Positioning system

A positioning system includes a plurality of actuators for driving an object to be positioned, a plurality of current output type amplifiers for amplifying drive signals from the actuators and for producing drive currents corresponding to the drive signals, and a plurality of acceleration sensors for detecting acceleration of the object in the neighborhood of the actuators. A feedback circuit negatively feeds back the outputs of the acceleration sensors to input sides of the amplifiers such that the actuators are driven in response to the drive currents.

FIELD OF THE INVENTION AND RELATED ART 
This invention relates to a positioning system with an actuator comprising 
a piezoelectric device or an electrostrictive device, for example, for 
positioning an article with precision of a submicron order. 
For fine positioning technique in the field of precise machining, 
assembling or adjustment, a positioning precision of a submicron order has 
been required. Particularly, in an ultra-precise positioning stage to be 
used for exposure of a fine pattern, a piezoelectric device or an 
electrostrictive device is used in many cases as an actuator, so as to 
attain higher driving resolution and a wider range of frequency response. 
An example is shown in FIG. 16, which is a fine-motion positioning device 
of a three-freedom type having a function of controlling translation (one 
freedom) in a vertical direction and tilt in a horizontal plane (two 
freedoms). This device is arranged so that a stage base 1 on which a 
semiconductor wafer 100 is placed is positioned by means of actuators 2M, 
2R and 2L for displacing the base in the vertical direction in response to 
applied voltage. The actuators 2M, 2R and 2L each include a piezoelectric 
device (driving element) and a displacement magnifying mechanism for 
magnifying the displacement of the piezoelectric device. Position sensors 
3M, 3R and 3L are disposed adjacent to the actuators 2M - 2L, for 
measuring displacement of the base 1 in the vertical z direction. The 
mechanism constituted by these components may be called a fine-motion 
positioning mechanism. 
Displacement signals measured by the position sensors 3M - 3L are 
transformed into electric signals by displacement amplifiers 4M, 4R and 
4L, respectively. These electric signals are then compared with 
instruction voltages applied to voltage input terminals 5M, 5R and 5L, 
respectively, and differential signals e.sub.M, e.sub.R and e.sub.L are 
produced. These differential signals are applied to preamplifiers 6M, 6R 
and 6L, respectively, for assuring predetermined sensitivity, and output 
of them are applied to programmable gain devices 7M, 7R and 7L adapted to 
adjust the characteristic of control loop. With the outputs of them, 
voltage amplifiers 8M, 8R and 8L are excited, whereby the actuators 2M - 
2L are moved upwardly or downwardly to cause translation of the base 1 
vertically or to incline it with respect to the z axis in the drawing. The 
closed loop control system related to the positioning may be called a 
feedback system, and a fine-motion positioning mechanism which is equipped 
with such feedback system may be called a fine-motion positioning system. 
The voltage amplifiers 8M - 8L are of the type that it outputs an electric 
current in response to an applied voltage. With this position control loop 
structure, convergence of steady-state deviation (error) to zero is 
assured. This is because: the piezoelectric device of the actuator 2M - 2L 
is electrically a capacitor, and the current output type voltage amplifier 
8M - 8L and the transfer function including corresponding piezoelectric 
device to be driven by the voltage amplifier include an integrator. Thus, 
the control loop is what is called a "1 type" and, according to the 
control theory, zero steady-state error is assured. 
The positioning system such as described above is discussed in papers such 
as "A six-degrees of freedom fine-motion mechanism" (second report), 
Henmi, Sato, Wada, Shimokawabe ("Journal of the Japan Society for 
Precision Engineering Association", 58/6/1992, pages 1035-1040), or "A 
six-axes motion control method for parallel linkage type fine-motion 
stage", Tomita et al ("Journal of the Japan Society for Precision 
Engineering Association", 58/4/1992, pages 684-690). 
SUMMARY OF THE INVENTION 
In the fine-motion positioning system such as shown in FIG. 16, the 
fine-motion positioning mechanism is generally manufactured as comprising 
three units each including a corresponding actuator 2M, 2R or 2L (having a 
piezoelectric device and a displacement magnifying mechanism) and a 
corresponding position sensor 3M, 3R or 3L. These units are disposed along 
a plane, for drive of the base 1. Here, even though each single unit has a 
superior damping characteristic, when the base 1 of a large mass is 
supported by these units, the damping coefficient of the mechanism as a 
whole becomes small. Therefore, when a feedback mechanism is added to 
provide a closed loop system, the response becomes vibratory. Namely, 
while the loop gain with respect to the position should be low for 
stabilization of the fine-motion positioning system, this would be an 
obstruction to a reduction in positioning time. 
Further, for suppressing disturbance to the fine-motion positioning 
mechanism, the loop gain of the closed loop system should be large. 
