Scale for sensing moving object, and apparatus for sensing moving object using same

Provided is an apparatus for accurately sensing the position of a moving object in two dimensions as well as a change in the attitude of the object when the object is moving. The apparatus includes a scale and at least one angle sensor. The scale is constituted by an angular grid, which is formed on the surface of a scale substrate inclusive of a planar surface and freely curved surface thereof, and which has an angle-related property that varies in two different directions (the x and y directions) in the form of a known function. The angle sensor is arranged so as to confront the angular grid surface of the scale. Either the scale or the angle sensor is mounted on a moving object and the position of the moving object in two-dimensional coordinates is sensed during relative movement between the scale and the angle sensor.

BACKGROUND OF THE INVENTION
 1. Field of the Invention
 This invention relates to a scale ideal for sensing the position and
 attitude of a moving object, as well as an apparatus for sensing a moving
 object by making use of this scale. More particularly, the invention
 relates to a sensing apparatus in which a scale is constituted by an
 angular grid which varies in two different directions in accordance with a
 known function, the angular grid being combined with an angle sensor,
 which moves relative to the surface of the angular grid, to thereby enable
 precise detection of the position and attitude of a moving object.
 2. Description of the Related Art
 When the position of a moving object such an XY table or cutter is sensed
 in a machine tool or the like, a measurement device such as a rotary
 encoder or linear encoder is necessary for each degree of freedom. When
 two-dimensional positioning is performed, for example, stages capable of
 being positioned along respective ones of x and y axes are stacked one
 above the other and a measurement device is provided for each stage to
 achieve positioning. Alternatively, a measurement device comprising a
 combination of a circumferential scale and a single-axis stage is used to
 measure rotational position and radial position independently to effect
 positioning.
 In a situation where position is determined in x and y directions using a
 laser interference-type displacement meter, position is sensed by a
 combination of two displacement meters and a device such as a highly
 precise straight edge the accuracy of the shape whereof is assured over a
 range of movement at right angles to the direction in which displacement
 is sensed.
 Further, in cases where attitude corresponding to pitching and yawing of a
 moving object is sensed in the prior art, use is made of an
 autocollimator. Though an autocollimator is capable of measuring pitching
 and yawing simultaneously with respect to linear movement along one axis,
 a high-precision straight edge is needed for detection of an object moving
 along the two x and y axes.
 Furthermore, though a leveling instrument is known as means for measuring
 the rolling of a moving object, problems arise in terms of speed of
 response and measurement accuracy and therefore a leveling instrument is
 not suitable for use as highly precise measurement equipment.
 Accordingly, the state of the art is such that two parallel straight edges
 are deployed and rolling angle is calculated from the difference between
 distances measured up to the straight edges, or such that rolling angle is
 sensed by an autocollimator using a single straight edge as a reference
 mirror surface.
 However, a measurement device such as the rotary encoder or linear encoder
 used in the conventional sensing apparatus described above is capable of
 performing positioning in one dimension only. At least two sets of the
 above-mentioned measurement apparatus are required to be combined for
 two-dimensional positioning, a fact that represents a major limitation
 when designing an apparatus for sensing moving objects.
 Further, in a situation where positioning is performed using the laser
 interference-type displacement meter, essentially only one-dimensional
 positioning is possible. If positioning in two dimensions is to be carried
 out, the highly precise straight edge is required. As a result, in a
 scenario where a machine tool or the like is provided with this type of
 apparatus for sensing moving objects, problems arise in terms of
 manufacturing limitations and higher cost.
 With the prior-art sensing apparatus, the encoder which determines position
 and the measurement circuitry which senses a change in attitude are
 separately arranged. Consequently, a sensing apparatus capable of sensing
 the two-dimensional position, pitching, rolling and yawing of a moving
 object would be complex and high in cost.
 With means such a photoelectric linear scale, a high degree of positional
 accuracy is required to correctly place the scale and the device that
 reads it. This makes it difficult to deploy a plurality of reading devices
 in an effort to enlarge the range of measurement beyond the length of the
 scale. The end result is that a long scale is required in order to sense
 movement along a long range of movement.
 SUMMARY OF THE INVENTION
 The present invention has been devised in view of the foregoing
 circumstances and its object is to provide a scale for sensing a moving
 object as well as an apparatus for sensing a moving object by using the
 scale, wherein the scale and apparatus make it possible to sense, in
 highly precise fashion, the position of a moving object in two dimensions
 as well as a change in the attitude of the object when the object moves.
 According to the present invention, the foregoing object is attained by
 providing a scale for sensing at least one of position and various
 attitudes of a moving object, the scale being formed from an angular grid,
 which is formed on or in a surface of a scale substrate inclusive of a
 planar surface and freely curved surface thereof, and which has an
 angle-related property that varies in two different directions in the form
 of known function.
 The angular grid comprises a multiplicity of peaks and valleys of a fixed
 amplitude wherein the angle-related property varies sinusoidally in two
 intersecting directions on or in the surface of the substrate.
 The angular grid is so adapted as to apply electromagnetic power to an
 electro-optic crystal or a liquid that fills the interior of a vessel and
 reacts to electromagnetic force or light, thereby subjecting the
 electro-optic crystal or liquid to a change in refractive index, wherein
 the change is in the form of a known function.
 The angular grid constructs orthogonal coordinates, cylindrical
 coordinates, polar coordinates or coordinates along a freely curved
 surface.
 According to the present invention, the foregoing object is attained by
 providing a sensing apparatus for sensing position of a moving object,
 comprising a scale constituted by an angular grid, which is formed on or
 in a surface of a scale substrate inclusive of a planar surface and freely
 curved surface thereof, and which has an angle-related property that
 varies in two different directions (x and y directions) in the form of a
 known function, and at least one two-dimensional angle sensor disposed to
 confront the angle-grid side of the scale, one of the scale and angle
 sensor being attached to a moving object and the position of the moving
 object in two-dimensional coordinates being detected in relative movement
 between the scale and the angle sensor.
