Abstract:
A damping apparatus that supports and dampens a stage apparatus that positions and drives a stage to a target position is provided herein. The damping apparatus including a support plate part, a support force generating means, and a first controlling means. The support plate part supports the stage apparatus. The support force generating means exerts a damping action by applying a support force to the support plate part in the vertical directions. The first controlling means uses the acceleration of the stage, which is derived from a target track, to the target position, to control the support force generated by the support force generating means so as to compensate for forces that both occur as a result of the acceleration of the stage and cause the support plate part to tilt. The present invention controls vibration and the tilt of a base plate with high precision.

Description:
RELATED INVENTIONS 
       [0001]    This application claims priority on U.S. Provisional Application Ser. No. 61/078,200, filed Jul. 3, 2008 and entitled “LITHOGRAPHY MACHINE PLATFORM CONTROL METHOD TO REDUCE STAGE MOTION AND WEIGHT INDUCED DEFORMATION”. As far as permitted, the contents of U.S. Provisional Application Ser. No. 61/078,200 are incorporated herein by reference. 
     
    
     FIELD OF THE INVENTION 
       [0002]    The present invention relates to a damping apparatus that dampens a positioning stage, and an exposure apparatus that uses the damping apparatus. 
       BACKGROUND 
       [0003]    An exposure apparatus that exposes a semiconductor substrate with a circuit pattern, which is formed in a mask, uses a stage apparatus to position the mask and the semiconductor substrate. Advances in circuit miniaturization have brought about an increasing demand for greater positioning performance of the stage apparatus. Typically, the stage apparatus is mounted on a base plate, which is a mechanism that prevents the transmission of vibrations from, for example, the floor to the stage; furthermore, actuators are used both to prevent vibration of the base plate and to correct any tilt of the base plate and the stage that results from the off-centered load that is applied to the base plate based on the position of the stage thereon; thereby the attitude of the base plate is controlled. 
         [0004]    With regard to a damping apparatus that uses actuators to control vibration as well as the tilt of the base plate, the conventionally adopted method measures the position of the base plate relative to the floor with a position sensor and performs feedback control to maintain zero displacement between the floor and the base plate. Nevertheless, the conventional controlling method cannot control either the vibration or the tilt of the base plate with sufficient accuracy, and, as a result, the positioning accuracy of the stage cannot be improved using such a method. 
         [0005]    The present invention considers the abovementioned points, and it is an object of the present invention to provide a damping apparatus capable of controlling both the vibration and the tilt of a base plate with high accuracy, and an exposure apparatus that uses the damping apparatus. 
       SUMMARY  
       [0006]    The present invention solves the abovementioned problems, is a damping apparatus that supports and dampens a stage apparatus, which positions and drives a stage to a target position, and comprises: a support plate part that supports the stage apparatus; a support force generating means that exerts a damping action by applying a support force to the support plate part in the vertical directions; and a first controlling means that, based on the acceleration of the stage, which is derived from the target track, to the target position, controls the support force generated by the support force generating means so as to compensate for forces that both occur as a result of the acceleration of the stage and cause the support plate part to tilt. 
         [0007]    According to this configuration, performing feedforward control, based on the stage&#39;s acceleration as derived from the target track of the stage to be positioned, makes it possible to perform control with good accuracy so that the support plate part (the base plate) does not tilt. 
         [0008]    In addition, in the abovementioned damping apparatus, the first controlling means controls the support force generated by the support force generating means so that torsional deformation does not occur in the support plate part. 
         [0009]    According to this configuration, it is possible to prevent torsion from occurring in the support plate part. 
         [0010]    In addition, the abovementioned damping apparatus comprises: a first position detecting means that detects the position of the support plate part in the vertical directions at a plurality of locations; a first deriving means that derives the center of gravity position, the inclination angle, and the amount of torsion of the support plate part based on its position at a plurality of locations as detected by the first position detecting means; and a second controlling means that, based on the center of gravity position, the inclination angle, and the amount of torsion derived by the first deriving means, controls the support force generated by the support force generating means so that inclination and torsional deformation do not occur in the support plate part. 
         [0011]    According to this configuration, it is possible to control the support plate part with greater accuracy by detecting the inclination and the torsion of the same and by adding feedback control so that inclination and torsion do not occur in the support plate part. 
         [0012]    In addition, the abovementioned damping apparatus comprises: a second position detecting means that detects the position of a structure, which is provided on the stage, at a plurality of locations; a second deriving means that derives the amount of torsion of the structure based on the position at a plurality of locations detected by the second position detecting means; and a third controlling means that controls the support force generated by the support force generating means so that the amount of torsion derived by the second deriving means is zeroed. 
         [0013]    According to this configuration, it is possible to perform control so that torsion does not occur in the structure on the stage. The structure on the stage is, for example, a lens that is used in an optical system of an exposure apparatus, which ultimately makes it possible to perform exposures with good accuracy. 
         [0014]    In addition, the abovementioned damping apparatus comprises: a fourth controlling means that, based on the position of the stage, which is derived from the target track, to the target position, controls the support force generated by the support force generating means so that the stage does not tilt. 
         [0015]    According to this configuration, it is possible to ensure that the stage does not tilt by performing feedforward control based on the position of the stage as derived from the target track of the stage to be positioned. According to the present invention, it is possible to control the vibration and the tilt of the base plate with high accuracy. 
     
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         [0016]      FIG. 1  is a schematic block diagram of a damping apparatus and a stage apparatus, which is installed on the damping apparatus, according to one embodiment of the present invention. 
           [0017]      FIG. 2  is a block diagram that shows the configuration of a control system of the damping apparatus. 
           [0018]      FIG. 3  is a view for explaining the calculating method wherein a movement position-based thrust calculating unit calculates a thrust instruction value F 2 . 
           [0019]      FIGS. 4(A) , (B) are views for explaining a calculating method wherein an acceleration state-based thrust calculating unit calculates a thrust instruction value F 1 . 
           [0020]      FIG. 5  is a view that shows a lens that is mounted to a base plate. 
           [0021]      FIG. 6  is a block diagram of an exposure apparatus whereto the damping apparatus and the stage apparatus are adapted. 
       
    
    