However, increasing the Loop gain is limited if the mechanism has small 
damping. For example, the graph of FIG. 17 illustrates the root locus of 
the three-freedom fine-motion positioning system of FIG. 16. The root of 
the closed loop system was plotted by x, as programmable gain devices 7M, 
7R and 7L were increased from zero for each 5/256. It is seen from this 
graph that the lowest-order one of the three complex-conjugate roots 
(i.e., z-axis translation motion) easily enters the right-hand side of the 
complex plane with the increase of the programmable gain and it become 
unstable. 
It is accordingly a first object of the present invention to provide a 
fine-motion positioning system in which the positioning operation is 
stabilized or the strength to disturbance and positioning speed thereof 
are improved. 
A fine-motion positioning system according to one preferred form of the 
present invention comprises: a planar base with which the positioning 
operation of one freedom of translation and two freedoms of rotation can 
be done; three actuators disposed approximately concentrically for the 
positioning operation, for moving the beset three position sensors each 
for measuring displacement of the base in the neighborhood of a 
corresponding actuator; and a feedback system comprising a preamplifier, a 
programmable gain device and a current output type voltage amplifier. The 
output of each position sensor is compared with a reference voltage to 
produce a differential signal on the basis of which the voltage amplifier 
is excited through the preamplifier and the programmable gain device, and 
wherein each actuator is driven in response to the output current of the 
voltage amplifier. The positioning system further comprises an 
acceleration sensor for detecting acceleration of the base, in the 
neighborhood of the actuator; and an acceleration feedback circuit 
including a lowpass filter having a suitable time constant and a gain, 
with which the output of the acceleration detector is negatively fed back 
to the input side of the voltage amplifier. 
The feedback system may comprise a preamplifier, a programmable gain 
device, a lowpass filter having a suitable time constant, and a current 
output type voltage amplifier. The output of each position sensor is 
compared with a reference voltage to produce a differential signal on the 
basis of which the voltage amplifier is excited through the preamplifier, 
the programmable gain device and the lowpass filter, and wherein each 
actuator is driven in response to the output current of the voltage 
amplifier. In that occasion, the acceleration feedback circuit may 
comprise an amplifier having a suitable gain, with which the output of the 
acceleration detector is negatively fed back to the input side of the 
lowpass filter. 
If the position control loop gain of the feedback system is increased for 
suppressing disturbance or reducing the positioning time, for example, 
without the acceleration feedback circuit, the position of the base easily 
becomes vibratory and unstable. With the acceleration feedback circuit of 
this preferred form of the present invention, the damping operation is 
strengthened and vibration is suppressed. Thus, no vibration occurs even 
if the loop gain for position control is increased. As a consequence, it 
is possible to increase the loop gain for position control without 
inconveniences, and the positioning operation can be accomplished in a 
reduced time without being disturbed. 
The performance of a fine-motion positioning system may be determined in 
respect to the positioning time and the position precision. When the 
actuators 2M - 2L and the position sensors 3M - 3L are suitably disposed 
with respect to the base 1 and independent feedback systems are added to 
these mechanisms, the improvement of the performance is limited due to the 
interference (interaction) among respective drive axes. Thus, for further 
reduction of positioning time or further enhancement of positioning 
precision, it is necessary to remove such interference. For example, in 
the case of precise positioning with a fine-motion positioning mechanism 
such as shown in FIG. 16, a control device by which the interference 
resulting from spatial disposition of the actuator and the position sensor 
is removed ant only a specified axis is made responsive has been proposed. 
This is discussed in the aforementioned paper, "A six-axes motion control 
method for parallel linkage type fine-motion stage", Tomita et al 
("Journal of Precise Engineering Association", 58/4/1992, pages 684-690). 
Simply stated, the transformation matrix from the displacement by actuator 
drive to the positioning point attitude and the transformation matrix from 
that attitude to the position sensor output are predicted, and respective 
inverse matrix operations are inserted into the pre-stage of the voltage 
amplifier and the post-stage of the position sensor, respectively, to 
thereby provide a closed loop system. Here, this is called a 
non-interacting (non-interfering) control, while the conventional control 
in which no such inverse matrix operation is inserted is called an 
independent control. 
With the non-interacting control, there is an advantage that the static 
interference (interaction) of the mechanism is released and 
non-interaction is accomplished. As proof of an advantageous effect of the 
non-interacting control, in that paper the positioning characteristic is 
pointed out. For example, it reports the results of an experiment that, 
when rotational motion was designated, excitation of another motion mode 
was extraordinarily suppressed. 
