Patent Publication Number: US-2017373580-A1

Title: Six-degree-of-freedom linear motor

Description:
TECHNICAL FIELD 
     The present invention relates to the field of photolithography and, in particular, to a linear motor with six degrees of freedom (DoFs). 
     BACKGROUND 
     With the advances in photolithography and the rapid development of the semiconductor industry, the following four basic metrics have been developed for measuring the performance of photolithography equipment: critical dimension (CD) uniformity, focus, overlay and throughput. In order to enhance the CD uniformity, increases in horizontal positioning accuracy of the wafer or mask stage are required. In order to enhance the focus accuracy, increases in vertical positioning accuracy of the wafer or mask stage are required. In order to obtain higher overlay accuracy, a photolithography tool has to improve the internal model of its wafer or mask stage for better dynamic positioning performance. In addition, in order to meet the necessary requirement for an increasing throughput of photolithography equipment, the wafer or mask stage is further required to be capable of fast movement, quick start-up and shutdown. The high-speed, high-acceleration and high-positioning-performance requirements on photolithography equipment conflict with one another, because a higher scanning speed requires a more powerful motor capable of long path movement at a higher speed with multiple DoFs for lithographic exposure and alignment. Linear motors are transmission devices that can directly convert electric power into linear mechanical movement without the aid of any intermediate conversion mechanism. They are advantageous over rotary motors in terms of structural simplicity, positioning accuracy, transmission efficiency, etc. and have therefore been widely used in high-precision positioning platforms for photolithography equipment. 
     As shown in  FIGS. 1 and 2 , a conventional linear motor generally includes a Halbach magnet array  101  and a group of three-phase coils  102   a ,  102   b ,  102   c . Upon the three-phase coils  102   a ,  102   b ,  102   c  energized with respective three-phase electrical currents, the Halbach magnet array  101  works with the three-phase coils  102   a ,  102   b ,  102   c  to produce actuating forces. The mechanism is that, assuming the currents introduced in the three-phase coils  102   a ,  102   b ,  102   c  are respectively ia, ib, ic, the linear motor is displaced by a distance of y in the Y-direction, a pole pitch of the permanent magnets is τ, and every two adjacent coils are spaced apart by 4τ/3, flux densities Bya, Byb, Byc of a magnetic field created by the three-phase coils  102   a ,  102   b ,  102   c  in the Y-direction and flux densities thereof Bza, Bzb, Bzc in the Z-direction are given as: 
     
       
         
           
             
               
                 
                   { 
                   
                     
                       
                         
                           Bya 
                           = 
                           
                             
                               B 
                               ^ 
                             
                              
                             
                                 
                             
                              
                             
                               y 
                               · 
                               
                                 cos 
                                  
                                 
                                   ( 
                                   
                                     
                                       π 
                                        
                                       
                                           
                                       
                                        
                                       y 
                                     
                                     τ 
                                   
                                   ) 
                                 
                               
                             
                           
                         
                       
                     
                     
                       
                         
                           Byb 
                           = 
                           
                             
                               B 
                               ^ 
                             
                              
                             
                                 
                             
                              
                             
                               y 
                               · 
                               
                                 cos 
                                  
                                 
                                   ( 
                                   
                                     
                                       
                                         π 
                                          
                                         
                                             
                                         
                                          
                                         y 
                                       
                                       τ 
                                     
                                     + 
                                     
                                       
                                         4 
                                          
                                         
                                             
                                         
                                          
                                         π 
                                       
                                       3 
                                     
                                   
                                   ) 
                                 
                               
                             
                           
                         
                       
                     
                     
                       
                         
                           Byc 
                           = 
                           
                             
                               B 
                               ^ 
                             
                              
                             
                                 
                             
                              
                             
                               y 
                               · 
                               
                                 cos 
                                  
                                 
                                   ( 
                                   
                                     
                                       
                                         π 
                                          
                                         
                                             
                                         
                                          
                                         y 
                                       
                                       τ 
                                     
                                     + 
                                     
                                       
                                         8 
                                          
                                         
                                             
                                         
                                          
                                         π 
                                       
                                       3 
                                     
                                   
                                   ) 
                                 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   1 
                   ) 
                 
               
             
             
               
                 
                   { 
                   
                     
                       
                         
                           Bza 
                           = 
                           
                             
                               - 
                               
                                 B 
                                 ^ 
                               
                             
                              
                             
                                 
                             
                              
                             
                               z 
                               · 
                               
                                 sin 
                                  
                                 
                                   ( 
                                   
                                     
                                       π 
                                        
                                       
                                           
                                       
                                        
                                       y 
                                     
                                     τ 
                                   
                                   ) 
                                 
                               
                             
                           
                         
                       
                     
                     
                       
                         
                           Bzb 
                           = 
                           
                             
                               - 
                               
                                 B 
                                 ^ 
                               
                             
                              
                             
                                 
                             
                              
                             
                               z 
                               · 
                               
                                 sin 
                                  
                                 
                                   ( 
                                   
                                     
                                       
                                         π 
                                          
                                         
                                             
                                         
                                          
                                         y 
                                       
                                       τ 
                                     
                                     + 
                                     
                                       
                                         4 
                                          
                                         
                                             
                                         
                                          
                                         π 
                                       
                                       3 
                                     
                                   
                                   ) 
                                 
                               
                             
                           
                         
                       
                     
                     
                       
                         
                           Bzc 
                           = 
                           
                             
                               - 
                               
                                 B 
                                 ^ 
                               
                             
                              
