3), xLM3(t) , yLM1(t) and yLM2(t) are measured by the linear encoder. Under the assumption that the orientation angle θc(t) in (2) is very small, the vector pc(t) in (2) can be determined by the following equation

pc(t)=Hc(t)pLM(t)T
(4)

where

Hc(t)=[1000L2xLM3(t)L1+L2L1+xLM3(t)L1+L201L1+L21L1+L2]
(5)

In (5), L1 and L2 are the distances from the centers of the linear motors LM1 and LM2 to the center of the coarse stage, respectively. In the sequel, the position and orientation angle of the coarse stage are represented as in (4).

media/image24.jpg

Figure 3.

Configuration of the fine stage.

2.3. Fine stage

The configuration of the fine stage is shown in Fig. 3. The fine stage is driven by the four VCMs, and uses the four air bearings as guide and a laser interferometer as position sensor. The four VCMs lie on the same plane so that the tilting forces that cause the roll and pitch motions of the fine stage are negligible. The VCM can be moved by the following Lorentz force

FVCM(t)=iVCM(t)dlVCM×BVCM
(6)

where FVCM(t) , iVCM(t) , lVCM and BVCM are force, current, coil length and flux density of VCM, respectively. The four VCMs generate the x-y-θ motion of the fine stage by the following equation

AFVCM(t)=uf(t)
(7)

where

A=[11000011b2b2a2a2]
(8)
FVCM(t)=[FVCM1(t)FVCM2(t)FVCM3(t)FVCM4(t)]T
(9)

and

uf(t)=[Ffx(t)Ffy(t)Tfθ(t)]T
(10)

In (8)–(10), a is the distance from the centre of VCM1 to the center of VCM3 (or from the center of VCM2 to the center of VCM4), b is the distance from the center of VCM1 to the center of VCM4 (or from the centre of VCM2 to the center of VCM3), FVCMi(t) , i=1,,4 are the forces of VCMi, i=1,,4 , respectively, Ffx(t) and Ffy(t) are the X-axis and Y-axis control forces for the fine stage, respectively, and Tfθ(t) is the control torque for the fine stage.

If we apply the current to coils of VCM1 and VCM2, the fine stage is driven in the X-axis direction. Similarly, we apply the current to coils of VCM3 and VCM4 for a driving in the Y-axis direction. In addition, the fine stage is driven in the θ direction if we make proper current and apply it to each coil. The VCM has the force of 220 N, and the mass of the fine stage is 36.5 kg. The position sensor of the fine stage is a laser interferometer with the resolution of 0.31 nm. The laser interferometer measures the x-y-θ motion of the fine stage by projecting laser beams onto the L-shaped plane mirror attached on top of the fine stage. The workspace of the fine stage is 5 5 mm2, and the range of the orientation angle of the fine stage is 0.05 deg.

Now, the kinematics of the fine stage is derived. Let the vector pf(t) be given by

pf(t)=[xf(t)yf(t)θf(t)]T
(11)

where xf(t) and yf(t) are the Xf and Yf positions of the fine stage, respectively, and uf(t) is the orientation angle of the fine stage. Let the vector pVCM(t) be given by

pVCM(t)=[xVCM(t)yVCM(t)θVCM(t)]T
(12)

where xVCM(t) and yVCM(t) are the X-axis and Y-axis displacements of VCM, respectively, and uVCM(t) is the orientation angle of VCM, which is equal to uf(t) . In (12), xVCM(t) , yVCM(t) and θVCM(t) are measured by the laser interferometer. Since the orientation angle θVCM(t) in (12) is not very small compared with θc(t) in (2), we should consider θVCM(t) to determine xf(t) and yf(t) in (11) and obtain the following equations by lengthy calculation.

xf(t)=[(xVCM(t)+r)cosθVCM(t)r]cosθVCM(t)[(yVCM(t)r)cosθVCM(t)+r]sinθVCM(t)
(13)
yf(t)=[(xVCM(t)+r)cosθVCM(t)r]sinθVCM(t)+[(yVCM(t)r)cosθVCM(t)+r]cosθVCM(t)
(14)

where r is the distance from the centre of the fine stage to the L-shaped plane mirror attached on top of the fine stage. Consequently, the position of the fine stage can be precisely determined by (13) and (14).