However, according to the studies made by the inventors of the subject 
application as to the applicability of that paper to the fine-motion 
positioning mechanism shown in FIG. 16, it has been confirmed that the 
performance cannot be improved constantly. FIGS. 18A-18C illustrate the 
differential signals e.sub.M, e.sub.R and e.sub.L as a voltage was applied 
only to the instruction voltage input terminal 5L. In the case of 
conventional independent control, there appeared differential signals also 
in the axes M and R, other than the axis L to which the instruction 
voltage was applied. However, when the non-interacting control was made, 
differential signals e.sub.M and e.sub.R other than the voltage-applied 
axis L did not appear. Thus, apparently it seemed that the non-interacting 
control assured the operation just intended. Namely, since any response 
other than the designated axis was not mixed into it, it seemed that 
reduction of positioning time and increase of positioning precision could 
be expected. However, the inventors wondered whether the superiority of 
the non-interacting control to the independent control could be held for 
every motion attitude to be designated to the base 1. 
Thus, the inventors made performance comparison to the independent control 
and the non-interacting control while changing the pattern of instruction 
voltage application. FIGS. 19A-19C illustrate differential signals 
e.sub.M, e.sub.R and e.sub.L as instruction voltages corresponding to +5, 
+5 and -5 (.mu.m) was applied stepwise to the voltage input terminals 5M, 
5R and 5L, respectively. In that case, the response is more deteriorated 
with the non-interacting control. Particularly, the differential signal 
e.sub.R is vibratory. Therefore, it can be said that the positioning 
performance of the non-interacting control is not always superior to that 
of the independent control. 
Further, the insertion of inverse matrix operation leads to complicatedness 
of the structure of the control system. Also, it requires some 
identification means for predicting the parameters of an inverse matrix 
operation to be inserted. This necessarily results in the necessity of a 
complicated adjustment for attaining satisfactory performance of the 
control device. In any case, the non-interacting control does not always 
assure improvement of positioning performance, and there is even a 
possibility of a deteriorated response. 
It is accordingly a second object of the present invention to provide a 
fine-motion positioning system of simple structure which assures improved 
non-interacting control and improved positioning performance. 
In accordance with an aspect of the present invention, the fine-motion 
positioning system does not use such complicated non-interacting control 
method that inverse matrix operations of transformation matrix based on 
the spatial disposition of actuators and position sensors are inserted 
into a closed loop. Rather, the fine-motion positioning mechanism itself 
has a statically and also dynamically non-interacting structure. Namely, 
any static and dynamic interference (interaction) within the fine-motion 
positioning mechanism itself is removed, and, to such fine-motion 
positioning mechanism, a feedback system for driving an actuator on the 
basis of the output of a position sensor is added. 
More specifically, referring to FIG. 16, now the mass of the stage base 1 
of the positioning mechanism is denoted by m, and the moments of inertia 
about x and y axes of x-y coordinates having an origin at the center of 
the principal axes of inertia are denoted by Jx and Jy. Actuators 2M - 2L 
are disposed concentrically along a circle of radius 1.sub.d. When the 
actuator 2M of the three actuators is disposed at the position (0, 
1.sub.d) of the x-y coordinates, the remaining two actuators 2R and 2L may 
preferably be disposed in the fourth and third quadrants of the x-y 
coordinates, in terms of balance, with the angles in this disposition 
being denoted by .theta..sub.d. Here, the fine-motion positioning 
mechanism satisfies equations (a) and (b), below: 
EQU 1.sub.d ={(Jx+Jy)/m}.sup.1/2 (a) 
EQU .theta..sub.d =sin.sup.-1 [J.sub.x /(J.sub.x +J.sub.y)] (b) 
To such fine-motion positioning mechanism based on the above equations, a 
feedback system is added to provide a fine-motion positioning system in 
which in the feedback system the outputs of three position sensors 3M, 3R 
and 3L are compared with a reference voltage to provide differential 
signals e.sub.M, e.sub.R and e.sub.L on the basis of which voltage 
amplifiers 8M, 8R and 8L are excited through preamplifiers 6M, 6R and 6L 
and compensators 7M, 7R and 7L to thereby drive the actuators 2M - 2L, 
respectively. 
In terms of dynamics, equations (a) and (b) mean that the driving points of 
actuators provide the "center of percussion" to the base 1. With this 
arrangement, the fine-motion positioning mechanism itself has a statically 
and also dynamically non-interfering structure. To such mechanism, a 
closed loop for driving each actuator on the basis of positional 
information from a corresponding position sensor is defined. Thus, as the 
control loop, three simple single-loops are provided. Also, since the 
fine-motion positioning mechanism itself has a non-interacting structure, 
the performance can be improved easily by increasing the loop gain of the 
control system. As compared therewith, if such three single-loops are 
provided to a conventional fine-motion positioning mechanism which does 
not satisfy the relations of equations (a) and (b), there is a limitation 
to increasing the loop gain due to interference components from the other 
axes. Thus, the positioning performance is limited. 
With the fine-motion positioning system of the present invention, improved 
non-interacting control is assured without the necessity of parameter 
identification or an adjusting operation. Thus, the productibility is 
improved and the cost of the device is reduced.