                             
                                 
                             
                              
                             
                               z 
                               · 
                               
                                 sin 
                                  
                                 
                                   ( 
                                   
                                     
                                       
                                         π 
                                          
                                         
                                             
                                         
                                          
                                         y 
                                       
                                       τ 
                                     
                                     + 
                                     
                                       
                                         8 
                                          
                                         
                                             
                                         
                                          
                                         π 
                                       
                                       3 
                                     
                                   
                                   ) 
                                 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   2 
                   ) 
                 
               
             
           
         
       
     
     where, {circumflex over (B)}y represents Y-directional flux density amplitude of the Halbach magnet array  101 , and {circumflex over (B)}z denotes Z-directional flux density amplitude of the Halbach magnet array  101 . 
     According to a vector control technique for three-phase motors, control is accomplished by D-Q decomposition, wherein vertical and horizontal (i.e., in the X-Y plane) forces are controlled by d- and q-axis currents, and the currents ia, ib, is for the three-phase coils  102   a ,  102   b ,  102   c  can be expressed as a function of the d- and q-axis currents id and iq: 
     
       
         
           
             
               
                 
                   { 
                   
                     
                       
                         
                           ia 
                           = 
                           
                             
                               iq 
                               · 
                               
                                 cos 
                                  
                                 
                                   ( 
                                   
                                     
                                       π 
                                        
                                       
                                           
                                       
                                        
                                       y 
                                     
                                     τ 
                                   
                                   ) 
                                 
                               
                             
                             - 
                             
                               id 
                               · 
                               
                                 sin 
                                  
                                 
                                   ( 
                                   
                                     
                                       π 
                                        
                                       
                                           
                                       
                                        
                                       y 
                                     
                                     τ 
                                   
                                   ) 
                                 
                               
                             
                           
                         
                       
                     
                     
                       
                         
                           ib 
                           = 
                           
                             
                               iq 
                               · 
                               
                                 cos 
                                  
                                 
                                   ( 
                                   
                                     
                                       
                                         π 
                                          
                                         
                                             
                                         
                                          
                                         y 
                                       
                                       τ 
                                     
                                     + 
                                     
                                       
                                         4 
                                          
                                         
                                             
                                         
                                          
                                         π 
                                       
                                       3 
                                     
                                   
                                   ) 
                                 
                               
                             
                             - 
                             
                               id 
                               · 
                               
                                 sin 
                                  
                                 
                                   ( 
                                   
                                     
                                       
                                         π 
                                          
                                         
                                             
                                         
                                          
                                         y 
                                       
                                       τ 
                                     
                                     + 
                                     
                                       
                                         4 
                                          
                                         
                                             
                                         
                                          
                                         π 
                                       
                                       3 
                                     
                                   
                                   ) 
                                 
                               
                             
                           
                         
                       
                     
                     
                       
                         
                           ic 
                           = 
                           
                             
                               iq 
                               · 
                               
                                 cos 
                                  
                                 
                                   ( 
                                   
                                     
                                       
                                         π 
                                          
                                         
                                             
                                         
                                          
                                         y 
                                       
                                       τ 
                                     
                                     + 
                                     
                                       
                                         8 
                                          
                                         
                                             
                                         
                                          
                                         π 
                                       
                                       3 
                                     
                                   
                                   ) 
                                 
                               
                             
                             - 
                             
                               id 
                               · 
                               
                                 sin 
                                  
                                 
                                   ( 
                                   
                                     
                                       
                                         π 
                                          
                                         
                                             
                                         
                                          
                                         y 
                                       
                                       τ 
                                     
                                     + 
                                     
                                       
                                         8 
                                          
                                         
                                             
                                         
                                          
                                         π 
                                       
                                       3 
                                     
                                   
                                   ) 
                                 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   3 
                   ) 
                 
               
             
           
         
       
     
     Therefore, the Y- and Z-forces Fy, Fz generated by the linear motor are: 
     
       
         
           
             
               
                 
                   
                     [ 
                     
                       
                         
                           Fy 
                         
                       
                       
                         
                           Fz 
                         
                       
                     
                     ] 
                   
                   = 
                   
                     
                       [ 
                       
                         
                           
                             Bya 
                           
                           
                             Byb 
                           
                           
                             Byc 
                           
                         
                         
                           
                             Bza 
                           
                           
                             Bzb 
                           
                           
                             Bzc 
                           
                         
                       
                       ] 
                     
                     × 
                     
                       [ 
                       
                         
                           
                             ia 
                           
                         
                         
                           
                             ib 
                           
                         
                         
                           
                             ic 
                           
                         
                       
                       ] 
                     
                   
                 
               
               
                 
                   ( 
                   4 
                   ) 
                 
               
             
           
         
       
     