     DESCRIPTION  
       [0022]      FIG. 1  is a schematic block diagram of a damping apparatus and a stage apparatus, which is installed on the damping apparatus, according to one embodiment of the present invention. 
         [0023]    In  FIG. 1 , the damping apparatus comprises a base plate  101  and four mounts (leg parts)  4 A,  4 B,  4 C,  4 D, which are provided to the four corners, respectively, of the base plate  101  and support such on the floor surface. Each of the mounts  4 A- 4 D is an air spring type support mechanism and holds the base plate  101  at a prescribed height above the floor surface. In addition, the pressure of the air spring of each of the mounts  4 A- 4 D can be controlled independent of that of the other mounts; furthermore, applying an appropriate pressure to the air spring of each of the mounts  4 A- 4 D in accordance with the state of the stage apparatus—e.g., the position and acceleration of a plate table  110  within a horizontal plane—makes it possible to maintain the base plate  101  in a horizontal state and to ensure that vibrations from the floor surface are not transmitted thereto. 
         [0024]    In  FIG. 1 , the stage apparatus, which is configured as described below, is installed on the upper surface of the base plate  101 . Namely, two X axis guides  102  (of the two, only the one on the near side is shown in the figure), which extend in the X directions, are laid parallel to each other on the upper surface of the base plate  101  so that they are spaced apart in the Y directions, and X carriages  103  are movably provided so that they straddle both of the side parts and the upper part of each of the X axis guides  102 . A Y axis guide  104 , which extends in the Y directions (those directions perpendicular to the paper surface), is securely fixed to the upper parts of the X carriages  103 ; furthermore, the Y axis guide  104  has a bridge-like suspension and connects the two X carriages  103 . A plurality of air bearings  105  is disposed between the X carriages  103  and the X axis guides  102 . The air bearings  105  are fixed to the X carriages  103 ; furthermore, the X carriages  103  and the Y axis guide  104  (in the explanation below, these two elements are collectively called an X stage for the sake of convenience) are noncontactually supported so that they float on the X axis guides  102 , are guided by the X axis guides  102 , and are configured movably in the X directions. 
         [0025]    A Y carriage  107  is mounted to the upper part of the Y axis guide  104 . A plurality of air bearings  108  (base bearings) is disposed between the Y carriage  107  and the upper surface of the Y axis guide  104 , and a plurality of air bearings  109  (side bearings) is disposed between the Y carriage  107  and both side surfaces of the Y axis guide  104 . The air bearings  109  are attached to the Y carriage  107  in the state wherein the air bearings  109  may move in the θ directions. All of the air bearings  108 ,  109  are fixed to the Y carriage  107 ; in addition, the Y carriage  107  (in the explanation below, called the Y stage for the sake of convenience), which is supported noncontactually so that it floats on the Y axis guide  104 , is guided by the Y axis guide  104 , and is configured movably in the Y directions and the θ directions. 
         [0026]    The plate table  110  is supported rotatably in the θ directions—rotational directions wherein the Z axis, which is perpendicular to the X and Y axes, is the rotational axis—on the Y carriage  107  by a plurality of air bearings  113 . A plate holder  111 , which vacuum chucks a glass substrate and the like, and a position sensor  112 , which is used for positional measurement, are provided on the plate table  110 . The position sensor  112  is, for example, a movable mirror that constitutes part of an interferometric measuring instrument (a laser interferometer); in addition, a laser beam from a light source that is provided externally (above the base plate  101 ) is radiated as a measuring beam to the movable mirror, is reflected thereby, and so is caused to interfere with a reference beam inside the abovementioned interferometric measuring instrument; thereby the position of the plate holder  111  is measured precisely using the base plate  101  as a reference. 
         [0027]    An X linear motor (not shown) is provided on each of both end parts (the near side and the far side of the paper surface) of the Y axis guide  104 , and the drive of these X linear motors moves a structure, which is mounted to the X carriages  103 , along the X axis guides  102  in the X directions. In addition, a Y linear motor  106  is provided to each side surface part of the Y carriage  107 , and the drive of the Y linear motors  106  moves the structure that is mounted to the Y carriage  107  along the Y axis guide  104  in the Y directions. Furthermore, in  FIG. 1 , while the movers of the Y linear motors  106  are shown, the stators are not. The stators may be provided to the X carriages  103 , but may also be provided to some other members. In addition, the drive of a θ linear motor (not shown) rotates the plate table  110  in the θ directions. Alternatively, creating differences in the magnitude of the thrusts generated by the left and right Y linear motors  106  may also rotate the plate table  110  in the θ directions. 
         [0028]      FIG. 2  is a block diagram that shows the configuration of a control system of the above mentioned damping apparatus. In  FIG. 2 , a damping apparatus  10  comprises: a track calculating unit  11 ; an acceleration state-based thrust calculating unit  12 ; a movement position-based thrust calculating unit  13 ; a lens torsion correcting unit  14 ; a pressure control unit  15 ; a thrust feedback control unit  16 ; a coordinates converting unit  17 ; a base plate torsion calculating unit  18 ; and a base plate torsion correcting unit  19 . 
         [0029]    The track calculating unit  11  inputs X coordinates X S , X E  of an operation start point and an operation end point, which are used to track and control the X stage, and Y coordinates Y S , Y E  of an operation start point and an operation end point, which are used to track and control the Y stage. The track calculating unit  11  calculates a target track for tracking and controlling each of the stages (the X stage and the Y stage) based on the operation start points and operation end points, in other words, the inputted X coordinates X S , X E  and the inputted Y coordinates Y S , Y E . The target track is represented by time series data for the X stage position X(t) and the Y stage position Y(t) associated with time t. The target track may be set arbitrarily in accordance with the required positioning performance (e.g., gradual or sharp acceleration). 
         [0030]    The acceleration state-based thrust calculating unit  12  first calculates, based on the target track calculated by the track calculating unit  11 , the acceleration X″ (X directions) of the X stage and the acceleration Y″ (Y directions) of the Y stage. The acceleration X″ and the acceleration Y″ can be derived by double differentiating the positional coordinates of the target track with respect to time. The acceleration state-based thrust calculating unit  12  next uses equations (8), (6-1), and (6-2) (discussed later) to calculate, based on the accelerations X″, Y″, a thrust instruction value F 1  that is applied to each of the mounts  4 A,  4 B,  4 C,  4 D to cancel the forces that cause the base plate  101  to tilt, which occurs as a result of the accelerative drive of the stages. Furthermore, the symbol ″ in the text of the present specification means the second derivative with respect to time. 
         [0031]    The movement position-based thrust calculating unit  13  substitutes the positional coordinates of the target track, which are calculated by the track calculating unit  11 , in equations (1-1)-(1-4) (discussed later) to calculate a thrust instruction value F 2  that is needed to cause each of the mounts  4 A,  4 B,  4 C,  4 D to generate a prescribed thrust for canceling the forces that tend to tilt the base plate  101 . The forces that tend to tilt the base plate  101  occur because the positions of the X stage and the Y stage, in both their moving and stationary states, are shifted from the center of the base plate  101 . 
         [0032]    The lens torsion correcting unit  14  calculates a thrust instruction value F 3  that is applied to each of the mounts  4 A,  4 B,  4 C,  4 D to correct torsion of a lens, which occurs when the lens is mounted to the base plate  101  via a prescribed support mechanism. If the stage apparatus and the damping apparatus of the present embodiment are adapted to an exposure apparatus in  FIG. 6  (discussed later), then the lens (not shown in  FIG. 1 ) may be considered part of the optical system for performing exposures. The base plate  101  is provided with a laser interferometer (not shown in  FIG. 1 ) that measures the precise position thereof in order to detect torsion of the lens, and the lens torsion correcting unit  14  inputs position measurement values a, b, c, d, which the laser interferometer measures at multiple locations of the lens. The lens torsion correcting unit  14  uses equation (14) (discussed later) to calculate the thrust instruction value F 3  based on the inputted measurement positions a, b, c, d. 
         [0033]    The pressure control unit  15  controls the thrust (which is proportional to the pressure of the air spring) in the Z directions (the height direction) that is generated by each of the mounts  4 A- 4 D in accordance with a total thrust instruction value, which is the sum of the following: the thrust instruction value F 1 , calculated by the acceleration state-based thrust calculating unit  12 ; the thrust instruction value F 2 , calculated by the movement position-based thrust calculating unit  13 ; the thrust instruction value F 3 , calculated by the lens torsion correcting unit  14 ; the thrust instruction value F 4 , calculated by the thrust feedback control unit  16 ; and the thrust instruction value F 5 , calculated by the base plate torsion correcting unit  19 . The base plate  101  is held in place by the aggregate of the thrusts F 1 -F 5 . 
         [0034]    Here, in order to detect the attitude of the base plate  101 , the mounts  4 A- 4 D are provided with position sensors (not shown in  FIG. 1 ), one position sensor for each mount, that detect the displacements Z A , Z B , Z C , Z D  thereof in the Z directions. The coordinates converting unit  17  uses equation (9) (discussed later) to transform the displacements Z A , Z B , Z C , Z D , which are detected by the position sensors, of each of the mounts  4 A- 5 D to the following: a center of gravity position Z of the base plate  101  in the Z directions; a rotational angle θ X  of the base plate  101  around the X axis; and a rotational angle θ Y  of the base plate  101  around the Y axis. 
         [0035]    The thrust feedback control unit  16  inputs the differences between the measurement values Z, θ X , θ Y , which the coordinates converting unit  17  outputs, and the respective target reference values Z 0 , θ X0 , θ Y0 . The thrust feedback control unit  16  first calculates, based on the differential values for Z, θ X , θ Y , the accelerations Z″, θ X ″, θ Y ″ that are needed to ensure that the actual attitude of the base plate  101 , i.e., the measurement values Z, θ X , θ Y , coincides with the reference values Z 0 , θ X0 , θ Y0 . It is possible to perform these calculations using a calculating method based on, for example, PID (proportional-integral-derivative) control. The thrust feedback control unit  16  next uses equation (11) (discussed later) to calculate, based on the accelerations Z″, θ X ″, θ Y ″, the thrust instruction value F 4  that is applied to each of the mounts  4 A,  4 B,  4 C,  4 D to cause the base plate  101  to undergo these accelerations. 
         [0036]    The base plate torsion calculating unit  18  uses equation (12) (discussed later) to calculate the torsion Z T  of the base plate  101  based on the displacements Z A , Z B , Z C , Z D  of the mounts  4 A- 4 D as detected by the position sensors. 
         [0037]    The base plate torsion correcting unit  19  inputs the differential between the torsion Z T  of the base plate  101 , which is output by the base plate torsion calculating unit  18 , and the target reference value Z T0  of the torsion Z T . The base plate torsion correcting unit  19  uses equation (13) (discussed later) to calculate, based on the differential values, the thrust instruction value F 5 , which is applied to each of the mounts  4 A,  4 B,  4 C,  4 D to make the torsion Z T  of the base plate  101  coincide with the reference value Z T0 . It is possible to perform this calculation using a calculating method that is based on, for example, PID (proportional-integral-derivative) control. 
         [0038]    The following explains in detail the controlling method that is implemented by each of the abovementioned parts.  FIG. 3  is a view that shows the arrangement of the base plate  101  and the Y stage (the X stage is not shown) and explains the calculating method wherein the movement position-based thrust calculating unit  13  calculates the thrust instruction value F 2 . If the position of the center of gravity of the Y stage is (X, Y), then the thrust instruction value F 2 , which is calculated by the movement position-based thrust calculating unit  13  to ensure that the base plate  101  does not tilt when the Y stage shifts from the origin (0, 0), is defined by equations (1-1)-(1-4) below. 
         [0000]    
       