DESCRIPTION OF THE PREFERRED EMBODIMENTS 
FIG. 1 is a block diagram of a fine-motion positioning system with an 
acceleration feedback function, which system can be incorporated into an 
exposure apparatus for the manufacture of microdevices, for example. In 
this embodiment, as compared with the fine-motion positioning device of 
FIG. 16 comprising a fine-motion positioning mechanism and a feedback 
system, an acceleration feedback loop is added. More specifically, 
acceleration sensors 9M, 9R and 9L are mounted at those positions very 
close to position sensors. The outputs of these acceleration sensors are 
transformed into electric signals by acceleration detectors 10M, 10R and 
10L, respectively, and through lowpass filters 11M, 11R and 11L each 
having a suitable time constant and an amplifying function, they are 
negatively fed back to pre-stages of current output type voltage 
amplifiers 8M, 8R and 8L, respectively. 
Referring to the block diagram of FIG. 5, the principle of acceleration 
feedback for one-freedom control will be explained. FIG. 5 illustrates one 
axis of the fine-motion positioning mechanism. The basic transfer function 
of a piezoelectric actuator is such as expressed by equation (1) below 
and, since the voltage amplifier for piezoelectric device drive is of a 
current output type, it is expressed as an integrator of a gain K.sub.I. 
In equation (1) below, M is the mass, K is the spring constant and D is 
the viscous friction coefficient: 
EQU (Ds+K)/(Ms.sup.2 +Ds+K) =(2.zeta..sub.0 T.sub.0 s+1)/(T.sub.0.sup.2 s.sup.2 
+2.zeta..sub.0 T.sub.0 s+1) (1) 
where 
EQU T.sub.0 =(M/K).sup.1/2 
EQU .zeta..sub.0 =D/2(MK).sup.1/2 
Here, while using a model of one freedom, two types of acceleration 
feedback to the fine-motion positioning mechanism will now be considered. 
Referring first to FIG. 6A, the transfer function from the input voltage v 
of the voltage amplifier to the displacement x can be expressed by 
equation (2) below: 
##EQU1## 
Here, if the time constant T is selected to satisfy T =2.zeta..sub.0 
T.sub.0, then equation (3) below is obtained: 
EQU X/V=K.sub.I (1+2.zeta..sub.0 T.sub.0 X)/[s{1+(2.zeta..sub.0 T.sub.0 
+K.sub.A K.sub.I)s+T.sub.0.sup.2 s.sup.2 }] (3) 
It is seen from equation (3) that, as a result of application of damping 
owing to the acceleration feedback, the root of the fine-motion 
positioning mechanism is set in the innermost part of the left half of the 
complex plane. Therefore, it is expected that the loop gain can be 
increased if a closed loop is defined. 
Next, referring to FIG. 6B, the transfer function from the input voltage v 
of the voltage amplifier to the displacement x is expressed as equation 
(4) below: 
##EQU2## 
Similarly to the derivation of equation (3), if the time constant T is 
selected to satisfy T=2.zeta..sub.0 T.sub.0, then equation (5) is 
obtained: 
EQU X/V=K.sub.I /[s{1+(2.zeta..sub.0 T.sub.0 +K.sub.A K.sub.I) s+T.sub.0.sup.2 
s.sup.2 }] (5) 
Similarly, in the case of equation (3), as result of application of damping 
owing to the acceleration feedback, the tool of the fine-motion 
positioning mechanism is set in the innermost part of the left half of the 
complex plane. Additionally, in this case, a transfer function without 
zero point is provided. 
Next, referring to FIGS. 7A-7C, description will be made to that, in a case 
where a closed loop of position is defined in a case where acceleration 
feedback is not provided, and in a case where acceleration feedback is 
provided, the loop gain can be increased as compared with the conventional 
control system. FIGS. 7A-7C correspond to the cases where acceleration 
feedback is provided in the cases of FIGS. 6A and 6B and the case of FIG. 
5. The results of stabilization analysis in the cases of FIGS. 7A-7C are 
such as expressed in equations (6), (7) and (8), where K.sub.loop 
=K.sub.loop 'K.sub.I. 
##EQU3## 
Here, while using the parameters of Table 1, below, that or those of the 
control structure of FIG. 7 whose loop gain can be easily increased are 
specified. Namely, each one whose basic transfer function has a complex 
root corresponds to this, and magnitude relation of equations (6)-(8) is 
checked. 
TABLE 1 
______________________________________ 
SYMBOL UNIT NUMERIC VALUE 
______________________________________ 
M Kg 10.5 
D Nsec/m 1.282 .times. 10.sup.3 
K N/m 1.723 .times. 10.sup.6 
K.sub.A,K.sub.I 
sec 1.0 .times. 10.sup.-4 
______________________________________ 
FIGS. 8A-8C illustrate the results in the cases of FIGS. 7A-7C, and they 
depict the position (mark x) of The root upon the complex plane as the 
K.sub.loop is increased from zero, at a rate of 10. Arrows in the drawings 
denote the direction of movement of the root due to the increase of 
K.sub.loop. The ranking of liability of unstableness with the increase of 
K.sub.loop is the FIG. 8C case, the FIG. 8B case and the FIG. 8A case, in 
that order. That is, the acceleration feedback such as of FIG. 6A or 6B is 
used to provide a closed loop of position, the loop gain of position can 
be enlarged significantly as compared with a case where the feedback is 
not provided. 