     The Y-directional horizontal force Fy and the Z-directional vertical force Fz make the linear motor able to move in the Y- and Z-directions, respectively. However, during the exposure and alignment, the photolithography equipment needs to move with six DoFs, i.e., the freedom of translational movement in the X-, Y- and Z-directions and slight rotational movement about the X-, Y- and Z-directions. Therefore, the traditional linear motor cannot meet the requirements of high-precision positioning. 
     In order to solve this problem, there has been also proposed in the prior art a linear motor incorporating four instances of the conventional Halbach linear motor so that they can cooperate with one another to support six-DoF movement needed by a photolithography platform. However, subject to limitations arising from the layout of magnet arrays, the linear motor has a limited movement path length in the horizontal direction. 
     SUMMARY OF THE INVENTION 
     The present invention seeks to overcome the drawbacks discussed above by presenting a linear motor capable of long-path movement with six degrees of freedom (DoFs). 
     In order to achieve the above goal, the present invention provides a linear motor with six DoFs, including a magnet array and a coil group disposed above the magnet array, wherein the magnet array includes a first Halbach magnet array and a second Halbach magnet array, the first Halbach magnet array has a parallelogrammic cross section in a plane defined by an X-axis and a Y-axis, the first Halbach magnet array has a first side parallel to the Y-axis and a second side forming an angle of θ with the Y-axis, the second Halbach magnet array is in mirror symmetry with the first Halbach magnet array with respect to the Y-axis, and wherein the coil group includes a first coil group, a second coil group, a third coil group and a fourth coil group, each having an axis inclined at an angle of θ with respect to the Y-axis, the first coil group is in mirror symmetry with the third coil group with respect to the Y-axis, the second coil group is in mirror symmetry with the fourth coil group with respect to the Y-axis. 
     Further, the first Halbach magnet array may include a plurality of Halbach units each consisting of four permanent magnets that are put together sequentially along the Y-axis and each have a parallelogrammic cross section in the plane. 
     Further, the four permanent magnets may include an S-magnet, an H-magnet, an N-magnet and another H-magnet disposed sequentially in this order along the Y-axis, the S-magnet being magnetized along a Z-axis perpendicular to the plane, the N-magnet is magnetized oppositely to the S-magnet along the Z-axis, the H-magnets is magnetized toward the N-magnet in the plane. 
     Further, the number of the Halbach units may be 6. 
     Further, the first coil group may consist of three coils arranged horizontally along the Y-axis, each of the second, third and fourth coil groups has a same structure as the first coil group, and wherein the first coil group and the second coil group are arranged sequentially along the Y-axis. 
     Further, center lines of the first and second coil groups along the Y-axis may be aligned with a same horizontal line. 
     Alternatively, center lines of the first and second coil groups along the Y-axis may also be staggered from each other along the X-axis by an interval. 
     Further, θ may be greater than 0 degree and smaller than 90 degrees. 
     Further, the coil group may be accommodated in a housing. 
     By means of proper distribution and control of a plurality of vertical forces along the Z-axis and plurality of horizontal forces in the plane defined by the X- and Y-axes generated from cooperation of the first and second Halbach magnet arrays each cross-sectionally parallelogrammic along the plane with the four coil groups, the proposed linear motor is capable of moving with six DoFs. In addition, properly designing the number of Halbach units in each of the first and second Halbach magnet arrays allows a long movement path in the Y-axis for the linear motor. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIG. 1  is a structural schematic of a conventional linear motor. 
         FIG. 2  is a cross-sectional view of the conventional linear motor. 
         FIG. 3  is a structural schematic of a linear motor with six degrees of freedom (DoFs) according to a first embodiment of the present invention. 
         FIG. 4  is a structural schematic of a Halbach unit according to the first embodiment of the present invention. 
         FIG. 5  schematically shows a distribution of coil groups according to the first embodiment of the present invention. 
         FIG. 6  schematically shows a distribution of coil groups according to a second embodiment of the present invention. 
     
    
    