         
           
             
               
                 
                   Equations 
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                    
                   1 
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   
                     F 
                     
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         [0039]    Therein, F 2A , F 2B , F 2C , F 2D  are thrust instruction values that are applied to the mounts  4 A,  4 B,  4 C,  4 D, W is the gravity that the X stage and the Y stage exert upon the base plate  101 , W Y  is the gravity that the Y stage alone exerts on the base plate  101 , L X  is the distance between mounts in the X directions, and L Y  is the distance between mounts in the Y directions. In addition, the positional coordinates of the target track, which are calculated by the track calculating unit  11 , are used as the center of gravity coordinates (X, Y). 
         [0040]    The objective for unit  13  is to calculate thrust forces F 2A (t), F 2B (t), F 2C (t), F 2D (t) using equations (1-1)-(1-4) given the measured center coordinates X(t) and Y(t) at any given time (t). The parameters W, Wy, Lx, Ly are to remain constant throughout operation. Any inaccuracy in W, Wy, Lx, Ly may thus result in sub-optimal result in the calculation of F 2A (t), F 2B (t), F 2C (t), F 2D (t). Such parameter inaccuracy may be due to difficulty in the identification of true center of gravity location. To remedy such situation, we can take advantage of the fact that equations (1-1)-(1-4) should hold true not only when X or Y stages are moving but also when both stages are not moving. When the stage is commanded to stay steady at a given location X(tp)=Xp and Y(tp)=Yp, the association between the steady states of F 2A (tp), F 2B (tp), F 2C (tp), F 2D (tp) and the steady states of X(tp), Y(tp) becomes accurately determined, because the four displacement sensors Z A , Z B , Z C , Z D  of the mounts  4 A- 4 D can serve as measurement reference for the determination of wheather steady state has been reached. After we repeat such procedure at least four different locations, the solvability of equations (1-1)-(1-4) becomes mathematically established and the true center of gravity location becomes accurately identified, reflected in the re-adjustment of parameters W, Wy, Lx, Ly. 
         [0041]    The air spring pressure instruction values that are applied to the mounts  4 A- 4 D to generate the thrust of the thrust instruction value F 2  mentioned above can be expressed by P 2A =F 2A /a A , P 2B =F 2B /a B , P 2C =F 2C /a C , and P 2D =F 2D /a D , respectively Therein, a A , a B , a C , a D  are the effective bearing surface areas of the mounts  4 A- 4 D, respectively. 
         [0042]    Furthermore, the above equations (1-1)-(1-4) can be derived as below. 
         [0043]    Namely, the equilibrium equation of the forces of the base plate  101  in the Z directions is defined by equation (2-1) below; the equilibrium equation of the moments around the X axis is defined by equation (2-2) below; and the equilibrium equation of the moments around the Y axis is defined by equation (2-3) below. 
         [0000]    
       
         
           
             
               
                 
                   Equations 
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                   2 
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   
                     
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         [0044]    If we define F 2A +F 2B ≡F α  and F 2C +F 2D ≡F β  and solve equations (2-1) and (2-2) for F α  and F β , we obtain equations (3-1), (3-2) below. 
         [0000]    
       
         
           
             
               
                 
                   Equations 
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                     F 
                     β 
                   
                   = 
                   
                     
                       W 
                       2 
                     
                     + 
                     
                       
                         
                           W 
                           Y 
                         
                         
                           L 
                           Y 
                         
                       
                        
                       Y 
                     
                   
                 
               
               
                 
                   ( 
                   
                     3 
                      
                     
                       - 
                     
                      
                     2 
                   
                   ) 
                 
               
             
           
         
       
     
         [0045]    At this point, if we establish equilibrium equations of the forces of the base plate  101  only on the mounts  4 A,  4 B side, then we obtain equations (4-1), (4-2) below. Therein, W α  defines the gravity that the X stage and the Y stage exert on the mounts  4 A,  4 B side of the base plate  101 . 
         [0000]    
       
         
           
             
               
                 
                   Equations 
                    
                   
                       
                   
                    
