Active anti-vibration apparatus and method of manufacturing the same

An active anti-vibration apparatus detects the motion of an anti-vibration table for supporting equipment with a plurality of sensors and controls actuators on the basis of detection outputs from the sensor. The sensors are arranged such that, when a motion parameter of the anti-vibration table is represented by a vector P, and an output signal group from the plurality of sensors is represented by a vector S, a condition number of a matrix A defined by an equation S=AP established between the vector P and the vector S in accordance with a geometrical arrangement of the plurality of sensors is minimized.

BACKGROUND OF THE INVENTION 
The present invention relates to an active anti-vibration apparatus and a 
method of manufacturing the same and, more particularly, to an active 
anti-vibration apparatus serving as a support mechanism for all mechanical 
systems including precision equipment represented by a semiconductor 
exposure apparatus, and a method of manufacturing the same. 
Along with an advance in precision of precision equipments such as an 
electron microscope, a step and repeat (scan) type semiconductor exposure 
apparatus, and semiconductor exposure apparatuses of other types, a higher 
performance is required of a precision anti-vibration apparatus for 
mounting the equipment. 
Particularly, in a semiconductor exposure apparatus, an anti-vibration 
table for removing vibrations transmitted from outside the apparatus, 
e.g., a vibration of the setting floor, is necessary for an appropriate 
and quick exposure operation. This is because, when each shot area on a 
semiconductor wafer is to be exposed, the X-Y stage on which the 
semiconductor wafer moves along the X-Y plane must be completely 
stationary. The X-Y stage has as its feature an intermittent operation 
called "step and repeat". Repeated step vibrations in this operation 
produce vibrations of the support table (anti-vibration table) itself 
which supports the equipment. 
Therefore, both anti-vibration performance for external vibrations and 
vibration damping performance for vibrations caused by the operation of 
the equipment itself are required of the anti-vibration table in a good 
balance. To meet this requirement, an active anti-vibration apparatus has 
been put in practice in which the vibration of the anti-vibration table is 
detected by a vibration sensor, and an output signal from the vibration 
sensor is compensated and fed back to an actuator, thereby actively 
performing vibration damping. The active anti-vibration apparatus allows 
realization of anti-vibration performance and vibration control 
performance in a good balance, which is difficult for a conventional 
passive anti-vibration apparatus constituted only by a support mechanism 
having spring and damper characteristics. 
An active anti-vibration apparatus is disclosed in, e.g., Japanese Patent 
Laid-Open No. 6-167414. In this prior art, a total of seven sensors are 
arranged to calculate translation and rotation motions of 
6-degree-of-freedom of the anti-vibration table. The detail of the sensor 
arrangement is as follows. On an orthogonal coordinate system wherein the 
long side direction of the rectangular anti-vibration table matches the 
X-axis while the vertical direction matches the Z-axis, one sensor is 
arranged in the X-axis direction, two are arranged in the Y-axis 
direction, and four are arranged in the Z-axis direction. 
In anti-vibration table control by the above active anti-vibration 
apparatus, motions in the directions of translation and rotation of the 
anti-vibration table, as the motions of a rigid body, are calculated from 
output signals from the sensors, and feedback loops are formed in units of 
motion directions. Therefore, the sensors must be arranged to allow 
detection of the motions in the directions of translation and rotation of 
the anti-vibration table as the motions of a rigid body. In addition, the 
sensors are preferably arranged to minimize any errors in calculation 
result, which are caused by observation noise included in the sensor 
output signals or the difference between the nominal value and the true 
value of a sensor position. 
With the sensor arrangement of the above Japanese Patent Laid-Open No. 
6-167414, the translation and rotation motions of 6-degree-of-freedom of 
the anti-vibration table can be calculated, though the influence of 
observation noise included in sensor output signals or the difference 
between the nominal value and the true value of a sensor position, which 
acts on the calculation result, is not considered at all. In addition, for 
a sensor arrangement for precisely calculating the translation and 
rotation motions of 6-degree-of-freedom of the anti-vibration table, no 
definite quantitative guideline is presented. 
Another active anti-vibration apparatus will be described below. FIG. 10 
shows the arrangement of an active anti-vibration apparatus. An 
anti-vibration table 105 for mounting precision equipment such as an X-Y 
stage 111 is supported by three anti-vibration units. The anti-vibration 
units are arranged at the three corners of the anti-vibration table 105. 
The anti-vibration units have the same arrangement and are discriminated 
by adding suffixes a, b, and c to the reference numeral. An anti-vibration 
unit 101a comprises an actuator 102a for applying a driving force to the 
anti-vibration table 105 in the horizontal direction, and an acceleration 
sensor 103a and a position sensor 104a, which are used to detect the 
horizontal vibration of the anti-vibration table 105. In practice, the 
anti-vibration unit 101a includes as constituent elements (not shown) 
mechanisms such as a mechanical spring, or a servo valve for supplying air 
as a working fluid if the actuator 102a consists of, e.g., an air spring. 
However, FIG. 10 only schematically shows the arrangement of the 
anti-vibration apparatus, and constituent elements essential for active 
vibration control of the anti-vibration table 105 are represented by 
elements shown in FIG. 10. This also applies to anti-vibration units 101b 
and 101c. 
Control in units of motion modes is disclosed in Japanese Patent Laid-Open 
No. 7-83276 as a method of controlling the active anti-vibration 
apparatus, in which motion modes associated with the acceleration of the 
anti-vibration table and motion modes associated with the position are 
extracted from signals from a plurality of acceleration sensors and 
position sensors, thereby performing optimum compensation in units of 
motion modes. This control in units of motion modes will be described in 
detail assuming that control in units of horizontal motion modes is 
applied to the active anti-vibration apparatus shown in FIG. 10. 
Regarding the anti-vibration table 105 as a rigid body, the horizontal 
rigid body motions of the anti-vibration table 105 are classified into 
motion modes of total 3-degree-of-freedom constituted by translations of 
2-degree-of-freedom and rotation of single-degree-of-freedom. An X-Y-Z 
orthogonal coordinate system is set on the anti-vibration table 105 such 
that the origin matches a center of gravity G of the anti-vibration table 
105, and the Z-axis direction matches the vertical direction. At this 
time, the horizontal rigid body motions of the anti-vibration table 105 
can be classified into three motion modes consisting of X-axis direction 
translation, Y-axis direction translation, and rotation about the Z-axis. 
When the acceleration sensors and position sensors incorporated in the 
respective anti-vibration units to detect the horizontal vibration of the 
anti-vibration table 105 are included in the X-Y plane defined by the 
X-Y-Z coordinate system or arranged near the X-Y plane, i.e., in other 
words, when the acceleration sensors and position sensors are included in 
a horizontal plane including the center of gravity G of the anti-vibration 
table 105 or arranged near the horizontal plane, the horizontal motion 
modes of 3-degree-of-freedom associated with the acceleration of the 
anti-vibration table 105 and the horizontal motion modes of 
3-degree-of-freedom associated with the position can be extracted from 
signals from the acceleration sensors and position sensors incorporated in 
the respective anti-vibration units. 