Note that, as shown in Fig. 4, the VCM consists of magnet, yokes and coil. The magnet and yoke of VCM are fixed on the fine stage. On the other hand, the coil of VCM is fixed on the coarse stage. In addition, the magnet sticks to the yokes and does not come into contact with the coil. Thus, the coarse and fine stages are not mechanically interconnected and can be controlled independently.

media/image57.jpg

Figure 4.

Configuration of the voice coil motor.

3. Precision motion control of hybrid stage with dual actuators

3.1. Overview

This section presents a precision motion control method of the x-y-θ motion of the hybrid stage. The motion control performance of the hybrid stage mainly depends on the motion control performance of the fine stage because the fine stage accomplishes the final alignment of an object. Therefore in this section the attention is focused on the precision motion control of the fine stage.

The block diagram of the hybrid stage control system is shown in Fig. 5. The coarse and fine stages are independently controlled under the common reference command. Let the reference command of the hybrid stage be given by

pref(t)=[xref(t)yref(t)θref(t)]T
(15)

where xref(t) , yref(t) and θref(t) are the X-axis position reference command, the Y-axis position reference command and the orientation angle reference command of the hybrid stage, respectively.

media/image62.jpg

Figure 5.

Block diagram of the hybrid stage control system.

3.2. Coarse stage control system

The error vector ec(t) for the coarse stage is defined as

ec(t)=pref(t)pc(t)
(16)

where pref(t) and pc(t) are defined in (15) and (2). Now the author explains each component of the coarse stage control system. First, the coarse stage controller consists of a position control loop, a velocity control loop and an antiwindup compensator. Specifically, as shown in Fig. 6, the position control loop has a proportional controller, and the velocity control loop has a proportional and integral controller. In addition, the velocity control loop is combined with an anti-windup compensator based on Bohn & Atherton, 1995 in order to eliminate the windup problem caused by the integral controller. In Fig. 6, ki(t) , i=1,,4 are positive scalars and s is the Laplace operator. The coarse stage controller generates the three control inputs FLM1(t) , FLM2(t) and FLM3(t) for control of the x-y-θ motion of the coarse stage where FLM1(t) , FLM2(t) and FLM3(t) are the control forces for the three linear motors LM1, LM2 and LM3, respectively.

Second, the coarse stage kinematics implies the transformation of pLM(t) in (3) into pc(t) in (2) by (4). Finally, the coarse stage inverse kinematics represents the transformation of ec(t) in (16) into the error vector eLM(t) , given by

eLM(t)=Hc1ec(t)=Hc1(pref(t)pc(t))=Hc1pref(t)pLM(t)
(17)

where Hc(t) is defined in (5) and its inverse matrix is given by

Hc1(t)=[10001L1xLM3(t)01L2xLM3(t)]
(18)

3.3. Fine stage control system

The error vector ef(t) for the fine stage is defined as

ef(t)=pref(t)pf
(19)

where pref(t) and pf(t) are defined in (15) and (11). The author describes each component of the fine stage control system. First, as shown in Fig. 6, the fine stage controller has the same structure that the coarse stage controller has. The fine stage controller produces the three control inputs Ffx(t) , Ffy(t) and Tfθ(t) for control of the x-y-θ motion of the fine stage.

Second, the precision position determiner means the equations of (13) and (14). Third, the generator of optimal force is proposed to make the optimal forces of the four VCMs. As shown in (7), after designing the three control inputs Ffx(t) , Ffy(t) and Tfθ(t) , we should determine the four forces FVCMi(t) , i=1,,4 of the four VCMs. In this case, (7) has infinitely many solutions for the four forces FVCMi(t) , i=1,,4 because it is underdetermined with three equations in four unknowns. Among many solutions to the above problem, the author presents a meaningful solution to (7) by considering a least squares problem. Before deriving a meaningful solution to (7), the following definition is given.

media/image91.jpg

Figure 6.

Block diagram of the coarse stage and fine stage controllers.

