Patent Publication Number: US-11397097-B2

Title: Displacement devices and methods and apparatus for detecting and estimating motion associated with same

Description:
RELATED APPLICATIONS 
     This application is a continuation of U.S. application Ser. No. 15/791,345 filed 23 Oct. 2017, which is in turn a continuation of U.S. application Ser. No. 15/016,027 filed 4 Feb. 2016, which is in turn a continuation of PCT application No. PCT/CA2014/050739 having an international filing date of 6 Aug. 2014. PCT application No. PCT/CA2014/050739 claims the benefit of the priority of (and the benefit under 35 USC 1.19 of) U.S. application No. 61/862,520 filed 6 Aug. 2013 and of U.S. application No. 62/008,519 filed 6 Jun. 2014. All of the applications referred to in this paragraph are hereby incorporated herein by reference. 
    
    
     TECHNICAL FIELD 
     The invention relates to displacement devices, their motion actuation, and measurement (e.g. detection and/or estimation) of their motion. Particular non-limiting embodiments provide displacement devices for use in the semiconductor fabrication industry and general automation industry and methods and apparatus for detecting and/or estimating their associated motion. 
     BACKGROUND 
     Motion stages (XY tables and rotary tables) are widely used in various manufacturing, product inspection, and assembling processes. A common solution currently in use achieves XY (i.e. planar) motion by stacking two linear stages (i.e. a X-stage and a Y-stage) together via connecting bearings. 
     A more desirable solution involves having a single movable stage capable of XY (i.e. planar) motion, eliminating additional bearings. It might also be desirable for such a movable stage to be able to provide at least some Z (out of plane) motion. Attempts have been made to design such displacement devices using the interaction between current-carrying coils and permanent magnets. Examples of efforts in this regard include the following: U.S. Pat. Nos. 6,003,230; 6,097,114; 6,208,045; 6,441,514; 6,847,134; 6,987,335; 7,436,135; 7,948,122; US patent publication No. 2008/0203828; W. J. Kim and D. L. Trumper, High-precision magnetic levitation stage for photolithography.  Precision Eng.  22 2 (1998), pp. 66-77; D. L. Trumper, et al, “Magnet arrays for synchronous machines”, IEEE Industry Applications Society Annual Meeting, vol. 1, pp. 9-18, 1993; and J. W. Jansen, C. M. M. van Lierop, E. A. Lomonova, A. J. A. Vandenput, “Magnetically Levitated Planar Actuator with Moving Magnets”, IEEE Tran. Ind. App., Vol 44, No 4, 2008. 
     There is a general desire to provide displacement devices having characteristics that improve upon those known in the prior art. 
     There is a general desire to estimate characteristics of the motion of such displacement devices. For example, there is a desire to estimate the position of (e.g. measure) the movable stage(s) of such devices relative to their stator(s). In some cases, there can be a desire to provide position estimation solutions that are independent of line of sight obstruction. 
     The foregoing examples of the related art and limitations related thereto are intended to be illustrative and not exclusive. Other limitations of the related art will become apparent to those of skill in the art upon a reading of the specification and a study of the drawings. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       Exemplary embodiments are illustrated in referenced figures of the drawings. It is intended that the embodiments and figures disclosed herein are to be considered illustrative rather than restrictive. 
         FIGS. 1A and 1B  are respectively partial schematic side and top views of a displacement device according to a particular embodiment of the invention. 
         FIGS. 2A, 2B and 2C  are respectively partial schematic side views of the  FIG. 1  displacement device according to particular embodiments showing different locations of the sensor array relative to the stator and/or the coils. 
         FIG. 3A  is a partial schematic isometric view of a displacement device according to a particular embodiment of the invention. 
         FIG. 3B  is a partial schematic cross-sectional view of the  FIG. 3A  displacement device along the line  1 B- 1 B showing the magnet arrays of the movable stage relative to the coils of the stator. 
         FIG. 3C  is a partial schematic cross-sectional view of the  FIG. 3A  displacement device along the line  1 C- 1 C. 
         FIG. 3D  shows additional detail of one of the Y-magnet arrays of the  FIG. 3A  displacement device in accordance with a particular embodiment. 
         FIG. 3E  shows additional detail of one of the X-magnet arrays of the  FIG. 3A  displacement device in accordance with a particular embodiment. 
         FIG. 3F  is a partial schematic cross-sectional view of the  FIG. 3A  displacement device along the line  1 F- 1 F showing the magnet arrays of the movable stage relative to the sensors of the stator. 
         FIGS. 4A and 4B  are schematic partial cross-sectional views of layouts of magnet arrays which may be used in connection with any of the displacement devices described herein and which are useful for showing a number of magnet array parameters. 
         FIGS. 5A-5L  show additional details of magnet arrays suitable for use with any of the displacement devices described herein in accordance with particular embodiments. 
         FIGS. 6A-6L  show additional details of magnet arrays suitable for use with any of the displacement devices described herein in accordance with particular embodiments. 
         FIGS. 7A and 7B  are schematic cross-sectional views of pairs of parallel adjacent magnet arrays according to particular embodiments suitable for use with any of the displacement devices described herein. 
         FIG. 8  is a schematic cross-sectional view of layouts of magnet arrays which may be used in any of the displacement devices described herein in accordance with other embodiments. 
         FIG. 9  is a schematic cross-sectional view of a sensor array which may be used with any of the displacement devices described herein in particular embodiments. 
         FIG. 10  is a schematic depiction of a particular sensor comprising a plurality of sensor sub-units according to a particular embodiment. 
         FIGS. 11A-11C  show various techniques for extracting measurement signals from sensors (e.g. hall-effect sensors) which may be used in particular embodiments. 
         FIGS. 12A-12B  illustrate arrays of field sensing elements with different embodiments of column summing/averaging operations and row summing/averaging operations implemented by analog circuits on some rows and columns. 
         FIGS. 13A-13B  illustrate different embodiments of a Y-magnet in relation to column sums/averages of column sensors extending in the Y direction. 
         FIG. 14A  illustrates another embodiment of a Y-magnet array in relation to column sums/averages of column sensors extending in the Y direction in which certain column sums/averages are removed.  FIG. 14B  illustrates the overall layout of sensors for the  FIG. 14A  embodiment. 
         FIG. 15  illustrates another embodiment of a Y-magnet array in relation to column sums/averages of column sensors extending in the Y direction in which additional column sums/averages are removed. 
         FIG. 16  illustrates another embodiment of a Y-magnet array in relation to column sums/averages of column sensors extending in the Y direction, the sensors having an alternative sensor pitch. 
         FIG. 17  illustrates an array of sensors according to a particular embodiment in which each sensor is offset away from equally spaced two dimensional grid points. 
         FIG. 18  illustrates an array of sensors having polygon-shaped independent sensing regions which each contain a sub-array of sensors according to a particular embodiment. 
         FIG. 19  is a schematic illustration of a method for estimating the position of a movable stage position according to a particular embodiment. 
         FIG. 20  schematically illustrates an apparatus for moving a plurality of movable stages through a plurality of different stators. 
         FIGS. 21A-21C  schematically depict displacement devices according to other embodiments having different relative orientations of coil traces and magnet arrays. 
         FIGS. 22A-22C  schematically depict cross-sectional views of magnet arrays having different numbers of magnetization directions within a particular magnetic spatial period which may be used in any of the displacement devices described herein according to particular embodiments. 
         FIGS. 23A-23C  show various embodiments of magnet arrays having offset or shifted sub-arrays which may be used in any of the displacement devices described herein according to particular embodiments. 
         FIGS. 24A, 24B and 24C  show a number of Y-magnet arrays which exhibit periodic spatial variation which extends in the X direction over their respective Y dimensions and which may be used in any of the displacement devices described herein according to particular embodiments. 
         FIGS. 25A and 25B  respectively depict a top view of a number of coil traces and a cross-sectional view of a coil trace which comprise multiple sub-traces which may be used in any of the displacement devices described herein according to particular embodiments. 
         FIGS. 26A and 26B  show various views of circular cross-section coil traces which may be used with any of the displacement devices described herein according to particular embodiments.  FIGS. 26C and 26D  show embodiments of how coil traces may comprise multiple sub-traces having circular cross-section. 
     
    
    