Referring to FIG. 10, a motion mode extraction unit 106 extracts horizontal 
motion modes a.sub.x, a.sub.y, and a.theta..sub.z associated with the 
acceleration from output signals a.sub.a to a.sub.c from acceleration 
sensors 103a to 103c incorporated in the three anti-vibration units 101a 
to 101c, respectively. In this case, a.sub.x represents the X-axis 
direction translation acceleration; a.sub.y represents the Y-axis 
direction translation acceleration, and a.theta..sub.z represents the 
angular acceleration about the Z-axis. Normally, the active anti-vibration 
apparatus has a function of controlling positioning of the anti-vibration 
table 105. A motion mode extraction unit 106' extracts horizontal motion 
modes e.sub.x, e.sub.y, e.theta..sub.z associated with the position from 
position deviation signals e.sub.a to e.sub.c which are obtained when 
output signals from position sensors 104a to 104c incorporated in the 
three anti-vibration units 101a to 101c, respectively, are compared with 
and subtracted from a position target signal. In this case, e.sub.x 
represents the X-axis translation position deviation, e.sub.y represents 
the Y-axis translation position deviation, and e.theta..sub.z represents 
the rotation angular deviation about the Z-axis. The motion mode 
extraction units 106 and 106' extract the horizontal motion modes of 
3-degree-of-freedom from the received three sensor signals. 
Anti-vibration table driving forces in units of motion modes associated 
with the acceleration and position are generated by appropriately 
compensating outputs from the motion mode extraction units 106 and 106'. 
As a compensator for compensating an output from the motion mode 
extraction unit 106 associated with the acceleration, a proportional gain 
is suitable assuming that the actuators use air springs. As a compensator 
for compensating an output from the motion mode extraction unit 106' 
associated with the position, a PI compensator is suitable to allow 
convergence of the position deviation in the steady state to zero. P of 
the PI compensator means a proportional operation, and I means an 
integrating operation. In FIG. 10, a proportional gain 109 is used to 
compensate the motion modes a.sub.x, a.sub.y, and a.theta..sub.z 
associated with the acceleration to generate anti-vibration table driving 
force F".sub.x, F".sub.y, and M".sub.z in units of motion modes associated 
with the acceleration. In addition, a PI compensator 108 is used to 
compensate the motion modes e.sub.x, e.sub.y, and e.theta..sub.z 
associated with the position to generate anti-vibration table driving 
forces F'.sub.x, F'.sub.y, and M'.sub.z in units of motion modes 
associated with the position. The anti-vibration table driving forces 
F".sub.x, F".sub.y, and M".sub.z in units of motion modes associated with 
the acceleration and the anti-vibration table driving forces F'.sub.x, 
F'.sub.y, and M'.sub.z in units of motion modes associated with the 
position are added by an adder 110, respectively, thereby generating final 
anti-vibration table driving forces F.sub.x, F.sub.y, and M.sub.z in units 
of motion modes. In this case, F.sub.x and F.sub.y represent the X-axis 
and Y-axis direction translation forces, respectively, and M.sub.z 
represents the moment about the Z-axis. 
The anti-vibration table driving forces F.sub.x, F.sub.y, and M.sub.z in 
units of motion modes are distributed to the actuators 102a to 102c, 
respectively, and applied to the anti-vibration table 105. When the 
actuators 102a to 102c incorporated in the respective anti-vibration units 
are included in the horizontal plane including the center of gravity of 
the anti-vibration table 105 or arranged near the horizontal plane, the 
actuators 102a to 102c can apply the anti-vibration table driving forces 
F.sub.x, F.sub.y, and M.sub.z in units of motion modes to the 
anti-vibration table 105 without affecting the vertical motion mode of the 
anti-vibration table 105. In other words, without affecting the vertical 
motion mode, the anti-vibration table driving forces F.sub.x, F.sub.y, and 
M.sub.z in units of motion modes can be distributed to the actuators 102a 
to 102c. In FIG. 10, outputs from a motion mode distribution unit 107 for 
distributing the anti-vibration table driving forces F.sub.x, F.sub.y, and 
M.sub.z in units of motion modes to the actuators 102a to 102c become 
actuator thrusts F.sub.a to F.sub.c generated from the actuators 102a to 
102c, respectively. 
As described above, when control in units of motion modes is applied to the 
active anti-vibration apparatus, posture control with optimum compensation 
in units of motion modes can be performed for the position, and optimum 
damping in units of motion modes is enabled for the acceleration. 
In the arrangement of the active anti-vibration apparatus shown in FIG. 10, 
the three actuators 102a to 102c are arranged in correspondence with the 
number of anti-vibration units serving as a support mechanism. The 
horizontal motion modes of the anti-vibration table 105 have a 
3-degree-of-freedom. Therefore, the calculation method for distributing 
the anti-vibration table driving forces in units of motion modes in the 
motion mode distribution unit 107 is limited to only one. This 
distribution calculation is uniquely determined in accordance with the 
geometrical positional relationship between the actuators 102a to 102c and 
the center of gravity of the anti-vibration table 105. 
In a conventional semiconductor exposure apparatus, generally, the 
anti-vibration table is supported by four anti-vibration units. The X-Y 
stage mounted on the anti-vibration table has as its feature an 
intermittent operation called "step and repeat". The repeat step operation 
is performed by setting the main step direction in the X direction or Y 
direction of the X-Y stage. When the anti-vibration table is supported by 
four anti-vibration units, actuators incorporated in the anti-vibration 
units to drive the anti-vibration table in the horizontal direction are 
normally arranged such that two of them are arranged to generate a thrust 
in the X direction of the X-Y stage, and the remaining two are arranged to 
generate a thrust in the Y direction of the X-Y stage. With this 
arrangement, the vibration of the anti-vibration table, which is excited 
by the step drive reaction in the step operation of the X-Y stage, can be 
effectively suppressed. More specifically, in the step operation of the 
X-Y stage, a large vibration is generated in the anti-vibration table 
along the step direction. When two actuators are arranged in the X and Y 
directions, respectively, an equal driving force can be applied to the 
anti-vibration table along the step direction in both the X-direction step 
and Y-direction step operation of the X-Y stage. 
To the contrary, for the arrangement wherein the anti-vibration table 105 
is supported by the three anti-vibration units 101a to 101c, as shown in 
FIG. 10, no definite guideline has been given yet as to the arrangement of 
the actuators 102a to 102c. For example, when two actuators are arranged 
to generate a thrust in the X-axis direction while the remaining one is 
arranged to generate a thrust in the Y-axis direction, and the three 
actuators can generate equal maximum thrusts, the Y-axis translation 
thrust F.sub.y acting on the anti-vibration table 105 is only 1/2 the 
X-axis translation thrust F.sub.x. When all the actuators 102a to 102c are 
arranged toward the center of gravity G of the anti-vibration table 105, 
almost equal translation thrusts can be applied to the anti-vibration 
table 105 in all directions in the horizontal plane. However, no moment 
M.sub.z is generated about the Z-axis at all, so the vibration of the 
anti-vibration table 105 in the direction of rotation about the Z-axis 
cannot be suppressed. To effectively suppress the vibration of the 
anti-vibration table, which is excited by the step operation of the X-Y 
stage in an arbitrary direction, the actuators 102a to 102c must be 
arranged to equally generate anti-vibration table driving forces F.sub.x, 
F.sub.y, and M.sub.z in units of motion modes. In a scan type exposure 
apparatus, the stage mounted on the anti-vibration table is mainly driven 
in the scanning direction, and it is desired to allow application of a 
large driving force to the anti-vibration table along the scanning 
direction. More specifically, it is desired to arrange the actuators to 
generate desired maximum driving forces for the respective anti-vibration 
table driving forces F.sub.x, F.sub.y, and M.sub.z in units of motion 
modes. However, a technique for realizing such an actuator arrangement has 
not been established yet. 