 In another aspect of the present invention, the foregoing object is
 attained by providing a sensing apparatus for sensing position and various
 attitudes of a moving object, comprising a scale constituted by an angular
 grid, which is formed on or in a surface of a scale substrate inclusive of
 a planar surface and freely curved surface thereof, and which has an
 angle-related property that varies in two different directions (x and y
 directions) in the form of a known function, and at least one pair of
 two-dimensional angle sensors disposed to confront the angle-grid side of
 the scale and spaced apart from each other by prescribed distances along
 the x and y directions, one of the scale and angle sensors being attached
 to a moving object and the position of the moving object in
 two-dimensional coordinates as well as pitching and rolling angle of the
 moving object being detected in relative movement between the scale and
 the angle sensor.
 In another aspect of the present invention, the foregoing object is
 attained by providing a sensing apparatus for sensing position and various
 attitudes of a moving object, comprising a scale constituted by an angular
 grid, which is formed on or in a surface of a scale substrate inclusive of
 a planar surface and freely curved surface thereof, and which has an
 angle-related property that varies in two different directions (x and y
 directions) in the form of a known function, and at least three
 two-dimensional angle sensors disposed to oppose the angle-grid side of
 the scale and spaced apart from each other prescribed distances along the
 x and y directions, one of the scale and angle sensors being attached to a
 moving object and the position of the moving object in two-dimensional
 coordinates as well as pitching, rolling angle and yawing angle of the
 moving object being detected in relative movement between the scale and
 the angle sensor.
 According to another aspect of the present invention, the foregoing object
 is attained by providing a sensing apparatus for sensing one of position
 or various attitudes of a moving object, comprising a scale constituted by
 an angular grid, which is formed on or in a surface of a scale substrate
 inclusive of a planar surface and freely curved surface thereof, and which
 has an angle-related property that varies along one axial direction (the x
 direction) in the form of a known function, and an angle sensor disposed
 to confront the angle-grid side of the scale, one of the scale and angle
 sensor being attached to a moving object and the position of the moving
 object along the one axial direction being detected in relative movement
 between the scale and the angle sensor.
 According to the present invention, the position of a moving object along
 one axial direction (direction of movement) and pitching angle of the
 moving object are sensed by a two-point method, which relates to an
 angularly shaped function, from angle of inclination along the one axial
 direction of the above-described angular grid sensed by a pair of angle
 sensors arranged with a prescribed distance between them along the
 one-axial direction of the angular grid.
 According to the present invention, the above-described angle sensor
 comprises a two-dimensional angle sensor for sensing a variation along one
 axial direction (direction of movement) and a variation along a direction
 at right angles to the direction of movement, wherein position along the
 one axial direction as well as pitching angle and rolling angle is sensed
 by this two-dimensional angle sensor.
 According to the present invention, angular variation of the angular grid
 is implemented in a form obtained by superimposing a plurality of sine
 waves having different frequencies.
 According to the present invention, the angle sensor comprises a plurality
 of displacement meters arrayed with a prescribed spacing among them,
 wherein the displacement meters are of optical type, of a type which
 senses an electro-optic quantity or of mechanical-contact type, a
 differential output from two mutually adjacent displacement meters serving
 as the output of the angle sensor.
 According to the present invention, the angle sensor is made to function as
 a distance sensor by applying a rotational angle of a known direction and
 known magnitude to the angle sensor, and distance between the angular grid
 surface of the scale and the angle sensor, or amount of change in the
 distance, is capable of being sensed in relative movement between the
 angular grid and the angle sensor.
 According to the present invention, the scale is constructed by causing an
 angular grid surface, whose angle-related property varies in the form of a
 well-known function, to be produced by standing waves obtained when
 periodic oscillation is applied to a resilient plate, a planar surface or
 curved surface having a resilient property, a crystal body, a liquid
 surface or a liquid filling a hermetically sealed vessel, wherein the
 angular grid is produced on or in the surface.
 According to the present invention, the scale comprises a plurality of
 divided scales each having an angular grid surface, and a plurality of the
 divided scales are arrayed intermittently or continuously in conformity
 with an area over which the moving object moves.
 According to the present invention, traveling waves are generated, an
 angular grid, in the surface of which an angular change is produced by the
 traveling waves, is formed and position in two dimensions is determined
 based upon a relationship between the angular grid and time.
 According to the present invention, the sensing apparatus further comprises
 means for correcting, based upon results of calibrating an error in the
 angular shape of the angular grid, results of measuring coordinate
 position and attitude angle by the angular grid.
 According to the present invention, the sensing apparatus further comprises
 means for applying a fixed amount of relative motion to the angle sensor
 along the x and y directions of the angular grid, and calculating data for
 calibrating error from the known ideal shape of the angular grid based
 upon each detection value from the angle sensor before and after relative
 movement and the difference between the values, and storage means for
 storing the calibration data calculated.
 In the present invention constructed as set forth above, the scale is
 formed from a two-dimensional angular grid representing an angular shape.
 As a result, the two-dimensional position of a moving object can be sensed
 as a matter of course, and so can the pitching angle, rolling angle and
 yawing angle of the moving object, merely by combining angle sensors with
 a simple scale. In addition, by adopting the angular grid as the scale, it
 is possible to sense position relating to two-dimensional coordinates such
 as orthogonal coordinates, cylindrical coordinates, polar coordinates or
 coordinates along a freely curved surface.
 In the apparatus for sensing a moving object according to the present
 invention, combining at least one two-dimensional angle sensor with a
 scale comprising a two-dimensional angular grid makes it possible to sense
 the position of a moving object in two-dimensional coordinates in relative
 movement between the scale and the angle sensor. By subjecting the angle
 sensor to a known prescribed change in angle, the distance between the
 scale and angle sensor can also be sensed.