Definition 1 (Leon, 1995): Let ARm×n have the rank of q < n. Then the singular value decomposition of A is given by

A=UΣVT=[U1U2]T[Σ10q×(nq)0(mq)×q0(mq)×(nq)][V1TV2T]=U1Σ1V1T
(20)

where U=[U1U2]Rm×m and V=[V1V2]Rn×n are orthogonal matrices with U1Rm×q , U2Rm×(mq) , V1Rn×q and V2Rn×(nq) . Moreover, 0m×n denotes the m × n zero matrix and Σ1Rq×q is a diagonal matrix given by

Σ1=[σ10000σ20000σ30000σq]
(21)

with the entries satisfying

σ1σ1σq0
(22)

Then the following theorem shows that the singular value decomposition provides the key to solve the least squares problem for design of the optimal forces of the four VCMs.

Theorem 1: Consider the equation (7). Let the singular value decomposition of A be UΣVT and define

where A+ denotes the pseudo-inverse of A. Then the following is a solution to (7):

FVCM(t)=A+uf(t)=[120ba2+b2120ba2+b2012aa2+b2012aa2+b2][Ffx(t)Ffy(t)Tfθ(t)]
(24)

Moreover, if h(t) is any other solution to (7), then we can guarantee

FVCM(t)2h(t)2
(25)

where   2 denotes the Euclidean norm.

Proof: Let FVCM(t)R4 and define

j(t)=UTu(t)f=[j1(t)j2(t)]
(26)
k(t)=VTF(t)VCM=[k1(t)k2(t)]
(27)

From the definition of singular value decomposition and (26) and (27), we can obtain

uf(t)AFVCM(t)22=UT22uf(t)AFVCM(t)22=UTuf(t)UT(UΣVTFVCM(t))22=UTuf(t)ΣVTFVCM(t)22=j(t)ΣVTFVCM(t)22=j(t)Σk(t)22=[j1(t)j2(t)][Σ1000][k1(t)k2(t)]22=[j1(t)Σ1k1(t)j2(t)]22=j1(t)Σ1k1(t)22+j2(t)22
(28)

Since j2(t) is independent of FVCM(t) , it follows that uf(t)AFVCM(t)22 will be minimal if and only if j1(t)Σ1k1(t)2=0 . Furthermore, uf(t)AFVCM(t)22 will be zero if and only if j1(t)Σ1k1(t)2=0 and j2(t)2=0 . Thus, FVCM(t) becomes a solution to (7) if and only if FVCM(t)=Vk(t) and j2(t)=0 where k(t) is a vector of the form

k(t)=[k1(t)k2(t)]=[Σ11j1(t)k2(t)]
(29)

Especially, FVCM(t)=A+uf(t) is a solution to (7) because

FVCM(t)=Vk(t)=V[Σ11j1(t)0]=V[Σ11000][j1(t)j2(t)]=VΣ+j(t)=VΣ+UTuf(t)=A+uf(t)=[120ba2+b2120ba2+b2012aa2+b2012aa2+b2][Ffx(t)Ffy(t)Tfθ(t)]
(30)

Next, if h(t) is any other solution to (7), h(t) must be of the form

h(t)=Vk(t)=V[Σ11j1(t)k2(t)]
(31)

where k2(t)0 . Then, for h(t) of (31), we can obtain the following result:

h(t)22=Vk(t)22=k(t)22=Σ11j1(t)22+k2(t)22Σ11j1(t)22=FVCM(t)22
(32)

This completes the proof.

Physically, FVCM(t) in (24) is the minimal norm solution to (7) such that the condition of (25) holds for any other solution to (7). Therefore, if we use FVCM(t) in (24), we can achieve the optimal performance in the sense of the control effort.

Finally, the author presents the feedforward and feedback perturbation observers by extending the study of Kwon et al., 2001. Specifically, the perturbation applied to the nominal dynamics of the fine stage can be expressed by

ψf(t)=Hfnp¨f(t)+Bfnp˙f(t)uf(t)
(33)

where Hfn and Bfn are the nominal inertia matrix and the nominal viscous damping coefficient of the fine stage, respectively, uf(t) is the control input of the fine stage defined in (10). Since one-step delay in signals is inevitable for the causality between input and output in practice, the perturbation observer is presented as follows

ψ^f(t)=Df{Hfnp¨f(ttc)+Bfnp˙f(ttc)}uf(ttc)
(34)

where Df is a diagonal matrix with scalar elements that plays the role of approximating ψ^f(t) to the real perturbation of ψf(t) , and tc is the control interval. If we apply ψ^f(t) in (34) to the nominal dynamics of the fine stage in order to compensate the perturbation, the nominal dynamic equation of the fine stage can be changed to