In addition, generally, an actuator does not always generate a thrust 
according to the command value from a controller. Some variations in 
thrust caused by factors such as torque ripples cannot be avoided. 
Furthermore, to design the motion mode distribution unit 107, it is 
essential to determine the positional relationship between the center of 
gravity G of the anti-vibration table 105 and the actuators 102a to 102c. 
However, it is difficult to accurately measure the positions of the 
actuators in fact. Some errors between the measured value and the true 
value cannot be avoided. For control in units of motion modes, it is 
preferable that the desired anti-vibration table driving forces F.sub.x, 
F.sub.y, and M.sub.z in units of motion modes be properly applied to the 
anti-vibration table 105 regardless of variations in actuator thrusts or 
measurement errors of the actuator position. Therefore, the actuators must 
be arranged such that the influence of variations in actuator thrust is 
minimized. However, a technique for realizing such an actuator arrangement 
has not been established yet. 
SUMMARY OF THE INVENTION 
The present invention has been made in consideration of the above 
situation, and has as its object to provide an active anti-vibration 
apparatus which enables highly precise anti-vibration control by precisely 
calculating the translation and rotation motions of an anti-vibration 
table as rigid body motions while minimizing the influence of error 
factors such as observation noise. 
It is another object of the present invention to provide a method of 
manufacturing an active anti-vibration apparatus, which enables to 
precisely calculate the translation and rotation motions of an 
anti-vibration table as rigid body motions while minimizing the influence 
of error factors such as observation noise. 
According to the present invention, there is provided an active 
anti-vibration apparatus having a plurality of sensors and actuators, 
which detects a motion of an anti-vibration table for supporting an 
equipment with the plurality of sensors and controls the actuators on the 
basis of output signals from the sensors to drive the anti-vibration 
table, thereby performing an anti-vibration operation for the 
anti-vibration table, wherein the plurality of sensors are arranged such 
that, when a motion parameter of the anti-vibration table is represented 
by a vector P, and an output signal group from the plurality of sensors is 
represented by a vector S, a condition number of a matrix A defined by an 
equation S=AP established between the vector P and the vector S in 
accordance with a geometrical arrangement of the plurality of sensors is 
minimized. 
In the active anti-vibration apparatus of the present invention, a 
relationship between the vector S having elements S.sub.1, . . . , and 
S.sub.n and the vector P having elements P.sub.x, P.sub.y, and 
P.theta..sub.z is represented as follows: 
##EQU1## 
where S.sub.n (n=1 to K (natural number equal to or larger than 2)) are 
the output signals from the plurality of sensors, P.sub.x is a motion 
parameter of a motion in an X direction associated with an X-Y plane of 
the anti-vibration table, P.sub.y is a motion parameter of a motion in a Y 
direction on the X-Y plane, and P.theta..sub.z is a motion parameter of a 
rotation motion in a .theta. direction on the X-Y plane. 
In the active anti-vibration apparatus of the present invention, the 
plurality of sensors are arranged such that, when the motion parameters 
P.sub.x, P.sub.y, and P.theta..sub.z are to be weighted, a condition 
number of a matrix AW obtained by multiplying the matrix A with a 
weighting matrix W is minimized. 
In the active anti-vibration apparatus of the present invention, the 
plurality of sensors are arranged such that angles formed by straight 
lines passing through the sensors and a center of gravity of the 
anti-vibration table, and motion detection directions of the sensors are 
substantially equal for the plurality of sensors. 
In the active anti-vibration apparatus of the present invention, the 
sensors comprise three sensors respectively arranged near vertexes of a 
regular triangle assumed on the X-Y plane. 
In the active anti-vibration apparatus of the present invention, the 
anti-vibration table has a substantially regular triangular structure 
having vertexes near positions where the three sensors are arranged. 
In the active anti-vibration apparatus of the present invention, the 
actuators are arranged near the vertexes of the anti-vibration table 
having the substantially regular triangular structure. 
In the active anti-vibration apparatus of the present invention, directions 
of action of the three actuators arranged near the vertexes of the 
anti-vibration table are substantially the same as the motion detection 
directions of the sensors arranged near the actuators. 
In the active anti-vibration apparatus of the present invention, the 
actuators are arranged near the plurality of sensors. 
In the active anti-vibration apparatus of the present invention, directions 
of action of the actuators arranged near the vertexes of the 
anti-vibration table are substantially the same as the motion detection 
directions of the sensors arranged near the actuators. 
According to the present invention, there is provided a method of 
manufacturing an active anti-vibration apparatus having a plurality of 
sensors and actuators, which detects a motion of an anti-vibration table 
for supporting an equipment with the plurality of sensors and controls the 
actuators on the basis of output signals from the sensors to drive the 
anti-vibration table, thereby performing an anti-vibration operation for 
the anti-vibration table, wherein the plurality of sensors are arranged 
such that, when a motion parameter of the anti-vibration table is 
represented by a vector P, and an output signal group from the plurality 
of sensors is represented by a vector S, a condition number of a matrix A 
defined by an equation S=AP established between the vector P and the 
vector S in accordance with a geometrical arrangement of the plurality of 
sensors is minimized. 
As described above, the present invention uses, as a quantitative guideline 
for sensor arrangement, the condition number of a coefficient matrix, 
which serves as an index representing the degree of influence of an error 
in equation acting on the translation and rotation motions of the 
anti-vibration table as the solution of simultaneous equations established 
between sensor output signals determined by the sensor arrangement and the 
translation and rotation motions of the anti-vibration table. When the 
sensor are arranged such that the condition number is minimized, the 
translation and rotation motions of the anti-vibration table are precisely 
calculated while minimizing the influence of observation noise. 
In addition, the translation and rotation motions as the rigid body motion 
of the anti-vibration table are weighted, and the sensors are arranged 
such that the condition number of the matrix as the product of the above 
coefficient matrix and a weighting matrix is minimized. With this 
arrangement, considering the difference between the physical dimensions of 
the translation and rotation motions, or placing importance to the 
calculation precision of a specific motion, the translation and rotation 
motions of the anti-vibration table can be precisely calculated while 
minimizing the influence of observation noise. 
According to the present invention, the sensor are arranged such that the 
condition number of the coefficient matrix is minimized. Therefore, the 
translation and rotation motions of the anti-vibration table can be highly 
precisely detected while minimizing the influence of error factors such as 
observation noise including in sensor output signals or the difference 
between the nominal value and the true value of a sensor position. 
It is still another object of the present invention to provide an active 
anti-vibration apparatus having a horizontal actuator arrangement for 
generating equal anti-vibration table driving forces in units of motion 
modes in any direction of the horizontal motion modes of the 
anti-vibration table, or generating desired maximum driving forces in any 
direction of the horizontal motion modes of the anti-vibration table, or a 
horizontal actuator arrangement for minimizing variations in 
anti-vibration table driving forces in units of motion modes caused due to 
variations in actuator thrust or measurement errors of the actuator 
position. 