     DESCRIPTION 
     Throughout the following description specific details are set forth in order to provide a more thorough understanding to persons skilled in the art. However, well known elements may not have been shown or described in detail to avoid unnecessarily obscuring the disclosure. Accordingly, the description and drawings are to be regarded in an illustrative, rather than a restrictive, sense. 
     Displacement devices are provided which comprise a stator and a movable stage. The stator comprises a two-dimensional sensor array and a plurality of coils shaped to provide pluralities of generally linearly elongated coil traces in one or more layers. Layers of coils may overlap in the Z direction. These layers of coils may also overlap with the sensor array in the Z direction. The movable stage comprises one or more magnet arrays. Each magnet array may comprise a plurality of magnetization segments generally linearly elongated in a corresponding direction. Each magnetization segment has a magnetization direction generally orthogonal to the direction in which it is elongated. The magnetization directions of the plurality of magnetization segments in each magnet array exhibit a magnetic spatial period λ in a direction across a width of the magnet array. One or more amplifiers may be selectively connected to drive current in the coil traces and to thereby effect relative movement between the stator and the movable stage. The sensors in the sensor array are configured to sense the position of the movable stage relative to the stator. A controller is connected to receive information based on outputs of the sensors in the array and is configured to use the information to determine a position of the movable stage (e.g. relative to the stator). The controller may also be configured to use information related to the current in coil traces to help determine the position of the movable stage (e.g. to compensate information from sensors to account for magnetic fields created by current in coil traces). 
     Particular Embodiment 
       FIGS. 1A and 1B  are respectively partial schematic side and top views of a displacement device  100  according to a particular embodiment. Displacement device  100  comprises a movable stage  110  and a stator  120 . Moveable stage  110  comprises one or more magnet arrays  112 . Stator  120  comprises a two-dimensional (2D) sensor array  500  and a plurality of coils  122 . Stator  120  may be located adjacent to and may be oriented generally parallel to movable stage  110 .  FIGS. 2A, 2B and 2C  are respectively partial schematic side views of the  FIG. 1  displacement device  100  according to particular embodiments showing different locations of sensor array  500  relative to stator  120  and/or coils  122 . Sensor array  500  may be located within stator  120  or may be attached to stator  120 . In particular, 2D sensor array  500  may be located on top of ( FIG. 2A ), below ( FIG. 2B ) or within ( FIG. 2C ) stator  120 . As explained in more detail below, each sensor  501  in sensor array  500  is a magnetic field sensor sensitive to variation in magnetic field. Sensors  501  may comprise, without limitation, Hall-effect magnetic field sensors, magneto-resistive sensors, and/or other suitable types of magnetic field sensors that can measure magnetic flux density. As explained in more detail below, each of coils  122  is elongated along a particular dimension, such that in a working region  124  of stator  120  (i.e. a region of stator  120  over which movable stage  110  can move), coils  122  effectively provide linearly elongated coil traces  126  (see  FIG. 3C ). 
     Displacement device  100  comprises a controller  504  connected to receive information based on an output from sensors  501  in array  500 . Controller  504  is configured (e.g. programmed) to use the received information to determine (e.g. estimate) a position of movable stage  110  in one or more non-parallel directions (e.g. directions X, Y and/or Z in the  FIG. 1  view). The position of movable stage  110  determined by controller  504  may be determined relative to stator  120 , relative to some reference on or associated with stator  120  and/or to some other reference (e.g. some other static reference). Controller  504  may comprise components of a suitable computer. In general, controller  504  comprise any suitably configured processor, such as, for example, a suitably configured general purpose processor, microprocessor, microcontroller, digital signal processor, field-programmable gate array (FPGA), other type of programmable logic device, pluralities of the foregoing, combinations of the foregoing, and/or the like. Controller  504  has access to software which may be stored in computer-readable memory (not expressly shown) accessible to controller  504  and/or in computer-readable memory that is integral to controller  504 . Controller  504  may be configured to read and execute such software instructions and, when executed by the controller  504 , such software may cause controller  504  to implement some of the functionalities described herein. 
     For purposes of describing the displacement devices disclosed herein, it can be useful to define a pair of coordinate systems—a stator coordinate system which is fixed to the stator (e.g. to stator  120 ) and does not move; and a stage coordinate system which is fixed to the movable stage (e.g. movable stage  110 ) and moves relative to the stator and the stator coordinate system. This description may use conventional Cartesian coordinates (X, Y, Z) to describe these coordinate systems, although it will be appreciated that other coordinate systems could be used. For convenience and brevity, in this description and the associated drawings, the directions (e.g. X, Y, Z directions) in the stator coordinate system and the directions in the stage coordinate systems are shown and described as being coincident with one another—i.e. the stator-X, stator-Y and stator-Z directions are shown and described as being coincident with the stage-X, stage-Y and stage-Z directions. Accordingly, this description and the associated drawings may refer to directions (e.g. X, Y and/or Z directions) to refer to directions in both or either of the stator and/or stage coordinate systems. However, it will be appreciated from the description herein that in some embodiments, the movable stage (e.g. stage  110 ) may move relative to the stator (e.g. stator  120 ) such that these directions are no longer coincident with one another. In such cases, this disclosure may adopt the convention of using the terms stator-X, stator-Y and stator-Z to refer to directions in the stator coordinate system and the terms stage-X, stage-Y and stage-Z to refer to directions in the stage coordinate system. 
       FIG. 2A  is a partial schematic side view of a displacement device  100  according to a particular embodiment, where sensor array  500  is distributed on the “top” surface of stator  120  (i.e. the surface of stator  120  located between coils  122  and movable stage  110 ). In the  FIG. 2A  embodiment, this “top” surface has a normal oriented in the positive z direction, although this is not necessary.  FIG. 2B  is a partial schematic side view of a displacement device  100  according to a particular embodiment, where sensor array  500  is distributed on the “bottom” surface of stator  120  (i.e. the surface of stator  120  opposite movable stage  110 ). In the  FIG. 2B  embodiment, this “bottom” surface has normal oriented in the negative z direction, although this is not necessary.  FIG. 2C  is a partial schematic side view of a displacement device  100  according to a particular embodiment, wherein sensor array  500  is located in stator  120  and is distributed in spaces between coils  122 . For example, coils  122  can be constructed using copper printed-circuit board manufacturing technology. Spaces can be created in the printed circuit board(s) between coil traces  122 . Each sensor  501  may be located in a corresponding space between coils  122 —i.e. where no coil traces are located. 
     In the embodiments of  FIGS. 2A-2C  magnetic field sensors  501  are distributed in array  500  in a plane extended in a first X direction and a second Y direction, with a normal direction in a third Z direction. As explained in more detail below and shown in  FIG. 3C , stator coils  122  may comprise a plurality of coil trace layers  128 . Each coil trace layer  128  may comprise a plurality of coil traces  126  shaped to be linearly elongated in one of the X, Y directions. As explained in more detail below and shown in  FIG. 3B , movable stage  110  may comprise one or more linearly elongated magnet arrays  112  (e.g. linearly elongated in the stage-X or stage-Y directions), each of which may comprise a corresponding plurality of linearly elongated magnetization segments  114  (e.g. linearly elongated in the stage-X or stage-Y directions). 
       FIGS. 3A-3F  (collectively  FIG. 3 ) show more detail of displacement device  100  according to a particular embodiment of the invention.  FIGS. 3A-3F  show a variety of different views of displacement device  100  and in some of  FIGS. 3A-3F , particular components of displacement device  100  are not shown to better illustrate other components. Displacement device  100  comprises a movable stage  110  and a stator  120 . Moveable stage  110  comprises one or more linearly elongated magnet arrays  112 . Movable stage  110  of the illustrated embodiment comprises a plurality (e.g. 4) of arrays of permanent magnets  112 A,  112 B,  112 C,  112 D (collectively, magnet arrays  112 ). Stator  120  comprises a plurality of coils  122  and a two-dimensional (2D) magnetic sensor array  500 . As explained in more detail below, each of coils  122  is elongated along a particular dimension, such that in a working region  124  of stator  120  (i.e. a region of stator  120  over which movable stage  110  can move), coils  122  effectively provide linearly elongated coil traces  126 . As explained in more detail below, each of coil traces  126  comprises a corresponding axis (e.g. X or Y-axis) along which it is linearly elongated. For clarity, only a portion of the working region  124  of stator  120  is shown in the views of  FIG. 3 . In general, the working region  124  of stator  120  may be significantly larger than that shown in  FIG. 3  and may be significantly larger than movable stage  110 . It will be appreciated that outside of the partial views of  FIG. 3 , coils  122  may have loops which are not linearly elongated. As explained in more detail below, each sensor  501  of 2D sensor array  500  may be located at a two-dimensional grid point. These grid points are the intersection points of two groups of parallel lines. Lines in one group are non-parallel (e.g. orthogonal) with lines in the other group. Each sensor  501  can measure magnetic field flux density in one direction, or in more than one direction. 
     In the illustrated  FIG. 3  embodiment (as best seen in  FIG. 3C ), stator  120  comprises a plurality (e.g. 4) of layers  128 A,  128 B,  128 C,  128 D (collectively, layers  128 ) of coil traces  126 , with each pair of coil trace layers  128  separated from one another by an electrically insulating layer  130 . It will be appreciated that the number of layers  128  in stator  120  may be varied for particular implementations and that the number of layers  128  shown in the illustrated embodiment is convenient for the purposes of explanation. In the illustrated embodiment, each layer  128  comprises coil traces  126  that are linearly elongated along axes that are parallel to one another. In the case of the illustrated embodiment, layers  128 A,  128 C comprise coil traces  126 Y which are generally linearly elongated in directions parallel to the Y direction (stator-Y direction) and layers  128 B,  128 D comprise coil traces  126 X which are generally linearly oriented in directions parallel to the X direction (stator-X direction). Coil traces  126 Y which are generally linearly oriented along the Y direction may be referred to herein as “Y-coils” or “Y-traces” and, as explained in more detail below, may be used to move movable stage  110  in the X and Z directions. Similarly, coil traces  126 X which are generally linearly oriented along the X direction may be referred to herein as “X-coils” or “X-traces” and, as explained in more detail below, may be used to move movable stage  110  in the Y and Z directions. 
     In the illustrated embodiment (as shown best in  FIG. 3B ), movable stage  110  comprises four magnet arrays  112 . In some embodiments, movable stage  110  may comprise more than four magnet arrays  112 . In other embodiments, movable stage  110  may comprise fewer than four magnet arrays. For example, movable stage  110  may comprise one or more magnet arrays. Each magnet array  112 A,  112 B,  112 C,  112 D of the  FIG. 3  embodiment comprises a plurality of corresponding magnetization segments  114 A,  114 B,  114 C,  114 D (collectively, magnetization segments  114 ) having different magnetization directions. In the illustrated embodiment, each magnetization segment  114  is generally linearly elongated along a corresponding axial dimension. The elongated shape of magnetization segments  114  of the illustrated embodiment is shown, for example, in  FIG. 3B . In some embodiments, the elongation direction length (i.e. dimension in the elongation direction) of each magnetization segment  114  is at least twice its width and height (i.e. the cross-sectional dimensions orthogonal to the elongation direction). In some embodiments, this ratio of elongation direction length of each magnetization segment  114  versus its orthogonal width and height dimensions may be at least four. This relationship between the elongation direction length of each magnetization segment  114  and its orthogonal width and height dimensions may hold independently of the dimensions of magnet arrays  112 . 
     It can be seen that in the case of the illustrated  FIG. 3  embodiment, magnetization segments  114 A of magnet array  112 A and magnetization segments  114 C of magnet array  112 C are generally elongated in directions parallel to the stage-X direction and magnetization segments  114 B of magnet array  112 B and magnetization segments  114 D of magnet array  112 D are generally elongated in directions parallel to the stage-Y direction. Because of the direction of elongation of their respective magnetization segments  114 : magnet arrays  112 A,  112 C may be referred to herein as “X-magnet arrays”  112 A,  112 C and their corresponding magnetization segments  114 A,  114 C may be referred to herein as “X-magnetization segments”; and magnet arrays  112 B,  112 D may be referred to herein as “Y-magnet arrays”  112 B,  112 D and their corresponding magnetization segments  114 B,  114 D may be referred to herein as “Y-magnetization segments”. This description uses a number of symbols to describe dimensions of magnet arrays  112 . As shown best in  FIGS. 3D, 3E and 4 , L xx  represents the stage-X direction length of an X-magnet array (e.g. X-magnet arrays  112 A,  112 C); W xy  represents the stage-Y direction width of an X-magnet array (e.g. X-magnet arrays  112 A,  112 C); H xz  represents the stage-Z direction height of an X-magnet array (e.g. X-magnet arrays  112 A,  112 C); L yy  represents the stage-Y direction length of a Y-magnet array (e.g. Y-magnet arrays  112 B,  112 D); W yx  represents the stage-X direction width of a Y-magnet array (e.g. Y-magnet arrays  112 B,  112 D); and H yz  represents the stage-Z direction height of an Y-magnet array (e.g. Y-magnet arrays  112 B,  112 D). 
     Magnetization segments  114  may be elongated. In some embodiments, the elongated lengths of magnetization segments  114  in their elongation directions are greater (e.g. twice, four times or more) than the widths and heights of magnetization segments  114  in directions orthogonal to their elongation directions—see  FIGS. 3B, 3D, 3E and 4 . In some embodiments, the elongation direction lengths of magnetization segments  114  (e.g. lengths L xx  of X-magnetization segments  114 A,  114 C and/or lengths L yy  of Y-magnetization segments  114 B,  114 D in the case of the illustrated  FIG. 3  embodiment, where the elongated direction lengths of magnetization segments  114  are the same as the corresponding dimensions of their respective magnet arrays  112 ) may be at least twice (and in some embodiments, at least four times) their respective widths (e.g. λ/(2Nt) or λ/N t , as explained in more detail below). Further, in some embodiments, the elongation direction lengths of magnetization segments  114  (e.g. lengths L xx  of X-magnetization segments  114 A,  114 C and/or lengths L yy  of Y-magnetization segments  114 B,  114 D in the case of the illustrated  FIG. 3  embodiment, where the elongated direction lengths of magnetization segments  114  are the same as the corresponding dimensions of their respective magnet arrays  112 ) may be at least twice (and in some embodiments, at least four times) their respective stage-Z direction heights (e.g. heights H xz  of X-magnetization segments  114 A,  114 C and heights H yz  of Y-magnetization segments  114 B,  114 D)—see  FIGS. 3B, 3D, 3E and 4 . It will be appreciated that the dimensions L xx  and L yy  are used above for convenience since in the illustrated embodiment of  FIG. 3 , the dimensions L xx , L yy  of magnet arrays  112  are the same the elongation direction lengths of magnetization segments  114 ; in general, however, these relationships between the elongation direction lengths of magnetization segments  114  and their orthogonal width and height dimensions may hold independently of the dimensions of magnet arrays  112 . 
       FIG. 3C  schematically shows the orientation of the magnetization of the various magnetization segments  114 B of Y-magnet array  112 B in accordance with a particular non-limiting example. More particularly, the schematically illustrated arrows in Y-magnet array  112 B of  FIG. 3C  show the magnetization directions of the various magnetization segments  114 B. Also, within each magnetization segment  114 B, the shaded regions represent the north poles of the magnets and the white regions represent the south poles of the magnets. 
       FIG. 3D  shows a cross-sectional view of Y-magnet array  112 B in more detail. It can be seen that Y-magnet array  112 B is divided into a number of mechanically contiguous magnetization segments  114 B along the stage-X direction and that the magnetization directions of the various segments  114 B are oriented in directions orthogonal to the stage-Y direction—i.e. the magnetization directions of the magnetization segments  114 B are orthogonal to the stage-Y direction along which magnetization segments  114 B are elongated. Mechanically contiguous magnetization segments  114 B of Y-magnet array  112 B that are adjacent to one another in the stage-X direction are in contact with one another. It may also be observed from  FIG. 3D  that the magnetization directions of magnetization segments  114 B have a spatial periodicity with a period (or wavelength) λ along the stage-X direction. This spatial periodicity λ of the magnetization directions of the magnetization segments  114  of a magnet array  112  may be referred to herein as the magnetic period λ, magnetic spatial period λ, magnetic wavelength λ or magnetic spatial wavelength λ. 
     In the illustrated  FIG. 3D  embodiment, Y-magnet array  112 B has a total stage-X direction width W yx  of 2λ—i.e. two periods of the magnetic period λ. This is not necessary. In some embodiments, Y-magnet array  112 B has a total stage-X direction width W yx  given by W yx =N m λ where N m  is a positive integer. In some embodiments, the stage-X direction width W yx , of Y-magnet arrays  112 B,  112 D is the same as the stage-Y direction width W xy  of X-magnet arrays  112 A,  112 C—i.e. W yx =W xy =W m . 
     In the case of the illustrated  FIG. 3D  embodiment, magnetization segments  114 B comprise four different magnetization directions: +Z, −Z, +X, −X (where Z refers to the stage-Z direction and X refers to the stage-X direction) which together provide a magnetic spatial period λ. This is not necessary. In some embodiments, magnetization segments  114 B may comprise as few as two magnetization directions to provide a magnetic spatial period λ and in some embodiments, magnetization segments  114 B may comprise more than four magnetization directions to provide a magnetic spatial period λ. The number of different magnetization directions of a magnet array  112  that make up a complete magnetic spatial period λ may be referred to herein as N t . Regardless of the number N t  of magnetization directions of magnetization segments  114 B, the magnetization direction of each segment  114 B is oriented generally orthogonally to the stage-Y direction.  FIG. 3D  also shows that, in the illustrated embodiment, the stage-X direction width of a magnetization segment  114 B is either: λ/(2Nt) or λ/N t . In the case of the  FIG. 3D  embodiment, where the number N t  of magnetization directions is N t =4, the stage-X direction width of magnetization sections  114 B is either λ/8 (as is the case for the edge segments labeled A, I) or λ/4 (as is the case for the interior segments labeled B,C,D,E,F,G,H). As discussed above, in some embodiments, the elongation direction lengths of Y-magnetization segments  114 B (e.g. L yy  in the illustrated embodiment) may be at least twice (and in some embodiments, at least four times) their respective widths (e.g. λ/(2Nt) or λ/N t ).  FIG. 3D  also shows the stage-Z direction height H yz  of Y-magnetization segments  114 B. As discussed above, in some embodiments, the elongation direction lengths of Y-magnetization segments  114 B (e.g. L yy  in the illustrated embodiment) may be at least twice (and in some embodiments, at least four times) their respective stage-Z direction heights (e.g. heights H yz ). 
     Another observation that may be made in the case of the illustrated  FIG. 3D  embodiment is that the magnetization of magnetization segments  114 B is mirror symmetric about a central stage Y-Z plane  118  (i.e. a plane  118  that extends in the stage-Y and stage-Z directions and that intersects magnet array  112 B at the center of its stage-X dimension W yx ). While not explicitly shown in  FIG. 3D , in some embodiments, magnet array  112 B may be provided with a non-magnetic spacer at the center of its stage-X direction dimension W yx . More particularly, magnetization segment  114 B at the center of the stage-X direction dimension W yx  of magnet array  112 B (i.e. the segment labeled E in the illustrated embodiment) may be divided into two segments of width λ/(2Nt)=λ/8 and a non-magnetic spacer may be inserted therebetween. As explained in more detail below, such a non-magnetic spacer can be used to cancel disturbance forces/torques generated by higher order harmonic magnetic fields. Another function of such a non-magnetic spacer is that such a non-magnetic spacer can be used to cancel/attenuate higher order harmonic magnetic fields detected by the sensor array  500 . Even with such non-magnetic spacer, magnet array  112 B and its magnetization segments  114 B will still exhibit the properties that: the magnetization directions of the various segments  114 B are oriented in directions orthogonal to the stage-Y direction; the stage-X direction widths of the various segments  114 B will be either: λ/(2Nt) (for the outer segments A,I and the two segments formed by dividing segment E) or λ/Nt (for the interior segments B,C,D,F,G,H); and the magnetization of magnetization segments  114 B is mirror symmetric about central Y-Z plane  118 . 
     Other than for its location on movable stage  110 , the characteristics of Y-magnet array  112 D and its magnetization segments  114 D may be similar to those of Y-magnet array  112 B and its magnetization segments  114 B. 
       FIG. 3E  shows a cross-sectional view of X-magnet array  112 A in more detail. It will be appreciated that X-magnet array  112 A is divided, along the stage-Y direction, into a number of mechanically contiguous magnetization segments  114 A which are generally linearly elongated in the stage-X direction. Mechanically contiguous magnetization segments  114 A of X-magnet array  112 A that are adjacent to one another in the stage-Y direction are in contact with one another. In the illustrated embodiment, the characteristics of X-magnet array  112 A and its magnetization segments  114 A may be similar to those of Y-magnet array  112 B and its magnetization segments  114 B, except that the stage-X and stage-Y directions are swapped. For example, the magnetization directions of magnetization segments  114 A have a spatial periodicity with a period (or wavelength) λ along the stage-Y direction; the width W xy  of X-magnet array  112 A in the stage-Y direction is given by W xy =N m λ where N m  is a positive integer; the magnetization directions of the various magnetization segments  114 A are oriented in directions orthogonal to the stage-X direction; the stage-Y direction widths of the various magnetization segments  114 A are either: λ/(2N t ) (for the outer segments A,I) or λ/N t  (for the interior segments B,C,D,E,F,G,H), where N t  represents the number of different magnetization directions in magnet array  112 A; and the magnetization of magnetization segments  114 A is mirror symmetric about central X-Z plane  118 . As discussed above, in some embodiments, the elongation direction lengths of X-magnetization segments  114 A (e.g. L xx  in the illustrated embodiment) may be at least twice (and in some embodiments, at least four times) their respective widths (e.g. λ/(2N t ) or λ/N t ).  FIG. 3E  also shows the stage-Z direction height H xz  of X-magnetization segments  114 A. As discussed above, in some embodiments, the elongation direction lengths of X-magnetization segments  114 A (e.g. L xx  in the illustrated embodiment) may be at least twice (and in some embodiments, at least four times) their respective stage-Z direction heights (e.g. heights H xz ). 
     Other than for its location on movable stage  110 , the characteristics of X-magnet array  112 C and its magnetization segments  114 C may be similar to those of X-magnet array  112 A and its magnetization segments  114 A. 
     Referring to  FIGS. 3B and 3C , the operation of displacement device  100  is now explained.  FIG. 3C  shows how movable stage  110  is spaced upwardly apart from stator  120  in the stator-Z direction. This space between stator  120  and movable stage  110  can be maintained (at least in part) by stator-Z direction forces created by the interaction of coils  122  on stator  120  with magnet arrays  112  on movable stage  110  as discussed below. In some embodiments, this space between stator  120  and movable stage  110  can be maintained using additional lifting and/or hoisting magnets, aerostatic bearings, roller bearings, sliding bearings and/or the like (not shown), as is known in the art. 
       FIG. 3B  shows four sets of active coil traces  132 A,  132 B,  132 C,  132 D (collectively, coil traces  132 ), each of which (when carrying current) is primarily responsible for interacting with a corresponding one of magnet arrays  112 A,  112 B,  112 C,  112 D to impart forces which cause movable stage  110  to move. More particularly: when coil traces  132 A are carrying current, they interact with X-magnet array  112 A to impart forces on movable stage  110  in the stator-Y and stator-Z directions; when coil traces  132 B are carrying current, they interact with Y-magnet array  112 B to impart forces on movable stage  110  in the stator-X and stator-Z directions; when coil traces  132 C are carrying current, they interact with X-magnet array  112 C to impart forces on movable stage  110  in the stator-Y and stator-Z directions; and when coil traces  132 D are carrying current, they interact with Y-magnet array  112 D to impart forces on movable stage  110  in the stator-X and stator-Z directions. 
     It will be appreciated that coil traces  132  shown in  FIG. 3B  can be selectively activated (e.g. by controller  504 ) to impart desired forces on movable stage  110  and to thereby control the movement of movable stage  110  with six degrees of freedom relating to the rigid body motion of movable stage  110 . As explained further below, coil traces  132  can also be controllably activated to control some flexible mode vibrating motion of movable stage  110 . When movable stage  110  is shown in the particular position shown in  FIG. 3B , coil traces other than coil traces  132  may be inactive. However, it will be appreciated that as movable stage  110  moves relative to stator  120 , different groups of coil traces will be selected to be active and to impart desired forces on movable stage  110 . 
     It may be observed that the active coil traces  132  shown in  FIG. 3B  appear to interact with other magnet arrays. For example, when carrying current, coil traces  132 C interact with X-magnet array  112 C as discussed above, but coil traces  132 C also pass under a portion of Y-magnet array  112 B. One might expect that, the current in coil traces  132 C might interact with the magnets in Y-magnet array  112 B and impart additional forces on movable stage  110 . However, because of the aforementioned characteristics of Y-magnet array  112 B, the forces that might have been caused by the interaction of coil traces  132 C and the magnetization segments  114 B of Y-magnet array  112 B cancel one another out, such that these parasitic coupling forces are eliminated or kept to a minimal level. More particularly, the characteristics of Y-magnet array  112 B that eliminate or reduce these cross-coupling forces include: Y-magnet array  112 B includes magnetization segments which are generally elongated in the stage-Y direction with varying magnetizations which are oriented orthogonally to the stage-Y direction; the stage-X direction width W yx  of Y-magnet array  112 B is W yx =N m λ where N m  is an integer and λ is the magnetic period λ described above; and Y-magnet array  112 B is mirror symmetric about a stage Y-Z plane that runs through the center of the stage-X dimension W yx  of Y-magnet array  112 B. 
     For example, the stage-X dimension width W yx  of Y-magnet array  112 B being an integer number of magnetic wavelengths (W yx =N m λ) minimizes force coupling with non-aligned coil traces  132 C, because the net force on magnet array  112 B will integrate to zero (i.e. will cancel itself out) over each wavelength λ of magnet array  112 B. Also, the mirror-symmetry of Y-magnet array  112 B about a stage Y-Z plane that is orthogonal to the stage-X direction and runs through the center of the stage-X dimension W yx  of Y-magnet array  112 B minimizes the net moment (about the Z axis and about the Y axis) due to the interaction of magnet array  112 B with stage-X-oriented coil traces  132 C. Similar characteristics of Y-magnet array  112 D eliminate or minimize cross-coupling from coil traces  132 A. 
     In an analogous manner, the characteristics of X-magnet array  112 A eliminate or reduce cross-coupling forces from coil traces  132 B. Such characteristics of X-magnet array  112 A include: X-magnet array  112 A includes magnetization segments which are generally elongated in the stage-X direction with varying magnetizations which are oriented orthogonally to the stage-X direction; the stage-Y dimension width W xy  of X-magnet array  112 A is W xy =N m λ where N m  is an integer and λ is the magnetic period λ described above; and X-magnet array  112 A is mirror symmetric about a stage X-Z plane that is orthogonal to the stage-Y direction and runs through the center of the stage-Y dimension W xy  of X-magnet array  112 A. Similar characteristics of X-magnet array  112 C eliminate or minimize cross coupling from coil traces  132 D. 
       FIG. 3F  shows a top view of the 2D array  500  magnetic field sensors  501  with respect to movable stage  110 . Each sensor  501  is sensitive to magnetic flux density in one, or two, or three non-parallel directions (e.g. in the stator-X, Y and/or Z directions). In the illustrated embodiment, sensors  501  are located at the intersection points between equally spaced, generally parallel lines  505  oriented in a first extension direction (e.g. the stator-X direction) and equally spaced, generally parallel lines  507  oriented in second extension direction (e.g. the stator-Y direction). Sensors  501  positioned generally on or suitably close to a line  505  may be said to be generally aligned with one another in a stator-X direction and may be said to belong to a stator-X oriented sensor row. Sensors  501  positioned generally on or suitably close to a line  507  may be said to be generally aligned with one another in a stator-Y direction and may said to belong to a stator-Y oriented sensor column. It may be desirable to have the first extension direction (lines  505 ) and the second extension direction (lines  507 ) of array  500  be orthogonal to each other. However, generally, the two extension directions of sensor array  500  can be in any non-parallel relation. In the illustrated  FIG. 3F  embodiment, sensors  501  are generally equally spaced along both the first extension (e.g. stator-X) direction and the second extension (e.g. stator-Y) direction with P x  being the sensor pitch along stator-X direction and P y  being the sensor pitch along stator-Y axis. These pitches P x , P y  or other references to the spacing or distance between sensors  501  may be interpreted to be the distance between the geometric center points of sensors  501 . 
     In some embodiments, these sensor pitches P x , P y  are set in general accordance with:
 