Hfnp¨f(t)+Bfnp˙f(t)=ηf(t)+ψ˜f(t)
(35)

where ηf(t)=uf(t)+ψ^f(t) is the tracking control input and ψ˜f(t)=ψf(t)ψ^f(t) is the perturbation compensation error. Since the reference command can be utilized in tracking control, the author presents the feedforward perturbation observer as follows

ψ^fff(t)=Dfff{Hfnp¨ref(ttc)+Bfnp˙ref(ttc)}uf(ttc)
(36)

where Dfff is a diagonal matrix with scalar elements. Also, the residue of the perturbation, given by Δψ˜f(t)=ψf(t)ψ^fff(t) , is compensated by the following feedback perturbation observer

ψ^ffb(t)=Δψ˜f(ttc)=Dffb{Hfnp¨f(ttc)+Bfnp˙f(ttc)}Dfff{Hfnp¨ref(t2tc)+Bfnp˙ref(t2tc)}uf(ttc)+uf(t2tc)
(37)

where Dffb is a diagonal matrix with scalar elements.

It is remarkable that the perturbation observers presented in this section are the generalization of the perturbation observers developed by Kwon et al., 2001 because their study can be regarded as a special case of the proposed method with Dfff=diag[111] and Dffb=diag[111] where diag means the diagonal matrix. Because of Dfff and Dffb , we have the extra freedom of designing the perturbation observers in real application. The perturbation observers presented in this section cannot help using delayed information because the current perturbation is monitored in discrete time with one-step delay. Using delayed information may result in bandwidth degradation and it is inevitable.

With a similar manner presented by Kwon et al., 2001, if we assume that the fine stage is time invariant during a control interval, the full state is available, the change of external disturbances during the control intervals is bounded, and the nominal inertia matrix of the fine stage Hfn satisfies the following condition for the real inertia matrix of the fine stage Hf(k) for all samples k

then the perturbation compensation error ψ˜f(t) in (35) is well bounded in a sufficiently small value.

4. Experimental results

As the hybrid stage control platform, the author uses the dSPACE system that features a power PC processor and is directly connected to all dSPACE I/O boards. The dSPACE system is an efficient and reliable engineering tool to develop and test control systems, and is in widespread use in many automotive industries. The graphical user interface software is programmed in order to control the hybrid stage by using the dSPACE system. By the graphical user interface software, we can give the target position command and target orientation angle command to the hybrid stage and can set all control parameters of the precision motion controller.

Then the performances of the precision motion controller of the hybrid stage are evaluated by experiment. The update rate of the dSPACE system is set to be 1 kHz. The author initially decides the gains k1, k2 and k3 of the coarse stage and fine stage controllers by adopting the Ziegler–Nichols method (Ogata, 1996), which is very useful to select the control gains of a proportional, integral and derivative-type controller for complex dynamic systems in practice, and then further tunes these gains in order to obtain a desired control performance in terms of the step response. Also, the author designs the gain k4 of the anti-windup compensator by an experimental method such that we make the overshot appearing in the step response, caused by windup, as small as possible. In the sequel, the undamped natural frequencies of the X-axis and Y-axis motions are decided to lie approximately at 117.909 and 118.448 rad/s, respectively. Also, the damping ratios of the X-axis and Y-axis motions are decided to lie approximately at 0.590 and 0.595, respectively.

media/image155.jpeg

Figure 7.

Step responses of the hybrid stage by a step input of 1000 nm magnitude: (a) X-axis step response, (b) Y-axis step response.