                   4 
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   
                     
                       F 
                       
                         2 
                          
                         A 
                       
                     
                     + 
                     
                       F 
                       
                         2 
                          
                         B 
                       
                     
                   
                   = 
                   
                     W 
                     α 
                   
                 
               
               
                 
                   ( 
                   
                     4 
                      
                     
                       - 
                     
                      
                     1 
                   
                   ) 
                 
               
             
             
               
                 
                   
                     
                       
                         W 
                         α 
                       
                        
                       X 
                     
                     + 
                     
                       
                         
                           L 
                           X 
                         
                         2 
                       
                        
                       
                         F 
                         
                           2 
                            
                           A 
                         
                       
                     
                   
                   = 
                   
                     
                       
                         L 
                         X 
                       
                       2 
                     
                      
                     
                       F 
                       
                         2 
                          
                         B 
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     4 
                      
                     
                       - 
                     
                      
                     2 
                   
                   ) 
                 
               
             
           
         
       
     
         [0046]    If we solve the abovementioned equations (4-1), (4-2) for F 2A  and F 2B , we obtain equations (5-1), (5-2) below. 
         [0000]    
       
         
           
             
               
                 
                   Equations 
                    
                   
                       
                   
                    
                   5 
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   
                     F 
                     
                       2 
                        
                       A 
                     
                   
                   = 
                   
                     
                       ( 
                       
                         
                           1 
                           2 
                         
                         - 
                         
                           X 
                           
                             L 
                             X 
                           
                         
                       
                       ) 
                     
                      
                     
                       W 
                       α 
                     
                   
                 
               
               
                 
                   ( 
                   
                     5 
                      
                     
                       - 
                     
                      
                     1 
                   
                   ) 
                 
               
             
             
               
                 
                   
                     F 
                     
                       2 
                        
                       B 
                     
                   
                   = 
                   
                     
                       ( 
                       
                         
                           1 
                           2 
                         
                         + 
                         
                           X 
                           
                             L 
                             X 
                           
                         
                       
                       ) 
                     
                      
                     
                       W 
                       α 
                     
                   
                 
               
               
                 
                   ( 
                   
                     5 
                      
                     
                       - 
                     
                      
                     2 
                   
                   ) 
                 
               
             
           
         
       
     
         [0047]    Accordingly, based on equations (3-1), (4-1) and the definitions of W α  and F α , we obtain equation (1-1) from the above equation (5-1), and equation (1-2) from the above equation (5-2). Equations (1-3), (1-4) are similarly obtained. The obtained equations (1-1)-(1-4) are correct solutions because they satisfy the relationship of the above equilibrium equation (2-3). 
         [0048]      FIGS. 4(A) , (B) are views for explaining the calculating method wherein the acceleration state-based thrust calculating unit  12  calculates the thrust instruction value F 1 , and shows aspects, a top view in  FIG. 4(A)  and a side view in  FIG. 4(B) , wherein the Y stage accelerates and moves toward the base plate  101  (the X stage is not shown). As shown in  FIG. 4(B) , the point of application of the acceleration force F Y , which acts on the Y stage as a result of the thrusts from the Y linear motors  106  discussed earlier, deviates from a center of gravity CG Y  of the Y stage by a height of d Y . Similarly, the point of application of the acceleration force F X , which acts on an X moving body, wherein the X stage and the Y stage are considered integral, as a result of the thrust from the X linear motors, deviates from a center of gravity CG X  of the X moving body by a height of d X . Thus, because the application points of the acceleration forces deviate from the centers of gravity, when the stage, driven by the motors, accelerates, these acceleration forces cause moments of force around the X and Y axes to act on the base plate  101 , which tilts as a result. The moments of force around the X axis M X  and the moments of force around the Y axis M Y  are defined by equations (6-1) and (6-2) below. 
         [0049]    Equations 6 
         [0000]        M   X   =−F   X   ·d   X =−( m   xy   ·X ″)· d   X =−( m   xy   ·d   X )· X″   (6-1) 
         [0000]        M   Y   =−F   Y   ·d   Y =−( m   y   ·Y ″)· d   Y =−( m   y   ·d   Y )· Y″   (6-2) 
         [0050]    Therein, m y  is the mass of the Y stage, m xy  is the mass of the X moving body, wherein the X stage and the Y stage are considered integral, and X″, Y″ are the accelerations of the centers of gravity of the Y stage in the X directions and the Y directions, respectively. The values obtained by double differentiating the Y stage center of gravity coordinates (X, Y), which are determined by the target track positional coordinates, with respect to time are used as the acceleration X″ in the X directions and the acceleration Y″ in the Y directions. 
         [0051]    At this time, the mounts  4 A,  4 B,  4 C,  4 D generate thrusts F 1A , F 1B , F 1C , F 1D , respectively, to the base plate  101 , whereon the abovementioned moments of force M X , M Y  act during acceleration, so that the base plate  101  does not tilt as a result of these moments of force M X , M Y . The equilibrium equation of the forces that act on the base plate  101  in the Z directions at this time is defined by equation (7-1) below, the equilibrium equation of the moments around the X axis is defined by the equation (7-2) below, the equilibrium equation of the moments around the Y axis is defined by the equation (7-3) below, and the equilibrium equation that must be satisfied to ensure the base plate  101  is not subject to torsion is defined by the equation (7-4) below. 
         [0000]    
       
         
           
             
               
                 
                   Equations 
                    
                   
                       
                   
                    
                   7 
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   
                     
                       F 
                       
                         1 
                          
                         A 
                       
                     
                     + 
                     
                       F 
                       
                         1 
                          
                         B 
                       
                     
                     + 
                     
                       F 
                       
                         1 
                          
                         C 
                       
                     
                     + 
                     
                       F 
                       
                         1 
                          
                         D 
                       
                     
                   
                   = 
                   0 
                 
               
               
                 
                   ( 
                   
                     7 
                      
                     
                       - 
                     
                      
                     1 
                   
                   ) 
                 
               
             
             
               
                 
                   
                     
                       
                         
                           L 
                           X 
                         
                         2 
                       
                        
                       
                         ( 
                         
                           
                             F 
                             
                               1 
                                
                               B 
                             
                           
                           + 
                           
                             F 
                             
                               1 
                                
                               C 
                             
                           
                         
                         ) 
                       
                     
                     - 
                     
                       
                         
                           L 
                           X 
                         
                         2 
                       
                        
                       
                         ( 
                         
                           
                             F 
                             
                               1 
                                
                               A 
                             
                           
                           + 
                           
                             F 
                             
                               1 
                                
                               D 
                             
                           
                         
                         ) 
                       
                     
                   
                   = 
                   
                     M 
                     X 
                   
                 
               
               
                 
                   ( 
                   
                     7 
                      
                     
                       - 
                     
                      
                     2 
                   
                   ) 
                 
               
             
             
               
                 
                   
                     
                       
                         
                           L 
                           Y 
                         
                         2 
                       
                        
                       
                         ( 
                         
                           
                             F 
                             
                               1 
                                
                               C 
                             
                           
                           + 
                           
                             F 
                             
                               1 
                                
                               D 
                             
                           
                         
                         ) 
                       
                     
                     - 
                     
                       
                         
                           L 
                           Y 
                         
                         2 
                       
                        
                       
                         ( 
                         
                           
                             F 
                             
                               1 
                                
                               A 
                             
                           
                           + 
                           
                             F 
                             
                               1 
                                
                               B 
                             
                           
                         