 P   X   =nλ   X   /N,  
 
 P   Y   =mλ   Y   /M,  
 
where λ X  is the stage-X oriented magnetic spatial period of Y-magnet arrays  112  (e.g. arrays  112 B,  112 D); λ Y  is the stage-Y oriented magnetic spatial period of X-magnet arrays  112  (e.g. arrays  112 A,  112 C); n, m, N and M are positive integer numbers; n, N selected such that
 
             n   N         
is not an integer; and m, M selected such that
 
             m   M         
is not an integer. As described in more detail below, this selection of the relationship between sensor pitches P x , P y  and magnetic periods λ X , λ Y  (together with suitable selection of widths of magnet arrays  112 —e.g. Y-magnet array width W yx =N my λ x  and X-magnet array width W xy =N mx λ y , N my  and N mx  are positive integers): may permit synchronous summing/averaging of sensor rows/columns which may aid in position detection; may permit sums/averages of stator-X oriented sensor rows to be insensitive to Y-magnet arrays (e.g. Y-magnet arrays  112 B,  112 D), and accordingly permit determination of the position of X-magnet magnet arrays (e.g. magnet arrays  112 A,  112 C) without impact from the fields of Y-magnet arrays; and may permit sums/averages of stator-Y oriented sensor columns to be insensitive to X-magnet arrays (e.g. X-magnet arrays  112 A,  112 C), and accordingly permit determination of the position of Y-magnet magnet arrays (e.g. magnet arrays  112 B,  112 D) without impact from the fields of X-magnet arrays.
 
Coil Array
 
     Additional detail of stator  120  and its coil arrays is now provided. As described above, stator  120  comprises a plurality of layers  128  of coil traces  126  which are generally linearly oriented in the working region  124 . Each layer  128  comprises coil traces  126  that are generally aligned with one another (e.g. generally linearly elongated in the same direction). In the illustrated embodiment of  FIGS. 3A-3E , vertically adjacent layers  128  (i.e. layers  128  next to one another in the stator-Z direction) comprise coil traces  126  that are orthogonally oriented with respect to one another. For example, coil traces  126 Y in layers  128 A,  128 C ( FIG. 3C ) are generally linearly oriented parallel to the stator-Y direction and coil traces  126 X in layers  128 B,  128 D are generally linearly oriented parallel to the stator-X direction. It will be appreciated that the number of layers  128  of coil traces  126  in stator  120  need not be limited to the four traces shown in the illustrated embodiment. In general, stator  120  may comprise any suitable number of layers  128  of coil traces  126 . Further, it is not a requirement that the orientations of coil traces  126  in vertically adjacent layers  128  be different from one another. Some embodiments may comprise a number of vertically adjacent layers  128  of Y-oriented traces  126 Y followed by a number of vertically adjacent layers  128  of X-oriented coil traces  126 X. 
     Stator  120  and its arrays of coils  122  may be fabricated using one or more printed-circuit boards (PCBs). PCBs can be manufactured using standard PCB fabrication, flat-panel display lithography, lithography and/or similar technology known in the art to provide coils  122  and coil traces  126 . Insulator layers  130  (such as FR4 core, prepreg, ceramic material and/or the like) may be fabricated or otherwise inserted between coil layers  128 . One or more coil layers  128  may be stacked together (i.e. in the stator-Z direction) in a single PCB board. In some embodiments, coil traces  126  generally elongated in the same direction (at different layers  128 ) may be connected in parallel or serially, depending on via design and/or connecting methods for the ends of coil traces  126 . In some embodiments, coil traces  126  generally elongated in the same direction (at different layers  128 ) are not connected to one another. 
     Coils  122  fabricated using PCB technology can accommodate sufficient current for controlling the motion of movable stage  110 . By way of non-limiting example, each coil  122  can be made from 6 oz copper (about 200-220 μm thick) or more. As discussed above, in working region  124 , each coil  122  is in the shape of a flat strip or coil trace  126 , which provides good thermal conductivity due to the high ratio of surface area to volume. The inventors have confirmed (via testing) that laminated copper can carry a sustained current density of 10 A/mm 2  with a 50° C. temperature rise above ambient without using an active heat sink. Another advantage of planar layers  128  of coils  122  and coil traces  126  is that the naturally stratified conductors that provide coils  122  make them ideally suitable for carrying AC current, because the self-generated alternating magnetic field can easily penetrate the conductor through top and bottom surfaces but generates only low self-induced eddy currents. 
     Multiple PCBs may be aligned side by side in both stator X and Y directions (similar to floor tiles) to provide the desired stator X-Y dimensions for working region  124 . Board-to-board lateral connections (in the stator-X and/or stator-Y directions) may be made at the edges by connecting pads, through-holes of edge-adjacent boards, copper wires and/or using other suitable bridging components of the like for electrically connecting conductors on adjacent PCB boards. In some embodiments, such bridging components may be located underneath the PCB boards (e.g. on the side opposite movable stage  110 ); in some embodiments, such bridging components may be additionally or alternatively located above the PCB boards or on the side(s) of the PCB boards. When PCBs are connected adjacent to one another in the stator-X and/or stator-Y directions, the end terminals (not shown) of coils  122  may be located at or near the perimeter of stator  120  for ease of wiring to the drive electronics. Connecting PCBs to one another in this manner allows displacement device  100  to be easily extended in both stator-X and stator-Y dimensions for various applications. When PCBs are connected to one another in the stator-X and/or stator-Y dimensions, the total number of coils  122  increases linearly with the stator X-Y dimensions of working region  124  of stator  120  (instead of quadratically, as is the case in some prior art techniques involving so-called “racetrack” coil designs). In some embodiments, coil traces  126  on stator X-Y adjacent PCB boards may be serially connected to one another to reduce the number of amplifiers (not shown) for driving current through coil traces  126 . In some embodiments, coil traces  126  on stator X-Y adjacent PCB boards may be individually controlled by separate amplifiers to increase the flexibility for multi-stage actuation and to reduce heat generation. 
     A single PCB board may be fabricated to have a thickness (in the stator-Z direction) of up to 5 mm (or more) using available PCB technology. When thicker boards are required for heavy-duty applications, multiple PCBs can be stacked vertically in the stator-Z direction. Another benefit of using PCB technology to fabricate stator  120  is the possibility of deploying large numbers of low-profile sensors (such as Hall-effect position sensor, capacitive position sensors and/or the like) directly on the board using daisy chain connections. 
     The PCB board of stator  120  may also include circuits to perform functions for column and row summing or averaging, as described below. These circuits may be on the same PCB board as the stator coil assembly or on a different PCB board attached to the stator coil assembly by epoxy, for example. 
     Magnet Arrays 
       FIGS. 4A and 4B  (collectively,  FIG. 4 ) are schematic partial cross-sectional views of layouts of magnet arrays  112  which may be used in movable stage  110  of the  FIG. 3  displacement device  100  and which are useful for showing a number of magnet array parameters. It can be observed that the layout of magnet arrays  112 A,  112 B,  112 C,  112 D in  FIG. 4A  is the same as that of magnet arrays  112 A,  112 B,  112 C,  112 D in  FIG. 3B . The layout of magnet arrays  112 A,  112 B,  112 C,  112 D in  FIG. 4B  is similar to that of magnet arrays  112 A,  112 B,  112 C,  112 D shown in  FIGS. 4A and 3B . The discussion in this section applies to both of the layouts shown in  FIGS. 4A and 4B . 
       FIG. 4  shows that X-magnet arrays  112 A,  112 C have widths W xy  and lengths L xx  and Y-magnet arrays  112 B,  112 D have widths W yx  and lengths L yy . In some embodiments, like the illustrated embodiments of  FIGS. 4A and 4B , W xy =W yx =W m  and/or L xx =L yy =L m , although this is not necessary. In the illustrated embodiment of  FIG. 4 , the lines associated with corresponding stage-Y oriented edges of X-magnet arrays  112 A,  112 C (i.e. the stage-Y oriented edges on the same relative sides of the arrays) are offset from one another in the stage-X direction by an offset O x  and the lines associated with proximate stage-X oriented edges of X-magnet arrays  112 A,  112 C (i.e. the stage X-oriented edges of the arrays that are most proximate to one another) are spaced apart from one another by a space T x . Similarly, in the illustrated embodiment of  FIG. 4 , the lines associated with corresponding stage-X oriented edges of Y-magnet arrays  112 B,  112 D (i.e. the stage-X oriented edges on the same relative sides of the arrays) are offset from one another in the stage-Y direction by an offset O y  and the lines associated with proximate stage-Y oriented edges of Y-magnet arrays  112 B,  112 D (i.e. the stage Y-oriented edges of the arrays that are most proximate to one another) are spaced apart from one another by a space T y . In some embodiments, like the illustrated embodiment O x =O y =O m  and/or T x =T y =T m , although this is not necessary. It can be observed that in the illustrated embodiment, movable stage  110  comprises a non-magnetic region  113  located in a center of its magnet arrays  112  and that the dimensions of non-magnetic region  113  are T x  by T y . In some embodiments, the dimensions T x , T y  are chosen to be greater or equal to the magnetic period λ, such that active coil traces  126  for two parallel magnet arrays  112  don&#39;t interfere with one another. As discussed above, for each magnet array  112 , the magnetization segments  114  and corresponding magnetization directions are uniform along their elongated dimensions L xx , L yy  and are oriented orthogonally to their elongated dimensions L xx , L yy . For each magnet array  112 , the magnetization segments  114  and corresponding magnetization direction vary along the direction of their widths W xy , W yx . While not expressly shown in the illustrated views, the magnet arrays  112  shown in  FIG. 4  may be mounted under a suitable table or the like which may be used to support an article (e.g. a semiconductor wafer) thereatop. 
     One particular non-limiting implementation of magnet arrays  112  is described above in connection with  FIG. 3D  (for Y-magnet array  112 B) and  3 E (for X-magnet array  112 A). In the description of magnet arrays  112  that follows, a comprehensive explanation is provided in the context of an exemplary Y-magnet array  112 B. X-magnet arrays may comprise similar characteristics where the X and Y directions and dimensions are appropriately interchanged. For brevity, in the description of Y-magnet array  112 B that follows, the alphabetic notation is dropped and Y-magnet array  112 B is referred to as magnet array  112 . Similarly, the magnetization segments  114 B of Y-magnet array  112 B are referred to as magnetization segments  114 . 
       FIG. 5A  shows an embodiment of a magnet array  112  substantially similar to magnet array  112 B described above in connection with  FIG. 3D . Magnet array  112  is divided, along the stage-X axis, into a number of magnetization segments  114  which are generally linearly elongated in the stage-Y axis direction. In the illustrated embodiment, the magnetization directions of magnetization segments  114  have a spatial periodicity with a period (or wavelength) λ along the stage-X axis; the width W yx  of magnet array  112  in the stage-X direction is given by W yx =N m λ where N m  is a positive integer (and N m =2 in the  FIG. 5A  embodiment); the magnetization directions of the various magnetization segments  114  are oriented in directions orthogonal to the stage-Y direction; the stage X-direction widths of the various magnetization segments  114  are either: λ/(2N t ) for the two outermost (edge) segments  114  or λ/N t  for the interior segments  114 , where N t  represents the number of different magnetization directions in magnet array  112  (and N t =4 in the  FIG. 5A  embodiment); and the magnetization of magnetization segments  114  is mirror symmetric about central Y-Z plane  118 . It will be appreciated that with W yx =N m λ and the magnetization of magnetization segments  114  being mirror symmetric about central Y-Z plane  118 , the outermost (edge) segments  114  have stage-X axis widths that are half the stage-X axis widths of interior segments  114  and that the outermost edge segments  114  have magnetizations that are oriented in along the stage-Z direction. 
       FIG. 5B  is another embodiment of a magnet array  112  suitable for use with the  FIG. 3  displacement device. The  FIG. 5B  magnet array  112  has characteristics similar to those of the  FIG. 5A  magnet array  112 , except that N m =1 and N t =4. It can be observed from  FIG. 5B  that the magnetic spatial period λ is defined even where the total stage-X axis width W yx  of the magnet array is less than or equal to λ. In the  FIG. 5B  case, the magnetization directions of magnetization segments  114  of magnet array  112  may be considered to be spatially periodic in the stage-X direction with a period λ, even though there is only a single period. 
     As discussed above, magnet arrays  112  that exhibit the properties of those shown in  FIGS. 5A and 5B  eliminate or reduce cross-coupling forces from coil traces  126  oriented in stator-X directions. Such characteristics of magnet arrays  112  shown in  FIGS. 5A and 5B  include: magnet arrays  112  including magnetization segments  114  which are generally elongated in the stage-Y direction with corresponding magnetizations oriented orthogonally to the stage-Y direction; the stage-X dimension width W yx  of magnet arrays  112  is W yx =N m λ where N m  is an integer and λ is the magnetic period λ described above; and magnet arrays  112  are mirror symmetric about a stage Y-Z axis that runs through the center of the stage-X dimension W yx  of magnet arrays  112 . 
       FIGS. 5C and 5D  show other embodiments of magnet arrays  112  suitable for use with the  FIG. 3  displacement device. In these embodiments, the magnetization directions of magnetization segments  114  have a spatial periodicity with a period (or wavelength) λ along the stage-X direction; the width W yx  of magnet array  112  in the stage-X direction is given by W m =(N m +0.5)λ where N m  is a non-negative integer (and N m =0 in the  FIG. 5C  embodiment and N m =1 in the  FIG. 5D  embodiment); the magnetization directions of the various magnetization segments  114  are oriented in directions orthogonal to the stage-Y direction; the magnetization of magnetization segments  114  is mirror anti-symmetric about central stage Y-Z plane  118 ; and the outermost (edge) segments  114  have magnetizations that are oriented in the Z direction and stage-X direction widths of λ/(2N t )=λ/8 (where N t =4 in the embodiments of both  FIGS. 5C and 5D ) which are half of the stage-X direction widths λ/N t =λ/4 for the interior segments  114 . In the  FIG. 5C  case, the magnetization directions of magnetization segments  114  of magnet array  112  may be considered to be spatially periodic in the stage-X direction with a period λ, even though magnet array  112  exhibits less than a single period λ. 
     When the width W yx , of magnet array  112  is a non-integer number of magnetic wavelengths λ (as in the case in the embodiments of  FIGS. 5C and 5D , for example), then there will be coupling of force or moment to magnet array  112  from current flow in non-aligned coil traces  126  that interact with the magnetic field of array  112 . For example, in the case of the Y-magnet arrays  112  shown in  FIGS. 5C and 5D  (which are mirror anti-symmetric about Y-Z plane  118 ), there will be coupling of moment in the rotational direction about Z to Y-magnet arrays  112  from current flow in coil traces oriented along the stator-X direction. This net moment can be compensated using suitable control techniques or using suitable arrangements of additional magnetic arrays  112  with different (e.g. opposite) magnetization patterns. 
       FIGS. 5E-5H  show other embodiments of magnet arrays  112  suitable for use with the  FIG. 3  displacement device. In these embodiments, the magnetization directions of magnetization segments  114  have a spatial periodicity with a period (or wavelength) λ along the stage-X direction; the width W yx  of magnet array  112  in the stage-X direction is given by W yx =N m λ/2, where N m  is a positive integer (and N m =1 in the  FIG. 5E  embodiment, N m =2 in the  FIG. 5F  embodiment, N m =3 in the  FIG. 5G  embodiment and N m =4 in the  FIG. 5H  embodiment); the magnetization directions of the various magnetization segments  114  are oriented in directions orthogonal to the stage-Y direction; and the outermost (edge) segments  114  have magnetizations that are oriented along the stage-X direction and stage-X direction widths of λ/(2N t )=λ/8 (where N t =4 in the embodiments of  FIGS. 5E and 5H ) which are half of the stage-X direction widths λ/N t =λ/4 for the interior segments  114 . Note that the central stage Y-Z plane  118  is not explicitly shown in  FIGS. 5E-5H . However, it will be appreciated that this stage Y-Z plane  118  divides the stage-X dimension W yx  of magnet array  112  in half. 
     In  FIGS. 5E and 5G , the magnetization of magnetization segments  114  is mirror symmetric about central stage Y-Z plane  118 , and the width W yx  of magnet array  112  in the stage-X direction is not an integer number of spatial periods λ. In the case of Y-magnet arrays  112  shown in  FIGS. 5E and 5G , there will be coupling of forces in the stage-Y direction to Y-magnet arrays  112  from current flow in coil traces  126  oriented along the stage-X direction. This net force can be compensated for using suitable control techniques or using suitable arrangements of additional magnetic arrays  112  with different (e.g. opposite) magnetization patterns. 
     In  FIGS. 5F and 5H , the magnetization of magnetization segments  114  is mirror anti-symmetric about central stage Y-Z plane  118 , and the width W ux  of magnet array  112  in the stage-X direction is an integer number of spatial periods λ. In the case of Y-magnet arrays  112  shown in  FIGS. 5F and 5H , there will be coupling of moment in the rotational direction around Z to Y-magnet arrays  112  from current flow in coil traces  126  oriented along the stator-X direction. This net moment can be compensated using suitable control techniques or using suitable arrangements of additional magnetic arrays  112  with different (e.g. opposite) magnetization patterns. 
       FIGS. 5I-5L  show other embodiments of magnet arrays  112  suitable for use with the  FIG. 3  displacement device. In these embodiments, the magnetization directions of magnetization segments  114  have a spatial periodicity with a period (or wavelength) λ along the stage-X direction; the width W yx  of magnet array  112  in the stage-X direction is given by W yx =N m λ/2, where N m  is a positive integer (and N m =1 in the  FIG. 5I  embodiment, N m =2 in the  FIG. 5J  embodiment, N m =3 in the  FIG. 5K  embodiment and N m =4 in the  FIG. 5L  embodiment); the magnetization directions of the various magnetization segments  114  are oriented in directions orthogonal to the stage-Y direction; and the stage-X direction widths of all of the magnetization segments  114  are λ/N t  (where N t =4 in the illustrated embodiments of  FIGS. 5I-5L . As the magnetization of magnetization segments in  FIGS. 5I-5L  is not mirror symmetric about central stage Y-Z plane  118 , there will be coupling of moment in the rotational direction around Z to Y-magnet arrays  112  from current flow in coil traces oriented along the stator-X direction. In addition, for the cases in  FIGS. 5I and 5K , as the width W yx  of magnet array  112  in the stage-X direction is not an integer number of spatial periods λ, there will be coupling of forces in the stator-Y direction to Y-magnet arrays  112  from current flow in coil traces  126  oriented along the stator-X direction. This net force and moment can be compensated using suitable control techniques or using suitable arrangements of additional magnetic arrays  112  with different (e.g. opposite) magnetization patterns. 
     In some embodiments, Y-magnet arrays  112  of  FIGS. 5A-5L  may be fabricated from a plurality of contiguous unit Y-magnetization segments  114 . Mechanically contiguous Y-magnetization segments  114  that are adjacent to one another in the stage-X direction are in contact with one another along their stage-Y dimensions. Unit Y-magnetization segments  114  may have stage-Y direction lengths L yy  and stage-X direction widths λ/(2N t ) or λ/(N t N) where N t  is the number of magnetization directions in a period λ as discussed above. In some embodiments, Y-magnetization segments  114  having stage-X direction widths λ/(N t ) may be fabricated from a pair of side-by-side magnetization segments  114  having stage-X direction widths λ(2N t ) and having their magnetization directions oriented in the same direction. In some embodiments, the stage-Z direction heights H yz  of the unit Y-direction magnetization segments  114  may be same as their stage-X direction widths—e.g. λ/(2N t ) or λ/(N t ). 
     As discussed above, a central non-magnetic spacer may be provided in magnet arrays  112 . In embodiments which are symmetric or mirror symmetric about central stage Y-Z plane  118 , such a non-magnetic spacer may divide the central magnetization segment  114  into a pair of “half-width” magnetization segments  114  (i.e. having stage-X direction widths similar to the stage-X direction widths of the edge segments  114 ). The resultant magnet arrays  118  remain symmetric or mirror symmetric about a central stage Y-Z plane  118 . In embodiments which are not symmetric about a central stage Y-Z plane  118 , different patterns may be used. 
       FIGS. 6A-6L  show magnet arrays  112  suitable for use with the  FIG. 3  displacement device  100  in accordance with particular embodiments. The magnet arrays  112  of  FIGS. 6A-6L  have features similar to those of magnet arrays  112  of  FIGS. 5A-5L , except that the magnet arrays  112  of  FIGS. 6A-6L  include non-magnetic spacers  136  centrally located (in their stage-X dimensions W yx ). Spacers  136  (of the Y-magnet arrays  112  shown in  FIGS. 6A-6L ) may be provided with a stage-X direction width g which is at least approximately equal to 
               g   =       (         N   g     5     +     1     1   ⁢   0         )     ⁢   λ       ,         
where N g  is a non-negative integer number. When the width g of spacers  136  exhibits this property, spacers  136  will have an attenuating (cancelling) effect on disturbance torques and/or forces created by the 5 th  order harmonic field of magnet array  112 . In general, the width g of the non-magnetic spacer  136  may be set to be at least approximately equal to
 