In the system of FIG. 1, of the two types of acceleration feedback systems 
shown in FIGS. 6A and 6B, the acceleration feedback of FIG. 6A is applied 
to a three-freedom fine-motion positioning mechanism. As shown in FIG. 1, 
acceleration sensors 9M, 9R and 9L are disposed at the same locations as 
displacement sensors 3M, 3R and 3L, respectively, and the outputs of them 
are negatively fed back to the pro-stage of current output type voltage 
amplifiers 8M, 8R and 8L, respectively. Here, the outputs of the 
acceleration sensors 9M - 9L are received by primary lowpass filters 11M, 
11R and 11L, respectively, each having a time constant effective to cancel 
the zero point determined by Ds+K in equation (1), whereby a suitable gain 
is imparted. In place of using physical parameters in Table 1, numerical 
experiments were carried out while using parameters of the three-freedom 
fine-motion positioning mechanism itself. 
FIGS. 9A-9I illustrate the results of an examination which was made on 
differences in step response caused with and without acceleration 
feedback, under the condition of the same loop gain of the position 
control system. FIGS. 9A-9C correspond to the cases of z-axis 
translational motion. FIGS. 9D-9F correspond to the cases of rotational 
motion about x axis, and FIGS. 9G-9I correspond to the cases of rotational 
motion about y axis. In the order from the left-hand sides of the figures, 
changes in voltage of the differential signals e.sub.M, e.sub.R and 
e.sub.L with time are illustrated. Reference character FB in the drawing 
denotes feedback. It is seen from the drawings that, with the acceleration 
feedback, in the response just after step application the differential 
signal somewhat expands as compared to that without acceleration feedback. 
However, in a portion as the difference becomes equal to zero with the 
lapse of time, the stabilization (settling) is better in the case where 
acceleration feedback is done, and reduction of positioning time is 
attained. 
FIGS. 10A-10F illustrate the results of a comparison which was made in 
respect to changes of step response, between cases with and without 
acceleration feedback, as the loop gain of position was increased 
successively. FIGS. 10A-10C correspond to the cases without acceleration 
feedback, and FIGS. 10D-10F correspond to the cases having acceleration 
feedback. In all the cases as illustrated, stepwise translational motion 
is applied in the direction of the z-axis. Like the examples in FIGS. 
9A-9I, in the order from the left-hand side of the figures, changes in 
voltage of the differential signals e.sub.M, e.sub.R and e.sub.L with time 
are illustrated. Numerical values designated in the drawings correspond to 
the set values of the programmable gain devices 7M, 7R and 7L. The loop 
gain was changed by controlling those gains. It is seen from the drawings 
that in the case without acceleration feedback the system easily becomes 
vibratory as the loop gain of position is enlarged. On the other hand, in 
the case with acceleration feedback, it does not become unstable even with 
similar increase of position loop gain. Thus, it is possible to increase 
the loop gain of position loop largely in the case where acceleration 
feedback is done. With this structure, therefore, the settling time for 
positioning can be made shorter and, additionally, the control system 
itself can be less sensitive to disturbance. 
From FIGS. 9A-9I and 10A-10F, it is confirmed that the acceleration 
feedback assures reduction of settling time and increase of loop gain. 
Now, the time constant of the lowpass filter of the acceleration feedback 
loop is considered. In derivation of equation (3), the time constant is 
set to satisfy T=2.zeta..sub.0 T.sub.0. This is at the same position as 
the zero point of the piezoelectric driving mechanism as determined by 
Ds+K and, actually, there may be some unreliability of setting resulting 
from an identification error. In consideration of thereof, the effect of 
any deviation of the time constant T of the lowpass filter was examined. 
FIGS. 11A-11C illustrate in the order from the left-hand side the voltages 
of differential signals e.sub.M, e.sub.R and e.sub.L, respectively, during 
z-axis translational motion as changes of increase/decrease by one digit 
were imparted to the center value T=7.4405.times.10.sup.-4 (sec) of the 
time constant of each of the lowpass filters 11M - 11L. Even with an 
increase of time constant T by one digit, the stabilization of vibration 
is good as compared with a case without acceleration feedback. It is seen 
therefore that the time constant setting of the filter does not have a 
large effect upon the performance. Of course, a time constant set in a 
high range provides a good result in respect to the stableness and 
performance of the control system. 