                         ) 
                       
                     
                   
                   = 
                   
                     M 
                     Y 
                   
                 
               
               
                 
                   ( 
                   
                     7 
                      
                     
                       - 
                     
                      
                     3 
                   
                   ) 
                 
               
             
             
               
                 
                   
                     
                       
                         L 
                         X 
                       
                        
                       
                         
                           L 
                           Y 
                         
                          
                         
                           ( 
                           
                             
                               F 
                               
                                 1 
                                  
                                 A 
                               
                             
                             + 
                             
                               F 
                               
                                 1 
                                  
                                 C 
                               
                             
                           
                           ) 
                         
                       
                     
                     - 
                     
                       
                         L 
                         X 
                       
                        
                       
                         
                           L 
                           Y 
                         
                          
                         
                           ( 
                           
                             
                               F 
                               
                                 1 
                                  
                                 B 
                               
                             
                             + 
                             
                               F 
                               
                                 1 
                                  
                                 D 
                               
                             
                           
                           ) 
                         
                       
                     
                   
                   = 
                   0 
                 
               
               
                 
                   ( 
                   
                     7 
                      
                     
                       - 
                     
                      
                     4 
                   
                   ) 
                 
               
             
           
         
       
     
         [0052]    Solving the above equations (7-1)-(7-4) results in equation (8), which defines the thrust instruction value F 1  as calculated by the acceleration state-based thrust calculating unit  12 . 
         [0000]    
       
         
           
             
               
                 
                   Equation 
                    
                   
                       
                   
                    
                   8 
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   
                     [ 
                     
                       
                         
                           
                             F 
                             
                               1 
                                
                               A 
                             
                           
                         
                       
                       
                         
                           
                             F 
                             
                               1 
                                
                               B 
                             
                           
                         
                       
                       
                         
                           
                             F 
                             
                               1 
                                
                               C 
                             
                           
                         
                       
                       
                         
                           
                             F 
                             
                               1 
                                
                               D 
                             
                           
                         
                       
                     
                     ] 
                   
                   = 
                   
                     [ 
                     
                       
                         
                           
                             
                               - 
                               
                                 
                                   M 
                                   X 
                                 
                                 
                                   2 
                                    
                                   
                                     L 
                                     X 
                                   
                                 
                               
                             
                             - 
                             
                               
                                 M 
                                 Y 
                               
                               
                                 2 
                                  
                                 
                                   L 
                                   Y 
                                 
                               
                             
                           
                         
                       
                       
                         
                           
                             
                               
                                 M 
                                 X 
                               
                               
                                 2 
                                  
                                 
                                   L 
                                   X 
                                 
                               
                             
                             - 
                             
                               
                                 M 
                                 Y 
                               
                               
                                 2 
                                  
                                 
                                   L 
                                   Y 
                                 
                               
                             
                           
                         
                       
                       
                         
                           
                             
                               
                                 M 
                                 X 
                               
                               
                                 2 
                                  
                                 
                                   L 
                                   X 
                                 
                               
                             
                             + 
                             
                               
                                 M 
                                 Y 
                               
                               
                                 2 
                                  
                                 
                                   L 
                                   Y 
                                 
                               
                             
                           
                         
                       
                       
                         
                           
                             
                               - 
                               
                                 
                                   M 
                                   X 
                                 
                                 
                                   2 
                                    
                                   
                                     L 
                                     X 
                                   
                                 
                               
                             
                             + 
                             
                               
                                 M 
                                 Y 
                               
                               
                                 2 
                                  
                                 
                                   L 
                                   Y 
                                 
                               
                             
                           
                         
                       
                     
                     ] 
                   
                 
               
               
                 
                   ( 
                   8 
                   ) 
                 
               
             
           
         
       
     
         [0053]    As is clear from the process used to derive equation (8), if the mounts  4 A- 4 D generate thrusts in accordance with the thrust instruction value F 1  defined above in equation (8), then the base plate  101  will be controlled such that neither tilt nor torsion occur during the acceleration of the Y stage. Furthermore, similar to the case of the thrust instruction value F 2  discussed above, the air spring pressure instruction value that is applied to each of the mounts  4 A- 4 D in order to generate the thrust defined by the abovementioned thrust instruction value F 1  can be defined so that P 1A =F 1A /a A , P 1B =F 1B /a B , P 1C =F 1C /a C , and P 1D =F 1D /a D . 
         [0054]    The following explains the calculating method wherein the thrust feedback control unit  16  calculates the thrust instruction value F 4 . Equation (9) below defines the conversion equation that converts the displacements Z A , Z B , Z C , Z D  of the mounts  4 A- 4 D, respectively, as detected by the position sensors to the following: a position Z of the center of gravity of the base plate  101  in the Z directions; a rotational angle θ X  of the base plate  101  around the X axis; and a rotational angle θ Y  of the base plate  101  around the Y axis. Therein, ΔX and ΔY are the center of gravity positions—which change when the X stage, the Y stage, or the like move—of the base plate  101  (including the X stage and the Y stage), and the L X , L Y  are the distances in the X directions and the Y directions, respectively, between the position sensors provided to the mounts  4 A- 4 D. 
         [0000]    
       
         
           
             
               
                 
                   Equation 
                    
                   
                       
                   
                    
                   9 
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   
                     [ 
                     
                       
                         
                           Z 
                         
                       
                       
                         
                           
                             θ 
                             X 
                           
                         
                       
                       
                         
                           
                             θ 
                             Y 
                           
                         
                       
                     
                     ] 
                   
                   = 
                   
                     
                       [ 
                       
                         
                           
                             
                               
                                 
                                   ( 
                                   
                                     
                                       L 
                                       Y 
                                     
                                     - 
                                     
                                       2 
                                        
                                       Δ 
                                        
                                       
                                           
                                       
                                        
                                       Y 
                                     
                                   
                                   ) 
                                 
                                  
                                 
                                   ( 
                                   
                                     
                                       L 
                                       X 
                                     
                                     - 
                                     
                                       2 
                                        
                                       Δ 
                                        
                                       
                                           
                                       
                                        
                                       X 
                                     
                                   
                                   ) 
                                 
                               
                               
                                 4 
                                  
                                 
                                   L 
                                   X 
                                 
                                  
                                 
                                   L 
                                   Y 
                                 
                               
                             
                           
                           
                             
                               
                                 
                                   ( 
                                   
                                     
                                       L 
                                       Y 
                                     
                                     - 
                                     
                                       2 
                                        
                                       Δ 
                                        
                                       
                                           
                                       
                                        
                                       Y 
                                     
                                   
                                   ) 
                                 
                                  
                                 
                                   ( 
                                   
                                     
                                       L 
                                       X 
                                     
                                     + 
                                     
                                       2 
                                        
                                       Δ 
                                        
                                       
                                           
                                       
                                        
                                       X 
                                     
                                   
                                   ) 
                                 
                               
                               
                                 4 
                                  
                                 
                                   L 
                                   X 
                                 
                                  
                                 
                                   L 
                                   Y 
                                 
                               
                             
                           
                           
                             
                               
                                 
                                   ( 
                                   
                                     
                                       L 
                                       Y 
                                     
                                     + 
                                     
                                       2 
                                        
                                       Δ 
                                        
                                       
                                           
                                       
                                        