     In  FIGS. 1 and 2, 101  denotes Halbach magnet array, and  102   a - 102   c , three-phase coils. 
     In  FIGS. 3 to 6, 201  denotes magnet arrays;  201   a , first Halbach magnet array;  2011   a , Halbach units;  201   b , second Halbach magnet array;  2012 , S-magnet;  2013 , N-magnet;  2014 , H-magnets;  202 , housing;  203 , first coil group;  203   a - 203   c , first coils;  204 , second coil group;  204   a - 204   c , second coils;  205 , third coil group;  205   a - 205   c , third coils;  206 , fourth coil group;  206   a - 206   c , fourth coils; θ, angle; and L, interval. 
     DETAILED DESCRIPTION 
     The present invention will be described below in detail with reference to the accompanying drawings. 
     Embodiment 1 
     As shown in  FIG. 3 , the present invention provides a linear motor with six degrees of freedom (DoFs) which includes magnet arrays  201  and coil groups. The magnet arrays  201  include a first Halbach magnet array  201   a  and a second Halbach magnet array  201   b . A cross section of the first Halbach magnet array  201   a  in the X-Y plane is a parallelogram. The first Halbach magnet array  201   a  has a pair of horizontal sides parallel to the Y-axis and a pair of inclined sides forming an angle θ with the Y-axis, where the angle θ is greater than 0° and smaller than 90°. The second Halbach magnet array  201   b  is structurally the same as the first Halbach magnet array  201   a  and is in mirror symmetry with the first Halbach magnet array  201   a  with respect to the Y-axis. The coil groups are disposed above the magnet arrays and so inclined that their center lines each form an angle θ with the Y-axis. That is, the inclined sides of the coil groups are each inclined at an angle θ relative to the Y-axis, where θ is greater than 0° and smaller than 90°. The coil groups include a first coil group  203 , a second coil group  204 , a third coil group  205  and a fourth coil group  206 . The first coil group  203  is mirror-symmetrical to the third coil group  205  and the second coil group  204  to the fourth coil group  206 , with respect to the Y-axis. The coil groups are accommodated within a housing  202 . The first Halbach magnet array  201   a  and the second Halbach magnet array  201   b , each having a parallelogrammic cross section in the X-Y plane, cooperate with the four coil groups to produce a plurality of vertical forces along the Z-axis and a plurality of horizontal forces in the X-Y plane. Distribution and control of the forces enable the linear motor to move with six DoFs. 
     Referring to  FIGS. 3 and 4 , the first Halbach magnet array  201   a  includes a number of Halbach units  2011   a  each consisting of four permanent magnets put together along the Y-axis. Each permanent magnet has a parallelogrammic cross-section in the X-Y plane. The number of the Halbach units  2011   a  is adjustable according to a practical length of movement along the Y-axis. In this embodiment, six Halbach units  2011   a  are provided. The second Halbach magnet array  201   b  is structurally the same as the first Halbach magnet array  201   a  and is in mirror symmetry with the first Halbach magnet array  201   a  with respect to the Y-axis. Long-path movement of the linear motor in the Y-axis is achievable by designing the number of Halbach units  2011   a  in the first Halbach magnet array  201   a  and the second Halbach magnet array  201   b.    
     Referring with emphasis to  FIG. 4 , the four permanent magnets include an S-magnet  2012 , an H-magnet  2014 , an N-magnet  2013  and another H-magnet  2014 , disposed sequentially in this order. The S-magnet  2012  is magnetized along the minus Z-axis, N-magnet  2013  along the plus Z-axis, and H-magnets  2014  always toward the N-magnet  2013  along the X-Y plane. Referring with emphasis to  FIG. 5 , the first coil group  203  consists of three first coils  203   a ,  203   b ,  203   c  arranged horizontal along the Y-axis. The second coil group  204  consists of three second coils  204   a ,  204   b ,  204   c  arranged horizontal along the Y-axis. The third coil group  205  consists of three third coils  205   a ,  205   b ,  205   c  arranged horizontal along the Y-axis. The fourth coil group  206  consists of three fourth coils  206   a ,  206   b ,  206   c  arranged horizontal along the Y-axis. The first coil  203   a ,  203   b ,  203   c  are inclined and their inclined sides are parallel to the inclined sides of the first Halbach magnet array  201   a . Each of the second, third and fourth coil groups  204 ,  205 ,  206  is structured identically to the first coil group  203 . The first coil group  203  and the second coil group  204  are arranged sequentially along the Y-axis and, the center lines of the first coil group  203  and the second coil group  204  along the Y-axis are aligned with the same horizontal line. 
     In the linear motor with six DoFs according to the present invention, the magnet arrays  201  work with the coil groups to produce actuating forces. The mechanism is that, for each coil group, assuming currents ia′, ib′, ic′ are introduced into the respective three coils, a pole pitch of the permanent magnets is τ′ the distance W between the centers of every two adjacent coils is 4τ′3 or 5τ′/3, with 4τ′/3 as an example, flux densities Bya′, Byb′, Byc′ of a magnetic field created by the first coils  203   a ,  203   b ,  203   c  and the third coils  205   a ,  205   b ,  205   c  along the y′-axis and flux densities thereof Bza′, Bzb′, Bzc′ along the Z-axis are given as: 
     
       
         
           
             
               
                 
                   { 
                   
                     
                       
                         
                           
                             Bya 
                             ′ 
                           
                           = 
                           
                             
                               B 
                               ^ 
                             
                              
                             
                                 
                             
                              
                             
                               
                                 y 
                                 ′ 
                               
                               · 
                               
                                 cos 
                                  
                                 
                                   ( 
                                   
                                     
                                       π 
                                        
                                       
                                           
                                       
                                        
                                       
                                         y 
                                         ′ 
                                       
                                     
                                     
                                       τ 
                                       ′ 
                                     
                                   
                                   ) 
                                 
                               
                             
                           
                         
                       
                     
                     
                       
                         
                           
                             Byb 
                             ′ 
                           
                           = 
                           
                             
                               B 
                               ^ 
                             
                              
                             
                                 
                             
                              
                             
                               
                                 y 
                                 ′ 
                               
                               · 
                               
                                 cos 
                                  
                                 
                                   ( 
                                   
                                     
                                       
                                         π 
                                          
                                         
                                             
                                         
                                          
                                         
                                           y 
                                           ′ 
                                         
                                       
                                       
                                         τ 
                                         ′ 
                                       
                                     
                                     + 
                                     
                                       
                                         4 
                                          
                                         
                                             
                                         
                                          
                                         π 
                                       
                                       3 
                                     
                                   
                                   ) 
                                 
                               
                             
                           
                         
                       
                     
                     
                       
                         
                           
                             Byc 
                             ′ 
                           
                           = 
                           
                             
                               B 
                               ^ 
                             
                              
                             
                                 
                             
                              
                             
                               
                                 y 
                                 ′ 
                               
                               · 
                               
                                 cos 
                                  
                                 
                                   ( 
                                   
                                     
                                       
                                         π 
                                          
                                         
                                             
                                         
                                          
                                         
                                           y 
                                           ′ 
                                         
                                       
                                       
                                         τ 
                                         ′ 
                                       
                                     
                                     + 
                                     
                                       
                                         8 
                                          
                                         
                                             
                                         
                                          
                                         π 
                                       
                                       3 
                                     
                                   
                                   ) 
                                 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   5 
                   ) 
                 
               
             
             
               
                 
                   { 
                   
                     
                       
                         
                           
                             
                               Bza 
                               ′ 
                             
                             = 
                             
                               
                                 - 
                                 
                                   B 
                                   ^ 
                                 
                               
                                
                               
                                   
                               
                                
                               
                                 