It is still another object of the present invention to provide a method of 
manufacturing an active anti-vibration apparatus having a horizontal 
actuator arrangement for generating equal anti-vibration table driving 
forces in units of motion modes in any direction of the horizontal motion 
modes of the anti-vibration table, or generating desired maximum driving 
forces in any direction of the horizontal motion modes of the 
anti-vibration table, or a horizontal actuator arrangement for minimizing 
variations in anti-vibration table driving forces in units of motion modes 
caused due to variations in actuator thrust or measurement errors of the 
actuator position. 
According to the present invention, there is also provided an active 
anti-vibration apparatus having a plurality of sensors and actuators, 
which detects a motion of an anti-vibration table for supporting an 
equipment with the plurality of sensors and controls the actuators on the 
basis of output signals from the sensors to drive the anti-vibration 
table, thereby performing an anti-vibration operation for the 
anti-vibration table, wherein the plurality of actuators are arranged such 
that, when thrusts generated by the plurality of actuators are represented 
by a vector x, and driving forces in units of motion modes, which act on 
the anti-vibration table, are represented by a vector b, a condition 
number of a matrix A defined by an equation Ax=b established between the 
vector x and the vector b in accordance with a geometrical arrangement of 
the plurality of sensors is minimized. 
In the active anti-vibration apparatus of the present invention, the 
plurality of actuators are arranged such that a condition number of a 
matrix WA used instead of the matrix A and obtained by multiplying the 
matrix A with a normalizing matrix W for normalizing the driving forces in 
units of motion modes, which are represented by the vector b, is 
minimized. 
In the active anti-vibration apparatus of the present invention, the 
plurality of actuators are arranged such that angles formed by straight 
lines passing through points of action of thrusts generated by the 
actuators and a center of gravity of the anti-vibration table or a center 
of gravity of a structure including the anti-vibration table and support 
members of the anti-vibration table, and lines of action of the thrusts 
generated by the actuators with respect to the anti-vibration table become 
substantially equal for the plurality of actuators. 
In the active anti-vibration apparatus of the present invention, the 
anti-vibration table has a substantially regular triangular structure. 
In the active anti-vibration apparatus of the present invention, the 
plurality of actuators are arranged near vertexes of the anti-vibration 
table having the substantially regular triangular structure, respectively. 
In the active anti-vibration apparatus of the present invention, a 
relationship between the vector b having elements F.sub.x, F.sub.y, and 
M.sub.z and the vector x having elements F.sub.a, F.sub.b, and F.sub.c is 
represented as follows: 
##EQU2## 
where F.sub.x is a translation force to be applied in an X-axis direction 
associated with an X-Y plane of the anti-vibration table, F.sub.y is a 
translation force to be applied in a Y-axis direction on the X-Y plane, 
M.sub.z is a moment to be applied to rotation about a Z-axis on the X-Y 
plane, and F.sub.a, F.sub.b, and F.sub.c are thrusts to be generated by 
the plurality of actuators. 
In the active anti-vibration apparatus of the present invention, the 
plurality of sensors are arranged in correspondence with the plurality of 
actuators. 
According to the present invention, there is also provided a method of 
manufacturing an active anti-vibration apparatus having a plurality of 
sensors and actuators, which detects a motion of an anti-vibration table 
for supporting an equipment with the plurality of sensors and controls the 
actuators on the basis of output signals from the sensors to drive the 
anti-vibration table, thereby performing an anti-vibration operation for 
the anti-vibration table, wherein the plurality of actuators are arranged 
such that, when thrusts generated by the plurality of actuators are 
represented by a vector x, and driving forces in units of motion modes, 
which act on the anti-vibration table, are represented by a vector b, a 
condition number of a matrix A defined by an equation Ax=b established 
between the vector x and the vector b in accordance with a geometrical 
arrangement of the plurality of sensors is minimized. 
In the active anti-vibration apparatus according to the present invention, 
the actuators are arranged such that the condition number of a matrix A of 
a fundamental equation for motion mode distribution given below is 
minimized, which fundamental equation is determined in accordance with the 
geometrical arrangement of the actuators and associates the anti-vibration 
table driving forces F.sub.x, F.sub.y, and M.sub.z in units of motion 
modes with the actuator thrusts F.sub.a, F.sub.b, and F.sub.c : 
##EQU3## 
Alternatively, in the anti-vibration apparatus according to the present 
invention, the actuators are arranged such that the condition number of 
the matrix WA obtained by multiplying the matrix A with the normalizing 
matrix W which normalizes the anti-vibration table driving forces F.sub.x, 
F.sub.y, and M.sub.z in units of motion modes by desired maximum driving 
forces F.sub.xmax, F.sub.ymax, and M.sub.zmax is minimized. 
According to the present invention, the actuators are arranged such that 
the condition number of the matrix A of the fundamental equation for 
motion mode distribution is minimized. With this arrangement, equal 
driving forces can be applied to the anti-vibration table in any direction 
of the horizontal motion modes of the anti-vibration table, so that 
vibrations of the anti-vibration table can be effectively suppressed. 
Alternatively, considering the difference between the anti-vibration table 
driving forces F.sub.x, F.sub.y, and M.sub.z in units of motion modes and 
the desired maximum driving forces F.sub.xmax, F.sub.ymax, and M.sub.zmax, 
the actuators are arranged such that the condition number of the produce 
WA of the normalizing matrix W for normalizing the driving forces F.sub.x, 
F.sub.y, and M.sub.z with the desired maximum driving forces and the 
matrix A is minimized. With this arrangement, desired maximum driving 
forces can be applied to the anti-vibration table in any direction of the 
horizontal motion modes of the anti-vibration table. 
In addition, with the actuator arrangement for minimizing the condition 
number of the matrix A or WA, variations in anti-vibration table driving 
forces in units of motion modes attributed to variations in actuator 
thrust or measurement errors of the actuator position can be minimized, so 
that vibrations of the anti-vibration table can be effectively suppressed. 
Further objects, features and advantages of the present invention will be 
apparent from the following detailed description of embodiments of the 
present invention with reference to the accompanying drawings.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS 
First Embodiment! 
FIG. 1 is a view showing the typical embodiment of an active anti-vibration 
apparatus according to the present invention, and the arrangement of 
sensors incorporated in the anti-vibration apparatus. FIG. 1 is a top view 
of the active anti-vibration apparatus. A support table (to be referred to 
as an anti-vibration table hereinafter) 4 with a regular triangular 
structure on which a precision equipment such as the X-Y stage of an 
exposure apparatus for manufacturing a semiconductor device is mounted is 
supported at its vertex portions by anti-vibration units 1a to 1c serving 
as a support mechanism. The anti-vibration units 1a to 1c have sensors 2a 
to 2c for measuring the motions, including an acceleration, a speed, and a 
moving amount, of the anti-vibration table 4, and actuators 3a to 3c for 
applying forces to the anti-vibration table 4, respectively. 