               =       (         N   g     k     +     1     2   ⁢   k         )     ⁢   λ       ,         
where N g  has the above described properties and k is the order of the harmonic of the magnetic field to be attenuated. In some embodiments, spacers  136  (of the Y-magnet arrays  112  shown in  FIGS. 6A-6L ) may be provided with a stage-X direction width g which is at least approximately equal to
 
               g   =           K   g     5     ⁢   λ     -     W   c         ,         
where K g  is a non-negative integer number and W c  is the stator-X direction width of coil traces  126  generally elongated in stator-Y direction. When the width g of spacers  136  exhibits this property, spacers  136  will have an attenuating (cancelling) effect on disturbance torques and/or forces created by the 5 th  order harmonic field of magnet array  112 . In general, the width g of the non-magnetic spacer  136  may be set to be at least approximately equal to
 
               =           K   g     k     ⁢   λ     -     W   c         ,         
where K g  and W c  have the above described properties and k is the order of the harmonic of the magnetic field to be attenuated.
 
     The magnet array  112  embodiments shown in  FIGS. 6A and 6B  have two sides arranged on either stage-X direction side of non-magnetic spacer  136 . Both the left and right sides (in the illustrated view) of the  FIG. 6A  magnet array  112  have magnetization patterns similar to those of magnet array  112  of  FIG. 5A ; and both the left and right sides of the  FIG. 6B  magnet array  112  have magnetization patterns similar to those of magnet array  112  of  FIG. 5B . The stage-X direction width W side  of each side of the magnet arrays  112  of  FIGS. 6A and 6B  (i.e. the stage-X direction distance between an edge of array  112  and the edge of non-magnetic spacer  136 ) is W side =N m λ where N m  is a positive integer and the total stage-X direction width of the magnet arrays  112  of  FIGS. 6A and 6B  is W yx =2N m λ+g, where N m =2 in  FIG. 6A  and N m =1 in  FIG. 6B . 
     The magnet array  112  embodiments shown in  FIGS. 6C and 6D  have two sides arranged on either stage-X direction side of non-magnetic spacer  136 . The left (in the illustrated view) sides of magnet arrays  112  shown in  FIGS. 6C and 6D  have magnetization patterns similar to those of magnet arrays  112  shown in  FIGS. 5C and 5D  respectively. The right (in the illustrated view) sides of magnet arrays  112  shown in  FIGS. 6C and 6D  have magnetization patterns that are opposite those of the left sides—i.e. as if the left side of the magnet array  112  was duplicated in the location of the right side of the magnet array  112  and then each individual magnetization segment  114  in the right side of the magnet array  112  was rotated 180° about its own central axis along which it is linearly elongated. The stage-X direction width W side  of each side of the magnet arrays  112  of  FIGS. 6C and 6D  is W side =(N m −0.5))λ, where N m  is a positive integer and the total stage-X direction width of the magnet arrays  112  of  FIGS. 6C and 6D  is W yx =(2N m −1)λ+g, where N m =1 in  FIG. 6C  and N m =2 in  FIG. 6D . 
     Similarly, the magnet array  112  shown in  FIGS. 6E, 6G, 6I, 6K  have two sides arranged on either stage-X direction side of non-magnetic spacer  136 , with their respective left (in the illustrated view) sides having magnetization patterns similar to  FIGS. 5E, 5G, 5I, 5K  magnet array  112  and their respective right (in the illustrated view) sides having magnetization patterns that are the opposite to those of the left (in the illustrated view) sides, where “opposite” has the same meaning as discussed above for the case of  FIGS. 6C and 6D . The stage-X direction widths W side  of each side of the magnet arrays  112  of  FIGS. 6E, 6G, 6I, 6K  is W side =(N m −0.5))λ, where N m  is a positive integer and the total stage-X direction width of the magnet arrays  112  of  FIGS. 6E, 6G, 6I, 6K  is W yx =(2N m −1)λ+g, where N m =1 in  FIG. 6E , N m =2 in  FIG. 6G , N m =1 in  FIG. 6I , N m =2 in  FIG. 6K . 
     The magnet arrays  112  shown in  FIGS. 6F, 6H, 6J, 6L  have two sides arranged on either stage-X direction side of non-magnetic spacer  136 , with both their left and right sides having magnetization patterns similar to those of magnet arrays  112  of  FIGS. 5F, 5H, 5J, 5L , respectively. The stage-X direction width W side  of each side of the magnet arrays  112  of  FIGS. 6F, 6H, 6J, 6L  is W side =N m λ where N m  is a positive integer and the total stage-X direction width of the magnet arrays  112  of  FIGS. 6F, 6H, 6J, 6L  is W yx =2N m λ+g, where N m =1 in  FIG. 6F , N m =2 in  FIG. 6H , N m =1 in  FIG. 6J , N m =2 in  FIG. 6L . The magnet arrays  112  shown in  FIGS. 6A-6L  may be fabricated in a manner similar to that described above for  FIGS. 5A-5L . 
     Layout of Magnet Arrays 
     As discussed above,  FIGS. 4A and 4B  show layouts of the magnet arrays  112  which may be used in movable stage  110  of displacement device  100  in accordance with particular embodiments. In accordance with particular embodiments, when arranging magnet arrays  112  on movable stage  110 , the spacing between corresponding Y-oriented edges of X-magnet arrays  112 A,  112 C (i.e. Y-oriented edges on the same respective sides of the arrays) may be given by W xy +T x  and (in the case of the  FIG. 4  embodiment), this spacing may be given by W xy +T x =N s λ/2 where N S  is a positive integer and λ is the magnetic period of the X-magnet arrays  112 A,  112 C. Similarly, in accordance with particular embodiments, the spacing between corresponding X-oriented edges of Y-magnet arrays  112 B,  112 D (i.e. X-oriented edges on the same respective sides of the arrays) may be given by W yx +T y  and (in the case of the  FIG. 4  embodiment), this spacing may be given by W yx +T y =N S λ/2 where N S  is a positive integer and λ is the magnetic period of the Y-magnet arrays  112 B,  112 D. Where the spacing of adjacent parallel magnet arrays  112  (e.g. a pair of X-magnet arrays  112 , such as X-magnet array  112 A and X-magnet array  112 C in the case of the  FIG. 4  embodiment and/or a pair of Y-magnet arrays  112 , such as Y-magnet arrays  112 B and Y-magnet arrays  112 D, in the case of the  FIG. 4  embodiment) are designed to have this feature, then the current distribution in the active coil traces  126  for each parallel magnet array  112  can be substantially similar in spatial distribution (i.e. in phase), provided that the parallel magnet arrays  112  have the same magnetization pattern ( FIG. 7A ) and N S  is even or the parallel magnet arrays  112  have opposite magnetization patterns ( FIG. 7B ) and N S  is odd. 
     In some embodiments, two parallel magnet arrays  112  on movable stage  110  (e.g. a pair of X-magnet arrays  112 , such as X-magnet arrays  112 A, 112 C in the case of the  FIG. 4  embodiment and/or a pair of Y-magnet arrays  112 , such as Y-magnet arrays  112 B,  112 D in the case of the  FIG. 4  embodiment) may comprise magnetization segments  114  with magnetization orientations that are the same as one another. This characteristic is shown, for example, in  FIG. 7A  where Y-magnet array  112 B and Y-magnet array  112 D comprise magnetization segments  114 B,  114 D with magnetization orientations that are the same as one another. In some embodiments, two parallel magnet arrays  112  on movable stage  110  may comprise magnetization segments  114  with magnetization orientations that are the opposites of one another—i.e. as if each magnetization segment  114  is individually rotated 180° about a corresponding central axis along which it is linearly elongated. This characteristic is shown, for example, in  FIG. 7B , where magnet array  112 B and magnet array  112 D comprise magnetization segments  114 B,  114 D with magnetization orientations that are opposite to one another. 
     In some embodiments, the elongated dimensions L xx , L yy  of magnet arrays  112  shown in  FIGS. 4A and 4B  is set at least approximately equal to L m =L xx =L yy =N L λ, where N L  is a positive integer number and λ is the magnetic period. Where magnet arrays  112  exhibit this characteristic, there will be a further reduction in the coupling force generated between a magnet array  112  and current flowing in coil traces  126  in directions orthogonal to the elongated dimension of magnet array  112 . 
     The layout of magnet arrays  112  shown in  FIGS. 4A and 4B  is not the only possible layout for magnet arrays  112  that could be used for movable stage  110  of the  FIG. 3  displacement device  100 . More particularly, another possible layout of magnet arrays  112  suitable for use in movable stage  110  of the  FIG. 3  displacement device  100  is shown in  FIG. 8 . 
       FIG. 8  shows a schematic cross-sectional view of layout of magnet arrays  112 A,  112 B,  112 C,  112 D which may be used for movable stage  110  of the  FIG. 3  displacement device  100  in accordance with a particular non-limiting embodiment. The  FIG. 8  layout of magnet arrays  112  differs from the  FIG. 4  layout of magnet arrays  112  because magnet arrays  112  are shaped (e.g. as squares) such that non-magnetic region  113  is eliminated and all of the undersurface area of movable stage  110  is occupied by magnet arrays  112 . In other words, the two X-magnet arrays  112 A,  112 C of the  FIG. 8  embodiment and the two Y-magnet arras  112 B,  112 D of the  FIG. 8  embodiment have no space between their proximate elongation direction oriented edges, so that T x =0 and T y =0, where T x  and T y  have the meanings described above in connection with  FIG. 4 . In the illustrated embodiment of  FIG. 8 , each magnet array  112  comprises a pattern of magnetization segments  114  having the characteristics of those shown in  FIG. 5A , although it will be appreciated that magnet arrays  112  of the  FIG. 8  layout could be provided with magnetization segments  114  exhibiting characteristics of any of the magnet arrays  112  and/or magnetization segments  114  described herein—e.g. exhibiting any of the magnetization patterns shown in  FIGS. 3, 4, 5A-5L and/or 6A-6L . 
     The characteristics of each individual magnet array  112  in the layouts of  FIG. 8  (e.g. the orientations of magnetization segments  114 , the elongated dimension lengths L xxx , L yy , the widths W xy , W yx  and the like) can be similar to any of those described herein—e.g. exhibiting any of the magnetization patterns shown in  FIGS. 5A-5L and 6A-6L . 
     2D Array of Magnetic Field Sensors 
       FIG. 9  depicts sensor array  500  and the distribution of magnetic field sensors  501  in more detail. Sensors  501  of sensor array  500  in the  FIG. 9  embodiment are generally located at the intersection points between equally spaced, generally parallel lines  505  oriented in a first extension direction (e.g. the stator-X direction) and equally spaced, generally parallel lines  507  oriented in a second extension direction (e.g. the stator Y-direction). It may be desirable to have the first extension direction (lines  505 ) and the second extension direction (lines  507 ) of array  500  be orthogonal to each other. However, generally, the two extension directions of sensor array  500  can be in any non-parallel relation. In the illustrated embodiment, sensors  501  in array  500  are arranged stator-X oriented sensor rows and stator-Y oriented sensor columns, where: sensors in a stator-X oriented sensor row are generally aligned with one another along a corresponding line  505  (e.g. along a corresponding stator-X direction) with each adjacent pair of sensors  501  in the stator X-oriented sensor row separated from one another by a pitch P x ; and sensors in a stator-Y oriented sensor column are generally aligned with one another along a corresponding line  507  (e.g. along a corresponding stator-Y direction) with each adjacent pair of sensors  501  in the stator Y-oriented sensor column separated from one another by a pitch P y . Sensors  501  in array  500  of the  FIG. 9  embodiment for a 2D array may be labelled E ij , where i is the row index (indicating a sensor&#39;s location in the stator-Y direction) and j is the column index (indicating the sensors&#39;s location in the stator-X direction). In some embodiments, sensors  501  may be distributed over the working region  124  of stator  120 . 
     In some embodiments, the pitches P x  and P y  are set in general accordance with
 
 P   X   =nλ   X   /N   (1a)
 
 P   Y   =mλ   Y   /M   (1b)
 
where λ X  is the stage-X oriented magnetic spatial period of Y-magnet arrays  112  (e.g. arrays  112 B,  112 D); λ Y  is the stage-Y oriented magnetic spatial period of X-magnet arrays  112  (e.g. arrays  112 A,  112 C); n, m, N and M are positive integer numbers; n, N selected such that n/N is not an integer; and m, M selected such that
 
             m   M         
is not an integer. For example, P X  can be set at λ 2 /2, or 3λ X /2, or 5λ X /2, or λ X /3, or 2λ X /3, or 4λ X /3, or 5λ X /3, or 7λ X /3, or λ X /4 or 3λ X /4, and so on; P Y  can be set at λ Y /2, or 3λ Y /2, or 5λ Y /2, or λ Y /3, or 2λ Y /3, or 4λ Y /3, or 5λ Y /3, or 7λ Y /3, or λ Y /4, or 3λ Y /4, and so on. As described in more detail below, this selection of the relationship between sensor pitches P x , P y  and magnetic periods λ X , λ Y  (together with suitable selection of widths of magnet arrays  112 —e.g. Y-magnet array width W yx =N my λ x  and X-magnet array width W xy =N mx λ y , N my  and N mx  are positive integers) may permit synchronous summing/averaging of sensor rows/columns which may aid in position detection; may permit sums/averages of stator-X oriented sensor rows to be insensitive to Y-magnet arrays (e.g. Y-magnet arrays  112 B,  112 D), and accordingly permit determination of the position of X-magnet magnet arrays (e.g. magnet arrays  112 A,  112 C) without impact from the fields of Y-magnet arrays; and may permit sums/averages of stator-Y oriented sensor columns to be insensitive to X-magnet arrays (e.g. X-magnet arrays  112 A,  112 C), and accordingly permit determination of the position of Y-magnet magnet arrays (e.g. magnet arrays  112 B,  112 D) without impact from the fields of X-magnet arrays. In some embodiments at least two of the stator-X direction rows (e.g. rows oriented along lines  505 ) are spaced apart from each other by a distance of ¼λ Y  in the stator-Y direction. Similarly, in some embodiments, at least two of the stator-Y direction columns (e.g. columns oriented along lines  507 ) are spaced apart from each other by a distance of ¼λ X  in the stator-X direction. In some embodiments at least two of the stator-X direction rows (e.g. rows oriented along lines  505 ) are spaced apart from each other by a distance of g¼λ Y  in the stator-Y direction where g is an odd integer greater than zero. Similarly, in some embodiments, at least two of the stator-Y direction columns (e.g. columns oriented along lines  507 ) are spaced apart from each other by a distance of h¼λ x  in the stator-X direction where h is an odd integer greater than zero.
 