                                   z 
                                   ′ 
                                 
                                 · 
                                 
                                   sin 
                                    
                                   
                                     ( 
                                     
                                       
                                         π 
                                          
                                         
                                             
                                         
                                          
                                         
                                           y 
                                           ′ 
                                         
                                       
                                       
                                         τ 
                                         ′ 
                                       
                                     
                                     ) 
                                   
                                 
                               
                             
                           
                         
                       
                       
                         
                           
                             
                               Bzb 
                               ′ 
                             
                             = 
                             
                               
                                 - 
                                 
                                   B 
                                   ^ 
                                 
                               
                                
                               
                                   
                               
                                
                               
                                 
                                   z 
                                   ′ 
                                 
                                 · 
                                 
                                   sin 
                                    
                                   
                                     ( 
                                     
                                       
                                         
                                           π 
                                            
                                           
                                               
                                           
                                            
                                           
                                             y 
                                             ′ 
                                           
                                         
                                         
                                           τ 
                                           ′ 
                                         
                                       
                                       + 
                                       
                                         
                                           4 
                                            
                                           
                                               
                                           
                                            
                                           π 
                                         
                                         3 
                                       
                                     
                                     ) 
                                   
                                 
                               
                             
                           
                         
                       
                       
                         
                           
                             
                               Bzc 
                               ′ 
                             
                             = 
                             
                               
                                 - 
                                 
                                   B 
                                   ^ 
                                 
                               
                                
                               
                                   
                               
                                
                               
                                 
                                   z 
                                   ′ 
                                 
                                 · 
                                 
                                   sin 
                                    
                                   
                                     ( 
                                     
                                       
                                         
                                           π 
                                            
                                           
                                               
                                           
                                            
                                           
                                             y 
                                             ′ 
                                           
                                         
                                         
                                           τ 
                                           ′ 
                                         
                                       
                                       + 
                                       
                                         
                                           8 
                                            
                                           
                                               
                                           
                                            
                                           π 
                                         
                                         3 
                                       
                                     
                                     ) 
                                   
                                 
                               
                             
                           
                         
                       
                     
                     , 
                   
                 
               
               
                 
                   ( 
                   6 
                   ) 
                 
               
             
           
         
       
     
     and flux densities Bya′, Byb′, Byc′ of a magnetic field created by the second coils  204   a ,  204   b ,  204   c  and the fourth coils  206   a ,  206   b ,  206   c  along the y′-axis and flux densities thereof Bza′, Bzb′, Bzc′ along the Z-axis are given as: 
     
       
         
           
             
               
                 
                   { 
                   
                     
                       
                         
                           
                             Bya 
                             ′ 
                           
                           = 
                           
                             
                               B 
                               ^ 
                             
                              
                             
                                 
                             
                              
                             
                               
                                 y 
                                 ′ 
                               
                               · 
                               
                                 cos 
                                  
                                 
                                   ( 
                                   
                                     
                                       π 
                                        
                                       
                                           
                                       
                                        
                                       
                                         ( 
                                         
                                           
                                             y 
                                             ′ 
                                           
                                           + 
                                           D 
                                         
                                         ) 
                                       
                                     
                                     
                                       τ 
                                       ′ 
                                     
                                   
                                   ) 
                                 
                               
                             
                           
                         
                       
                     
                     
                       
                         
                           
                             Byb 
                             ′ 
                           
                           = 
                           
                             
                               B 
                               ^ 
                             
                              
                             
                                 
                             
                              
                             
                               
                                 y 
                                 ′ 
                               
                               · 
                               
                                 cos 
                                  
                                 
                                   ( 
                                   
                                     
                                       
                                         π 
                                         ( 
                                         
                                             
                                         
                                          
                                         
                                           
                                             y 
                                             ′ 
                                           
                                           + 
                                           D 
                                         
                                         ) 
                                       
                                       
                                         τ 
                                         ′ 
                                       
                                     
                                     + 
                                     
                                       
                                         4 
                                          
                                         
                                             
                                         
                                          
                                         π 
                                       
                                       3 
                                     
                                   
                                   ) 
                                 
                               
                             
                           
                         
                       
                     
                     
                       
                         
                           
                             Byc 
                             ′ 
                           
                           = 
                           
                             
                               B 
                               ^ 
                             
                              
                             
                                 
                             
                              
                             
                               
                                 y 
                                 ′ 
                               
                               · 
                               
                                 cos 
                                  
                                 
                                   ( 
                                   
                                     
                                       
                                         π 
                                         ( 
                                         
                                             
                                         
                                          
                                         
                                           
                                             y 
                                             ′ 
                                           
                                           + 
                                           D 
                                         
                                         ) 
                                       
                                       
                                         τ 
                                         ′ 
                                       
                                     
                                     + 
                                     
                                       
                                         8 
                                          
                                         
                                             
                                         
                                          
                                         π 
                                       
                                       3 
                                     
                                   
                                   ) 
                                 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   7 
                   ) 
                 
               
             
             
               
                 
                   { 
                   
                     
                       
                         
                           
                             
                               Bza 
                               ′ 
                             
                             = 
                             
                               
                                 - 
                                 
                                   B 
                                   ^ 
                                 
                               
                                
                               
                                   
                               
                                
                               
                                 
                                   z 
                                   ′ 
                                 
                                 · 
                                 
                                   sin 
                                    
                                   
                                     ( 
                                     
                                       
                                         π 
                                          
                                         
                                             
                                         
                                          