Next, limitation of input which might result from the execution of 
acceleration feedback is considered. Since the acceleration feedback loop 
operates only in response to occurrence of acceleration, it does not 
relate to steady characteristics. Namely, as readily understood from 
equation (3), no change occurs in the direct current term. Therefore, the 
addition of acceleration feedback will not cause some input limitation 
such as saturation, for example. FIGS. 12A-12C illustrate the results of 
numerical experiments which were made for confirmation, and it depicts 
changes in input voltage of the voltage amplifiers 8M, 8R and 8L in 
response to increase of gain of the acceleration feedback loop. More 
specifically, it illustrates, in the order from the left-hand side, the 
changes with time of the input voltages of the voltage amplifiers 8M - 8L 
caused in response to application of stepwise translational motion, 
equivalent to 5 (.mu.m), in the z-axis direction. It is seen from the 
drawing that the execution of acceleration feedback does not cause 
application of excessive drive to the voltage amplifiers 8M - 8L. Thus, no 
input limitation is caused. 
Next, the physical dimension and sensitivity of acceleration sensor are 
considered. As in the numerical experiments, the control system of the 
present invention uses high-sensitivity sensors to control minute 
acceleration. Also, such a sensor should be small since it is to be 
mounted on the base 1. Preferable examples of such sensor may be a 
piezoelectric resistance type acceleration sensor based on silicon 
micro-machining technique and an acceleration sensor which uses ceramic 
series bimorph beam, each being small in size and high in sensitivity. For 
example, an acceleration sensor of latter type may have a sensitivity 1000 
(mV/g), a resolution 0.0005 (gpK), a frequency response 1-2000 (Hz) and a 
size of 16 mm (diameter) and 13 mm (height). This is sufficient for 
accomplishing acceleration feedback to be added to the fine-motion 
positioning mechanism. 
While in the preceding embodiment the invention has been described with 
reference to the addition of acceleration feedback to a fine-motion 
positioning mechanism including three actuators and having a function for 
controlling three freedoms (one freedom of translation and two freedoms of 
rotation), the invention is not limited to such three-freedom fine-motion 
positioning mechanism. The invention is applicable to a fine-motion 
positioning mechanism of a larger number of freedoms (e.g. six freedoms). 
Also, in the preceding embodiment, the acceleration sensors 9M - 9L are 
disposed at the same locations as the position sensors 3M - 3L, 
respectively. However, it is well known that a sensor when disposed at the 
same location as the actuator driving point (called "co-location") 
provides good controllability, and the acceleration sensors 9M - 9L may of 
course be disposed at the same locations as the driving points of the 
actuators 2M - 2L, respectively. Since a sensor has a finite physical 
dimension and if non-contact position measurement is to be done, it is 
difficult to place the position sensors 3M - 3L exactly at the same 
positions as the actuators 2M - 2L, respectively. However, as regards the 
acceleration sensors 9M - 9L, since they may be adhered to respective 
positions of driving points, the condition of co-location can be easily 
accomplished. 
While in the three-freedom fine-motion positioning mechanisms of FIG. 1, 
the acceleration feedback of FIG. 6A is added with respect to each axis, 
as an alternative the acceleration feedback of FIG. 6B may be applied to 
the three-freedom fine-motion positioning mechanism. In FIG. 2, primary 
lowpass filters 12M, 12R and 12L each having a suitable time constant are 
inserted into the pre-stage of current output type voltage amplifiers 8M, 
8R and 8L. The output of acceleration sensors 9M, 9R and 9L are 
transformed into electric signals by acceleration detectors 10M, 10R and 
10L and, thereafter, while being passed through amplifiers 13M, 13R and 
13L each having a suitable amplification rate, they are negatively fed 
back to the pro-stage of the lowpass filters 12M - 12L. 
In accordance with this aspect of the present invention, insufficient 
damping of a fine-motion positioning mechanism which uses piezoelectric 
devices, for example, can be compensated by the execution of acceleration 
feedback. Therefore, there is an advantage of stableness of positioning 
operation. Also, since the mechanism is stabilized with the acceleration 
feedback, in a closed loop of position it is possible to enlarge its loop 
gain as compared with a case without acceleration feedback. Thus, it is 
possible to provide a fine-motion positioning system which is less 
sensitive to disturbance and which assures quick positioning operation. 
FIG. 13 is a block diagram of a fine-motion positioning system according to 
another embodiment of the present invention. In this embodiment, the 
disposition of actuators ant sensors, constituting the fine-motion 
positioning mechanism, is optimized in accordance with equations (3) and 
(4) set forth hereinbefore. 