     In some embodiments, each magnetic field sensor E i,j  may contain several sub-units.  FIG. 10  shows one particular embodiment of a sensor  501  comprising a plurality (e.g. 4) of sub-units  531   a ,  531   b ,  531   c ,  531   d  (collectively and individually, sub-units  531 ). Sensor  501  of  FIG. 10  may form one of the sensors  501  in array  500  of  FIGS. 3F and 9 , for example. Each sub-unit  531  is capable of measuring magnetic field flux density in one to three non-parallel directions. In some embodiments, the stator-Y direction distance S 2  between sub-units  531   a  and  531   d  may be selected to conform generally with 
                 S   2     =       λ   Y     ⁡     (       v   5     +     1   10       )         ,         
where v is a suitable non-negative integer. In some embodiments, the stator-X direction distance S 1  between sub-units  531   b  and  531   a  may be selected to conform generally with
 
                 S   1     =       λ   X     ⁡     (       v   5     +     1   10       )         ,         
where v is a suitable non-negative integer number. The output of the  FIG. 10  sensor  501  may comprise a sum and/or average of the outputs from its sub-units  531 —it will be appreciated that determining an average typically comprises determining a sum. This means that the output of the  FIG. 10  sensor  501  provides a sum/average value of magnetic field intensity at a plurality (e.g. 4) discrete points. The summing/averaging operation can be either performed by a digital computing device (e.g. controller  504 ) or by an analog circuit. By setting S 1  to conform generally with
 
                 S   1     =       λ   X     ⁡     (       v   K     +     1     2   ⁢   K         )         ,         
the sum/average value of four sub-units  531  will filter off the magnetic field distortion of K th  order harmonics from Y-magnet arrays, where v is a non-negative integer and K is an integer greater than 1; and by setting S 2  to conform generally with
 
                 S   2     =       λ   Y     ⁡     (       v   K     +     1     2   ⁢   K         )         ,         
the sum/average value of four sub-units  531  will filter off the magnetic field distortion of K th  order harmonics from X-magnet arrays, where v is a non-negative integer and K is an integer greater than 1. Where sensors  501  comprise multiple sub-units  531 , the pitches P x , P y  or other references to the spacing or distance between sensors may be interpreted to be the distance between the geometric center points of the plurality of the sub-units  531  that make up sensors  501 .
 
       FIGS. 11A-11C  show various techniques for extracting measurement signals from sensors  501  which may be used in particular embodiments. In  FIG. 11A , sensor  501  comprises a raw hall-effect sensor element  502  excited with a suitable excitation current I s . The output voltage V o  of the  FIG. 11A  magnetic field sensor  501  is a voltage potential difference (differential format) of two terminals of the hall-effect sensor element  502 . Hall-effect sensor element  502  comprise four terminals, I+, I−, V+, V−. The I+ and I− are connected to a voltage or current source so that a bias current I s  can flow from I+ to I−. The V+ and V− are output voltage terminals, and their voltage potential difference V o  is proportional to externally applied magnetic field flux density. In  FIG. 11B , magnetic field sensor  501  comprises a raw hall-effect sensor element  502  (similar to that of  FIG. 11A ), a suitable amplifier  503  and other suitable electronics circuits such as excitation current supply (not shown) for hall-effect sensor element  502  and voltage supply for amplifier  503  (not shown). The output V o  of the  FIG. 11B  magnetic field sensor  501  is the output voltage of amplifier  503  reference to a ground (or to some other suitable reference) Amplifier  503  can be a differential amplifier or an instrument amplifier or other type of suitable amplifier. In  FIG. 11C , a general structure of a magnetic field sensor  501  is shown. Magnetic field sensor  501  comprises a magnetic field sensing element  502  and a suitable processing circuit  503 A. The output of the  FIG. 11C  sensor  501  can be in a format of analog voltage (a differential voltage or a ground-referenced voltage), analog current, or a digital signal according to a transmission protocol such as but not limited to SPI or I2C. In some particular embodiment, processing circuit  503 A can be as simple as two lines, as shown in  FIG. 11A . Magnetic field sensing element  502  can be a hall-effect sensor element, a magneto-resistive sensing element, or a magneto-strictive sensing element, or any suitable sensor element that is sensitive to magnetic field flux density. 
     One consideration associated with the 2-D grid layout  500  of sensors  501  (e.g. of the embodiments shown in  FIGS. 3F and 9 ) is the potentially large number of outputs to be processed. By way of non-limiting example only, a 30 by 30 sensor array  500  contains 900 sensors and 900 corresponding outputs and each output can be a scalar or a vector of length 2 to 3, depending on the number of magnetic field directions that each sensor  501  can measure. To reduce the number of outputs to be processed and simplify the output signal processing and/or to otherwise aggregate outputs from various sensors  501 , the sum and/or average of sensors  501  in each stator-X oriented sensor row and each stator-Y oriented sensor column can be used, instead of processing the output of each sensor  501  directly. It will be appreciated that determining an average typically comprises determining a sum. In some embodiments, controller  504  is configured (e.g. programmed) to determine the stator-X direction position of movable stage  110  based on the sum and/or average of sensors  501  in each of a plurality of stator-Y oriented sensor columns. Controller  504  may be additionally or alternatively configured to determine the stator-Y direction position of movable stage  110  based on the sum and/or average of sensors  501  in each of a plurality of stator X-oriented sensor rows. 
     Mathematically, we can convert the outputs of sensors  501  outputs according to 
                     column   ⁢           ⁢   averaging   ⁢           ⁢     A     Y   ,   j         =       1     N   Y       ⁢       ∑     i   =   1       N   Y       ⁢           ⁢     E     i   ,   j                   (     2   ⁢   a     )                 row   ⁢           ⁢   averaging   ⁢           ⁢     A     X   ,   i         =       1     N   X       ⁢       ∑     j   =   1       N   X       ⁢           ⁢     E     i   ,   j                   (     2   ⁢   b     )               
where A X,i  is the average of the outputs of an i th  group sensors  501  distributed along the first extension direction (e.g. the average of the outputs of sensors  501  in the i th  stator-X oriented sensor row), N X  is the number of sensors  501  in the i th  group/row, A Y,j  is the average of the outputs of a j th  group of sensors  501  distributed along the second extension direction (e.g. the average of the outputs of sensors  501  in the j th  stator-Y oriented sensor column), and N Y  is the number of sensors  501  in the j th  group/column. It will be appreciated that summing these outputs is involved in computing these averages and may be performed using similar equations, but without dividing by N x  or N y . One non-limiting method of setting N x  and N y  comprises selecting N x *P x  to be greater than or equal to the stage-X direction length L xx  of a X-magnet array, and N y *P y  is greater than the stage-Y direction length L yy  of a Y-magnet array, as described in more detail below. In the example described above, this summing and/or averaging reduces the original 900 outputs E i,j  (i=1, . . . 30, j−1, . . . 30) to 60 outputs: A X,i  (i=1, . . . , 30) and A Y,j  (j=1, . . . , 30). As a result, the number of outputs to be processed is significantly reduced. It should be noted that there can be any number of groups of sensors  501  distributed along the first extension direction (e.g. stator-X oriented sensor rows) and/or any number of groups of sensors  501  distributed along the second extension direction (e.g. stator-Y oriented sensor columns) and that the number of groups in each extension direction do not have to be equal to one another.
 
     For brevity, the remainder of this description refers to the summing/averaging over groups of sensors in the first and second extension directions as summing/averaging over rows and columns, without loss of generality. The column and row summing/averaging operation for sensors  501  can be implemented either digitally (e.g. by controller  504 ) or by suitable analog circuitry.  FIG. 12A  shows one non-limiting embodiment of such column and row summing/averaging operations implemented by analog circuitry. For clarity,  FIG. 12A  only expressly depicts the circuits for the 5 th  and 6 th  stator-Y oriented sensor columns and the 3 rd  and 4 th  stator-X oriented sensor rows, it being understood that other stator-X oriented rows and other stator-Y oriented columns can be summed/averaged in a similar way. Each resistor value R may have slightly different value in order to precisely compensate the response (i.e. calibrate) for non-uniformity among sensors  501 . Each sensor  501  may be connected to (or comprise) an associated analog circuit, for purposes of: buffering its output, adjusting its offset, or adjusting its scaling factor, and/or converting its output between a differential voltage signal and a single-ended voltage signal. It should be noted that the sensors  501  and associated circuitry of  FIG. 12A  can be implemented on a rigid or flexible printed circuit board (where all the sensors  501 , resistors, and operational amplifiers are installed); such a printed circuit board can be one of the same printed circuit board(s) used for implementing the stator coil assembly or a different printed circuit board bonded to one of the printed circuit board(s) used to implement the stator coil assembly, with epoxy, for example. 
     In one embodiment, each stator-Y oriented column summing trace  514  in  FIG. 12A  is coincident with or close to a stator-Y oriented line associated with each stator-Y oriented column; each stator-X oriented row summing trace  515  in  FIG. 12A  is coincident with or close to a stator-X oriented line associated with each stator-X oriented row. In the particular case of  FIG. 12A , the stator-Y oriented column summing trace  514  for the sensors E i,5  (i=1, . . . 30) in the 5 th  stator-Y oriented column may go through or close to the centers of the sensing elements of the sensors E i,5  (i=1, . . . 30) in the 5 th  stator-Y oriented column; and the stator-X oriented row summing trace  515  for the sensors E 3,j  (j=1, . . . 30) in the 3 rd  stator-X oriented row may go through or close to the centers of sensing elements of the sensors E 3,j  (j=1, . . . 30) in the 3 rd  stator-X oriented row. Here the centers of the sensing elements of the sensors may be understood to mean the geometric center of a raw hall-effect sensor element. 
     When a Y-magnet array  112 B,  112 D travels in stator-X direction at high speeds, a column sum/average result as processed in  FIG. 12A  not only contains the magnetic field from the Y-magnet array  112 B,  112 D, but also includes the back-emf induced voltage. When a column summing trace  514  goes through the stator-X dimension center of a sensing element, the back emf voltage and the sensor output are proportional. This means the back-emf voltage only creates a scaling error. Due to identical scaling errors for all column average values, there will be no error in calculating stator-X position of a Y-magnet array  112 B,  112 D by using the algorithm discussed later. However, for stator-Z position calculation, suitable compensation (e.g. scaling and/or offsetting) of back-emf may be used to increase the accuracy of the result. 
       FIG. 12B  shows another embodiment of such column/row summing/averaging operation. Each sensor  501  comprises a raw hall-effect sensor element. The column summing/averaging operation for sensors  501  along 5 th  stator-Y oriented sensor column is implemented through vertical summing traces  510  and  511  and a summing operational amplifier  530  to produce a sum/average value A Y,5 . The row summing/averaging operation for sensors  501  along 4 th  stator-X oriented sensor row is implemented through horizontal summing traces  512  and  513  and a summing operational amplifier  531  to produce sum/average value A X,4 . The row/column summing/averaging operation for sensors  501  along other rows or columns may be implemented in a similar way. 
     To simplify the signal processing associated with the outputs of sensors  501 , it may be desirable: to minimize the sensitivity of the sums/averages of the stator-Y oriented sensor columns A Y,j  to motion of the X-magnet arrays (for example,  112 A and  112 C in  FIG. 3F ); to minimize the sensitivity of the sums/averages of the stator-Y oriented sensor columns A Y,j  to stator-Y direction motion of Y-magnet arrays (for example  112 B and  112 D in  FIG. 3F ); to minimize the sensitivity of the sums/averages of the stator-X oriented sensor rows A X,i  to motion of the Y-magnet arrays (for example,  112 B and  112 D in  FIG. 3F ); and/or to minimize the sensitivity of the sums/averages of the stator-X oriented sensor rows A X,i  to stator-X direction motion of X-magnet arrays (for example  112 A and  112 C in  FIG. 3F ). To achieve these desires, the stage-Y direction length L yy  of the Y-magnet arrays  112 B,  112 D may be set at an integer multiple of the pitch P Y ; and/or the stage X-direction length L xx  of the X-magnet arrays  112 A,  112 C may be set at an integer multiple of the pitch P X . Additionally, in some embodiments, the stage-Y direction width W xy  of each X-magnet array  112 A,  112 C may be set to be an integer multiple of its magnetic spatial period λ Y (e.g. W xy =N mx λ y , N mx  is a positive integer); and/or stage-X direction width W yx  of each Y-magnet array  112 B,  112 D may be set to be an integer multiple of its magnetic spatial period λ X  (e.g. W yx =N my λ x , N my  is a positive integer). Further, in some embodiments, the relationship between the sensor pitch P x  and the magnetic spatial period λ X  of the Y-magnet arrays  112 B,  112 D conforms with equation (1a) described above; and/or the relationship between the sensor pitch P y  and the magnetic spatial period λ Y  of the X-magnet arrays  112 A,  112 C conforms with equation (1b) described above. With this configuration, the stator-X and stator-Z positions of Y-magnet arrays  112 B,  112 D can be derived from the sums/averages A Y,j  of the stator-Y oriented sensor columns and the stator-Y and stator-Z positions of X-magnet arrays  112 A,  112 C can be derived from the outputs A X,i  of the stator-X oriented sensor rows, as described in more detail below. 
     The remaining problem is to determine stator-X and stator-Z position of a Y-magnet array  112 B,  112 D from the available sums/averages A Y,j  (j=1, 2, 3, . . . , N) of the stator-Y-oriented sensor columns; and to determine stator-Y and stator-Z position of an X-magnet array  112 A,  112 C from the available sums/average A X,j  (j=1, 2, 3, . . . , M) of the stator-X oriented sensor rows. As these two issues can be solved in a similar way, the following discussion only focuses on the first one: to derive stator-X and stator-Z motion of a Y-magnet array  112 B,  112 D from the available sums/averages A Y,j =1, 2, 3, . . . , N) of the stator-Y-oriented sensor columns. 
       FIG. 13A  shows one non-limiting example embodiment, where a Y-magnet array  112 B with spatial period λ X =λ is shown in relation to (e.g. above) a plurality of sums/averages of stator-Y oriented sensor columns. Each of A 1 , B 1 , A 1 ′, B 1 ′, A 2 , B 2 , . . . represents the sum/average A Y,j  of the sensors  501  in a corresponding stator-Y oriented sensor column. The stator-x direction pitch P x  of the stator-Y oriented sensor columns is set at P X =λ/4. 
     Stator-Y oriented sensor columns which are separated from one another by a stator-X direction distance λ or an integer multiple of λ may be referred to herein as synchronous stator-Y oriented sensor columns or, for brevity, synchronous columns. The sum/average of the outputs of sensors  501  in each synchronous column may be referred to as a corresponding synchronous stator-Y oriented column value or, or brevity, a synchronous column value. In an analogous manner, stator-X oriented sensor rows which are separated from one another by a stator-Y direction distance λ or an integer multiple of λ may be referred to herein as synchronous stator-X oriented sensor rows or, for brevity, synchronous rows. The sum/average of the outputs of sensors  501  in each synchronous row may be referred to as a corresponding synchronous stator-X oriented row value or, or brevity, a synchronous row value. 
     For the particular configuration shown in  FIG. 13A  embodiment, sets of synchronous column values include: {A 1 , A 2 , A 3 , A 4 , . . . }, {B 1 , B 2 , B 3 , B 4 , . . . }, {A 1 ′, A 2 ′, A 3 ′, A 4 ′, . . . } and {B 1 ′, B 2 ′, B 3 ′, B 4 ′, . . . }. We may then define synchronous sum values and/or synchronous average values to be the sum/average of synchronous column values. For example, for the particular embodiment of  FIG. 13A , the synchronous average values may be defined to be: 
                   A   =       1     n   1       ⁢     (       A   1     +     A   2     +     A   3     +     A   4     +   ⋯   +     A     n   1         )               (     3   ⁢   a     )                 A   ′     =       1     n   2       ⁢     (       A   1   ′     +     A   2   ′     +     A   3   ′     +     A   4   ′     +   ⋯   +     A     n   2         )               (     3   ⁢   b     )               B   =       1     n   3       ⁢     (       B   1     +     B   2     +     B   3     +     B   4     +   ⋯   +     A     n   3         )               (     3   ⁢   c     )                 B   ′     =       1     n   4       ⁢     (       B   1   ′     +     B   2   ′     +     B   3   ′     +     B   4   ′     +   ⋯   +     A     n   4         )               (     3   ⁢   d     )               
Where n 1 , n 2 , n 3 , n 4  are the number of synchronous column values in each corresponding set. It will be appreciated that synchronous sum values may be determined in a manner similar to that of determining synchronous average values, except that it is not necessary to divide the sums by n 1 , n 2 , n 3 , n 4 .
 