                                         
                                           ( 
                                           
                                             
                                               y 
                                               ′ 
                                             
                                             + 
                                             D 
                                           
                                           ) 
                                         
                                       
                                       
                                         τ 
                                         ′ 
                                       
                                     
                                     ) 
                                   
                                 
                               
                             
                           
                         
                       
                       
                         
                           
                             
                               Bzb 
                               ′ 
                             
                             = 
                             
                               
                                 - 
                                 
                                   B 
                                   ^ 
                                 
                               
                                
                               
                                   
                               
                                
                               
                                 
                                   z 
                                   ′ 
                                 
                                 · 
                                 
                                   sin 
                                    
                                   
                                     ( 
                                     
                                       
                                         
                                           π 
                                           ( 
                                           
                                               
                                           
                                            
                                           
                                             
                                               y 
                                               ′ 
                                             
                                             + 
                                             D 
                                           
                                           ) 
                                         
                                         
                                           τ 
                                           ′ 
                                         
                                       
                                       + 
                                       
                                         
                                           4 
                                            
                                           
                                               
                                           
                                            
                                           π 
                                         
                                         3 
                                       
                                     
                                     ) 
                                   
                                 
                               
                             
                           
                         
                       
                       
                         
                           
                             
                               Bzc 
                               ′ 
                             
                             = 
                             
                               
                                 - 
                                 
                                   B 
                                   ^ 
                                 
                               
                                
                               
                                   
                               
                                
                               
                                 
                                   z 
                                   ′ 
                                 
                                 · 
                                 
                                   sin 
                                    
                                   
                                     ( 
                                     
                                       
                                         
                                           π 
                                            
                                           
                                               
                                           
                                            
                                           
                                             ( 
                                             
                                               
                                                 y 
                                                 ′ 
                                               
                                               + 
                                               D 
                                             
                                             ) 
                                           
                                         
                                         
                                           τ 
                                           ′ 
                                         
                                       
                                       + 
                                       
                                         
                                           8 
                                            
                                           
                                               
                                           
                                            
                                           π 
                                         
                                         3 
                                       
                                     
                                     ) 
                                   
                                 
                               
                             
                           
                         
                       
                     
                     , 
                   
                 
               
               
                 
                   ( 
                   8 
                   ) 
                 
               
             
           
         
       
     
     where {circumflex over (B)}y′ represents flux density amplitude of the magnet arrays  201  along the y′-axis, {circumflex over (B)}z′ denotes flux density amplitude of the magnet arrays  201  along the Z-axis, y′=(Y-displacement of the linear motor) *sin θ, * is the mathematical multiplication symbol, and D is the distance between the centers of the first coil group  203  and the second coil group  204  along the y′-axis. 
     According to a vector control technique for three-phase motors, control is accomplished by D-Q decomposition, wherein vertical and horizontal (i.e., in the X-Y plane) forces are controlled by d- and q-axis currents, with the currents ia′, ib′, ic′ for the first coils  203   a ,  203   b ,  203   c  and the third coils  205   a ,  205   b ,  205   c  expressed as a function of the d- and q-axis currents id and iq: 
     
       
         
           
             
               
                 
                   { 
                   
                     
                       
                         
                           
                             
                               ia 
                               ′ 
                             
                             = 
                             
                               
                                 iq 
                                 · 
                                 
                                   cos 
                                    
                                   
                                     ( 
                                     
                                       
                                         π 
                                          
                                         
                                             
                                         
                                          
                                         
                                           y 
                                           ′ 
                                         
                                       
                                       
                                         τ 
                                         ′ 
                                       
                                     
                                     ) 
                                   
                                 
                               
                               - 
                               
                                 id 
                                 · 
                                 
                                   sin 
                                    
                                   
                                     ( 
                                     
                                       
                                         π 
                                          
                                         
                                             
                                         
                                          
                                         
                                           y 
                                           ′ 
                                         
                                       
                                       
                                         τ 
                                         ′ 
                                       
                                     
                                     ) 
                                   
                                 
                               
                             
                           
                         
                       
                       
                         
                           
                             
                               ib 
                               ′ 
                             
                             = 
                             
                               
                                 iq 
                                 · 
                                 
                                   cos 
                                    
                                   
                                     ( 
                                     
                                       
                                         
                                           π 
                                            
                                           
                                               
                                           
                                            
                                           
                                             y 
                                             ′ 
                                           
                                         
                                         
                                           τ 
                                           ′ 
                                         
                                       
                                       + 
                                       
                                         
                                           4 
                                            
                                           
                                               
                                           
                                            
                                           π 
                                         
                                         3 
                                       
                                     
                                     ) 
                                   
                                 
                               
                               - 
                               
                                 id 
                                 · 
                                 
                                   sin 
                                    
                                   
                                     ( 
                                     
                                       
                                         
                                           π 
                                            
                                           
                                               
                                           
                                            
                                           
                                             y 
                                             ′ 
                                           
                                         
                                         
                                           τ 
                                           ′ 
                                         
                                       
                                       + 
                                       
                                         
                                           4 
                                            
                                           
                                               
                                           
                                            
                                           π 
                                         
                                         3 
                                       
                                     
                                     ) 
                                   
                                 
                               
                             
                           
                         
                       
                       
                         
                           
                             
                               ic 
                               ′ 
                             
                             = 
                             
                               
                                 iq 
                                 · 
                                 
                                   cos 
                                    
                                   
                                     ( 
                                     
                                       
                                         
                                           π 
                                            
                                           
                                               
                                           
                                            