Derivation of these equations will be explained. FIG. 14 illustrates 
coordinates of a stage base 10 as seen from the above, on which a 
semiconductor wafer 100 is placed. Denoted in the drawing by painted dots 
2M, 2R and 2L are actuators which are disposed at the coordinates 
illustrated. The center of coordinates coincides with the center of 
inertia, and the x-y coordinates are so set as illustrated. Here, the 
kinetic equation is expressed as equation (9) below: 
EQU MX+DX+KK=KJ.sub.xd AU+DJ.sub.xd AU (9) 
where used characters denote the following: 
X=[z, .theta..sub.x, .theta..sub.y ].sup.T : displacement vector of 
principal axes of inertia 
z [m]: z-axis displacement of principal axis of inertia of base 1 
.theta..sub.x [rad]: angle of rotation of base 1 about x axis 
.theta..sub.y [rad]: angle of rotation of base 1 about y axis 
M=diag (m, Jx, Jy): inertia matrix 
m [Kg]: mass of the base 1 
Jx [Kgm.sup.2 ]: moment of inertia of base 1 about x axis 
Jy [Kgm.sup.2 ]: moment of inertia of base 1 about y axis 
[Z.sub.dM, Z.sub.dR, Z.sub.dL ].sup.T [m]: z-axis displacement of actuators 
K [N/m]: spring constant of 2M, 2R and 2L 
d [Nsec/m]: viscous friction coefficient of 2M, 2R and 2L 
A=diag (a.sub.M, a.sub.R, a.sub.L) [m/V]: voltage-to-displacement 
conversion coefficient 
U=[u.sub.M, u.sub.R, u.sub.L ].sup.T [V]: applied voltage vector to 
piezoelectric device 
.theta..sub.d [rad]: angle of disposition of actuator 
1.sub.d ]m]: radius 
superscript T: transposed matrix 
(.): differential with time 
s: Laplace operator 
J.sub.xd : transformation matrix from actuator drive displacement 
[Z.sub.dM, Z.sub.dR, Z.sub.dL ] to displacement X, being expressed by 
equation (10), below: 
##EQU4## 
D: attenuation coefficient matrix as expressed by equation (11), below: 
##EQU5## 
K: rigidity coefficient matrix as expressed by equation (12), below: 
##EQU6## 
The relation from applied voltage vector U to displacement vector X is 
expressed by equation (13), below: 
EQU X=(Ms.sup.2 +Ds+K).sup.-1 (Ds+K)J.sub.xd AU (13) 
In this equation, the portion representing the relation from U to X 
corresponds to the transfer function matrix G(s) of the fine-motion 
positioning mechanism. Here, the elements are denoted by characters in 
equation (14), below: 
##EQU7## 
Here, the polynomials that provide zero points of G.sub.31 (s), G.sub.12 
(s) and G.sub.32 (s) are such as equations (15)-(17), below: 
EQU (ds+k)(Jx-ml.sub.d.sup.2 sin .theta..sub.d)s.sup.2 (15) 
EQU (ds+k)(Jx-ml.sub.d.sup.2 sin .theta..sub.d)s.sup.2 (Jys.sup.2 
+2dl.sub.d.sup.2 cos.sup.2 .theta..sub.ds +2kl.sub.d.sup.2 cos.sup.2 
.theta..sub.d) (16) 
##EQU8## 
Thus, in these polynomials the condition for making all the coefficients of 
s equal to zero, can be easily determined, such as expressed in equations 
(18) and (19), below: 
EQU Jx+Jy=ml.sub.d.sup.2 (18) (18) 
EQU sin .theta..sub.d =Jx/(Jx+Jy) (19) 
Namely, when equations (18) and (19) are satisfied, the transfer functions 
of non-diagonal term in equation (14) all become equal to zero, and only 
diagonal components G.sub.11 (s), G.sub.22 (s) and G.sub.33 (s) remain as 
non-zero. This means that the fine-motion positioning mechanism is made 
non-interacting, statically and dynamically. 
While equations (18) and (19) define conditions for making, equal to zero 
at once, the coefficients of s of the polynomials that provide the zero 
points of G.sub.31 (s), G.sub.12 (s) and G.sub.32 (s), simultaneously they 
also define the conditions for making, equal to zero, the coefficients of 
s of the polynomials that provide zero points of G.sub.13 (s), G.sub.21 
(s) and G.sub.23 (s). Therefore, in accordance with equations (18) and 
(19), all the non-diagonal terms of equation (14) become equal to zero. 
It is to be noted that equation (a) corresponds to the result of solving 
equation (18) in respect to 1.sub.d, while taking the mass m of the base 
1, the inertia moment Jx about the x axis and the inertia moment Jy about 
the y axis as being unchangeable (i.e. predetermined). Equation (b) 
corresponds to the result of solving equation (19) in respect to 
.theta..sub.d. Namely, since modifying the design in terms of m, Jx and Jy 
largely is not practical, solutions are detected in terms of 1.sub.d and 
.theta..sub.d. Of course, any combination of m, Jx and Jy satisfying 
equations (18) and (19) may be adopted while maintaining 1.sub.d and 
.theta..sub.d unchanged. In any case, by designing the fine-motion 
positioning mechanism so as to satisfy equations (18) and (19), 
interacting (interfering) components from those other than the designated 
drive axis can be avoided. 