Fig. 7 shows the experimental result of the X-axis and Y-axis step responses of the hybrid stage by a step input of 1000 nm magnitude. From Fig. 7, we see that the maximum overshoots of the X-axis and Y-axis motions are 10.07% and 9.76%, respectively, the delay times of the X-axis and Y-axis motions are 0.014 and 0.014 s, respectively, the rise times of the X-axis and Y-axis motions are 0.026 and 0.025 s, respectively, the peak times of the X-axis and Y-axis motions are 0.033 and 0.033 s, respectively, and the 5% settling times of the X-axis and Y-axis motions are 0.071 and 0.072 s, respectively. Also, Fig. 7 demonstrates that the hybrid stage effectively responds to a step input of 1000 nm magnitude in the X-axis and Y-axis motions. Specifically, when the coarse and fine stages are operated for a step input of 1000 nm magnitude, the X-axis and Y-axis steady-state errors of the fine stage after 0.071 and 0.072 s rising periods remain within 50 nm, respectively. On the other hand, the X-axis and Y-axis steady-state errors of the coarse stage reach more than 100 and 80 nm although the time elapses 0.08 and 0.12 s after the step input, respectively. Therefore it is concluded that the hybrid stage has remarkable advantages in terms of the response time and positioning accuracy. Although there is no direct contact between the coarse and fine stages, the motion errors or vibrations of the coarse stage may lead to variations of the interaction force between the two stages. These variations present themselves as disturbances to the fine stage, and the resolution of the fine stage degrades unless the bandwidth of the fine stage is substantially higher than that of the coarse stage. This observation explains the fluctuation of the fine stage at steady state in Fig. 7.

Note that FVCM(t) in (24) is the minimal norm solution to (7). In order to demonstrate the result, the Euclidean norm histories of current inputs of the four VCMs is shown in Fig. 8. As shown in Fig. 8, the generator of optimal force yields the control effort of about 0.05 A. On the other hand, we need the control effort of about 2.8 A if we do not use the generator of optimal force. In this case, we can save the control effort significantly for the operation of the four VCMs by adopting the generator of optimal force.

media/image157.jpeg

Figure 8.

Euclidean norm histories of current inputs of the four VCMs.

Since the perturbation gives rise to a vibration of the hybrid stage in practice, the author determines the gains of the perturbation observers by the experimental method such that we make the position stability of the hybrid stage as small as possible. Then Fig. 9 shows the experimental result for the X-axis and Y-axis position stabilities of the hybrid stage. From Fig. 9, the X-axis and Y-axis position stabilities by the perturbation observers are about 10 nm, respectively. On the other hand, the X-axis and Y-axis position stabilities are about 30 nm, respectively, if we do not use the perturbation observers. Consequently, we see that the perturbation observers have the function of observing the perturbation and compensating it effectively about 66%.

media/image158.jpeg

Figure 9.

Position stabilities of the hybrid stage: (a) X-axis position stability, (b) Y-axis position stability.

Now the X-axis and Y-axis incremental step responses and orientation angle responses of the hybrid stage are evaluated by applying some step input to the hybrid stage. Specifically, the X-axis and Y-axis target positions are increased by 10 nm from 0 to 50 nm, and then decreased by 10 nm from 50 to 10 nm, respectively. Note that the coarse stage is on operation when the author conducts the X-axis and Y-axis incremental step responses and orientation angle responses.

media/image159.jpeg

Figure 10.

X-axis incremental step response and orientation angle response of the hybrid stage: (a) X-axis incremental step response, (b) Orientation angle response during the X-axis incremental step motion.

Then Fig. 10 shows the experimental results of the X-axis incremental step response and the orientation angle response during the X-axis incremental step motion, and Fig. 11 shows the experimental results of the Y-axis incremental step response and the orientation angle response during the Y-axis incremental step motion. From Figs. 10 and 11, we see that the resolutions of the X-axis and Y-axis motions are about 10 nm, respectively, and the fluctuations of the orientation angle during the X-axis and Y-axis incremental step motions are about 0.02 and 0.04 arcsec, respectively. Note that 1 arcsec is equal to 1/3600 deg.

Finally, the X-axis and Y-axis bidirectional repeatabilities of the hybrid stage are tested. Note that the repeatability is the error between a number of successive attempts to move the machine to the same position (Slocum, 1992). And the bidirectional repeatability is the repeatability achieved when the target position is approached from two different directions (Slocum, 1992).

media/image160.jpeg

Figure 11.

Y-axis incremental step response and orientation angle response of the hybrid stage: (a) Y-axis incremental step response, (b) Orientation angle response during the Y-axis incremental step motion.

In order to evaluate the X-axis bidirectional repeatability, the X-axis target position is increased by 40 mm from 0 to 400 mm, and then decreased by 40 mm from 400 to 0 mm. Also, in order to evaluate the Y-axis bidirectional repeatability, the Y-axis target position is increased by 30 mm from 0 to 300 mm, and then decreased by 30 mm from 300 to 0 mm. Then Fig. 12 shows the experimental results of the X-axis and Y-axis bidirectional repeatabilities. As shown in Fig. 12, the X-axis and Y-axis bidirectional repeatabilities are about 48.9 and 40.7 nm(6), respectively.

media/image161.jpeg

Figure 12.