 In the apparatus for sensing a moving object according to the present
 invention, combining at least one two-dimensional angle sensors with a
 scale comprising a two-dimensional angular grid makes it possible to sense
 the position of a moving object in two-dimensional coordinates, as well as
 the pitching and rolling angles of the moving object, in relative movement
 between the scale and the angle sensors. By subjecting the angle sensor to
 a known prescribed change in angle, the distance between the scale and
 angle sensors can also be sensed.
 In the apparatus for sensing a moving object according to the present
 invention, combining at least three two-dimensional angle sensors with a
 scale comprising a two-dimensional angular grid makes it possible to sense
 the position of a moving object in two-dimensional coordinates, as well as
 the pitching, rolling and yawing angles of the moving object, in relative
 movement between the scale and the angle sensors. In addition, if the
 angle sensors are subjected to a known change in attitude, such as a
 change in pitching angle or rolling angle, the distance between the scale
 and angle sensors can also be sensed at the same time.
 In the apparatus for sensing a moving object according to the present
 invention, position along one axis, pitching angle and rolling angle can
 be sensed by combining at least one pair of angle sensors with a scale
 constructed from an angular grid which varies along one axis (the x axis)
 in the form of a known function.
 In the present invention, an arrangement is adopted in which the angular
 variation of the angular grid has a form obtained by superimposing a
 plurality of sine waves having different frequencies. As a result, when,
 by way of example, a sinusoidal angular variation one period of which is
 the full length of the angular grid surface in the x direction and the
 angular change of a sine wave having a frequency which is M times that of
 the first-mentioned angular change are superposed to form one angular
 grid, the output of an angle sensor at a certain position will include two
 frequency components of the angular grid surface if a constant oscillation
 is applied to this angle sensor in the x direction at an amplitude larger
 than the period of a high frequency. Since the low-frequency component
 gives an angular grid component in which the full length is one period,
 the position of the angle sensor with respect to the full length is sensed
 from this angular grid component. Since the high-frequency component gives
 an angular grid component of a high frequency, it is possible to sense
 position precisely from this angular grid component. It is possible to
 select a component in which the angular shape varies linearly or a
 component in which the differential of the angular shape varies linearly.
 Furthermore, if the origin is provided on the angular grid surface,
 movement of the angle sensor after restoration to the origin will give the
 absolute coordinates from the origin.
 Further, in the present invention, the angle sensor comprises a plurality
 of displacement meters arrayed with a prescribed spacing among them,
 wherein the displacement meters are of optical type, of a type which
 senses an electro-optic quantity or of mechanical-contact type. By sensing
 a change in the angle of inclination of the shape of the angular grid
 surface, which change has been applied by a differential output from the
 displacement meters in the form of a change in height and shape, this
 sensed change can be utilized instead of angle information. If two
 displacement meters arrayed in each of the x and y directions, for a total
 of four displacement meters, or three displacement vectors arrayed at the
 apices of a triangle are arranged with prescribed distances between them,
 the displacement meters will function as a two-dimensional angle sensor.
 Further, in the present invention, the scale is capable of forming an
 angular grid surface, whose angle-related property varies spatially, by
 standing waves obtained when periodic oscillation is applied to a
 resilient plate, a planar surface or curved surface having a resilient
 property, a crystal body, a liquid surface of a liquid filling a
 hermetically sealed vessel, wherein the angular grid is produced on or in
 the surface. The scale can be utilized as the angular grid surface only
 while the oscillation is being applied.
 Further, in the present invention, in accordance with the present
 invention, the scale is constructed from a plurality of divided scales
 each having an angular grid surface, wherein a plurality of the divided
 scales or angle sensors which read the scale are arrayed intermittently or
 continuously in conformity with the area over which the moving object
 moves. As a result, even if an angle sensor departs, in relative terms,
 from one angular grid surface, information indicative of the position of
 the angular grid surface can be sensed by the adjacent divided scale or
 angle sensor. This makes it possible to enlarge the range of relative
 movement between the angle sensor and the angular grid.
 Further, in the present invention, correction means is provided for
 correcting, based upon results of calibrating and error in the angular
 shape of the angular grid, results of measuring coordinate position and
 attitude angles by the angular grid. Accordingly, in situations where the
 angular grid cannot be fabricated to a high precision, the calibration
 data is stored in memory beforehand and data between known items of data
 is approximated by interpolation, thereby making it possible to correct
 measurement data based upon results of calibration.
 Further, in the present invention, a fixed amount of known relative motion
 is applied to an angle sensor in the x and y directions of the angular
 grid, and data for calibrating deviation from a known ideal shape of the
 angular grid, based upon each detection value from the angle sensor before
 and after relative movement and the difference between these values, can
 be obtained autonomously.
 Other features and advantages of the present invention will be apparent
 from the following description taken in conjunction with the accompanying
 drawings, in which like reference characters designate the same or similar
 parts throughout the figures thereof.

DESCRIPTION OF THE PREFERRED EMBODIMENTS
 Preferred embodiments of the present invention will now be described in
 detail with reference to the accompanying drawings.
 FIG. 1 is a fundamental structural view illustrating dispositional
 relationship in a case where a sensing apparatus according to a first
 embodiment of the present invention is constituted by a scale and one
 two-dimensional angle sensor.
 A scale is constituted by a waveform the height of which varies
 periodically, and an angle sensor is assumed to be an optical sensor for
 sensing the angle of inclination of the inclined surface of the scale.
 Further, it is assumed that the function of angular variation is such that
 inclinations in the x and y directions are represented by f(x,y) and
 g(x,y), respectively.
 Shown in FIG. 1 are a scale 10 disposed on a stationary side and an angle
 sensor 20 provided on a sensor mount disposed on a moving side, not shown.