     It will be appreciated by those skilled in the art that the synchronous sum/average values A and A′ are 180 degrees out of phase with one another, the synchronous sum/average values B and B′ are 180 degrees out of phrase with one another and that α=A−A′ and β=B−B′ are quadrature signals 90 degrees out of phase with one another. In particular, α and β represent two sinusoidal functions of the stator-X position of Y-magnet array  112 B, whose amplitudes are exponentially related to the magnet array stator-Z motion, as follows: 
     
       
         
           
             
               
                 
                   α 
                   = 
                   
                     
                       A 
                       - 
                       
                         A 
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                     = 
                     
                       
                         C 
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                       ⁢ 
                       
                         e 
                         
                           - 
                           
                             
                               2 
                               ⁢ 
                               π 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               z 
                             
                             λ 
                           
                         
                       
                       ⁢ 
                       
                         cos 
                         ⁡ 
                         
                           ( 
                           
                             
                               2 
                               ⁢ 
                               π 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               x 
                             
                             λ 
                           
                           ) 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     4 
                     ⁢ 
                     a 
                   
                   ) 
                 
               
             
             
               
                 
                   β 
                   = 
                   
                     
                       B 
                       - 
                       
                         B 
                         ′ 
                       
                     
                     = 
                     
                       
                         C 
                         0 
                       
                       ⁢ 
                       
                         e 
                         
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                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               z 
                             
                             λ 
                           
                         
                       
                       ⁢ 
                       
                         sin 
                         ⁡ 
                         
                           ( 
                           
                             
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     From α=A−A′ and β=B−B′, we can derive the stator-X direction and stator-Z direction positions of Y-magnet array  112 B at very high resolution, such as 100 to 1000 times smaller than the spatial period λ X =λ of Y-magnet array  112 B. Particularly, the stator-X direction position of a Y-magnet array  112 B can be derived with the well-known quadrature decoding method: 
                   x   =     atan   ⁢           ⁢   2   ⁢     (       B   -     B   ′       ,     A   -     A   ′         )     ⁢     λ     2   ⁢   π                 (   5   )               
where atan 2 is the arctangent function with two arguments. In particular, a tan 2(y,x) may be defined (in radians) over the range (−π, π] as
 
                     atan   ⁢           ⁢   2   ⁢     (     x   ,   y     )       =     {             arctan   ⁡     (     y   x     )       ⁢                     x   &gt;   0     ⁢                         arctan   ⁡     (     y   x     )       +   π             y   ≥   0     ,     x   &lt;   0                   arctan   ⁡     (     y   x     )       -   π             y   &lt;   0     ,     x   &lt;   0                   +     π   2       ⁢                     y   &gt;   0     ,     x   =   0                   -     π   2       ⁢                     y   &lt;   0     ,     x   =   0                 undefined   ⁢                     y   =   0     ,     x   =   0                       (   6   )               
and can be mapped to the range [0,2π) by adding 2π to any negative results.
 
     In some embodiments, controller  504  may be configured to use synchronous sum/average values to determine a stator-Z position of movable stage  110 . In the  FIG. 13A  example, the stator-Z direction position of Y-magnet array  112 B can be derived according to: 
                   z   =       λ     2   ⁢   π       ⁢     ln   (       C   0             (     B   -     B   ′       )     2     +       (     A   -     A   ′       )     2           )               (   7   )               
where the constant C 0  can be calibrated when magnet array  112 B is at position z=0 (i.e. magnet array  112 B sits on top of sensors  501 ) or at another convenient reference position.
 
     Although the sum/average of the outputs of sensors  501  in each individual sensor-Y oriented column (e.g. each individual column value A 1 , B 1 , A 1 ′, B 1 ′, A 2 , B 2 , . . . ) is not a sinusoidal function of stator-X direction position of Y-magnet array  112 B due to the finite stage-X-width of Y-magnet array  112 B, the synchronous sum/average values (e.g. A, A′, B, B′) are sinusoidal functions. These synchronous sum/average values can effectively eliminate the fringing field effect of magnet array  112 B and help produce accurate position information. 
     As shown in  FIG. 13B , the stator-X direction scope or range R x  of stator-Y oriented sensor columns used to obtain synchronous sum/average values (e.g. A, A′, B, B′) in the above-described synchronous averaging operation is greater than the stator-X direction width W yx  of Y-magnet array  112 B by a suitable distance L ext  on both sides of magnet array  112 B—that is R x =W yx +2L ext . Synchronous column values beyond this range R x  can also be included to calculate synchronous sum/average values, but synchronous column values outside of this range R x  make little contribution to the accuracy of the position determination due to the fact that the magnetic field strength beyond the range R x  is relatively weak. A typical choice of L ext  is a value between λ/4 and λ. Accordingly, the spacing T y  between two Y-magnet arrays  112 B,  112 D may be chosen to be at least T y =2L ext . Similarly, the spacing T x  between two X-magnet arrays  112 A,  112 C may be chosen to be at least T x =2L ext . For larger values of L ext , more accurate results may be obtained, but at the cost of computational expense and larger spacing T x , T y  between magnet arrays  112 . For example, L ext  can be set at λ/2, or λ/4, or 3λ/4. In some embodiments, L ext  is set at λ/2. 
     In some embodiments, sensor array  500  disclosed herein can be used to determine an absolute position of a magnet array  112  without requiring a homing operation during the start of system operation or calibration. Referring to  FIG. 13A , we may define groups of stator-Y oriented sensor columns to be the stator-Y oriented sensor columns in a particular magnetic period λ. For example, in the  FIG. 13A  embodiment, groups of stator-Y oriented sensor columns may be defined to be sensor group  1 ={A 1 ,B 1 ,A 1 ′,B 1 ′}; sensor group  2 ={A 2 ,B 2 ,A 2 ′,B 2 ′}; sensor group  3 ={A 3 ,B 3 ,A 3 ′,B 3 ′}; sensor group  4 ={A 4 ,B 4 ,A 4 ′,B 4 ′}, and so on. Using such defined groups and the sum/average of the outputs of sensors  501  in each individual sensor-Y oriented column (e.g. each individual column value A 1 , B 1 , A 1 ′, B 1 ′, A 2 , B 2 , . . . ), a suitable algorithm can be used to determine in which group of sensor-Y oriented columns, the center of magnet array  112 B is located. The exact position of the center of magnet array  112 B within a particular sensor group can be derived from the quadrature signals α=A−A′ and β=B−B′, as described above. As the quadrature signal amplitudes are exponential functions of stator-Z motion of magnet array  112 B, the stator-Z motion of magnet array  112 B can be determined in accordance with equation (7). 
     It should be noted that electrical current inside coils near each sensor  501  can also influence the output of a sensor  501 . However, current in each coil trace is known to controller  504  (e.g. because controller  504  calculates the driving currents associated with moving movable stage  110 ) and the influence of the current in each coil trace on corresponding sensors  501  can be pre-calibrated and thus be subtracted or otherwise removed from the sensor outputs—e.g. from the output of each sensor  501 , from the output (e.g. sum/average) of each stator-Y oriented sensor column value, from the output of each synchronous sum/average value and/or the like. 
     It will be appreciated that the methods described herein for determining stator-X and stator-Z positions of Y-magnet array  112 B from the output (e.g. sum/average) of each stator-Y oriented sensor column (e.g. A Y,j  (j=1, 2, 3, . . . , N)) can be applied for determining stator-X and stator-Z positions of any Y-magnet array (e.g. Y-magnet array  112 D), and can also be applied for determining stator-Y and stator-Z positions of X-magnet arrays  112 A,  112 C from the output (e.g. sum/average) of each stator-X oriented sensor row (e.g. A X,i  (i=1, 2, 3, . . . , N)). 
     It is understood that outputs of sensors  501  and/or outputs corresponding to sums/averages of the outputs of sensors  501  in sensor rows/columns (e.g. as shown in the embodiments of  FIG. 12A, 12B ) may be converted to digital values by suitable analog-to-digital convertor(s), and the converted values can be processed by controller  504  to derive the position of each magnet array  112 . Position information for all magnet arrays  112  of a movable stage  110  may be used to calculate 6-dimensional positions of movable stage  110 . For example, for the case in  FIG. 3F , the stator-X and stator-Z positions of Y-magnet arrays  112 B,  112 D may be determined from the output (e.g. sum/average) of each stator-Y oriented sensor column (e.g. A Y,j  (i=1, 2, 3, . . . , N)), the stator-Y and stator-Z positions of X-magnet arrays  112 A,  112 C may be determined from the output (e.g. sum/average) of each stator-X oriented sensor row (e.g. A X,i  (i=1, 2, 3, . . . , N)), and this position information may be combined together to determine the three translational positions and three rotational positions of movable stage  110 . 
       FIGS. 14A and 14B  (together  FIG. 14 ) show cross-sectional and top views of another non-limiting embodiment of the invention. The embodiment of  FIG. 14  differs from the embodiment of  FIGS. 9 and 13A  in that the stator-Y oriented sensor columns corresponding to Ai′ and Bi′ (i=1, 2, . . . ) are removed in the embodiment of  FIG. 14 . The  FIG. 14  configuration may continue to provide information desirable for determining the stator-X and stator-Y positions of Y-magnet array  112 B because the synchronous sum/average values A′ and A (equations (3a), (3b)) are nearly opposite to each other as they have 180 degree phase difference, and thus have redundant information. Similarly, the synchronous sum/average values B′ and B (equations (3c), (3d)) are nearly opposite to each other and thus have redundant information. Comparing the layout of sensor array  500  in  FIGS. 9 and 14 , it can be seen that some stator-Y oriented sensor columns and/or some stator-X oriented sensor rows are removed from the  FIG. 14B  embodiment. Such removal has a periodic repetitive pattern, leaving the sensor array  500  of  FIG. 14B  with a period pattern. In  FIG. 14B  implementation, the previously mentioned methods of summing/average over each stator-Y oriented sensor column and each stator-X oriented sensor row and determining synchronous sum/average values can still be applied. Accordingly, for the case of synchronous averaging, the synchronous averaging values for the embodiment of  FIG. 14  can be modified as: 
                   A   =       1     n   5       ⁢     (       A   1     +     A   2     +     A   3     +     A   4     +   ⋯   +     A     n   5         )               (     8   ⁢   a     )               B   =       1     n   6       ⁢     (       B   1     +     B   2     +     B   3     +     B   4     +   ⋯   +     A     n   6         )               (     8   ⁢   b     )               
Where n 5 , n 6  are the number of synchronous column values in each synchronous average value. It will be appreciated that synchronous sum values may be determined in a manner similar to that of determining synchronous average values, except that it is not necessary to divide the sums by n 5 , n 6 . The stator-X and stator-Z direction positions of Y-magnet array  112 B can be determined according to
 
                   A   =       C   0     ⁢     e     -       2   ⁢   π   ⁢           ⁢   x     λ         ⁢     cos   ⁡     (       2   ⁢   π   ⁢           ⁢   x     λ     )                 (     9   ⁢   a     )               B   =       C   0     ⁢     e     -       2   ⁢   π   ⁢           ⁢   z     λ         ⁢     sin   ⁡     (       2   ⁢   π   ⁢           ⁢   x     λ     )                 (     9   ⁢   b     )               x   =     atan   ⁢           ⁢   2   ⁢     (     B   ,   A     )     ⁢     λ     2   ⁢   π                 (     9   ⁢   c     )               z   =       λ     2   ⁢   π       ⁢     ln   (       C   0           B   2     +     A   2           )               (     9   ⁢   d     )               
As before, the stator-X and stator-Z positions of Y-magnet array  112 D and the stator-Y and stator-Z positions of X-magnet arrays  112 A,  112 C can be determined in an analogous manner.
 
       FIG. 15  shows a cross-sectional view of another non-limiting embodiment of the invention. The embodiment of  FIG. 15  differs from the embodiment of  FIGS. 9 and 13A  in that the stator-Y oriented sensor columns corresponding to Ai′, Bi, and Bi′ (i=1, 2, . . . ) are removed in the embodiment of  FIG. 14 . The  FIG. 15  configuration may continue to provide information desirable for determining the stator-X and stator-Y positions of Y-magnet array  112 B when each sensor  501  can sense magnetic field in two directions Z and X, because the flux density in stator-Z and stator-X are separated by a 90 degree phase different and thus can be used to interpolate the position of magnet array  112 B accurately. In the case of the  FIG. 15  embodiment, the stator-X direction field measurement outputs from sensors  501  may be column summed/averaged and also synchronously summed/averaged; and the stator-Z direction field measurement outputs from sensors  501  may be column summed/averaged and also synchronously summed/averaged. Then, the two synchronous sum/average outputs (i.e. corresponding to stator-X and stator-Z direction field sensitivity) may be used to determine the stator-X and stator-Z positions of Y-magnet array  112 B. In the  FIG. 15  embodiment, the synchronous sum/average values can be determined according to: 
                     stator   ⁢     -     ⁢   X   ⁢           ⁢   field   ⁢           ⁢   synchronous   ⁢           ⁢   average   ⁢     :     ⁢           ⁢     A   x       =       1     n   7       ⁢     (       A     1   ⁢   x       +     A     2   ⁢   x       +     A     3   ⁢   x       +     A     4   ⁢   x       +   ⋯   +     A     n   7         )               (     10   ⁢   a     )                 stator   ⁢     -     ⁢   Z   ⁢           ⁢   field   ⁢           ⁢   synchronous   ⁢           ⁢   average   ⁢     :     ⁢           ⁢     A   z       =       1     n   8       ⁢     (       A     1   ⁢   z       +     A     2   ⁢   z       +     A     3   ⁢   z       +     A     4   ⁢   z       +   ⋯   +     A     n   8         )               (     10   ⁢   b     )               
Where: A ix  (i=1, 2, . . . ) represents the average of over a stator-Y oriented sensor column of the sensor outputs corresponding to the stator-X direction magnetic field sensitivity; A iz  (i=1, 2, . . . ) represents the average of over a stator-Y oriented sensor column of the sensor outputs corresponding to the stator-Z direction magnetic field sensitivity; A x  is synchronous average value of A ix  (i=1, 2, . . . ), A z  is synchronous average value of A iz  (i=1, 2, . . . ); and n 7 , n 8  are the number of synchronous column values in each synchronous average value. It will be appreciated that synchronous sum values may be determined in a manner similar to that of determining synchronous average values, except that it is not necessary to divide the sums by n 7 , n 8 . The stator-X and stator-Z direction positions of Y-magnet array  112 B can be determined according to:
 
                     A   x     =       C   0     ⁢     e     -       2   ⁢   π   ⁢           ⁢   x     λ         ⁢     sin   ⁡     (       2   ⁢   π   ⁢           ⁢   z     λ     )                 (     11   ⁢   a     )                 A   z     =       C   0     ⁢     e     -       2   ⁢   π   ⁢           ⁢   z     λ         ⁢     cos   ⁡     (       2   ⁢   π   ⁢           ⁢   x     λ     )                 (     11   ⁢   b     )               x   =     atan   ⁢           ⁢   2   ⁢     (       A   x     ,     A   z       )     ⁢     λ     2   ⁢   π                 (     11   ⁢   c     )               z   =       λ     2   ⁢   π       ⁢     ln   (       C   0           A   z   2     +     A   x   2           )               (     11   ⁢   d     )               
As before, the stator-X and stator-Z positions of Y-magnet array  112 D and the stator-Y and stator-Z positions of X-magnet arrays  112 A,  112 C can be determined in an analogous manner.
 
       FIG. 16  shows a cross-sectional view of another non-limiting embodiment of the invention. The embodiment of  FIG. 16  differs from the embodiment of  FIGS. 9 and 13A  in that instead of four stator-Y oriented sensor columns per spatial period λ (as is the case in the embodiment of  FIG. 13A ), the embodiment of  FIG. 16  comprise three generally equally-spaced stator-Y oriented sensor columns per spatial period λ (i.e. P x =λ/3). Each of Ai, Bi, Ci (i=1, 2, 3, . . . ) represents sum/average sensors  501  distributed along a stator-Y oriented sensor column (e.g. aligned along a line oriented in the stator-Y direction). Synchronous sum/average values A, B, C may be used to determine the stator-X and stator-Z position of a Y-magnet array according to: 
                   A   =       1     n   9       ⁢     (       A   1     +     A   2     +     A   3     +     A   4     +   ⋯   +     A     n   9         )               (     12   ⁢   a     )               B   =       1     n   10       ⁢     (       B   1     +     B   2     +     B   3     +     B   4     +   ⋯   +     B     n   10         )               (     12   ⁢   b     )               c   =       1     n   11       ⁢     (       C   1     +     C   2     +     C   3     +     C   4     +   ⋯   +     C     n   11         )               (     12   ⁢   c     )               
Where n 9 , n 10 , n 11  are the number of synchronous column values in each synchronous average value. It will be appreciated that synchronous sum values may be determined in a manner similar to that of determining synchronous average values, except that it is not necessary to divide the sums by n 9 , n 10 , n 11 .
 
