Jing Wang, Ming Zhang,, Yu Zhu, Xin Li and Leijie Wang

(1.State Key Laboratory of Tribology, Department of Mechanical Engineering, Tsinghua University, Beijing 100084, China; 2.Beijing Key Lab of Precision/Ultra-precision Manufacturing Equipments and Control, Tsinghua University, Beijing 100084, China)

Abstract: The null space of the 6-DOF gain decoupling matrix of actuators and a modified velocity feedback controller are adopted to suppress the vibration of the wafer stage during exposure. To deal with varying flexibilities at different performance locations, the vibration controller is designed to be a time-variant linear quadratic regulator, using the conventional gain scheduling method, which could provide good vibration control for each field under exposure. This control method can guarantee the stability of the closed-loop system and will not deteriorate the rigid modes control of the wafer stage. To minimize the control spillover caused by the higher uncontrolled modes, actuator placement is optimized to minimize their controllability grammians in modal coordinates. An unconstrained rectangular plate is used to represent the fine stage of the wafer stage. Effectiveness of the proposed method is verified on the plate through a closed-loop simulation.

Key words: ultra-precision motion stage; vibration control; time-variant performance location

In lithographic equipment, one of the key motion systems is the wafer stage; it positions the wafer with respect to the imaging optics and enables the subsequent exposure on a single wafer[1-2]. The wafer has to be positioned extremely accurately. Meanwhile, higher-speed movements are desired in order to increase throughput. The mainstream positioning stage consists of a fine stage and a coarse stage. The coarse stage provides movement in long strokes. The fine stage is measured and controlled in 6-degrees-of-freedom (6-DOF) to provide the ultimate positioning accuracy. Next-generation motion systems are inevitably lightweight, which leads to a pronounced dynamic behavior in frequency ranges that are relevant for control[3-4]. However, conventional control is designed based on the rigid modes of the stage, which does not take the vibration caused by mode flexibility into account. Over-actuation is an effective way to reduce the vibration, which aims to prevent the actuators from exciting the specified flexible modes[5-6]. However, it also means that the actuators have no control over these flexible modes, thus losing the ability to suppress the vibration caused by these flexible modes due to disturbances and model uncertainties[7]. Vibration feedback control is considered another effective way to suppress the vibration[8-10]. Using a local closed loop is widely employed method for vibration control in addition to the rigid modes control. However, vibration controllers designed to minimize the vibration of one certain point only exhibit a local effect[11-12]. The wafer stage is a time-variant system since the performance location (field under exposure) varies with time. The scanning performance of each field has to be guaranteed instead of just one local point. One way to deal with this is to treat the variance of the field under exposure as the model uncertainty and achieve the desired control performance by using a robust controller[13]. However, since the field under exposure varies from individual to individual throughout the wafer, the flexibility to be controlled will vary greatly, which would deteriorate the performance and introduce instability into the closed loop[14]. An alternative is to apply spatialH2controllers or spatialH∞controllers that minimize the vibration over the entire structure in an average global sense[15-18]. However, these controllers are a great waste of control efforts when applied in the wafer stage, since it is only the scanning performance of the field under exposure that is of interest, thus the control efforts should always be targeted at the field under exposure while leaving other fields out of concern.

Fig.2 Block diagram of the fine stage

In this paper,a new vibration control method utilizing the null space of the 6-DOF gain decoupling matrix of actuators to decouple the vibration control from the rigid modes control is introduced. The direct velocity feedback controller and the phase controller are combined to actively dampen the dominant flexible modes while ensuring the global stability of the closed loop. The time-variant linear quadratic regulator (LQR) method and the conventional gain-scheduling method are adopted to derive the optimal control gain of the controller. Effectiveness of the proposed method is verified through numerical simulation on an unconstrained plate.

1 Null Space of the 6-DOF Gain Decoupling Matrix

A schematic sketch of the wafer stage is shown in Fig.1.The block diagram of the fine stage control is illustrated in Fig.2. The gain balancing matrix H converts the desired global force Fdinto individual actuator forces f

f=H×Fd

(1)

Fig.1 Schematic sketch of the wafer stage

The 6-DOF gain decoupling matrixΓconverts the individual actuator forces f into the global force F, which is defined as

F=Γ×f

(2)

When the number of the actuators exceeds the number of rigid modes, the extra design freedoms of the actuators can be exploited to actively control the vibration by adding another local closed loop to the rigid modes control, as shown in Fig.2. Great efforts can be conserved in the design of these two closed loops if they can be decoupled from each other. In this paper, decoupling is achieved by exploiting the null space of the 6-DOF gain decoupling matrix, which is denoted by x. According to the definition of the null space, the following relationship holds true

Γ×x=0

(3)

Assume that the gain balancing is conducted to minimize the heat dissipation of the actuators. Then, the gain balancing matrix is determined as the pseudo inverse of the 6-DOF gain decoupling matrix[19], which is

H=Γ+

(4)

Then, the global forces F provided for rigid modes control are calculated by

F=Γ×(H×Fd+x×u)=
Γ×(Γ+×Fd+x×u)=
Γ×Γ+×Fd+Γ×x×u=
Fd+0=Fd

(5)

where u is an arbitrary constant vector denoting the control efforts used to control the vibration. Eq. (5) shows that no matter how u varies, it does not change the control forces provided for the rigid modes, which indicates that the flexible modes control and the rigid modes are decoupled. Meanwhile, the individual actuator forces f have changed accordingly since f is related to u

f=Γ+×Fd+x×u

(6)

and x×u does not necessarily have to be zero. Based on this conclusion, a decoupled local control loop that could suppress the vibration is added onto the original rigid modes control loop without affecting it.