Now, the effectiveness of the present invention will be described on the 
basis of comparison of the step response of the fine-motion positioning 
system of the present invention with a fine-motion positioning system of 
independent control type. FIGS. 15A-15C illustrate response waveforms of 
differential signals e.sub.M, e.sub.R and e.sub.L produced as a voltage 
equivalent to +5 (.mu.m) is applied only to the voltage input terminal 5L. 
With the non-interacting control of the present invention, the response of 
the differences e.sub.R and e.sub.L is completely zero. Thus, the 
advantageous effect of the present invention is clear. Of course, even if 
any instruction is applied to the voltage input terminal 5M, 5R or 5L, the 
non-interacting control operates correctly. As compared therewith, in the 
conventional non-interacting control which is static, depending on the 
pattern of voltage application to the terminals 5M - 5L, such as shown in 
FIG. 19, there may be cases wherein the performance is lower than that of 
the independent control. This is not the case with the present invention. 
While in the foregoing the invention has been described with reference to a 
fine-motion positioning system wherein three actuators 2M - 2L are 
disposed on the same plane and wherein non-interacting control is executed 
to its fine-motion positioning mechanism for controlling three freedoms 
(one freedom of translation and two freedoms of rotation) through vertical 
z-axis displacement of the actuators, the invention is not limited to such 
three-freedom fine-motion positioning mechanism. It is applicable to a 
fine-motion positioning mechanism of a larger number of freedoms. This 
will be readily understood, since equations (14) -(17) are the conditions 
for mechanism parameters, making the coefficients of polynomials, 
providing zero points, all equal to zero at once. In terms of dynamics, 
equations (18) and (19) mean that each driving point is at the "center of 
percussion". Thus, it is within the present invention that: a fine-motion 
positioning system which includes, with respect to a rigid member, 
actuators of a number at least corresponding to kinetic freedoms to be 
controlled, position sensors of a number corresponding to the kinetic 
freedoms, and a feedback system for driving a corresponding actuator on 
the basis of feedback of corresponding position sensor, wherein the 
driving points of the actuators are located at the center of percussion. 
In accordance with the embodiment of the present invention described 
hereinbefore, the fine-motion positioning mechanism itself has a 
non-interacting structure, avoiding static and dynamic interaction 
(interference). To such mechanism, a closed loop for controlling the 
actuators on the basis of positional information from the position sensors 
is added. Thus, the control loop can be provided by simple three single 
loops. This makes the structure very simple. Also, since the fine-motion 
positioning mechanism itself has non-interacting structure, increasing the 
loop gain of the control system to improve the performance is very easy. 
Further, parameter identification or adjustment operation is not 
necessary. Therefore, the productibility is high and cost of the system is 
low. 
Now, description will be made on the manufacture of semiconductor devices 
by using an exposure apparatus into which a fine-motion positioning system 
of the present invention is incorporated. 
FIG. 3 is a flow chart of the sequence of manufacturing a semiconductor 
device such as a semiconductor chip (e.g. IC or LSI), a liquid crystal 
panel or a CCD, for example. Step 1 is a design process for designing the 
circuit of a semiconductor device. Step 2 is a process for manufacturing a 
mask on the basis of the circuit pattern design. Step 3 is a process for 
manufacturing a wafer by using a material such as silicon. 
Step 4 is a wafer process which is called a pre-process wherein, by using 
the so prepared mask ant wafer, circuits are practically formed on the 
wafer through lithography. Step 5 subsequent to this is an assembling step 
which is called a post-process wherein the wafer processed by step 4 is 
formed into semiconductor chips. This step includes assembling (dicing and 
bonding) and packaging (chip sealing). Step 6 is an inspection step 
wherein operability check, durability check and so on of the semiconductor 
devices produced by step 5 are carried out. With these processes, 
semiconductor devices are finished and they are shipped (step 7). 
FIG. 4 is a flow chart showing details of the wafer process. Step 11 is an 
oxidation process for oxidizing the surface of a wafer. Step 12 is a CVD 
process for forming an insulating film on the wafer surface. Step 13 is an 
electrode forming process for forming electrodes on the wafer by vapor 
deposition. Step 14 is an ion implanting process for implanting ions to 
the wafer. Step 15 is a resist process for applying a resist 
(photosensitive material) to the wafer. Step 16 is an exposure process for 
printing, by exposure, the circuit pattern of the mask on the wafer 
through the exposure apparatus described above. Step 17 is a developing 
process for developing the exposed wafer. Step 18 is an etching process 
for removing portions other than the developed resist image. Step 19 is a 
resist separation process for separating the resist material remaining on 
the wafer after being subjected to the etching process. By repeating these 
processes, circuit patterns are superposedly formed on the wafer. 
While the invention has been described with reference to the structures 
disclosed herein, it is not confined to the details set forth and this 
application is intended to cover such modifications or changes as may come 
within the purposes of the improvements or the scope of the following 
claims.