     Each of the synchronous sum/average values A, B, C will be a sinusoidal function of the stator-X position of Y-magnet array  112 B separated by a 120 degree phase difference, where phase is the stator-X position of Y-magnet array  112 B divided by spatial period λ and multiplied by 2π. The sinusoidal amplitude is related to the stator-Z position of Y-magnet array  112 B exponentially and can be used to derive the stator-Z position of Y-magnet array  112 B. 
     Mathematically, A, B, C can be represented as: 
                   A   =       C   0     ⁢     e     -       2   ⁢   π   ⁢           ⁢   z     λ         ⁢     cos   ⁡     (       2   ⁢   π   ⁢           ⁢   x     λ     )                 (     13   ⁢   a     )               B   =       C   0     ⁢     e     -       2   ⁢   π   ⁢           ⁢   z     λ         ⁢     cos   ⁡     (         2   ⁢   π   ⁢           ⁢   x     λ     -       2   ⁢   π     3       )                 (     13   ⁢   b     )               C   =       C   0     ⁢     e     -       2   ⁢   π   ⁢           ⁢   z     λ         ⁢     cos   ⁡     (         2   ⁢   π   ⁢           ⁢   x     λ     -       4   ⁢   π     3       )                 (     13   ⁢   c     )               
These three values are sufficient to solve the two unknowns (the stator-X and stator-Z positions of Y-magnet array  112 B) with a suitable method/algorithm, where constant C 0  can be either pre-calibrated by experiments or be pre-calculated with an aid of simulation software.
 
     One non-limiting method to determine stator-X and stator-Z position of a Y-magnet array  112 B from synchronous average values A, B, C is as follows: 
                     A   90     =         [     A   ,   B   ,   C     ]     ⁡     [         0             sin   ⁡     (       2   ⁢   π     3     )                 -     sin   ⁡     (       2   ⁢   π     3     )               ]       =     1.5   ⁢     C   0     ⁢     e     -       2   ⁢   π   ⁢           ⁢   z     λ         ⁢     sin   ⁡     (       2   ⁢   π   ⁢           ⁢   x     λ     )                   (     14   ⁢   a     )                 A   0     =         [     A   ,   B   ,   C     ]     ⁡     [         1             -   0.5               -   0.5           ]       =     1.5   ⁢     C   0     ⁢     e     -       2   ⁢   π   ⁢           ⁢   z     λ         ⁢     cos   ⁡     (       2   ⁢   π   ⁢           ⁢   x     λ     )                   (     14   ⁢   b     )               x   =     atan   ⁢           ⁢   2   ⁢     (       A   90     ,     A   0       )     ⁢     λ     2   ⁢   π                 (     14   ⁢   c     )               z   =       λ     2   ⁢   π       ⁢     ln   (       1.5   ⁢     C   0             A   0   2     +     A   90   2           )               (     14   ⁢   d     )               
As before, the stator-X and stator-Z positions of Y-magnet array  112 D and the stator-Y and stator-Z positions of X-magnet arrays  112 A,  112 C can be determined in an analogous manner.
 
     Another variation of  FIG. 16  implementation is to change the sensor spacing from λ/3 to 2λ/3 or 4λ/3, to reduce the number of sensors  501  in the system. 
       FIG. 17  shows a top view of a sensor array  500 ′ according to another non-limiting embodiment of the invention. In comparison to sensor array  500  in the  FIG. 9  embodiment, each sensor  501  of array  500 ′ of the  FIG. 17  embodiment is offset away from equally spaced 2-dimensional grid points  509  (i.e. offset from the intersection points  509  of stator-X oriented lines  505  and stator-Y oriented lines  507 ). The pitches of the grid intersection points are P x  in the stator-X direction and P y  in the stator-Y direction. The actual location of each sensor  501  is shifted away in the stator-X direction by an amount of either ±S x , and shifted in the stator-Y direction by an amount of either ±S y . Spatially, the shifts exhibit a periodic pattern in the stator-X and stator-Y directions. In particular, each sensor  501  in a stator-Y oriented sensor column, while offset (by an offset 2S x ) from adjacent sensors  501  in the stator-Y oriented sensor column, is still generally aligned with the other sensors  501  in the stator-Y oriented sensor column and pairs of adjacent sensors  501  in the stator-Y oriented sensor column are still separated by a stator-Y direction pitch P y . Similarly, each sensor  501  in a stator-X oriented sensor row, while offset (by an offset 2S y ) from adjacent sensors  501  in the stator-X oriented sensor row, is still generally aligned with the other sensors  501  in the stator-X oriented sensor row and pairs of adjacent sensors  501  in the stator-X oriented sensor row are still separated by a stator-X direction pitch P x . 
     The shifted pattern of array  500 ′ may cause some filtering out of the higher order harmonic fields of magnet arrays  112  during the summing/averaging operation along each stator-X oriented sensor row and/or each stator-Y oriented sensor column. The offsets S x , S y  may be generally smaller than the grid pitch (i.e. S x &lt;P x  and S y &lt;P y ). To filter off the 5 th  order harmonic fields, S x , S y  can be set at S X =λ X /20, and S Y =λ Y /20, where λ X  is the stage-X direction magnetic spatial period of Y-magnet arrays (for example, Y-magnet arrays  112 B,  112 D in  FIG. 3F ); λ Y  is the stage-Y direction magnetic spatial period of X-magnet magnet arrays (for example X-magnet arrays  112 A,  112 C in  FIG. 3F ). Other options include, for example, S X =λ X /36, and S Y =λ Y /36, to filter off the 9 th  order harmonic field effect on the sums/averages of the stator-X oriented sensor rows and stator-Y oriented sensor columns (A X,i  and A Y,j ). By setting S x  to conform generally with S X =λ X /(4*K), the stator-Y oriented sensor column sum/average will filter out K th  order harmonic fields of the Y-magnet arrays; and by setting S y  to conform generally with S Y =λ Y /(4*K), the stator-X oriented sensor row sum/average will filter out K th  order harmonic fields of X-magnet array. Typically, the minimum order of harmonic distortion of concern for a magnet array  112  is K=3 when N t =2; consequently, in some embodiments, S x  and/or S y  are set with K being a positive integer and K≥3. 
     In some cases, the determined stator-X and stator-Z positions for a Y-magnet array  112 B,  112 D may contain some periodic error due to some practical imperfectness such as magnet array manufacturing error or material non-uniformity. One non-limiting embodiment of removing such systematic error is a correction process: 
               X   c     =     X   -       a   1     ⁢           ⁢     cos   ⁡     (       2   ⁢   π   ⁢           ⁢   x     λ     )         -       b   1     ⁢           ⁢     sin   ⁡     (       2   ⁢   π   ⁢           ⁢   x     λ     )         -       a   2     ⁢           ⁢     cos   ⁡     (       4   ⁢   π   ⁢           ⁢   x     λ     )         -       b   2     ⁢           ⁢     sin   ⁡     (       4   ⁢   π   ⁢           ⁢   x       λ   ⁢               )                         Z   c     =     Z   -       c   1     ⁢           ⁢     cos   ⁡     (       2   ⁢   π   ⁢           ⁢   x     λ     )         -       d   1     ⁢           ⁢     sin   ⁡     (       2   ⁢   π   ⁢           ⁢   x     λ     )         -       c   2     ⁢           ⁢     cos   ⁡     (       4   ⁢   π   ⁢           ⁢   x     λ     )         -       d   2     ⁢           ⁢     sin   ⁡     (       4   ⁢   π   ⁢           ⁢   x     λ     )                 
where X and Z are the stator-X and stator-Z positions determined using the methods described above, X c  and Z c  are the stator-X and stator-Z positions determined after error correction, and a 1 , a 2 , b 1 , b 2 , c 1 , c 2 , d 1 , d 2  are constant coefficients that can be experimentally calibrated. Similar correction methods can be applied for stator-X and stator-Z positions of X-magnet arrays  112 A,  112 C.
 
     In some embodiments, the overall stator X/Y plane may be split into multiple independent sensing regions. The overall sensor array  500  may then be split into many polygon-shaped independent sensing regions  520 . As shown in  FIG. 18 , each polygon-shaped independent sensing region  520  is a square, which contains a sub-array of sensors  501 , for example 12 by 12 sensors. The column and row summing/averaging operations described above may be confined to each region  520  independently (e.g. using suitable hardware wiring or suitable software algorithms). That is row/column sums/averages are only determined using sensors inside one region  520 . Independent sensing regions  520  can be in other shapes such as, for example, rectangles or hexagons. 
     In some embodiments, the overall stator X/Y plane is split into multiple independent column sensing regions; and the overall stator X/Y plane is split into multiple independent row sensing regions. Such a split or partition may allow motion measurement of multiple movable stages simultaneously and independently. Each independent column sensing region may have a polygon shape, such as but not limited to a rectangle, a square, or a hexagon. Each independent row sensing region may have a polygon shape, such as but not limited to a rectangle, a square, or a hexagon. Column summing/averaging operations described above may be confined to each column sensing region. Row summing/averaging operations described above may be confined to each row sensing region. It should be noted, the overall partition of independent row sensing region and the overall partition of independent column sensing region are not necessarily identical, and they may be identical in some regions and may be totally different in some other regions. 
     As described above, in some embodiments sensors are mounted on printed circuit boards. The overall sensor array  500  of the whole working region  124  may comprise a plurality of printed circuits boards. Each printed circuit board may include one or a plurality of independent sensing zones. 
       FIG. 19  shows one non-limiting embodiment of a flow chart for movable stage position determination. At each sampling event, the whole process in the flow chart will be executed to produce 6-axis position information in real time. Some steps in the flow chart in  FIG. 19  may be omitted to save computation time. For example, system error correction and/or coil trace current effect removal may not be necessary for low precision application.  FIG. 19  depicts a series of steps for each column and a similar series of steps for each row. The steps for the rows and columns can be executed in parallel or in series. 
     Each sensor  501  may have a non-zero output bias: when there is no external magnetic field, the output voltage/signal may not quite be zero. Such output bias may change with time or with environmental factors, such as temperature and/or the like. One way to minimize the impact of such output bias is to subtract a calibrated bias value from the sensor output signal. One way to calibrate the bias value is to record the sensor output when there is no external magnetic field. Such calibration values can be determined in a range of environmental conditions and stored in a look up table, for example. In some embodiments, such sensor bias value(s) may be repeatedly refined by recording the sensor output when no movable stage is close to the sensors. As a result, even the magnetic field sensor bias value may change over time or with environmental conditions. Such a continually refined output bias calibration procedure may help to compensate for such drifting output bias. In one particular implementation, if there is no movable stage close to the sensors  501  in a stator-Y oriented sensor column or a stator-X oriented sensor row, then the column/row sum/average output value can be recorded as a newly calibrated output for bias removal used later. 
     Multiple Movable Stages 
     In certain applications, such as photo-lithography, automated assembly systems and/or the like, there can be a desire to simultaneously and independently control more than one movable stage. This may be achieved, for example, by providing a corresponding plurality of independently controllable stators and controlling the movement of one movable stage on each stator. In some circumstances, it is desirable to interchange the movable stages (e.g. to move a movable stage from one stator to another stator). In some applications, it may be desirable to move movable stages  110  through a number of different stages.  FIG. 20  schematically illustrates an apparatus  460  suitable for this purpose. In the illustrated embodiment, movable stages  110 A- 110 D move between several stators  120 A- 120 F and, in some applications, may stop at each stator  120  for some operation. In general, there may be any suitable number of movable stages  110  and any suitable number (greater than the number of movable stages  110 ) of stators  120 . On each stator  120 A- 120 F, a position estimating system of the type described herein may be used as part of a control system to control positions of the corresponding movable stage  110 A- 110 D. In some embodiments, precision position control may only be required inside stators  120 A- 120 F. Consequently, stator-to-stator motion (e.g. motion of movable stage  110 A- 110 D between stators  120 A- 120 F) may be guided by relatively inexpensive positions measurement systems, such as indoor GPS, stereo camera and/or the like. 
     Other Layouts and Configurations 
       FIG. 21A  schematically depicts a displacement device  600  according to another embodiment. Displacement device  600  comprises a movable stage (not explicitly shown) which comprises a plurality of magnet arrays  612 . In the illustrated embodiment, displacement device  600  comprise three magnet arrays  612  (labeled  612 A,  612 B,  612 C). Each magnet array  612 A,  612 B,  612 C comprises a corresponding plurality of magnetization segments  614 A,  614 B,  614 C which are generally linearly elongated at a particular orientation in the stage X-Y plane for example, magnetization segments  614 A of magnet array  612 A have one orientation of linear elongation, magnetization segments  614 B of magnet array  612 B have a second orientation of linear elongation and magnetization segments  614 C of magnet array  612 C have a third orientation of linear elongation. As is the case with the other displacement devices described herein, the magnetization directions of magnetization segments  614 A,  614 B,  614 C may be generally orthogonal to the direction that they are physically elongated. Other than for their relative orientations, the characteristics of magnet arrays  612  and magnetization segments  614  may be similar to those discussed above for magnet arrays  112  and magnetization segments  114 . 
     Displacement device  600  also comprises a stator (not explicitly shown) that comprises a plurality of generally linearly elongated coil traces  626 . In the illustrated embodiment, displacement device  600  comprise three sets of coil traces  626  (labeled  626 A,  626 B,  626 C) which may be located on corresponding layers (not explicitly shown) of the stator. Each layer of coil traces  626 A,  626 B,  626 C may comprise coil traces  626 A,  626 B,  626 C that are generally linearly elongated at a particular orientation in a corresponding stator X-Y plane. Such layers and their corresponding coil traces  626 A,  626 B,  626 C may overlap one another (in the stator-Z direction) in the working region of displacement device  600 . Other than for their relative orientations, the characteristics of coil traces  626  may be similar to those of coil traces  126  discussed above. 
     Displacement device  600 ′ shown in  FIG. 21B  is similar to displacement device  600 , except that the orientations of the linearly elongated coil traces  626 A′,  626 B′,  626 C′ are different than the orientations of the linearly elongated traces  626 A,  626 B,  626 C and the orientations at which magnetization segments  614 A′,  614 B′ and  614 C′ extend are different than the orientations at which magnetization segments  614 A,  614 B,  614 C extend. 
       FIG. 21C  schematically depicts a displacement device  700  according to another embodiment. Displacement device  700  comprises a movable stage (not explicitly shown) which comprises a plurality of magnet arrays  712 . In the illustrated embodiment, displacement device  700  comprises two magnet arrays  712  (labeled  712 A,  712 B). Each magnet array  712 A,  712 B comprises a corresponding plurality of magnetization segments  714 A,  714 B which are generally linearly elongated at a particular orientation in the stage-X-Y plane—for example, magnetization segments  714 A of magnet array  712 A have one orientation of linear elongation and magnetization segments  714 B of magnet array  712 B have a second orientation of linear elongation. As is the case with the other displacement devices described herein, the magnetization directions of magnetization segments  714 A,  714 B may be generally orthogonal to the direction that they are physically elongated. Other than for their relative orientations, the characteristics of magnet arrays  712  and magnetization segments  714  may be similar to those discussed above for magnet arrays  112  and magnetization segments  114 . 
     Displacement device  700  also comprises a stator (not explicitly shown) that comprises a plurality of generally linearly elongated coil traces  726 . In the illustrated embodiment, displacement device  700  comprise two sets of coil traces  726  (labeled  726 A,  726 B) which may be located on corresponding layers (not explicitly shown) of the stator. Each layer of coil traces  726 A,  726 B may comprise coil traces  726 A,  726 B that are generally linearly elongated at a particular orientation in a corresponding stator-X-Y plane. Such layers and their corresponding coil traces  726 A,  726 B may overlap one another (in the stator-Z direction) in the working region of displacement device  700 . Other than for their relative orientations, the characteristics of coil traces  726  may be similar to those of coil traces  126  discussed above. 
     It will be appreciated that displacement device  700  of the  FIG. 21C  embodiment will not be able to provide all six degrees of freedom. With suitable control techniques, the embodiment of  FIG. 21C  may be capable of providing motion with 4 degrees of freedom. 
       FIGS. 21A-21C  are useful to demonstrate a feature of one aspect and particular embodiments of the invention. Some of the herein-described embodiments include relatively large numbers of magnet arrays. While this can achieve over-actuation which may enhance the ability to control the movement of the movable stage relative to the stator, this is not necessary. Particular embodiments may comprise movable stages having any suitable number (as few as one) magnet array, wherein each such magnet array comprises a plurality of magnetization sections that are generally linearly elongated along a corresponding direction. While in some embodiments, the preferred direction(s) of linear elongation may comprise at least two orthogonal directions (which may make control calculations relatively more simple), this is not necessary. In the case where the magnet arrays are aligned in a single movable stage XY plane, any two or more non-parallel directions of linear elongation will span the stage XY plane. Further, some embodiments involve the use of only one magnet array. In some embodiments where six degrees of freedom are desired, three of more magnet arrays are provided with at least two of the magnet arrays being linearly elongated in non-parallel directions and with the force-centers of the three magnet arrays being non-co-linear. In addition, the directions of magnetization of the magnetization segments in each magnet array are generally orthogonal to the direction in which the magnetization segments are linearly elongated. Within a magnet array, the magnetization of the magnetization segments may have characteristics similar to any of those described herein—see  FIGS. 5 and 6  for example. 
     Similarly, particular embodiments may comprise stators having coil traces elongated in any suitable number of (one or more) directions. While in some embodiments, the directions of linear elongation may comprise at least two orthogonal directions (which may make control calculations relatively more simple), this is not necessary. Any two or more non-parallel directions of linear elongation will span the notional stator XY plane of the stator. Further, some embodiments involve the use of only one coil elongation direction. The stator XY plane of the stator may be referred to as a notional XY plane, since coil traces having different directions of linear elongation may be provided on different layers as discussed above. Such layers may have different locations in the stator-Z direction. Accordingly, the notional XY plane of the stator may be thought of as though the coil traces in each such layer were notionally brought to a single XY plane having a corresponding single location along the stator-Z axis. 
     The description set out herein describes that there may be different numbers N t  of magnetization directions within a magnetic spatial period λ. However, N t =4 for the illustrated embodiments described above.  FIGS. 22A-22C  schematically depict magnet arrays  802 A,  802 B,  802 C having different values of N t —i.e. different numbers of magnetization directions within a particular magnetic period λ. Magnet array  802 A of  FIG. 22A  has N t =4, magnet array  802 B of  FIG. 22B  has N t =2 and magnet array  802 C of  FIG. 22C  has magnet array N t =8. The number N t  may be selected to be any suitable number, with the advantage of having relatively large N t  is that relatively large N t  provides the corresponding magnet array with a relatively large fundamental harmonic and relatively small higher order harmonics at the expense of possibly greater cost and complexity in fabricating the magnet array. When N t =4, there exists a 5 th  order harmonic field in a magnet array  112 ; when N t =2, there exist a 3 rd  order harmonic field in a magnet array  112 . 
     In some embodiments, magnet arrays  112  may be provided with different numbers of sub-arrays.  FIG. 23A  shows a particular embodiment where the stage-Y dimension L m  of Y-magnet array  112  comprises a pair of sub-arrays  112 A,  112 B, each having a stage-Y dimension of L m /2 and offset from one another by a distance O m  in the stage-X direction. The offset distance O m  of the  FIG. 23A  sub-arrays  112 A,  112 B can the offset O m  may be set at least approximately equal to O m =(N m /5− 1/10)λ, where N m  is any positive integer number. Setting O m  to have this characteristic will tend to attenuate or cancel the effects of the interaction of the 5 th  order harmonic of the magnet field of magnet array  112  with coil traces  126  that carry current in the stator-Y direction, thereby reducing or minimizing associated force ripples. Setting O m  have this characteristic may tend to attenuate or cancel the effects of the interaction of the 5 th  order harmonic of the magnet field of magnet array  112  with stator-Y oriented sensor column sum/average values of 2D array  500  of sensors  501 , thereby reducing or minimizing associated position estimation errors. In some embodiments, the offset O m  may be set at least approximately equal to O m  is set at 
                 (         N   m     9     -     1   18       )     ⁢   λ     ,         
to attenuate the areas of the interaction of the 9 th  order harmonic of the magnetic field of magnet array  112  with coil traces  126  that carry current in the stator-Y direction and to attenuate or cancel the effects of the interaction of the 9 th  order harmonic of the magnet field of magnet array  112  with stator-Y oriented sensor column sum/average values of 2D array  500  of sensors  501 . In some embodiments, the offset O m  may be set at least approximately equal to
 