                                           
                                             y 
                                             ′ 
                                           
                                         
                                         
                                           τ 
                                           ′ 
                                         
                                       
                                       + 
                                       
                                         
                                           8 
                                            
                                           
                                               
                                           
                                            
                                           π 
                                         
                                         3 
                                       
                                     
                                     ) 
                                   
                                 
                               
                               - 
                               
                                 id 
                                 · 
                                 
                                   sin 
                                    
                                   
                                     ( 
                                     
                                       
                                         
                                           π 
                                            
                                           
                                               
                                           
                                            
                                           
                                             y 
                                             ′ 
                                           
                                         
                                         
                                           τ 
                                           ′ 
                                         
                                       
                                       + 
                                       
                                         
                                           8 
                                            
                                           
                                               
                                           
                                            
                                           π 
                                         
                                         3 
                                       
                                     
                                     ) 
                                   
                                 
                               
                             
                           
                         
                       
                     
                     , 
                   
                 
               
               
                 
                   ( 
                   9 
                   ) 
                 
               
             
           
         
       
     
     and the currents ia′, ib′, for the second coils  204   a ,  204   b ,  204   c  and the fourth coils  206   a ,  206   b ,  206   c  expressed as a function of the d- and q-axis currents id and iq: 
     
       
         
           
             
               
                 
                   { 
                   
                     
                       
                         
                           
                             
                               ia 
                               ′ 
                             
                             = 
                             
                               
                                 iq 
                                 · 
                                 
                                   cos 
                                    
                                   
                                     ( 
                                     
                                       
                                         π 
                                          
                                         
                                             
                                         
                                          
                                         
                                           ( 
                                           
                                             
                                               y 
                                               ′ 
                                             
                                             + 
                                             D 
                                           
                                           ) 
                                         
                                       
                                       
                                         τ 
                                         ′ 
                                       
                                     
                                     ) 
                                   
                                 
                               
                               - 
                               
                                 id 
                                 · 
                                 
                                   sin 
                                    
                                   
                                     ( 
                                     
                                       
                                         π 
                                          
                                         
                                             
                                         
                                          
                                         
                                           ( 
                                           
                                             
                                               y 
                                               ′ 
                                             
                                             + 
                                             D 
                                           
                                           ) 
                                         
                                       
                                       
                                         τ 
                                         ′ 
                                       
                                     
                                     ) 
                                   
                                 
                               
                             
                           
                         
                       
                       
                         
                           
                             
                               ib 
                               ′ 
                             
                             = 
                             
                               
                                 iq 
                                 · 
                                 
                                   cos 
                                    
                                   
                                     ( 
                                     
                                       
                                         
                                           π 
                                           ( 
                                           
                                               
                                           
                                            
                                           
                                             
                                               y 
                                               ′ 
                                             
                                             + 
                                             D 
                                           
                                           ) 
                                         
                                         
                                           τ 
                                           ′ 
                                         
                                       
                                       + 
                                       
                                         
                                           4 
                                            
                                           
                                               
                                           
                                            
                                           π 
                                         
                                         3 
                                       
                                     
                                     ) 
                                   
                                 
                               
                               - 
                               
                                 id 
                                 · 
                                 
                                   sin 
                                    
                                   
                                     ( 
                                     
                                       
                                         
                                           π 
                                            
                                           
                                               
                                           
                                            
                                           
                                             ( 
                                             
                                               
                                                 y 
                                                 ′ 
                                               
                                               + 
                                               D 
                                             
                                             ) 
                                           
                                         
                                         
                                           τ 
                                           ′ 
                                         
                                       
                                       + 
                                       
                                         
                                           4 
                                            
                                           
                                               
                                           
                                            
                                           π 
                                         
                                         3 
                                       
                                     
                                     ) 
                                   
                                 
                               
                             
                           
                         
                       
                       
                         
                           
                             
                               ic 
                               ′ 
                             
                             = 
                             
                               
                                 iq 
                                 · 
                                 
                                   cos 
                                    
                                   
                                     ( 
                                     
                                       
                                         
                                           π 
                                            
                                           
                                               
                                           
                                            
                                           
                                             ( 
                                             
                                               
                                                 y 
                                                 ′ 
                                               
                                               + 
                                               D 
                                             
                                             ) 
                                           
                                         
                                         
                                           τ 
                                           ′ 
                                         
                                       
                                       + 
                                       
                                         
                                           8 
                                            
                                           
                                               
                                           
                                            
                                           π 
                                         
                                         3 
                                       
                                     
                                     ) 
                                   
                                 
                               
                               - 
                               
                                 id 
                                 · 
                                 
                                   sin 
                                    
                                   
                                     ( 
                                     
                                       
                                         
                                           π 
                                            
                                           
                                               
                                           
                                            
                                           
                                             ( 
                                             
                                               
                                                 y 
                                                 ′ 
                                               
                                               + 
                                               D 
                                             
                                             ) 
                                           
                                         
                                         
                                           τ 
                                           ′ 
                                         
                                       
                                       + 
                                       
                                         
                                           8 
                                            
                                           
                                               
                                           
                                            
                                           π 
                                         
                                         3 
                                       
                                     
                                     ) 
                                   
                                 
                               
                             
                           
                         
                       
                     
                     . 
                   