A method of optimally arranging the sensors 2a to 2c to precisely measure 
the horizontal vibration of the anti-vibration table 4, and an 
anti-vibration apparatus manufactured by this method will be described 
below. Regarding the anti-vibration table 4 as a rigid body, the 
horizontal motions of the anti-vibration table 4 are classified into 
motion modes of total 3-degree-of-freedom constituted by translation 
motions of the center of gravity in a horizontal plane, which have 
2-degree-of-freedom, and a rotation motion about the vertical axis 
including the center of gravity, which has a single-degree-of-freedom. 
The center of gravity of the anti-vibration table 4 is defined as G, and an 
X-Y-Z coordinate system having its origin at G is fixed on the 
anti-vibration table. The X-Y plane matches the horizontal plane. In this 
case, the horizontal motions of the anti-vibration table 4 can be 
represented by translation motions in the x and Y directions and rotation 
motion in the .theta..sub.Z direction about the vertical axis (Z-axis) 
including the center of gravity G. These directions of motions are defined 
as motion modes of the anti-vibration table 4, and parameters such as the 
displacement and acceleration of the anti-vibration table in units of 
motion modes are defined as motion parameters. The horizontal motion modes 
of the anti-vibration table 4 are X, Y, and .theta.Z, and motion 
parameters in units of motion modes are expressed as P.sub.x, P.sub.y, and 
P.theta..sub.z. 
The sensors 2a to 2c can measure mechanical uniaxial rectilinear motions 
including an acceleration, a velocity, and a moving position. The 
measurement points of the sensors 2a to 2c are at the vertex positions of 
the anti-vibration table 4. The measurement direction vectors of the 
sensors 2a to 2c and the action direction vectors of forces generated by 
the actuators 3a to 3c are assumed to be in the horizontal plane (X-Y 
plane) including the center of gravity G. Under these conditions, the 
arrangement of the sensors 2a to 2c has a single-degree-of-freedom; 
rotates about the vertical axis passing through the vertexes of the 
anti-vibration table 4. Since the anti-vibration table 4 has a regular 
triangular structure, the distance from the center of gravity G to each 
vertex of the anti-vibration table 4, i.e., the distance from the center 
of gravity G to each of the sensors 2a to 2c is r. 
As shown in FIG. 1, the coordinate point of the sensor 2a on the X-Y 
coordinate system fixed at the center of gravity G is defined as 
(x.sub.a,y.sub.a), and the angle formed by the vector from the coordinate 
point of the sensor 2a in the positive direction along the measurement 
direction and the X-axis is defined as .theta..sub.2ax. When the 
anti-vibration table 4 is in a motion represented by the motion parameters 
P.sub.x, P.sub.y, and P.theta..sub.z, the output signal from the sensor 2a 
is s.sub.a. In this case, the relationship between the motion parameters 
and the sensor output signal s.sub.a is represented by the following 
equation (1): 
##EQU4## 
As for the sensors 2b and 2c, similarly, the coordinate points of the 
sensors 2b and 2c on the X-Y coordinate system fixed at the center of 
gravity G are defined as (x.sub.b,y.sub.b) and (x.sub.c,y.sub.c), 
respectively, and angles formed by the vectors from these coordinate 
points in the positive direction along the measurement directions and the 
X-axis are defined as .theta..sub.2bx and .theta..sub.2cx, respectively. 
At this time, the relationships between three sensor output signals 
s.sub.a to s.sub.c and the motion parameters P.sub.x, P.sub.y, and 
P.theta..sub.z are represented by linear equation (2) below. 
A in equations (2) and (3) below is a coefficient matrix that is determined 
by the positions of the sensors 2a to 2c and the measurement directions. 
The coordinates of the sensors 2a to 2c are determined by the radius r of 
a circle passing through the three vertexes of the anti-vibration table 4, 
as represented by equation (4) below. Therefore, the coefficient matrix A 
is expressed as a function of the radius r and the measurement directions 
of the sensors (.theta..sub.2ax, .theta..sub.2bx, and .theta..sub.2cx) By 
using linear equation (2) below, the motion parameters P.sub.x, P.sub.y, 
and P.theta..sub.z of the anti-vibration table 4 can be obtained from the 
three sensor output signals s.sub.a to s.sub.c. 
##EQU5## 
As a method of controlling the active anti-vibration apparatus, closed 
control loops are formed by the sensors 2a to 2c and the actuators 3a to 
3c incorporated in the anti-vibration units 1a to 1c, respectively, 
thereby independently controlling the respective anti-vibration units. 
Alternatively, control loops are formed in units of motion modes of the 
anti-vibration table 4. In this case, the motion parameters of the 
anti-vibration table 4 are measured from sensor information to determine 
the operating forces to be applied to the anti-vibration table 4 in units 
of motion modes. The actuators 3a to 3c are driven to realize the 
operating forces in units of motion modes. 
To measure the motion modes of the anti-vibration table 4 from sensor 
information, equation (2) above is used. Solving linear simultaneous 
equations represented by equation (2) yields, the motion parameters 
P.sub.x, P.sub.y, and P.theta..sub.z. Since the left-hand side member of 
equation (2) represents output signals from the sensors 2a to 2c, it 
includes measurement noise. As for the coefficient matrix A determined by 
the sensor arrangement as well, an error may be present. 
To quantitatively evaluate the influence of the error included in the 
observation data, which affects the solution, i.e., the motion parameters, 
the condition number of the coefficient matrix A must be checked. The 
condition number is a positive number and its minimum value is 1. When the 
condition number is large (bad condition), the solution, i.e., the motion 
parameters largely change due to a small difference in sensor output 
signals. Alternatively, a small error between the measured values and the 
true values of the sensor positions and directions largely degrades the 
reliability of the solution, i.e., the motion parameters. When the 
condition number becomes infinitely large, a drop in rank occurs in the 
coefficient matrix A. That is, the motion parameters cannot be obtained. 
When the sensors 2a to 2c are arranged to set the condition number as close 
to 1 (best condition) as possible, an optimum sensor arrangement for 
measuring the motion parameters can be realized. This sensor arrangement 
also allows to most easily measure the horizontal motions of 
3-degree-of-freedom of the anti-vibration table 4. Therefore, the optimum 
sensor arrangement for measuring the motion parameters in control of the 
anti-vibration table 4 in units of motion modes can also be optimally 
applied to independently control the anti-vibration units. 
As described above, when the measurement directions of the sensors 2a to 2c 
are determined such that the condition number of the coefficient matrix A 
of equation (3) is minimized, an optimum sensor arrangement is obtained. 
The condition number can be used as a quantitative guideline for the 
sensor arrangement. 
FIG. 1 shows the sensor arrangement for minimizing the condition number. 
The straight line in the measurement direction of each of the sensors 2a 
to 2c and a line connecting each vertex of the anti-vibration table 4 at 
which the sensor is positioned to the center of gravity G form the same 
acute angles .theta.. The sensors are arranged such that the acute angle 
is formed on the left side of the line from the center of gravity G to the 
vertex. 
As shown in FIG. 2, the condition number can also be minimized with a 
sensor arrangement in which the same acute angle .theta. is formed on the 
right side of the line from the center of gravity G to the vertex of the 
anti-vibration table 4. Only the arrangements shown in FIGS. 1 and 2 can 
minimize the condition number of the coefficient matrix A of equation (3). 