 An angular grid 102 is formed on the planar surface of a substrate 101
 constructing the scale 10. The angular grid 102 comprises a collection of
 sinusoidal peaks and valleys which vary in two perpendicularly
 intersecting directions (x and y directions) on the planar surface in the
 form of a known function. The angular grid 102 constructs a scale for
 sensing position in two directions.
 The single angle sensor 20 is disposed on the side of the scale 10 at which
 it confronts the angular grid 102, is spaced a prescribed distance away
 from the angular grid surface and is capable of translational movement.
 The angle sensor 20 irradiates the angular grid 102 with emitted light
 rays and senses, along the x and y axes, the direction of light reflected
 from the angular grid 102. The position of a moving object in
 two-dimensional coordinates when the angle sensor 20 has been moved
 relative to the scale 10 along the x and y axes is sensed.
 For example, sine waves obtained when periods in two perpendicularly
 intersecting directions, namely x and y directions, of the angular grid
 102 are represented by Tx, Ty and amplitudes are represented by a, b are
 as follows:
EQU f(x,y)=a sin(2.pi.x/Tx) (A)
EQU g(x,y)=b sin(2.pi.y/Ty) (B)
 When the angle sensor 20 for sensing direction of reflection in the x and y
 directions is moved with respect to the plane of this two-dimensional
 angular grid, the angles in the two directions differ owing to the
 inclined surfaces of the peaks even though the height of the sensor with
 respect to the peaks of the angular grid 102 is the same. As a result,
 position in two dimensions can be determined distinctly by this
 difference. This makes it possible to sense the position of the moving
 object in two-dimensional coordinates.
 Various techniques for effecting interpolation between wavelengths using a
 conventional interferometer can be employed to perform interpolation
 between the wavelengths of-the individual sine waves expressed by
 Equations (A) and (B) above. By oscillating the angle sensor mechanically
 or, in the case of a photoelectric angle sensor, oscillating only the
 light beam, in the x and y directions only when performing interpolation
 between wavelengths, it is possible to obtain two signals respectively
 phase-shifted by Tx/4, Ty/4, namely by .pi./2, in the x and y directions,
 respectively. Of course, sensors for sensing the position phase-shifted by
 Tx/4, Ty/4 may be additionally provided and the two signals phase-shifted
 by .pi./2 may be sensed simultaneously.
 Further, it is permissible to use an angle sensor which, by applying a
 technique relying upon the reading of a photoelectric scale, reduces the
 influence of error in scale graduation spacing by reading the average
 value of the positions of a plurality of scale graduations.
 FIG. 2 is a fundamental structural view illustrating dispositional
 relationship in a case where a sensing apparatus according to a second
 embodiment of the present invention is constituted by a scale and two
 two-dimensional angle sensors. This illustrates an instance in which
 position in two dimensions, pitching angle and rolling angle can be
 sensed.
 In a manner similar to that shown in FIG. 1, the angular grid 102, which
 varies in two perpendicularly intersecting directions (x and y directions)
 on the planar surface of the substrate 101 in the form of a known
 function, is formed on the planar surface of the substrate 101
 constructing the scale 10. Here a pair of angle sensors 20A, 20B are
 disposed on the side of the scale 10 at which they confront the angular
 grid 102 and are spaced a prescribed distance away from the angular grid
 surface. The angle sensors 20A, 20B irradiate the angular grid 102 with
 emitted light rays and sense, along the x and y axes, the direction of
 light reflected from the angular grid 102. The angle sensors 20A, 20B are
 supported on a plate-shaped sensor mount 201 which lies parallel to the
 plane of the angular grid and are arrayed so as to be spaced apart from
 each by dx, dy along x and y axes, respectively. As a result, the angle
 sensor 20A senses angle of inclination along the x and y axes at a
 position given by coordinates (x,y), and the angle sensor 20B senses angle
 of inclination along the x and y axes at a position given by coordinates
 (x+dx,y+dy).
 According to the embodiment shown in FIG. 2, x-direction angle outputs ma1,
 ma2 of the angle sensors 20A, 20B and y-direction angle outputs mb1, mb2
 of the angle sensors 20A, 20B, respectively, are given by the following
 equations, where pe(x,y) represents the pitching angle (inclination in the
 x direction) of the sensor mount and re(x,y) represents the rolling angle
 (inclination in the y direction) of the sensor mount:
EQU ma1=f(x,y)+pe(x,y) (1)
EQU mb1=g(x,y)+re(x,y) (2)
 ma2=f(x+dx,y+dy)+pe(x,y) (3)
EQU mb2=g(x+dx,y+dy)+re(x,y) (4)
 From Equations (1).about.(4), we have
EQU ma2-ma1=f(x+dx,y+dy)-f(x,y) (5)
EQU mb2-mb1=f(x+dx,y+dy)-g(x,y) (6)
 If f and g are known functions, then x and y can be determined from
 Equations (5) and (6).
 For example, if f and g are given by the periodic functions
EQU f(x,y)=a cos(2.pi.x/Tx) (7)
EQU g(x,y)=b cos(2.pi.y/Ty) (8)
 then x and y can be determined from Equations (5) and (6) provided that a,
 b, dx, dy, Tx and Ty are known. If the number of periodic changes (the
 number of waves) of the output signal is counted, no problems will arise
 even if x and y are larger than Tx and Ty, respectively. As a result, both
 pitching angle pe and rolling angle re are determined by Equations (1) and
 (2).
 It should be noted that x, y in the foregoing equations do not take into
 account the pitching angle pe, the rolling angle re and a disparity
 between the true position X,Y of the angle sensor and the sensed position
 x,y on the angular grid surface, which disparity is caused by a distance
 dz from the surface of the angular grid to the angle sensor. The
 relationships are given by the following equations:
EQU X=x-pe(x,y)dz (9)
EQU Y=y-re(x,y)dz (10)
 Accordingly, once x, y, pe and re have been obtained, X and Y can be found
 using Equations (9), (10) and dz, which is known.
