## 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.

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

where
*x-y-θ* motion of the coarse stage. Specifically, the stators of the two linear motors LM_{1} and LM_{2} are fixed to the base plate and parallel to each other. And the sliders of the two linear motors LM_{1} and LM_{2} are connected by the stator of the linear motor LM_{3} that floats on the base plate. Thus, the movements of the three linear motors LM_{1}, LM_{2}, and LM_{3} 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 mm^{2}.

Next, the kinematics of the coarse stage is derived. For the coarse stage shown in Fig. 2, let the vector

where

where
_{3}, LM_{1} and LM_{2}, respectively. In (3),

where

In (5),
_{1} and LM_{2} 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).

### 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

where
*x-y-θ* motion of the fine stage by the following equation

where

and

In (8)–(10), *a* is the distance from the centre of VCM_{1} to the center of VCM_{3} (or from the center of VCM_{2} to the center of VCM_{4}), *b* is the distance from the center of VCM_{1} to the center of VCM_{4} (or from the centre of VCM_{2} to the center of VCM_{3}),
_{i},
*X*-axis and *Y*-axis control forces for the fine stage, respectively, and

If we apply the current to coils of VCM_{1} and VCM_{2}, the fine stage is driven in the X-axis direction. Similarly, we apply the current to coils of VCM_{3} and VCM_{4} 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 mm^{2}, 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

where

where

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.

## 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

where

### 3.2. Coarse stage control system

The error vector

where
*x-y-θ* motion of the coarse stage where
_{1}, LM_{2} and LM_{3}, respectively.

Second, the coarse stage kinematics implies the transformation of

where

### 3.3. Fine stage control system

The error vector

where
*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

*Definition 1* (Leon, 1995): Let

where

with the entries satisfying

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

where

Moreover, if

where

*Proof:* Let

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

(28) |

Since

Especially,

(30) |

Next, if

where

This completes the proof.

Physically,

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

where

where

where

where

(37) |

where

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

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

then the perturbation compensation error

## 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 k_{1}, k_{2} and k_{3} 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 k_{4} 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.

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

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%.

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.

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).

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.

## 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.