2 Combination of Direct Velocity Feedback Control with Phase Control

As illustrated previously, it is only the scanning performance of the field under exposure that contributes to the exposure performance. Therefore, all control efforts should be used to improve the scanning performance of the field under exposure. Due to the time-variant dynamics of performance location, a simple and robust controller is preferred for vibration control, such as the direct velocity feedback controller, which multiplies the velocity output signals through gains and directly feeds them back to actuators[20]. This is equivalent to the attachment of a viscous dashpot to the structural DOF of the sensor[21〗. This controller is quite simple and can guarantee stability in the presence of spillover with the residual modes[20]. One disadvantage is that the controller maintains the assumption that the sensors and actuators are collocated, which is not the case on the wafer stage during exposure since the locations of the actuators are fixed while the field under exposure varies with time. To deal with this problem, phase control is applied in combination with the direct velocity feedback controller to guarantee the global stability of the closed-loop system.

During exposure, the vibration on each field cannot be measured directly but can be inferred from the measured variables. This requires system identification to deliver models that are suitable for high performance inferential control[21]. The wafer stage can be equipped with additional sensors on each field during the identification step. Based on the identified model and measurements during exposure, the vibration on each field can be derived and used for feedback control.

(7)

(8)

(9)

Fig.3 Closed loop transformed into the modal space

(10)

Therefore, the control forceFis calculated by

(11)

whereγ=diag[sgn(φ1) … sgn(φN)] is called the phase controller.

Assume that the mass density is uniform, the energy in the closed-loop system is calculated by[20]

(12)

(13)

whereζ=diag[ζ1…ζN] and its elements are the damping ratios of the controlled flexible modes and B is the input matrix for actuator forces.

Combining Eqs.(11)-(13) with Eq.(7) yields

(14)

The optimal feedback gain can be determined by using the LQR method since it supplies the optimal control and is simple for design[23]. To guarantee the scanning performance of each field under exposure, the conventional gain-scheduling method is utilized due to its simplicity in application[24]. A finite grid composed of operating points is chosen to design the LQR controllers. These controllers are then interpolated to generate a continuously parameterized family of controllers[25].

3 Optimal Actuator Placement to Minimize the Control Spillover Effect

Due to the limitation on the number of actuator numbers, only a few critical flexible modes can be actively controlled through over-actuation. However, the excess control force may excite the higher uncontrolled modes[27], which is called the control spillover effect and should be minimized. In this paper, the control spilllover effect is reduced by minimzing the modal controllability of the higher residual modes through optimizing the actuator placement. The modal controllability can be quantified by the closed-form diagonal entries of the controllability grammians in modal coordinates[22]as given below

(15)

Thus, the cost function is given by

(16)

whereαris the coefficient to weight the relative importance of ther-th residual flexible mode, andwris the modal controllability of ther-th residual flexible mode. The choice of the weight coefficientαris difficult since different performance locations may correspond to a different relative importance of ther-th residual flexible mode. As the performance location moves all over the working area Ω, a new cost function is introduced to minimize the control spillover effect in the global sense

(17)

whereφr(x,y) is the mass-normalized eigenvector of ther-th residual flexible mode evaluated at location (x,y) in working area Ω. During optimization of the actuator placement, design constraints of actuator locations and the limitations on actuator peak forces should be involved. The optimal actuator placement can be derived by solving the following problem as

(18)

where SLand SHdenote the lower bound and the upper bound on the actuator locations, fHdenote the upper bound on actuator peak force.

4 Closed Loop Simulation

For ease of understanding and demonstration, a simple model of an unconstrained plate representing the fine stage is adopted in this paper to validate the effectiveness of the proposed method. The dimension of the plate is 450 mm×450 mm×20 mm. Only the first three flexible modes are considered since it is commonly agreed that lower flexibility modes contribute more to the vibration. The plate is assumed to have uniform mass distribution and a constant cross section. Four voice coil actuators are adopted to drive the plate in theZ,RxandRydirections. The extra design freedom is utilized to actively control the first flexible mode with a natural frequency of 313 Hz. The actuator placement is optimized to minimize the control spillover effect caused by the second and the third flexible mode. By solving the problem in Eq.(17), the optimal actuator placement is derived as shown in Fig.4. Actuators are placed at the nodes of the second flexible mode and the third flexible mode, while away from the node of the first flexible mode. A virtual displacement sensor measuring in theZdirection is placed near the node of the third flexible mode to make this mode not observable. This virtual sensor moves along a circle with the diameter of 340 mm at a velocity of 1 m/s on the plate, which is intended to simulate the variance of the field under exposure.