Bidirectional repeatabilities of the hybrid stage: (a) X-axis bidirectional repeatability, (b) Y-axis bidirectional repeatability.

5. Conclusion

In this chapter, the author presented a three degrees-of-freedom precision hybrid stage that can move and align an object on it for the measurement of its three-dimensional image using the confocal scanning microscope. Since the hybrid stage consists of two individually operating x-y-θ stages, it has not only a long operation travel but also a fine position stability. In order to control the hybrid stage, the author proposed a precision motion controller. The author evaluated the performances of the precision motion controller by experiment with a hardware setup. The experimental results showed that the precision motion controller provided the hybrid stage with desirable advantages in terms of the response time, positioning accuracy, control effort and perturbation compensation.

References

1 - B. A. Awabdy, W.-C. Shin, D. M. Auslander, 1998 Nanometer positioning of a linear motion stage under static loads. IEEE/ASME Trans. Mechatron., 3 2 113 119
2 - C. Bohn, D. P. Atherton, 1995 An analysis package comparing PID anti-windup strategies. IEEE Control Syst. Mag., 15 2 34 40
3 - C. C. Chung, C. W. Seo, S. H. Lee, 2000 Two degree-of-freedom dual-stage actuator controller design for hard disk drives. IEEE Trans. Magn., 36 5 2255 2257
4 - C. Du, G. Guo, D. Wu, 2005 Low-hump sensitivity function design for dual-stage HDD systems with different microactuators. IEE Proc., Control Theory Appl., 152 6 655 661
5 - L. S. Fan, H. H. Ottesen, T. C. Reiley, R. W. Wood, 1995 Magnetic recording head positioning at very high track densities using a microactuator-based two-stage servo system. IEEE Trans. Ind. Electron., 42 3 222 233
6 - Y.-H. Kim, S.-H. Lee, 2004 An approach to dual-stage servo design in computer disk drives. IEEE Trans. Control Syst. Technol., 12 1 12 20
7 - S. Kwon, W. K. Chung, Y. Youm, 2001 On the coarse/fine dual-stage manipulators with robust perturbation compensator, Proc. IEEE Int. Conf. Robot. Autom., 121 126, Seoul, Korea, May 2001
8 - C.-W. Lee, S.-W. Kim, 1997 An ultraprecision stage for alignment of wafers in advanced microlithography. Precision Eng., 21, 2-3, 113 122
9 - S.-H. Lee, Y.-H. Kim, 2004 Minimum destructive interference design of dual-stage control systems for hard disk drives. IEEE Trans. Control Syst. Technol., 12 4 517 531
10 - S. J. Leon, 1995 Linear Algebra With Application, Prentice Hall, New Jersey
11 - Y. Li, R. Horowitz, 2001 Mechatronics of electrostatic microactuators for computer disk drive dual-stage servo system. IEEE/ASME Trans. Mechatron., 6 2 111 121
12 - Y. Li, R. Horowitz, R. Evans, 2003 Vibration control of a PZT actuated suspension dual-stage servo system using a PZT sensor. IEEE Trans. Magn., 39 2 932 937
13 - K. Ogata, 1996 Modern Control Engineering, Prentice Hall, New Jersey
14 - H. J. Pahk, D. S. Lee, J. H. Park, 2001 Ultra precision positioning system for servo motor-piezo actuator using the dual servo loop and digital filter implementation. Int. J. Mach. Tools Manuf., 41 1 51 63
15 - X. Shan, S.-K. Kuo, J. Zhang, C.-H. Menq, 2002 Ultra precision motion control of a multiple degrees of freedom magnetic suspension stage. IEEE/ASME Trans. Mechatron., 7 1 67 78
16 - A. H. Slocum, 1992 Precision Machine Design, Prentice Hall, New Jersey
17 - D. Wu, G. Guo, T. C. Chong, 2002 Midfrequency disturbance suspension via microactuator in dual-stage HDDs. IEEE Trans. Magn., 38 5 2189 2191