                 O   m     =           N   m     5     ⁢   λ     -     W   c         ,         
where N m  is any integer number and W c  is the stator-X direction width of coil traces  126  generally elongated in stator-Y direction. Setting O m  to have this characteristic will tend to attenuate or cancel the effects of the interaction of the 5 th  order harmonic of the magnet field of magnet array  112  with coil traces  126  that carry current in the stator-Y direction, thereby reducing or minimizing associated force ripples. In some embodiments, the offset O m  may be set at least approximately equal to O m  is set at
 
                     N   m     9     ⁢   λ     -     W   c       ,         
to attenuate the effects of the interaction of the 9 th  order harmonic of the magnetic field of magnet array  112  with coil traces  126  that carry current in the stator-Y direction. While magnet array  112  shown in the illustrated embodiment of  FIG. 23A  comprises two sub-arrays, magnet arrays  112  may generally be provided with any suitable number of sub-arrays having characteristics similar to those shown in  FIG. 23A .
 
       FIGS. 23B and 23C  show a number of embodiments of magnet arrays  112  which may be used to attenuate the effects of multiple spatial harmonics of their corresponding magnetic fields.  FIGS. 23B and 23C  show one embodiment of a Y-magnet array  112 , which comprises six sub-arrays having stage-Y direction lengths 
               L   m     8         
(labeled a,b,c,f,g,h in  FIG. 23C ), and one sub-array having stage-Y direction length
 
               L   m     4         
(labeled d-e in  FIG. 23C ), where L m  is the total stage-Y direction length of magnet array  112 .  FIG. 23D  shows how some of sub-arrays (a, b, c, d-e, f, g, h) are shifted or offset (in the stage-X direction) relative to one another. In the embodiment of  FIGS. 23B and 23C , sub-arrays b and g are aligned in the stage-X direction, sub-arrays a and h are shifted (rightwardly in the illustrated view) relatively to sub-arrays b and g by an amount O m2 , sub-arrays d and e (together sub-array d-e) are shifted (rightwardly in the illustrated view) relatively to sub-arrays b and g by an amount O m1  and sub-arrays c and f are shifted (rightwardly in the illustrated view) relatively to sub-arrays b and g by an amount 2O m2 +O m1 . Each sub-array a,b,c,d-e,f,g,h of the illustrated embodiment has a stage-X dimension width W m . Mirror symmetry on line A-A (at the center of the stage-Y dimension L m  of magnet array  112 ) reduces or minimizes moment and/or force disturbance on the  FIGS. 23C, 23D  magnet array  112 . The harmonics attenuated by the  FIGS. 23B, 23C  arrangement have spatial wavelengths equal to 2O m1  and 2O m2 . For example, by setting O m1 =λ/10 and O m2 =λ/26, the 5 th  and 13 th  harmonics of the magnetic field are attenuated in connection with both force generation using coil traces and also in column/row sum/average values determined from sensor array  500 . In general, setting O m1 =λ(M−0.5)/p, O m2 =λ(M−0.5)/q will significantly minimize disturbance moment/force resulting from harmonic magnetic fields of wavelength (spatial period) both λ/p and λ/q, where M and N are arbitrary integer numbers.
 
     The techniques illustrated in  FIGS. 23B-23C  can be extrapolated so that field-induced disturbance moment and/or force effects associated with any suitable number of harmonics may be simultaneously attenuated using a suitable variation of these techniques and field-induced position estimation errors associated with any suitable number of harmonics may be simultaneously attenuated using a suitable variation of these techniques. It is also possible to attenuate the field-induced effects of one harmonic order, but retain some level of net moment disturbance (such as shown in  FIG. 23A ). 
     Magnet arrays  112  of particular embodiments can be skewed or provided with spatial periodicity along the direction that their respective magnetization segments  114  are generally linearly elongated. Such skewing and/or spatial periodicity of magnet arrays  112  may be used to reduce or minimize the effects of higher order harmonics of the magnetic fields of these magnet arrays  112 .  FIG. 24A  shows a Y-magnet array  112  which is generally linearly elongated in the stage-Y direction, but which is skewed by an amount O p  in the stage-X direction over its stage-Y dimension length L m . Assuming that the  FIG. 24A  magnet array  112  is configured to interact with coil traces  126  having a rectangular geometry with a coil width W c  as defined above, then the skew amount may be set to be at least approximately equal to a non-negative value O p =kΛ f −W c , where Λ f  is the wavelength of the spatial harmonic of the magnetic field that is to be attenuated and k is a positive integer number. For example, if it desired to attenuate the effects of the 5 th  order harmonic filed of the  FIG. 24A  magnet array  112 , then O p  can be set to be kλ/5−W c  where k is a positive integer number. 
       FIGS. 24B and 24C  show spatially periodic Y-magnet arrays  112 , wherein an edge of each array  112  varies in the stage-X direction by an amount O p  over it stage-Y dimension length L m . The magnet arrays  112  of  FIGS. 24B and 24C  are periodic with a spatial period τ m  where τ m =L m  in the  FIG. 24B  array and τ m =L m /2 in the  FIG. 24C  array. Like the case of the spatially periodic coil traces discussed above, the spatial period τ m  may generally be set to be an integer factor of the stage-Y dimension length L m . Also, similar to the case of the spatially periodic coil traces discussed above, spatially periodic magnet arrays may be provided with spatially periodic waveforms other than triangular waveforms, such as square waves, sinusoidal waveforms or superposed waveforms. The peak-to-peak amplitude parameter O p  can have the characteristics of the term O p  discussed above in connection with  FIG. 24A . 
     In some embodiments, a combination of skewed coil traces and slanted magnet arrays may also be usefully implemented to eliminate internal stresses in the magnetic arrays while reducing or minimizing the effects of the interaction of current carrying coil traces with higher order harmonics of the magnetic fields of the magnet arrays. 
     Certain implementations of the invention comprise controllers, computers and/or computer processors which execute software instructions which cause the controllers, computers and/or processors to perform a method of the invention. For example, one or more processors in a controller or computer may implement data processing steps in the methods described herein by executing software instructions retrieved from a program memory accessible to the processors. The invention may also be provided in the form of a program product. The program product may comprise any medium which carries a set of computer-readable signals comprising instructions which, when executed by a data processor, cause the data processor to execute a method of the invention. Program products according to the invention may be in any of a wide variety of forms. The program product may comprise, for example, physical (non-transitory) media such as magnetic data storage media including floppy diskettes, hard disk drives, optical data storage media including CD ROMs, DVDs, electronic data storage media including ROMs, flash RAM, or the like. The instructions may be present on the program product in encrypted and/or compressed formats. 
     Where a component (e.g. a software module, controller, processor, assembly, device, component, circuit, etc.) is referred to above, unless otherwise indicated, reference to that component (including a reference to a “means”) should be interpreted as including as equivalents of that component any component which performs the function of the described component (i.e., that is functionally equivalent), including components which are not structurally equivalent to the disclosed structure which performs the function in the illustrated exemplary embodiments of the invention. 
     While a number of exemplary aspects and embodiments are discussed herein, those of skill in the art will recognize certain modifications, permutations, additions and sub-combinations thereof. For example:
         The magnet arrays shown in the embodiments of  FIGS. 23 and 24  are Y-magnet arrays, but it will be appreciated that X-magnet arrays could be provided with similar characteristics. Also, the magnet arrays shown in the embodiments of  FIGS. 23 and 23  have a particular pattern of magnetization. In general, these magnet arrays may be provided with any suitable magnetization pattern, such as any of those shown in  FIGS. 5 and 6 , for example.   For the purpose of minimizing or reducing eddy currents induced by the motion of magnet arrays  112  on movable stage  110 , coil traces  126  may be made relatively narrow. In some embodiments, each coil trace  126  may comprise a plurality of sub-traces  126 ′. Such an embodiment is shown schematically in  FIGS. 25A  (in top view) and in  25 B (in cross-section). In coil traces  126 A,  126 B,  126 C of  FIG. 25A , each coil trace  126 A,  126 B,  126 C comprises a plurality of corresponding sub-traces  126 A′,  126 B′,  126 C′ (collectively, sub-traces  126 ′) where each sub-trace  126 ′ has a width T c  that is a fraction of the width W c  of its corresponding coil  126 . Each sub-trace  126 ′ only carries a portion of the current flowing through its corresponding trace  126 . Each sub-trace  126 ′ in the  FIG. 25A  embodiment is insulated from its adjacent sub-trace  126 ′ by an insulator of width T f , although it is not generally necessary for the insulator width T f  to be uniform within a coil trace  126  and there is a desire to minimize Tf, to achieve high surface fill factor. In general, any suitable number of sub-traces  126 ′ may be provided in each trace  126  depending on the trace width W c , the sub-trace width T c  and the insulation with T f . The sub-traces  126 ′ of each corresponding coil trace  126  may be electrically connected in parallel at their ends (e.g. at their stator-Y dimension ends in the case of the illustrated embodiment). The regions where sub-traces  126 ′ are connected to one another may be outside of the working region of device  100 —i.e. outside of the range of motion of movable stage  110 , although this is not necessary. In other embodiments, sub-traces  126 ′ may be serially connected with one another. Coil sub-traces  126 ′ may be fabricated using known PCB fabrication technology.  FIG. 25B  shows a cross-sectional view of one particular trace  126  and its corresponding sub-traces  126 ′.   Coil traces  126  may be fabricated using techniques other than PCB technology. Any conductor that is or may be shaped to be generally linearly elongated may be used to provide coil traces  126 .  FIGS. 26A and 26B  show one example with coils  122  in a working region  124  of stator  120  comprising coil traces  126  having round cross-sections.  FIG. 26B  shows detail of how traces  126  are generally linearly elongated in the X and Y directions to provide alternating layers  128  of traces X-oriented traces  126 X and Y-oriented traces  126 Y. Each trace  126  shown in  FIGS. 26A and 26B  may be made up of further sub-traces of various cross-sections.  FIG. 26C  shows one example, wherein a trace  126  having circular cross-section comprises a plurality of sub-traces  126 ′ having circular cross-section. One common method for implementing this trace would be to use standard multi-filament wire with an external insulator.  FIG. 26D  shows one example of a coil trace  126  having rectangular cross-section with sub-traces  126 ′ of circular cross-section.   In the illustrated embodiments, coil traces  126  on different layers  128  are shown as being the same as one another. In some embodiments, coil traces  126  on different layers  128  and/or coil traces  126  with different orientations (e.g. X-orientations and Y-orientations) may have properties that are different from one another. By way of non-limiting example, X-oriented coil traces  126  may have a first coil width W c1  and/or coil pitch P c1  and Y-oriented coil traces  126  may have a second coil width W c2  and/or coil pitch P c2  which may be the same or different from those of the X-oriented coil traces  126 . Other properties of coil traces  126  could additionally or alternatively be different from one another. Similarly, magnet arrays  112  (e.g. magnet arrays  112  of different orientations (e.g. X-magnet arrays and Y-magnet arrays  112 ) or even magnet arrays  112  with the same orientations) are shown as being the same as one another. In some embodiments, different magnet arrays  112  may have properties that are different from one another. By way of non-limiting example, X-magnet arrays could have first widths W m1  and/or spatial periods λ 1  and Y-magnet arrays may have second widths W m2  and/or spatial periods λ 2 . Other properties of magnet arrays  112  could additionally or alternatively be different from one another.   In this description and the accompanying claims, elements (such as layers  128 , coil traces  126 , movable stages  110 , magnet arrays  112  and/or sensors  501  of 2D sensor array  500 ) are said to overlap one another in or along a direction. For example, coil traces  126  from different layers  128  may overlap one another in or along the stator-Z direction. When it is described that two or more objects overlap in or along a direction (e.g. the stator-Z direction), this usage should be understood to mean that a line extending in that direction (e.g. a stator-Z direction-oriented line) could be drawn to intersect the two or more objects. In this description and the accompanying claims, elements (such as sensors  501 ) are said to be aligned with one another in or along a direction. For example, sensors  501  in a stator-Y oriented sensor column may be described as being aligned with one another in a stator-Y direction. When it is described that two or more objects are aligned in or along a direction (e.g. the stator-Y direction), this usage should be understood to mean that a line in that direction (e.g. a stator-Y direction-oriented line) could be drawn to intersect the two or more objects.   In the description and drawings provided herein, movable stages are shown as being static with their X, Y and Z axes being the same as the X, Y and Z axes of the corresponding stator. This custom is adopted in this disclosure for the sake of brevity. It will of course be appreciated from this disclosure that a movable stage can (and is designed to) move with respect to its stator, in which case the stage-X, stage-Y and stage-Z directions/axes of the movable stage may no longer be the same as (or aligned with) the stator-X, stator-Y and stator-Z directions/axes of its stator. Accordingly, in this description and the claims that follow, the X, Y and Z axes of the stator may be referred to as the stator-X axis, the stator-Y axis and the stator-Z axis and the X, Y and Z axes of the movable stage may be referred to as the stage-X axis, the stage-Y axis and the stage-Z axis. Corresponding directions may be referred to as the stator-X direction (parallel to the stator-X axis), the stator-Y direction (parallel to the stator-Y axis), the stator-Z direction (parallel to the stator-Z axis), the stage-X direction (parallel to the stage-X axis), the stage-Y direction (parallel to the stage-Y axis) and the stage-Z direction (parallel to the stage-Z axis). Directions, locations and planes defined in relation to the stator axes may generally be referred to as stator directions, stator locations and stator planes and directions, locations and planes defined in relation to the stage axes may be referred to as stage directions, stage locations and stage planes.   In the description above, stators comprise current carrying coil traces and 2D sensor arrays and movable stages comprise magnet arrays. It is of course possible that this could be reversed—i.e. stators could comprise magnet arrays and movable stages could comprise current carrying coil traces and 2D sensor arrays. Also, whether a component (e.g. a stator or a movable stage) is actually moving or whether the component is actually stationary will depend on the reference frame from which the component is observed. For example, a stator can move relative to a reference frame of a movable stage, or both the stator and the movable stage can move relative to an external reference frame. Accordingly, in the claims that follow, the terms stator and movable stage and references thereto (including references to stator and/or stage X, Y, Z directions, stator and/or stage X, Y, Z-axes and/or the like) should not be interpreted literally unless the context specifically requires literal interpretation Moreover, unless the context specifically requires, it should be understood that the movable stage (and its directions, axes and/or the like) can move relative to the stator (and its directions, axes and/or the like) or that the stator (and its directions, axes and/or the like) can move relative to a movable stage (and its directions, axes and/or the like).   In the description above, a number of steps include calculating average values. For example, some steps require calculating an average value of the outputs in a row or a column while other steps require calculating a synchronous average value of a plurality of rows or columns. It should be understood that instead of calculating average values, it is sufficient to calculate sum values instead in each of these steps. That is to say, it is unnecessary to divide by the number of values being summed, if all other averages in the step are also only summed instead of averaged.   It will be appreciated by those skilled in the art that because of the spatially periodic nature of the magnet arrays and the sensor rows and columns described herein, there may be various mathematically equivalent manners in which concepts can be described, where such mathematical equivalence is attributable to the spatial periodicity of the magnet arrays, sensor rows and/or sensor columns. Unless the context dictates otherwise, mathematical descriptions of particular features used herein should be considered to incorporate mathematically equivalent features which are expressed differently, because of the spatial periodicity of the magnet arrays, sensor rows and/or sensor columns.       

     While a number of exemplary aspects and embodiments have been discussed above, those of skill in the art will recognize certain modifications, permutations, additions and sub-combinations thereof. It is therefore intended that the following appended claims and claims hereafter introduced are interpreted to include all such modifications, permutations, additions and sub-combinations as are within their true spirit and scope.