                                       Y 
                                     
                                   
                                   ) 
                                 
                                  
                                 
                                   ( 
                                   
                                     
                                       L 
                                       X 
                                     
                                     + 
                                     
                                       2 
                                        
                                       Δ 
                                        
                                       
                                           
                                       
                                        
                                       X 
                                     
                                   
                                   ) 
                                 
                               
                               
                                 4 
                                  
                                 
                                   L 
                                   X 
                                 
                                  
                                 
                                   L 
                                   Y 
                                 
                               
                             
                           
                           
                             
                               
                                 
                                   ( 
                                   
                                     
                                       L 
                                       Y 
                                     
                                     + 
                                     
                                       2 
                                        
                                       Δ 
                                        
                                       
                                           
                                       
                                        
                                       Y 
                                     
                                   
                                   ) 
                                 
                                  
                                 
                                   ( 
                                   
                                     
                                       L 
                                       X 
                                     
                                     - 
                                     
                                       2 
                                        
                                       Δ 
                                        
                                       
                                           
                                       
                                        
                                       X 
                                     
                                   
                                   ) 
                                 
                               
                               
                                 4 
                                  
                                 
                                   L 
                                   X 
                                 
                                  
                                 
                                   L 
                                   Y 
                                 
                               
                             
                           
                         
                         
                           
                             
                               - 
                               
                                 1 
                                 
                                   2 
                                   · 
                                   
                                     L 
                                     Y 
                                   
                                 
                               
                             
                           
                           
                             
                               - 
                               
                                 1 
                                 
                                   2 
                                   · 
                                   
                                     L 
                                     Y 
                                   
                                 
                               
                             
                           
                           
                             
                               1 
                               
                                 2 
                                 · 
                                 
                                   L 
                                   Y 
                                 
                               
                             
                           
                           
                             
                               1 
                               
                                 2 
                                 · 
                                 
                                   L 
                                   Y 
                                 
                               
                             
                           
                         
                         
                           
                             
                               1 
                               
                                 2 
                                 · 
                                 
                                   L 
                                   X 
                                 
                               
                             
                           
                           
                             
                               - 
                               
                                 1 
                                 
                                   2 
                                   · 
                                   
                                     L 
                                     X 
                                   
                                 
                               
                             
                           
                           
                             
                               - 
                               
                                 1 
                                 
                                   2 
                                   · 
                                   
                                     L 
                                     X 
                                   
                                 
                               
                             
                           
                           
                             
                               1 
                               
                                 2 
                                 · 
                                 
                                   L 
                                   X 
                                 
                               
                             
                           
                         
                       
                       ] 
                     
                      
                     
                       [ 
                       
                         
                           
                             
                               Z 
                               A 
                             
                           
                         
                         
                           
                             
                               Z 
                               B 
                             
                           
                         
                         
                           
                             
                               Z 
                               C 
                             
                           
                         
                         
                           
                             
                               Z 
                               D 
                             
                           
                         
                       
                       ] 
                     
                   
                 
               
               
                 
                   ( 
                   9 
                   ) 
                 
               
             
           
         
       
     
         [0055]    The differentials between the measurement values Z, θ X , θ Y  of the center of gravity positions and the rotational angles of the base plate  101  as derived from the abovementioned equation (9) and the corresponding target reference values Z 0 , θ X0 , θ Y0  define the tracking errors for controlling the attitude of the base plate  101 . Typical PID control and other techniques can be used to derive the accelerations Z″, θ X ″, θ Y ″ to be applied to the base plate  101 , and thereby keep these tracking errors small; in other words, to make the attitude (Z, θ X , θ Y ) of the base plate  101  approach the state of the reference values. Thrusts F 4A , F 4B , F 4C , F 4D  are generated from the mounts  4 A,  4 B,  4 C,  4 D, respectively, so that the base plate  101  is subject to these derived accelerations Z″, θ X ″, θ Y ″. The equilibrium equation of the forces of the base plate  101  in the Z directions at this time is defined by the equation (10-1) below, the equilibrium equation of the moments around the X axis is defined by the equation (10-2) below, the equilibrium equation of the moments around the Y axis is defined by the equation (10-3) below, and the equilibrium equation that is needed to ensure that the base plate  101  does not undergo torsion is defined by the equation (10-4) below. 
         [0000]    
       
         
           
             
               
                 
                   Equations 
                    
                   
                       
                   
                    
                   10 
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   
                     
                       F 
                       
                         4 
                          
                         A 
                       
                     
                     + 
                     
                       F 
                       
                         4 
                          
                         B 
                       
                     
                     + 
                     
                       F 
                       
                         4 
                          
                         C 
                       
                     
                     + 
                     
                       F 
                       
                         4 
                          
                         D 
                       
                     
                   
                   = 
                   
                     m 
                     · 
                     
                       Z 
                       ″ 
                     
                   
                 
               
               
                 
                   ( 
                   
                     10 
                      
                     
                       - 
                     
                      
                     1 
                   
                   ) 
                 
               
             
             
               
                 
                   
                     
                       
                         
                           L 
                           X 
                         
                         2 
                       
                        
                       
                         ( 
                         
                           
                             F 
                             
                               4 
                                
                               B 
                             
                           
                           + 
                           
                             F 
                             
                               4 
                                
                               C 
                             
                           
                         
                         ) 
                       
                     
                     - 
                     
                       
                         
                           L 
                           X 
                         
                         2 
                       
                        
                       
                         ( 
                         
                           
                             F 
                             
                               4 
                                
                               A 
                             
                           
                           + 
                           
                             F 
                             
                               4 
                                
                               D 
                             
                           
                         
                         ) 
                       
                     
                   
                   = 
                   
                     
                       I 
                       YY 
                     
                     · 
                     
                       θ 
                       Y 
                       ″ 
                     
                   
                 
               
               
                 
                   ( 
                   
                     10 
                      
                     
                       - 
                     
                      
                     2 
                   
                   ) 
                 
               
             
             
               
                 
                   
                     
                       
                         
                           L 
                           Y 
                         
                         2 
                       
                        
                       
                         ( 
                         
                           
                             F 
                             
                               4 
                                
                               C 
                             
                           
                           + 
                           
                             F 
                             
                               4 
                                
                               D 
                             
                           
                         
                         ) 
                       
                     
                     - 
                     
                       
                         
                           L 
                           Y 
                         
                         2 
                       
                        
                       
                         ( 
                         
                           
                             F 
                             
                               4 
                                
                               A 
                             
                           
                           + 
                           
                             F 
                             
                               4 
                                
                               B 
                             
                           
                         
                         ) 
                       
                     
                   
                   = 
                   
                     
                       I 
                       XX 
                     
                     · 
                     
                       θ 
                       X 
                       ″ 
                     
                   
                 
               
               
                 
                   ( 
                   
                     10 
                      
                     
                       - 
                     
                      
                     3 
                   
                   ) 
                 
               
             
             
               
                 
                   
                     
                       ( 
                       
                         
                           F 
                           
                             4 
                              
                             A 
                           
                         
                         + 
                         
                           F 
                           
                             4 
                              
                             C 
                           
                         
                       
                       ) 
                     
                     - 
                     
                       ( 
                       
                         
                           F 
                           
                             4 
                              
                             B 
                           
                         
                         + 
                         
                           F 
                           
                             4 
                              
                             D 
                           
                         
                       