 Furthermore, if these relations are utilized, an unknown dz can be found
 from relations similar to those of Equations (9) and (10) by providing the
 angle sensor side with a mechanism which rotates the angle sensor solely,
 or together with the sensor mount, through a known angle .alpha.0 or
 .beta.0.
 More specifically, the following equations hold, where x0, y0 represent
 displacements, in the x and y directions, corresponding to a change in
 angle sensor output caused by rotation through the angles .alpha.0,
 .beta.0:
EQU dz=x0/.alpha.0 (11)
EQU dz=y0/.beta.0 (12)
 Further, dz can be found by measuring the angles .alpha.0, .beta.0 which
 prevail when values corresponding to the x- and y-axis displacements x0,
 y0 indicated in Equations (11), (12) are made fixed values, as in the
 manner of the intervals of scale graduations.
 This doubles also as a sensor for sensing the distance between the angle
 sensor and the place of the angular grid.
 FIG. 3 illustrates an instance where the sensor according to a third
 embodiment of the invention is constituted by three two-dimensional
 sensors. This illustrates a configuration capable of sensing position of a
 moving object in two dimensions as well as pitching angle, rolling angle
 and yawing angle of the moving object.
 In the third embodiment, as depicted in FIG. 3, three angle sensors 20A,
 20B and 20C are arranged at the apices of an isosceles triangle in the
 planar surface of the plate-shaped sensor mount 202 lying parallel to the
 angular grid surface in a manner similar to that shown in FIG. 2. Let
 angle sensor 20C be situated at a position (dx,-dy) along the x and y axes
 from the angle sensor 20, which is located at the origin, let ma3, mb3
 represent the angle outputs of the angle sensor 20C along the x and y
 axes, and let ma2, mb2 represent the angle outputs of the angle sensor 20B
 along the x and y axes. If yawing angle .gamma., the center of rotation of
 which is the angle sensor 20A at the origin, is considered, the following
 relations are obtained:
EQU ma2=f(x+dx+.gamma.dy, y+dy+.gamma.dx)+pe(x,y) (13)
EQU mb2=g(x+dx+.gamma.dy, y+dy+.gamma.dx)+re(x,y) (14)
EQU ma3=f(x+dx-.gamma.dy, y-dy+.gamma.dx)+pe(x,y) (15)
EQU mb3=g(x+dx-.gamma.dy, y-dy+.gamma.dx)+re(x,y) (16)
 When pe is known or is negligibly small, x can be determined from Equation
 (1) and .gamma. can be determined from Equation (13), whereby y and re may
 be found using Equations (14) and (16). Conversely, after y and .gamma.
 have been obtained from Equations (2) and (14), similar to a case where re
 is negligible or known, x and pe may be found using Equations (13) and
 (15).
 When both pe and re are unknown and are not negligibly small, it is
 required that .gamma. be obtained solely from a differential output of the
 angle sensors.
 Since there is no loss of generality even in
EQU f(x,y)=f(x,y+dy)=f(x,y-dy) (17)
EQU g(x,y)=g(x+dx,y) (18)
 the following equations are obtained:
EQU ma2-ma1=f(x+dx+.gamma.dy,y)-f(x,y) (19)
EQU ma3-ma1=f(x+dx-.gamma.dy,y)-f(x,y) (20)
 Letting .gamma. be a minute quantity and expressing partial differentials
 of f, g with respect to x, y by fx, fy using the suffixes x, y, we have
EQU ma2-ma3=2.gamma.dyfx(x+dx,y) (21)
 Since the function fx(x+dx,y) and dy are known, .gamma. is obtained.
 Similarly, we have
EQU mb2-mb3=.gamma.dx{gy(x,y+dy)-gy(x,y-dy)} (22)
 and the minute quantity .gamma. is obtained also from the angle output in
 the y direction. Since x, y, .alpha. and .beta. may be found when .gamma.
 has been obtained, these are recorded. If the minute quantity .gamma.,
 which is a slowly varying quantity, is accumulated while .gamma. is
 sequentially obtained, the final .gamma. at the required position is
 obtained.
 This is similar to a case where x, y vary slowly or only by a minute
 amount. For example, .gamma. may be simply obtained, providing that y is
 known and constant, from the amount of change in the differential output
 given by Equation (22). Therefore, if .gamma. is found on the assumption
 that the change in y once determined is small and the other quantities are
 decided based upon this, then it is obvious that a case in which y varies
 slowly or minutely can be dealt with as well.
 In general, it is difficult to conceive of a situation in which all degrees
 of freedom vary at the same speed and by the same magnitude in precision
 equipment. Accordingly, if any one degree of freedom whose variation is
 small or slow is selected and processing similar to that in the case of
 .gamma. described above is executed, then the amounts of change in all
 degrees of freedom, namely position relating to two-dimensional
 coordinates, distance from the angular grid surface to the angle sensor
 and pitching angle, rolling angle and yawing angle of the moving object,
 can be determined.
 It should be noted that the angle sensors 20B and 20C may be placed on a
 right-angled triangle at positions (dx,o) and (o,dy) in the x and y
 directions or on a more general triangle.
 The principle described above is applicable not only to detection of the
 position of a moving object but also to the repetitive resetting of the
 same moving object to the same desired position and attitude. Accordingly,
 if an arrangement is adopted in which marks predetermined on an object by
 a two-dimensional angular change are used to enable detection of the
 relative position and attitude between the object and angular sensor, then
 the principle of the invention can be utilized effectively in the
 positioning of a wafer in a semiconductor manufacturing apparatus.
 FIG. 4 is a fundamental structural view of a fourth embodiment of the
 invention so adapted that position along the x direction, pitching angle
 and rolling angle can be sensed by a planar angular grid, which applies a
 sinusoidal angular variation only in the x direction, and angle sensors.