                 
               
               
                 
                   ( 
                   10 
                   ) 
                 
               
             
           
         
       
     
     From these formulas, the horizontal and vertical forces Fh, Fz generated by the linear motor can be calculated as: 
     
       
         
           
             
               
                 
                   
                     [ 
                     
                       
                         
                           Fh 
                         
                       
                       
                         
                           Fv 
                         
                       
                     
                     ] 
                   
                   = 
                   
                     
                       [ 
                       
                         
                           
                             
                               Bya 
                               ′ 
                             
                           
                           
                             
                               Byb 
                               ′ 
                             
                           
                           
                             
                               Byc 
                               ′ 
                             
                           
                         
                         
                           
                             
                               Bza 
                               ′ 
                             
                           
                           
                             
                               Bzb 
                               ′ 
                             
                           
                           
                             
                               Bzc 
                               ′ 
                             
                           
                         
                       
                       ] 
                     
                     × 
                     
                       [ 
                       
                         
                           
                             
                               ia 
                               ′ 
                             
                           
                         
                         
                           
                             
                               ib 
                               ′ 
                             
                           
                         
                         
                           
                             
                               ic 
                               ′ 
                             
                           
                         
                       
                       ] 
                     
                   
                 
               
               
                 
                   ( 
                   11 
                   ) 
                 
               
             
           
         
       
     
     The horizontal force Fh and the vertical force Fv can be converted into forces Fx, Fy, Fz along the X-, Y- and Z-axis according to: 
     
       
         
           
             
               
                 
                   
                     [ 
                     
                       
                         
                           Fx 
                         
                       
                       
                         
                           Fy 
                         
                       
                       
                         
                           Fz 
                         
                       
                     
                     ] 
                   
                   = 
                   
                     
                       [ 
                       
                         
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               θ 
                             
                           
                           
                             0 
                           
                         
                         
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               θ 
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             1 
                           
                         
                       
                       ] 
                     
                     × 
                     
                       [ 
                       
                         
                           
                             Fh 
                           
                         
                         
                           
                             Fv 
                           
                         
                       
                       ] 
                     
                   
                 
               
               
                 
                   ( 
                   12 
                   ) 
                 
               
             
           
         
       
     
     As shown in  FIG. 5 , the first coil group  203  generates a vertical force Fv 1  along the Z-axis and a horizontal force Fh 1  in the X-Y plane, the second coil group  204  generates a vertical force Fv 2  along the Z-axis and a horizontal force Fh 2  in the X-Y plane, the third coil group  205  generates a vertical force Fv 3  along the Z-axis and a horizontal force Fh 3  in the X-Y plane, and the fourth coil group  206  generates a vertical force Fv 4  along the Z-axis and a horizontal force Fh 4  in the X-Y plane. Y-directional components of the horizontal forces Fh 1 , Fh 2  are exerted along a same horizontal line, and Y-directional components of the horizontal forces Fh 3 , Fh 3  along another same horizontal line. According to the force distribution and control principles, the horizontal forces Fh 1 , Fh 2 , Fh 3 , Fh 4  in the X-Y plane allow translation along the X- and Y-axis and slight rotation about the Z-axis of the linear motor. Additionally, the vertical forces Fv 1 , Fv 2 , Fv 3 , Fv 4  allow slight translation along the Z-axis and slight rotation about the X- and Y-axis of the linear motor. That is, the linear motor is capable of movement with six DoFs with high positioning accuracy. 
     Embodiment 2 
     Differing from Embodiment 1, the center line along the Y-axis of the second coil group  204  is staggered from that of the first coil group  203  along the X-axis by an interval L, in accordance with this embodiment. Since the third coil group  205  is in mirror symmetry with the first coil group  203  and the fourth coil group  206  is in mirror symmetry with the second coil group  204  with respect to the Y-axis, there is also an interval L along the X-axis between the center lines of the fourth coil group  206  and the third coil group  205 , as shown in  FIG. 6 . The first coil group  203  produces a vertical force Fv 1  along the Z-axis and a horizontal force Fh 1  in the X-Y plane, the second coil group  204  produces a vertical force Fv 2  along the Z-axis and a horizontal force Fh 2  in the X-Y plane, the third coil group  205  produces a vertical force Fv 3  along the Z-axis and a horizontal force Fh 3  in the X-Y plane, and the fourth coil group  206  produces a vertical force Fv 4  along the Z-axis and a horizontal force Fh 4  in the X-Y plane. The components of the horizontal forces Fh 1 , Fh 2  along the Y-axis are staggered from each other along the X-axis by an interval L, and the components of the horizontal forces Fh 3 , Fh 3  along the Y-axis are also staggered from each other along the X-axis by an interval L. According to the force distribution and control principles, the horizontal forces Fh 1 , Fh 2 , Fh 3 , Fh 4  within the X-Y plane allow translation along the X- and Y-axis and slight rotation about the Z-direction of the linear motor. Additionally, the vertical forces Fv 1 , Fv 2 , Fv 3 , Fv 4  allow slight translation along the Z-axis and slight rotation about the X- and Y-axis of linear motor. That is, the linear motor is capable of movement with six DoFs. 
     The above arrangement of the coil groups allows for Y-directional redundant actuation of the linear motor with six DoFs according to this embodiment. As used herein, the term “redundant actuation” refers to using additional actuators to those necessary for the six DoFs to reduce vibration during movement, improve the control bandwidth or allow less mass of the linear motor at the same bandwidth so as to enhance the performance thereof. 
     Although several embodiments of the present invention have been described above herein, they are intended to be merely illustrative but not to limit the scope of the invention. All omissions, substitutions and alterations made without departing from the spirit of the present invention are all embraced within the scope thereof.