                       ) 
                     
                   
                   = 
                   
                     
                       Δ 
                        
                       
                           
                       
                        
                       F 
                     
                     = 
                     0 
                   
                 
               
               
                 
                   ( 
                   
                     10 
                      
                     
                       - 
                     
                      
                     4 
                   
                   ) 
                 
               
             
           
         
       
     
         [0056]    Therein, m is the mass of the base plate  101 , which includes the X stage and the Y stage, I YY  is the rotating moment around the Y axis of the base plate  101 , and I XX  is the rotating moment around the X axis of the same. In addition, the difference in the forces that operate on the base plate  101  along the two diagonal lines is defined by the torsional force ΔF; here, it is assumed that ΔF=0 so that the base plate  101  is not subject to torsion because of the control performed by the thrust feedback control unit  16 . Solving the above equations (10-1)-(10-4) results in equation (11) below, which defines the thrust instruction value F 4  calculated by the thrust feedback control unit  1   6  (here, the second term on the right side of equation (11) is 0 because ΔF=0). 
         [0000]    
       
         
           
             
               
                 
                   Equation 
                    
                   
                       
                   
                    
                   11 
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   
                     [ 
                     
                       
                         
                           
                             F 
                             
                               4 
                                
                               A 
                             
                           
                         
                       
                       
                         
                           
                             F 
                             
                               4 
                                
                               B 
                             
                           
                         
                       
                       
                         
                           
                             F 
                             
                               4 
                                
                               C 
                             
                           
                         
                       
                       
                         
                           
                             F 
                             
                               4 
                                
                               D 
                             
                           
                         
                       
                     
                     ] 
                   
                   = 
                   
                     
                       
                         [ 
                         
                           
                             
                               
                                 m 
                                 4 
                               
                             
                             
                               
                                 - 
                                 
                                   
                                     I 
                                     XX 
                                   
                                   
                                     2 
                                      
                                     
                                       L 
                                       Y 
                                     
                                   
                                 
                               
                             
                             
                               
                                 - 
                                 
                                   
                                     I 
                                     YY 
                                   
                                   
                                     2 
                                      
                                     
                                       L 
                                       X 
                                     
                                   
                                 
                               
                             
                           
                           
                             
                               
                                 m 
                                 4 
                               
                             
                             
                               
                                 - 
                                 
                                   
                                     I 
                                     XX 
                                   
                                   
                                     2 
                                      
                                     
                                       L 
                                       Y 
                                     
                                   
                                 
                               
                             
                             
                               
                                 
                                   I 
                                   YY 
                                 
                                 
                                   2 
                                    
                                   
                                     L 
                                     X 
                                   
                                 
                               
                             
                           
                           
                             
                               
                                 m 
                                 4 
                               
                             
                             
                               
                                 
                                   I 
                                   XX 
                                 
                                 
                                   2 
                                    
                                   
                                     L 
                                     Y 
                                   
                                 
                               
                             
                             
                               
                                 
                                   I 
                                   YY 
                                 
                                 
                                   2 
                                    
                                   
                                     L 
                                     X 
                                   
                                 
                               
                             
                           
                           
                             
                               
                                 m 
                                 4 
                               
                             
                             
                               
                                 
                                   I 
                                   XX 
                                 
                                 
                                   2 
                                    
                                   
                                     L 
                                     Y 
                                   
                                 
                               
                             
                             
                               
                                 - 
                                 
                                   
                                     I 
                                     YY 
                                   
                                   
                                     2 
                                      
                                     
                                       L 
                                       X 
                                     
                                   
                                 
                               
                             
                           
                         
                         ] 
                       
                        
                       
                         [ 
                         
                           
                             
                               
                                 Z 
                                 ″ 
                               
                             
                           
                           
                             
                               
                                 θ 
                                 X 
                                 ″ 
                               
                             
                           
                           
                             
                               
                                 θ 
                                 Y 
                                 ″ 
                               
                             
                           
                         
                         ] 
                       
                     
                     + 
                     
                       
                         [ 
                         
                           
                             
                               
                                 1 
                                 4 
                               
                             
                           
                           
                             
                               
                                 - 
                                 
                                   1 
                                   4 
                                 
                               
                             
                           
                           
                             
                               
                                 1 
                                 4 
                               
                             
                           
                           
                             
                               
                                 - 
                                 
                                   1 
                                   4 
                                 
                               
                             
                           
                         
                         ] 
                       
                        
                       Δ 
                        
                       
                           
                       
                        
                       F 
                     
                   
                 
               
               
                 
                   ( 
                   11 
                   ) 
                 
               
             
           
         
       
     
         [0057]    Thus, if the position sensors measure the positions of the mounts  4 A- 4 D in the Z directions and if the mounts  4 A- 4 D generate thrusts in accordance with the thrust instruction value F 4  defined in the equation (11) above, then feedback control is achieved such that the base plate  101  neither tilts nor undergoes torsion. Furthermore, similar to the case of the thrust instruction value F 2  discussed above, the air spring pressure instruction values that are applied to the mounts  4 A- 4 D to generate thrusts in accordance with the abovementioned thrust instruction value F 4  can be defined so that P 4A =F 4A /a A , P 4B =F 4B /a B , P 4C =F 4C /a C , and P 4D =F 4D /a D . 
         [0058]    The following explains the calculating method wherein the base plate torsion correcting unit  19  calculates the thrust instruction value F 5 . If the base plate  101  is subject to the torsion Z T  defined by equation (12) below, then the base plate torsion correcting unit  19  performs control, causing the mounts  4 A- 4 D to generate thrusts that correct this torsion. Namely, it is possible to use a technique, such as typical PID control, to derive the torsional force ΔF that zero the differential between the torsion Z T  of the base plate  101 , which was calculated using equation (12) below based on the position of each of the mounts  4 A- 4 D in the Z directions as measured by the position sensors, and the target reference value Z T0  (=0). The mounts  4 A,  4 B,  4 C,  4 D generate thrusts F 5A , F 5B , F 5C , F 5D , respectively—equation (13) below is equivalent to the second term on the right side of equation (11), also below—according to the thrust instruction value F 5  as defined by equation (13) below so that the derived torsional force ΔF acts on the base plate  101 . 
         [0000]    
       
         
           
             
               
                 
                   Equation 
                    
                   
                       
                   
                    
                   12 
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   
                     Z 
                     T 
                   
                   = 
                   
                     
                       
                         1 
                         2 
                       
                        
                       
                         ( 
                         
                           
                             Z 
                             A 
                           
                           + 
                           
                             Z 
                             C 
                           
                         
                         ) 
                       
                     
                     - 
                     
                       
                         1 
                         2 
                       
                        
                       
                         ( 
                         
                           
                             Z 
                             B 
                           
                           + 
                           
                             Z 
                             D 
                           
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   12 
                   ) 
                 
               
             
             
               
                 
                   Equation 
                    
                   
                       
                   
                    
                   13 
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   
                     [ 
                     
                       
                         
                           
                             F 
                             
                               5 
                                
                               A 
                             
                           
                         
                       
                       
                         
                           
                             F 
                             
                               5 
                                
                               B 
                             
                           
                         
                       
                       
                         
                           
                             F 
                             
                               5 
                                
                               C 
                             
                           
                         
                       
                       
                         
                           
                             F 
                             
                               5 
                                
                               D 
                             
                           
                         