The sensor angle .theta. is uniquely determined within the range from 
0.degree. to 90.degree. in accordance with the radius r of the 
anti-vibration table 4. Each arrow in FIG. 1 or 2, which extends through 
the sensor, indicates the positive direction of the sensor measurement 
direction. The sensor measurement direction may be set in any direction as 
far as it is parallel to the straight line having the angle .theta. with 
respect to the line connecting the vertex of the anti-vibration table 4 to 
the center of gravity G. 
To minimize the condition number of the matrix A, the apparent fact that 
the sensor angle .theta. increases as the radius r becomes smaller must be 
considered. A sensor arrangement for most easily detecting the rotation 
motion in the .theta..sub.Z direction is an arrangement along the 
direction (circumferential direction) perpendicular to the line connecting 
the center of gravity to the vertex, as shown in FIG. 3. To the contrary, 
a sensor arrangement for most easily detecting the X- and Y-direction 
translation motions is an arrangement along the direction toward the 
center of gravity G (radial direction), as shown in FIG. 4. As the radius 
r of the anti-vibration table 4 becomes smaller, detection of the rotation 
motion in the .theta..sub.z direction becomes more difficult. Therefore, 
the sensors 2a to 2c must be set close to the circumferential direction. 
That is, the sensor angle .theta. increases. As described above, the 
sensor arrangement for minimizing the condition number of the coefficient 
matrix A changes depending on the radius r. 
For example, when the radius r=1, the condition number of the matrix A of 
equation (2) is minimized with a sensor angle .theta.=45.degree.. This is, 
the middle direction between the circumferential direction and the radial 
direction. In this case, the condition number is 1. 
In the embodiment shown in FIGS. 1 and 2, the action directions of the 
actuators 3a to 3c are set in the same direction as the measurement 
directions of the sensors 2a to 2c, respectively. With this arrangement, 
both methods of controlling the active anti-vibration apparatus, i.e., a 
method of forming control loops in units of motion modes of the 
anti-vibration table 4 and a method of forming closed control loops in 
units of anti-vibration units, can be realized. 
In equation (2) as a fundamental equation for calculating the optimum 
sensor arrangement in the above embodiment, the physical dimensions of the 
motion parameters P.sub.x, P.sub.y, and P.theta..sub.z are different. The 
dimension of the translation parameters P.sub.x and P.sub.y is different 
from that of the rotation parameter P.theta..sub.z. In the above 
arrangement, the optimum sensor arrangement is calculated by equally 
handling quantities with different physical dimensions. When the motion 
parameters are represented by using the unit of length m! and the unit of 
rotation rad!, and the actual motion of the anti-vibration table 4 can be 
represented by the translation parameters P.sub.x and P.sub.y and the 
rotation parameter P.theta..sub.z, all of which have the close value, 
equation (2) can be used as a fundamental equation. 
Otherwise, weighting is needed between the translation parameters and the 
rotation parameters. In addition, even when the measurement precision for 
the remaining parameters are sacrificed to some extent to precisely 
measure a specific motion parameter, the motion parameters must be 
weighted. Equations (5) and (6) below are used as fundamental equations 
for obtaining an optimum sensor arrangement in consideration of weighting 
between the motion parameters: 
##EQU6## 
The sensor arrangement for minimizing the condition number of a coefficient 
matrix AW of simultaneous linear equations (5) and (6) above is an optimum 
sensor arrangement. To weight the parameters, a weighting value w.sub.i 
(i=x, y, or .theta.z) of a motion mode to be precisely measured is set to 
be large. Assuming that the motion of the anti-vibration table 4 is 
represented in units of motion modes, when the motion parameters P.sub.x, 
P.sub.y, and P.theta..sub.z have the close value, and all the motion 
parameters are to be measured at the same precision, the same weighting 
value is used (w.sub.x =w.sub.y =w.theta..sub.z =1). 
When the weighting values w.sub.x and w.sub.y of the translation motion 
parameters P.sub.x and P.sub.y are the same value, the three sensors 2a to 
2c are set in the direction of angle .theta. with respect to the lines 
connecting the vertexes of the anti-vibration table 4, where the sensors 
2a to 2c are positioned, to the center of gravity G, as in the above 
embodiment. 
The value of the sensor angle .theta. changes depending on the ratio of the 
weighting value w.sub.x (=w.sub.y) to the weighting value w.theta..sub.z 
and the radius r of the anti-vibration table 4. When the weighting values 
w.sub.x and w.sub.y are different, the three sensors 2a to 2c are set at 
different sensor angles. For example, the optimum sensor arrangement 
placing importance to the precision of the Y-direction translation motion 
parameter P.sub.y is shown. This arrangement is calculated while setting 
the weighting values w.sub.y =2 and w.sub.x =w.theta..sub.z =1 and the 
radius r of the anti-vibration table 4=1. Two sensor arrangements shown in 
FIGS. 5A and 5B are obtained as an arrangement for minimizing the 
condition number of the coefficient matrix AW. Since the measurement 
precision in the Y direction is given importance, the sensor measurement 
direction becomes close to the Y direction. 
In the above embodiment, an anti-vibration operation for the anti-vibration 
table with a three-point support mechanism and horizontal motions of 
3-degree-of-freedom has been described. However, the present invention 
which minimizes the condition number of a coefficient matrix in 
simultaneous equations that associate the motion parameters with sensor 
signal outputs is not limited to the anti-vibration table with the 
three-point support mechanism and is not limited to the horizontal motions 
of 3-degree-of-freedom, either. The present invention can be applied to 
any case of arrangement of sensors which are the same in number as the 
motion parameters to be calculated. 
As described above, according to the anti-vibration apparatus of the 
present invention, the vibration of the anti-vibration table with, e.g., 
horizontal motions of 3-degree-of-freedom can be precisely calculated by 
the vibration sensors incorporated in the anti-vibration apparatus serving 
as a support mechanism while minimizing the influence of observation 
noise. In addition, a quantitative guideline for the sensor arrangement 
can be provided. 
In addition, according to the method of arranging the sensors of the 
anti-vibration apparatus of the present invention, an anti-vibration 
apparatus which can precisely calculate the vibration of an anti-vibration 
table with, e.g., horizontal motions of 3-degree-of-freedom by using 
vibration sensors incorporated in the anti-vibration table serving as a 
support mechanism while minimizing the influence of observation noise, can 
be manufactured. 
Second Embodiment! 
Anti-vibration table driving forces in units of motion modes as inputs to a 
motion mode distribution unit 107 in FIG. 10 are defined as b, and 
actuator thrusts as outputs are defined as x. Note that b and x are vector 
quantities. With the arrangement of the active anti-vibration apparatus in 
FIG. 10, b and x are represented by the following equations (7) and (8), 
respectively: 
##EQU7## 
T on the right-hand side of equations (7) and (8) represents transposition 
of a matrix. The relationship between b and x is represented by equation 
(9) below in accordance with the arrangement of actuators 102a to 102c 
with respect to the center of gravity of an anti-vibration table 105: 
EQU Ax=b (9) 
A matrix A multiplied with x on the left-hand side of equation (9) is a 
constant matrix determined by the arrangement of the actuators 102a to 
102c. Equation (9) is a fundamental equation for motion mode distribution. 
To apply desired driving forces b in units of motion modes to the 
anti-vibration table 105, actuator thrusts x as a solution of equation (9) 
must be generated by the actuators 102a to 102c. 