 As shown in FIG. 4, a one-dimensional angular grid 103 the angle of which
 varies along the axis is formed on the planar surface of the substrate 101
 constructing the scale 10. The variation in the angle of the angular grid
 103 is represented by a known function f(x). Two angle sensors 20A, 20B
 are attached to a plate-shaped sensor mount 203 so as to arrayed along the
 x axis with a prescribed distance dx between them. When the angle sensors
 20A and 20B are moved together with the sensor mount 203 along the x axis
 relative to the angular grid, position along the x axis, pitching angle
 and rolling angle can be sensed on the moving side.
 In this embodiment, outputs m1, m2 of the two angle sensors 20A, 20B are
 expressed by the following equations, where pe(x) represents the pitching
 angle:
EQU m1=f(x)+pe(x) (23)
EQU m2=f(x+dx)+pe(x) (24)
 If one angle sensor is used and moved in the x direction in a situation
 where the pitching angle is negligible, then position along the x axis can
 be found from the change in the output f(x).
 Further, if Pe(x) is not negligible, then, taking the difference between
 the outputs of the two angle sensors, we obtain the following:
EQU m2-m1=f(x+dx)-f(x) (25)
 This difference between known functions is of course a known function. The
 x position of the angle sensor, therefore, can be determined from the
 change in the differential output (m2-m1). If x is known, then the
 pitching angle Pe(x) can be obtained, by calculation, from m1.
 If the two angle sensors mentioned above are made two-dimensional angle
 sensors for sensing angles in both the x and y directions, then the
 angular shapes (where the change is ideally zero) of the scale in the x
 and y directions are found in the same manner as in the two-point method
 of the pitching angle described above. If the angle of this angled scale
 in the y direction thereof is known, then the rolling angle re(x) of the
 sensor mount, i.e. the moving object, can be sensed from the y-direction
 angle output of the angle sensor.
 In another feasible embodiment, calibration data of the angular grid can be
 obtained by equipping the sensor mount of FIG. 4 with a mechanism for
 moving the mount infinitesimally a known amount D (not shown) in the x and
 y direction. In such case, let f(x) represent the ideal sinusoidal shape
 (the designed shape), and let e(x) represent a deviation from f(x) of the
 actual angular grid. Pitching due to movement of the angle sensor will be
 considered negligible. First x is determined utilizing the function f(x)
 and the angle sensor D is shifted at this position [which actually
 includes an unknown error .delta. owing to the effect of the error e(x)]
 by D in the x direction using a piezoelectric element or the like. Let m1,
 m1.sub.D represent the outputs of the angle sensor before and after the
 shift, respectively. The following equation is obtained from the
 difference between these two outputs:
EQU m1.sub.D -m1=f(x+.delta.x+D)-f (x+.delta.x)+e(x+.delta.x+D)-e(x+.delta.x)
 (26)
 Making .delta.x infinitesimally small, the following equation is obtained
 as an approximate derivative of e(x):
EQU e'(x)={e(x+D)-e(x)}/D
EQU .apprxeq.[m1.sub.D -m1-{f (x+D)-f(x)}]/D (27)
 Since {f(x+D)-f(x)} is a known function, the right side of this equation is
 a known function. Accordingly, if this approximate derivative e'(x) is
 numerically integrated by some method, the approximate function e.sub.c
 (x) of e(x) will be calculated.
 Formula error owing to the approximation of the derivative and the
 numerical integration at this time is a percentage decided for every
 frequency and therefore can be corrected by way of Fourier transform and
 inverse Fourier transform.
 Further, if the numerical integration is performed again after the x
 position evaluated by f(x) is corrected using the approximate curve ec(x)
 of e(x) obtained above, the accuracy of the approximate curve of e(x)
 obtained anew can be improved. If this correction of the x position is
 repeated until the correction quantity .delta.x becomes sufficiently
 small, a calibration curve of the required accuracy will be obtained.
 Described next will be calibration by a method of aligning two angle
 sensors, which are arrayed in the x direction at a spacing dx, in one row
 in the x direction in order to eliminate the effects of pitching when the
 angle sensors are moved in the x direction. The outputs m1, m2 of the two
 angle sensors are represented by the following equations:
EQU m1(x)=f(x)+e(x)+pe(x) (28)
EQU m2(x)=f(x+dx)+e(x+dx)+pe(x) (29)
 where p(x) represents the pitching of the sensor at the x position.
 In order to eliminate the effects of pitching, use is made of the
 differential output of the two sensors. The differential output is
 represented by the following equation:
EQU m2(x)-m1(x)={f(x+dx)-f(x)}+{e(x+dx)-e(x)}
EQU =f1(x)+e1(x) (30)
 where we write:
EQU f1(x)=f(x+dx)-f(x) (31)
 e1(x)=e(x+dx)-e(x) (32)
 Here f1(x) is a function having the same period as that of the original
 angular grid in the x direction and can be regarded as the ideal function
 of the grid. If dx is known and the ideal shape of f(x) (the average
 sensitivity in terms of a displacement meter) has been ascertained, then
 the ideal shape of f1(x) can also be determined and this can be used to
 infer x.
 If a configuration is adopted in which the mount having the two attached
 angle sensors is shifted by D along the x axis and the outputs of the
 angle sensors are read before and after the shift, then an approximate
 value of the derivative of e1(x) will be obtained as indicated by the
 following equation in the same manner as described above:
EQU e1'(x)={e1(x+D)-e1(x)}/D
EQU .apprxeq.[m2.sub.D (x)-m2(x)-{m1.sub.D (x)-m1(x)}-{f1(x+D)-f1(x)}/D (33)
 where m1.sub.D (x), m2.sub.D (x) are the outputs of the angle sensors when
 the sensor mount has been shifted by D at the x position.
 Since {f1(x+D)-f1(x)} is a known function, the left side of this equation
 is a known function. Accordingly, if this approximate derivative e1'(x) is
 numerically integrated by some method, the approximate function e1.sub.C
 (x) of e1(x) will be calculated.