                       
                     
                     ] 
                   
                   = 
                   
                     
                       [ 
                       
                         
                           
                             
                               1 
                               4 
                             
                           
                         
                         
                           
                             
                               - 
                               
                                 1 
                                 4 
                               
                             
                           
                         
                         
                           
                             
                               1 
                               4 
                             
                           
                         
                         
                           
                             
                               - 
                               
                                 1 
                                 4 
                               
                             
                           
                         
                       
                       ] 
                     
                      
                     Δ 
                      
                     
                         
                     
                      
                     F 
                   
                 
               
               
                 
                   ( 
                   13 
                   ) 
                 
               
             
           
         
       
     
         [0059]    Furthermore, if the servo operator of the torsion Z T  is defined as servo (Z T ), then the equation that derives the torsional force ΔF based on the differential between Z T  and Z T0  is ΔF=−Z T ·servo(Z T ). In addition, similar to the case of the thrust instruction value F 2  discussed above, the air spring pressure instruction values applied to the mounts  4 A- 4 D in order to generate the thrusts according to the abovementioned thrust instruction value F 5  can be defined so that P 5A =F 5A /a A , P 5B =F 5B /a B , P 5C =F 5C /a C , and P 5D =F 5D /a D . 
         [0060]    The following explains the calculating method wherein the lens torsion correcting unit  14  calculates the thrust instruction value F 3 .  FIG. 5  is a view that shows a lens  60  that is mounted to the base plate  101  via a prescribed support mechanism (not shown). Laser beams L 1 -L 4  from laser interferometers are reflected at four points A, B, C, D on the lens  60  and the positions a, b, c, d of the four points A-D are measured by interferometric measurement wherein those reflected beams are used. Here, the torsion of the lens  60  is defined by L=(a+c)−(b+d). When the lens  60  undergoes torsion L, the torsional force ΔF is applied to the base plate  101  to cancel the torsion L. A proportional relationship (with a proportional constant of a) exists between the torsional force ΔF and the torsion L of the lens  60 , and therefore the lens torsion correcting unit  14  causes the mounts  4 A,  4 B,  4 C,  4 D to generate thrusts F 3A , F 3B , F 3C , F 3D , respectively, in accordance with the thrust instruction value F 3  as defined by equation (14) below; thereby, control that cancels the torsion L, which arises in the lens  60 , is achieved. 
         [0000]    
       
         
           
             
               
                 
                   Equation 
                    
                   
                       
                   
                    
                   14 
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   
                     [ 
                     
                       
                         
                           
                             F 
                             
                               3 
                                
                               A 
                             
                           
                         
                       
                       
                         
                           
                             F 
                             
                               3 
                                
                               B 
                             
                           
                         
                       
                       
                         
                           
                             F 
                             
                               3 
                                
                               C 
                             
                           
                         
                       
                       
                         
                           
                             F 
                             
                               3 
                                
                               D 
                             
                           
                         
                       
                     
                     ] 
                   
                   = 
                   
                     
                       
                         [ 
                         
                           
                             
                               
                                 1 
                                 4 
                               
                             
                           
                           
                             
                               
                                 - 
                                 
                                   1 
                                   4 
                                 
                               
                             
                           
                           
                             
                               
                                 1 
                                 4 
                               
                             
                           
                           
                             
                               
                                 - 
                                 
                                   1 
                                   4 
                                 
                               
                             
                           
                         
                         ] 
                       
                        
                       Δ 
                        
                       
                           
                       
                        
                       F 
                     
                     = 
                     
                       [ 
                       
                         
                           
                             
                               - 
                               
                                 
                                   a 
                                   · 
                                   L 
                                 
                                 4 
                               
                             
                           
                         
                         
                           
                             
                               
                                 a 
                                 · 
                                 L 
                               
                               4 
                             
                           
                         
                         
                           
                             
                               - 
                               
                                 
                                   a 
                                   · 
                                   L 
                                 
                                 4 
                               
                             
                           
                         
                         
                           
                             
                               
                                 a 
                                 · 
                                 L 
                               
                               4 
                             
                           
                         
                       
                       ] 
                     
                   
                 
               
               
                 
                   ( 
                   14 
                   ) 
                 
               
             
           
         
       
     
         [0061]    Furthermore, similar to the case of the thrust instruction value F 2  discussed above, the air spring pressure instruction values that are applied to the mounts  4 A- 4 D in order to generate thrusts in accordance with the abovementioned thrust instruction value F 3  can be defined so that P 3A =F 3A /a A , P 3B =F 3B /a B , P 3C =F 3C /a C , and P 3D =F 3D /a D . 
         [0062]    The above text explained an embodiment of the present invention based on the drawings, but the specific constitution of the invention is not limited to these embodiments, and it is understood that variations and modifications may be effected without departing from the spirit and scope of the invention. 
         [0063]    The thrust that is generated by each of the mounts  4 A- 4 D is not necessarily controlled in accordance with a total thrust instruction value, which is the sum of the following: the thrust instruction value F 1 , calculated by the acceleration state-based thrust calculating unit  12 ; the thrust instruction value F 2 , calculated by the movement position-based thrust calculating unit  13 ; the thrust instruction value F 3 , calculated by the lens torsion correcting unit  14 ; the thrust instruction value F 4 , calculated by the thrust feedback control unit  16 ; and the thrust instruction value F 5 , calculated by the base plate torsion correcting unit  19 ; for example, the thrusts of the mounts  4 A- 4 D may be controlled in accordance with all or some combination of the F 1 , F 2 , F 3 , F 4 , F 5 . 
         [0064]    In addition, the damping apparatus and the stage apparatus, according to the embodiments discussed above, can be adapted to an exposure apparatus that prints a fine circuit pattern on, for example, a glass substrate or a semiconductor substrate.  FIG. 6  is a block diagram of the exposure apparatus whereto the damping apparatus and the stage apparatus discussed above are adapted. An exposure apparatus  601  comprises the following: an illumination optical system  602 ; a mask stage apparatus  603 , which holds and moves a mask M; a projection optical system PL; a substrate stage apparatus  605 , which holds and moves a glass substrate P; and a damping apparatus  606 . 
         [0065]    The illumination optical system  602  comprises a light source unit, a shutter, an optical system for forming a two dimensional light source, a beam splitter, a condenser lens system, a reticle blind, and an imaging lens system, none of which are shown; furthermore, the illumination optical system  602  illuminates a prescribed illumination area (which includes the circuit pattern) on the mask M, which is held by the mask stage apparatus  603 , with illumination light IL of a uniform luminous flux intensity. 
         [0066]    The projection optical system PL is an optical system (e.g., a dioptric system) that has multiple lens elements disposed along the directions of an optical axis AX at prescribed spacings; furthermore, when the illumination area of the mask M is illuminated with the illumination light IL from the illumination optical system  602 , the illumination light that transmits through the mask M is used to project an erect image (with a prescribed magnification) of the circuit pattern in the illumination area of the mask M through the projection optical system PL and onto the glass substrate P, thereby exposing photoresist that is coded on the front surface of the glass substrate P. 
         [0067]    The substrate stage apparatus  605  is installed so that it is supported by the damping apparatus  606 , which performs the control discussed above. Furthermore, a configuration may be adopted so that the mask stage apparatus  603  is supported by the damping apparatus  606 .