If the matrix A has an inverse matrix, the solution of equation (9) is 
obtained. Normally, the actuators 102a to 102c are arranged such that the 
solution of equation (9) is present. In fact, since the thrust generated 
by an actuator is finite, the anti-vibration table driving forces b in 
units of motion modes which can be realized are also limited. To 
effectively suppress the vibration of the anti-vibration table 105, 
elements F.sub.x, F.sub.y, and M.sub.z of the anti-vibration table driving 
forces b in units of motion modes are preferably equally realizable. For 
this purpose, the actuators 102a to 102c must be arranged such that the 
condition number of the matrix A of equation (9) is minimized. This will 
be described below. 
When the actuator thrust x satisfies a constraint as represented by 
equation (10) below, the anti-vibration table driving forces b in units of 
motion modes which can be realized from x form an elliptical body 
represented by equation (11) below in a three-dimensional space defined by 
F.sub.x, F.sub.y, and M.sub.z. This is apparent from equation (12) below: 
##EQU8## 
The principal axis of the elliptical body of equation (11) and the length 
of the principal axis are obtained by singular point degeneration of the 
matrix A. Assume that singular point degeneration of the matrix A is 
represented by the following equations (13) and (14): 
EQU A=UST.sup.T (13) 
##EQU9## 
Since the matrix A has an inverse matrix, 
.sigma.1.gtoreq..sigma.2.gtoreq..sigma.3. When the vector of the ith row 
of U is defined as u.sub.i.sup.T, the principal axis of the elliptical 
body is given by .sigma..sub.1 u.sub.1, .sigma..sub.2 u.sub.2, and 
.sigma..sub.3 u.sub.3. When orthogonal transformation as represented by 
equation (15) below is considered for b, equation (16) below is obtained 
from equations (11) and (15): 
##EQU10## 
As is apparent from equation (16), the direction of coordinate axis for b, 
i.e., u.sub.1, u.sub.2, and u.sub.3 indicates the direction of principal 
axis of the elliptical body, and the radii in this direction are .sigma.1, 
.sigma.2, and .sigma.3. 
In the three-dimensional space of the anti-vibration table driving forces 
F.sub.x, F.sub.y, and M.sub.z in units of motion modes, a large driving 
force can be applied to the anti-vibration table 105 along the direction 
of the long radius of the principal axis of the elliptical body. However, 
only a small driving force can be applied along the direction of the short 
radius. Therefore, to generate equal F.sub.x, F.sub.y, and M.sub.z, the 
elliptical body is preferably close to a sphere. Of the radii of principal 
axis of the elliptical body, the longest is .sigma.1, and the shortest is 
.sigma.2. For this reason, when the ratio of .sigma.1 to .sigma.3, i.e., 
.sigma.1/.sigma.3 comes close to 1, the elliptical body becomes close to a 
sphere. The ratio .sigma.1/.sigma.3 is defined as the condition number of 
the matrix A. As the condition number of the matrix A becomes smaller, the 
elliptical body becomes close to a sphere. 
In the above description, the actuator thrust x is assumed to satisfy the 
constraint that the sum of squares of the elements of x is smaller than 1, 
as is represented by equation (10). However, as represented by equation 
(17) below, it is more practical to consider a constraint that limits the 
maximum thrusts as the elements of x, which can be generated by the 
respective actuators: 
EQU .vertline.F.sub.a .vertline.,.vertline.F.sub.b .vertline.,.vertline.F.sub.c 
.vertline..ltoreq.1 (17) 
Equation (17) forms a cube as shown in FIG. 9 in a three-dimensional space 
defined by F.sub.a, F.sub.b, and F.sub.c. Equation (10) represents a 
sphere inscribed with the respective planes of the cube shown in FIG. 9. 
When a constraint as represented by equation (17) is given to x, the 
anti-vibration table driving forces b in units of motion modes which can 
be realized from x form a hexahedron in the three-dimensional space 
defined by F.sub.x, F.sub.y, and M.sub.z. As the condition number of the 
matrix A becomes close to 1, this hexahedron becomes close to a cube. 
Therefore, even when the constraint as represented by equation (17) is 
considered, all of F.sub.x, F.sub.y, and M.sub.z equal can be generated as 
the condition number of the matrix A becomes smaller. 
In the above description, the difference between physical dimensions of the 
translation thrusts F.sub.x and F.sub.y and the moment M.sub.z has been 
ignored. Depending on the driving conditions for an equipment mounted on 
the anti-vibration table 105, a larger driving force must be generated as 
one of F.sub.x, F.sub.y, and M.sub.z in some cases. To handle the thrusts 
and the moment placing equal weight on them and to give consideration to 
the difference between the desired maximum driving forces F.sub.x, 
F.sub.y, and M.sub.z, F.sub.x, F.sub.y, and M.sub.z can be normalized, as 
will be described below. The desired maximum driving forces are defined as 
F.sub.xmax, F.sub.ymax, and M.sub.zmax, and the normalized anti-vibration 
table driving forces b in units of motion modes are defined as represented 
by the following equations (18) and (19): 
##EQU11## 
Substituting equation (18) into equation (9) yields the following equation 
(20): 
EQU WAx=b (20) 
To consider weighting between F.sub.z, F.sub.y, and M.sub.z, the actuators 
102a to 102c are arranged such that the condition number of matrix WA of 
equation (20) is minimized. F.sub.xmax, F.sub.ymax, and M.sub.zmax need 
not be values having physical dimensions and can be relative ratios as 
dimensionless quantities. 
Generally, an actuator does not always generate a thrust corresponding to a 
command value from a controller. Some variations in thrust due to a factor 
such as a torque ripple cannot be avoided. In addition, to obtain the 
matrix A of equation (9), it is essential to determine the positional 
relationship between a center of gravity G of the anti-vibration table 105 
and the actuators 102a to 102c. However, it is difficult to accurately 
measure the positions of the actuators in fact, and some errors between 
the measured value and the true value cannot be avoided. To quantitatively 
evaluate the error between the desired driving forces and the 
anti-vibration table driving forces in units of motion modes actually 
acting on the anti-vibration table 105, which is produced due to 
variations in actuator thrust or measurement errors of the actuator 
position, the condition number of the matrix A of equation (9) must be 
checked. 
In the linear equation such as equation (9), the condition number of the 
matrix A indicates the sensitivity of the solution x with respect to an 
error included in A and b. As the condition number becomes larger, the 
solution x largely changes with respect to a fine variation in A and b. 
Using an inverse matrix A.sup.-1 of the matrix A, equation (9) is 
represented by the following equation (21): 
EQU A.sup.-1 b=x (21) 
The condition number of the matrix A is the same as that of the inverse 
matrix A.sup.-1. Regarding equation (21) as a new linear equation 
replacing equation (9), the condition number of the matrix A indicates the 
sensitivity coefficient representing the influence of an error included in 
A.sup.-1 and x on b. More specifically, the condition number of the matrix 
A becomes a sensitivity coefficient representing the error between the 
desired driving forces and the anti-vibration table driving forces in 
units of motion modes acting on the anti-vibration table 105, which is 
generated due to variations in actuator thrust or measurement errors of 
the actuator position. When the condition number is smaller, desired 
anti-vibration table driving forces in units of motion modes can be 
applied to the anti-vibration table without any influence of error 
factors. Therefore, to minimize the influence of error factors, the 
actuators 102a to 102c must be arranged such that the condition number of 
the matrix A is minimized. Similarly, when the anti-vibration table 
driving forces F.sub.x, F.sub.y, and M.sub.z in units of motion modes are 
normalized, the actuators 102a to 102c must be arranged such that the 
condition number of the matrix WA of equation (20) is minimized. 