 Further, if the numerical integration is performed again after the x
 position evaluated by f1(x) is corrected (adopting .delta.x as the
 correction quantity) using the approximate curve e1 c(x) of e1(x) obtained
 above, the accuracy of the approximate curve of e1(x) obtained anew can be
 improved. If this correction of the x position is repeated until the
 correction quantity .delta.x becomes sufficiently small, a calibration
 curve of the required accuracy will be obtained.
 If the final result of e1(x) is integrated one more time, e(x) is found,
 the angular shape of the angular grid is obtained from e(x) and position
 along the x axis and pitching are sensed by the two x-direction angle
 sensors. This completes the calibration of the angular grid.
 FIG. 5 is a fundamental structural view of a fifth embodiment of the
 invention so adapted that positioning by polar coordinates is made
 possible.
 As shown in FIG. 5, a scale 52 for polar coordinates is constructed by
 forming a two-dimensional angular grid 51, the angle of which varies in
 the radial and circumferential directions in accordance with a known
 function, on a circular disk 50. An angle sensor 53, which moves relative
 to the scale 52 along the surface of the two-dimensional angular grid, is
 disposed so as to confront the angular grid. Positioning based upon polar
 coordinates is made possible by the angle sensor 53.
 In the fifth embodiment, the two-dimensional angular grid 51 is not limited
 to that having the shape shown in FIG. 5 but may be one which provides an
 angular change along a spiral.
 FIG. 6 is a fundamental structural view of a sixth embodiment of the
 invention so adapted that positioning by spherical coordinates is made
 possible.
 As shown in FIG. 6, a scale 61 for cylindrical coordinates is constructed
 by forming an angular grid 61, the angle of which varies in the direction
 of the generating lines on the outer circumferential surface of a cylinder
 60 and in the circumferential direction orthogonal to the generating lines
 in accordance with a known function, on the outer circumferential surface
 of the cylinder 60. An angle sensor 63, which moves relative to the scale
 62 along the surface of the angular grid, is disposed so as to confront
 the angular grid. Positioning based upon cylindrical coordinates is made
 possible by the angle sensor 63.
 In the sixth embodiment, the two-dimensional angular grid 61 is not limited
 to that having the shape shown in FIG. 6 but may be one of the type in
 which the angular shape varies along a helix.
 In the embodiments shown in FIGS. 1 through 6, the apparatus may be one
 separated into a light sensor and a source of light rays or the like for
 applying angle information to the angle sensor, with the light sensor and
 source being disposed on opposite sides of the angular grid so that a
 change in the angle of transmitted light rays or the like may be sensed.
 The angular grid in such a configuration may rely upon a change in
 refractive index or may cause a change in the direction of transmitted
 light by roughness on the underside of the light-transmitting plate.
 FIG. 7 is a fundamental structural view of a seventh embodiment of the
 invention so adapted that positioning by spherical coordinates is made
 possible.
 As shown in FIG. 7, a scale 72 for spherical coordinates is constructed by
 forming a two-dimensional angular grid 71 on the inner surface of a
 spherical body 70. The position of a moving object can be determined by
 three angle sensors 73 provided on a moving object. Further, minute
 oscillations in three directions at the center of rotation can be sensed
 by sensing the attitude of a rotating object in three directions at the
 center of the spherical body 70.
 FIG. 8 is a fundamental view in which, according to an eighth embodiment of
 the present invention, it is possible to perform two-dimensional
 positioning by constructing a contact-type angle sensor utilizing the
 principle of a sensor in a microscope that employs interatomic force, and
 adopting a microgrid such as a crystal as the angular grid.
 As shown in FIG. 8, a crystal surface 81 of a crystal 80 is utilized as a
 two-dimensional angular grid for a scale. A contactor 82 has its position
 on the crystal surface 81 decided by the interatomic force of the crystal
 or by contact pressure, and microlevers 83, 84 for two directions are
 successively connected to the contactor 82. The flexure of the microlevers
 83, 84 varies depending upon the direction of the surface normal line at
 the point of contact. The microlevers 83, 84 are irradiated with light
 from light sources (not shown), and the direction of light reflected from
 the microlevers 83, 84 is sensed by optical sensors such as a
 semiconductor light-position sensor. This makes it possible to sense a
 change in the angular shape and to achieve positioning in two dimensions.
 In this embodiment, the alignment of the atoms of the crystal surface 81
 can be utilized as the two-dimensional angular grid for the scale.
 It should be noted that another sensor for sensing a fundamental change may
 be used instead of the light sensor for sensing flexure of the microlevers
 83, 84, and that strain gauges may be affixed to the microlevers.
 The present invention is not limited to the arrangements described in the
 foregoing embodiments.
 By way of example, an arrangement may be adopted in which the angular
 change of the angular grid has a form obtained by superimposing a
 plurality of sine waves having different frequencies.
 Further, the angular grid may be implemented utilizing a change in
 refractive index brought about by a change in the composition of a
 material within a transparent plate. The angular grid may employ a
 material the refractive index of which is changed by an externally applied
 electromagnetic or mechanical force. The material may be used alone or
 sealed within a vessel.
 Further, the scale of the present invention may be constructed by causing
 an angular grid surface, whose angle-related property varies in the form
 of a known function, to be produced by standing waves obtained when a
 periodic excitation force is applied to a resilient plate, a planar
 surface having a resilient property, a crystal body or a liquid surface,
 wherein the angular grid is produced on or in the surface.
 Further, according to the present invention, the scale may be constructed
 from a plurality of divided scales each having an angular grid, wherein a
 plurality of the divided scales can be arrayed intermittently or
 continuously in conformity with the area over which a moving object moves.
 An arrangement can be adopted in which, rather than increasing the number
 of angular grid surfaces, plural sets of angle sensors having the same
 function are deployed at intervals smaller than the size of the angular
 grid surface.