Fig.4 Unconstrained plate in closed loop simulation with corresponding flexible mode shapes

Classical PID regulators are utilized to control the rigid modes of the plate with the control bandwidth set to be 250 Hz. The sampling time of the control system is selected as 0.2 ms. The disturbance is assumed to be white Guassian noise in the frequency range of 0-2 500 Hz. The closed loop of the plate is illustrated in Fig.5.

To provide control over the rigid modes in theZ,RxandRydirections, the actuator forces should satisfy

(17)

where the vector (f1,f2,f3,f4)Tdenotes the force generated by actuator #1, #2, #3 and #4 , respectively. The termFzddenotes the desired control force in theZdirection, the termTxddenotes the

Fig.5 Closed loop of the unconstrained plate

desired control force in theRxdirection and the termTyddenotes the desired control force in theRydirection. The positioning errors in theZ,RxandRydirections are calculated and provided in Fig.6, Fig.7 and Fig.8, respectively. The vibration in theZdirection caused by the first flexible mode along the circle is calculated and given in Fig.9, while the positioning error along the circle measured by the virtual sensor is given in Fig.10.

Fig.6 Positioning error in Z direction

Fig.7 Positioning error in Rx direction

Fig.8 Positioning error in Ry direction

Fig.9 Vibration in Z direction caused by the first flexible mode along the circle without and with vibration control

Fig.10 Positioning error measured by the virtual sensor along the circle without and with vibration control

The proposed vibration control method is then applied to control the vibration on the plate. The four actuators are assumed to generate point forces for simplification. The orthogonal basis of the null space of the rigid modes gain decoupling matrix can be determined as [-0.50.5-0.50.5]T. The mass-normalized eigenvector of the first flexible mode evaluated at the locations of the actuators is [-0.275 1 0.275 1 -0.275 1 0.275 1]T. It can be found that positive velocity (when the displacement sensor moves to the position where the eigenvector of the first flexible mode is positive) will always yield negative feedback control, while negative velocity (when the displacement sensor moves to the position where the eigenvector of the first flexible mode is negative) will always yield positive feedback control. From this aspect, the phase controller is designed as shown below to guarantee the global stability of the closed loop

(18)

wherexandystand for positions along the plate in theXandYdirection, respectively, with the origin at the center of the plate. For a continuous-time linear system described by

(19)

with a quadratic cost function defined as

(20)

where Q is a positive definite or a semi-positive definite weight matrix and R is a positive definite weight matrix. The feedback control law that minimizes the value of the cost function is

u=-k×x

(21)

where k is given by

k=R-1×BT×P

(22)

and P is found by solving the continuous time algebraic Riccati equation[20]

AT×P+P×A-P×B×R-1×BT×P+Q=0

(23)

As seen in Fig.4,along the circle, the first flexible mode dominates the vibration when the performance location is located away from its node area. While in this area the second flexible mode dominates the vibration. Therefore, a high-performance vibration controller that takes into account the time-varying performance locations into account is desired. For systems with position-dependent dynamics, gain-scheduling control is an appropriate solution[28]. Traditional gain-scheduling control and linear parameter varying (LPV) control are two main approaches[29]that are widely used. LPV control is a design method that guarantees closed loop stability for all possible time-varying parameter trajectories[28]. Thus, the traditional gain-scheduling control method is adopted in combination with the LQR method in this paper to realize high-performance vibration control. A thousand discrete positions on the plate are chosen to design the optimal feedback gains withRselected as 0.000 2. These optimal feedback gains at these positions are then interpolated to derive the function between the optimal feedback gain and the location (x,y), which is given as

(24)

The positioning error measured by the virtual sensor in the presence of the proposed vibration controller is given in Fig.10. It is found that the magnitude of the positioning error has been reduced significantly from 8×10-9m to 3×10-9m due to the existence of the proposed vibration controller, since the contribution of the first flexible mode has greatly decreased, as shown in Fig.9. This conclusion can be further validated by the PSD of the positioning error shown in Fig.11.

Fig.11 PSD of the positioning error measured by the virtual sensor along the circle without and with vibration control

5 Conclusions

To further improve the performance of the ultra-precise motion systems, vibration caused by flexibility should be actively controlled. However, traditional vibration controllers are not feasible for high-performance vibration control due to the time-variant performance location. In this paper, a new control method utilizing the null space of the 6-DOF gain decoupling matrix of actuators is proposed. This control method combines the direct velocity feedback controller with a phase controller to guarantee the global stability of the closed loop. Then, the LQR method and the conventional gain-scheduling method are used to guarantee the vibration control performance over each field under exposure. Meanwhile, actuator placement is optimized to minimize the control effect caused by residual flexible modes. This control method also has the advantage that the vibration controller is independent of the rigid modes control, which could greatly save the effort in the design and evaluation of these two closed loops. An unconstrained plate with an extra design freedom in actuator placement is adopted to verify the proposed method. The results indicate that the proposed method could realize high-performance vibration control at all performance locations.

Since the high-precision model used to calculate the vibration of each field under exposure is never exact, future research will aim to optimize the vibration control method in order to improve final performance.