As described above, when the actuators 102a to 102c are arranged such that 
the condition number of the matrix A of equation (9) as the fundamental 
equation for motion mode distribution, or the product WA of the matrix A 
and the weighting matrix W is minimized, equal anti-vibration table 
driving forces F.sub.x, F.sub.y, and M.sub.z in units of motion modes, or 
desired maximum driving forces can be applied to the anti-vibration table. 
In addition, any error between the desired driving forces and the 
anti-vibration table driving forces in units of motion modes, which is 
produced due to variations in actuator thrust or measurement errors of the 
actuator position, can be minimized. 
FIG. 6 is a view showing a typical arrangement of the present invention. 
FIG. 6 is a view of anti-vibration units 101a to 101c and the 
anti-vibration table 105, which are viewed from the top. The 
anti-vibration table 105 with a regular triangular structure is supported 
at its vertex portions by the anti-vibration units 101a to 101c. The 
anti-vibration units 101a to 101c have the actuators 102a to 102c for 
applying driving forces to the anti-vibration table 105, respectively. The 
lines of action of thrusts generated by the actuators pass through the 
vertexes of the anti-vibration table 105 and are in a horizontal plane 
including the center of gravity G of the anti-vibration table 105. 
The X-Y-Z coordinate system has its Z-axis along the vertical direction 
such that the origin matches the center of gravity G of the anti-vibration 
table 105. In addition, the X-Y-Z coordinate system is fixed on the 
anti-vibration table 105 while setting its X-axis parallel to the base of 
the anti-vibration table 105 viewed from the top as in FIG. 6. 
Referring to FIG. 6, the arrangement of the actuators 102a to 102c has a 
single-degree-of-freedom; rotates about the vertical axis passing through 
the vertexes of the anti-vibration table 105. The angle of rotation is 
represented by angles .theta..sub.a, .theta..sub.b, and .theta..sub.c 
formed by lines connecting the respective vertexes to the center of 
gravity G and the lines of action of thrusts generated by the actuators. 
The distance from each vertex to the center of gravity G is defined as r. 
At this time, equations (22) to (24) below are obtained as fundamental 
equations for motion mode distribution in correspondence with equation 
(9): 
##EQU12## 
FIG. 6 shows the arrangement o f the actuators 102a to 102c for minimizing 
the condition number of the matrix A of equation (22). The lines of action 
of thrusts generated by the actuators 102a to 102c and the lines 
connecting the respective vertexes of the anti-vibration table 105 to the 
center of gravity G form the same acute angles .theta. (.theta..sub.a 
=.theta..sub.b =.theta..sub.c). The acute angles .theta. do not oppose 
each other along the respective sides of the anti-vibration table 105 with 
the triangle structure. FIG. 7 also shows an arrangement for satisfying 
the above conditions. Only the actuator arrangements shown in FIGS. 6 and 
7 can minimize the condition number of the matrix A of equation (22). 
Depending on the distance r, the angle .theta. is uniquely determined 
within the range of 0.degree. to 90.degree.. As the distance r becomes 
smaller, the angle .theta. becomes larger. This is because, when the 
actuators 102a to 102c come closer to the center of gravity G of the 
anti-vibration table 105, hardly any moment M.sub.z is generated about the 
Z-axis. For example, when r=1, the condition number of the matrix A is 
minimized at .theta.=45.degree.. The minimum value is 1. 
When the difference between the desired driving forces F.sub.x, F.sub.y, 
and M.sub.z is taken into consideration, the actuators 102a to 102c are 
arranged such that the condition number of the product WA of the 
normalizing matrix W of equation (19) and the matrix A is minimized. When 
the desired maximum values F.sub.xmax and F.sub.ymax of the translation 
thrusts F.sub.x and F.sub.y have the same value, the lines of action of 
thrusts generated by the actuators 102a to 102c are set in directions 
forming the same angles .theta. with respect to the lines connecting the 
respective vertexes of the anti-vibration table 105 to the center of 
gravity G. The angle .theta. changes in accordance with the ratio of 
F.sub.xmax (=F.sub.ymax) to M.sub.zmax, and the distance r. As M.sub.zmax 
is set to be larger, and as r is set to be smaller, the angle .theta. 
becomes larger. When F.sub.xmax is different from F.sub.ymax, the angles 
.theta..sub.a, .theta..sub.b, and .theta..sub.c are different. 
In a scan type exposure apparatus, the stage mounted on the anti-vibration 
table is mainly driven in the scanning direction. For this reason, it is 
required to apply a large driving force to the anti-vibration table along 
the scanning direction. FIGS. 8A and 8B show actuator arrangements for 
applying in the Y-axis direction a translation thrust twice that in the 
X-axis direction assuming that the scanning operation is performed along 
the Y-axis direction. F.sub.xmax, F.sub.ymax, and M.sub.zmax may not be 
actual desired maximum driving forces but may be relative ratios as 
dimensionless quantities. When F.sub.xmax =1, F.sub.ymax =2, M.sub.zmax 
=1, and r=1, the condition number of the matrix WA is minimized with the 
arrangement of the actuators 102a to 102c as in FIG. 8A or 8B. Since 
F.sub.ymax is set to be large, the actuator 102b points in the Y-axis 
direction, unlike that in FIGS. 6 and 7. As described above, according to 
the present invention, the actuators can be arranged while placing 
importance to the vibration damping performance in a specific direction, a 
large vibration damping effect can be obtained as an anti-vibration 
apparatus for a scan type exposure apparatus. 
In the above arrangement, an anti-vibration table with an regular 
triangular structure has been described. However, the present invention 
which arranges the actuators such that the condition number of the matrix 
A or WA is minimized is not limited to an anti-vibration table with a 
regular triangular structure. The present invention can be applied to any 
actuator arrangement in the horizontal direction as far as the 
anti-vibration table is supported by three anti-vibration units. 
In addition, the center of gravity G may be either the center of gravity of 
the anti-vibration table 105 itself or the center of gravity of the entire 
structure including the anti-vibration table 105 and the support members 
therefor. 
As has been described above, according to the present invention, equal 
driving forces can be applied to the anti-vibration table in any direction 
of the horizontal motion modes of the anti-vibration table. In addition, 
desired maximum driving forces can be applied to the anti-vibration table 
in any direction of the horizontal motion modes of the anti-vibration 
table. Furthermore, a variation in anti-vibration table driving forces in 
units of motion modes, which is attributed to a variation in actuator 
thrust or measurement error of the actuator position, can be minimized. 
With this arrangement, the vibration of the anti-vibration table can be 
effectively suppressed. 
The present invention is not limited to the above embodiments and various 
changes and modifications can be made within the spirit and scope of the 
present invention. Therefore, to apprise the public of the scope of the 
present invention, the following claims are made.