 Further, according to the present invention, means may be provided for
 applying a fixed amount of relative motion to an angle sensor along the x
 and y directions of the angular grid, and calculating data for calibrating
 error from the known ideal shape of the angular grid based upon each
 detection value from the angle sensor before and after relative movement
 and the difference between the values. Further, it is permissible to adopt
 an arrangement having storage means for storing the calculated calibration
 data or calibration data obtained by an ordinary comparison calibration,
 and correction means for correcting, on the basis of the calibration data,
 results of measuring positional coordinates and various attitude angles by
 the angular grid.
 Further, the present invention can be adapted to generate traveling waves,
 form an angular grid in which a uniform change in the surface whereof is
 produced by the standing waves, and determine position two-dimensionally
 based upon a relationship between the angular grid and time.
 In the present invention, an arrangement may be adopted in which the
 angular grid is so adapted as to apply electromagnetic power to an
 electro-optic crystal or a liquid that fills the interior of a vessel and
 reacts to electromagnetic force or light, thereby subjecting the
 electro-optic crystal or liquid to a change in refractive index, wherein
 the change is in the form of a known function.
 The present invention exhibits a number of outstanding effects.
 Specifically, in accordance with the present invention as described above,
 the scale used to sense the position or various attitudes of a moving
 object is formed from a two-dimensional angular grid representing an
 angular shape. As a result, the two-dimensional position of a moving
 object can be sensed as a matter of course, and so can the pitching angle,
 rolling angle and yawing angle of the moving object, merely by combining
 angle sensors with a simple scale. In addition, by adopting the angular
 grid as the scale, it is possible to sense position relating to
 two-dimensional coordinates such as orthogonal coordinates, cylindrical
 coordinates, polar coordinates or coordinates along a freely curved
 surface.
 Further, by combining at least one two-dimensional angle sensor with a
 scale comprising a two-dimensional angular grid, the position of a moving
 object in two-dimensional coordinates can be sensed in relative movement
 between the scale and the angle sensor. If the angle sensor is subjected
 to a known change in attitude, such as a change in pitching angle or
 rolling angle, the distance between the scale and angle sensor can also be
 sensed at the same time.
 Further, in accordance with the present invention, combining at least one
 pair of two-dimensional angle sensors with a scale comprising a
 two-dimensional angular grid makes it possible to sense the position of a
 moving object in two-dimensional coordinates, as well as the pitching and
 rolling angles of the moving object, in relative movement between the
 scale and the angle sensors. If the angle sensor is subjected to a known
 change in attitude, such as a change in pitching angle or rolling angle,
 the distance between the scale and angle sensors can also be sensed at the
 same time.
 Further, in accordance with the present invention, combining at least three
 two-dimensional angle sensors with a scale comprising a two-dimensional
 angular grid makes it possible to sense the position of a moving object in
 two-dimensional coordinates, as well as the pitching, rolling and yawing
 angles of the moving object, in relative movement between the scale and
 the angle sensors. In addition, by subjecting the angle sensors to a known
 prescribed change in angle, the distance between the scale and angle
 sensors can also be sensed.
 Further, in accordance with the present invention, position along one axis,
 pitching angle and rolling angle can be sensed by combining at least one
 pair of angle sensors with a scale constructed from an angular grid which
 varies along one axis (the x axis) in the form of a known function.
 Further, in accordance with the present invention, it is possible to sense
 position precisely by adopting an arrangement in which the angular
 variation of the angular grid has a form obtained by superimposing a
 plurality of sine waves having different frequencies.
 Further, in accordance with the present invention, the angle sensor is
 constructed from a plurality of displacement meters arrayed with a
 prescribed spacing among them, wherein the displacement meters are of
 optical type, a type which senses an electro-optic quantity or of
 mechanical-contact type. By sensing a quantity corresponding to a change
 in the angle of inclination of the shape of the angular grid surface,
 which change has been applied by a differential output from the
 displacement meters in the form of a change in height and shape, this
 sensed quantity can be utilized instead of angle information. If three or
 more displacement meters are arranged two-dimensionally with prescribed
 distances between them, the displacement meters can function as a
 two-dimensional angle sensor.
 Further, in accordance with the present invention, the scale is capable of
 forming an angular grid surface, whose angle-related property varies
 spatially, by standing waves obtained when periodic oscillation is applied
 to a resilient plate, a planar surface or curved surface having a
 resilient property, a crystal body, a liquid surface of a liquid filling a
 hermetically sealed vessel, wherein the angular grid is produced on or in
 the surface. The scale can be utilized as the scale surface only while the
 oscillation is being applied.
 Further, in accordance with the present invention, the scale is constructed
 from a plurality of divided scales each having an angular grid surface,
 wherein a plurality of the divided scales are arrayed intermittently or
 continuously in conformity with the area over which the moving object
 moves. As a result, even if the angle sensor departs, in relative terms,
 from one angular grid surface, information indicative of the position of
 the angular grid surface of the adjacent divided scale can be sensed. This
 makes it possible to enlarge the range of relative movement between the
 angle sensor and the angular grid.
 Further, in accordance with the present invention, correction means is
 provided for correcting, based upon results of calibrating an error in the
 angular shape of the angular grid, results of measuring coordinate
 position and attitude angles by the angular grid. Accordingly, in
 situations where the angular grid cannot be fabricated to a high
 precision, the calibration data is stored in memory beforehand and data
 between known items of data is approximated by interpolation, thereby
 making it possible to correct measurement data based upon results of
 calibration.
 Further, in accordance with the present invention, a deviation from the
 ideal shape of an angular grid can be calibrated autonomously by applying
 a known amount of motion to an angle sensor. This makes it possible to
 compensate for the error component when the accuracy of the angular grid
 is poor.
 As many apparently widely different embodiments of the present invention
 can be made without departing from the spirit and scope thereof, it is to
 be understood that the invention is not limited to the specific
 embodiments thereof except as defined in the appended claims.