Abstract:
An image-forming apparatus has: a transfer belt transferring a sheet; an encoder outputting a pulse signal in response to a rotational motion of the transfer belt; a belt reference sensor detecting a belt reference mark provided on the transfer belt; a first accumulator measuring a rotational position of the transfer belt against the reference mark by counting a pulse signal; a storage unit storing a table of printing timing correction values for one rotational cycle of the transfer belt; a second accumulator generating a printing timing signal by shifting a phase of the pulse signal by a printing timing correction value read from the table; and a image-forming unit controlling ink heads to eject inks on the sheet in synchronization with the printing timing signal, wherein an error in transfer amount of the sheet is calculated from a fluctuation in an interval of the inks ejected from the ink heads.

Description:
BACKGROUND OF THE INVENTION 
     1. Field of the Invention 
     The present invention relates to image-forming apparatuses with ink-eject heads that enable to eject inks to form images on image-formed media. 
     2. Description of Related Art 
     There has been widely used an ink-jet recording apparatus as recording means for recording images on image-formed media, such as paper, based on image information. The ink-jet recording apparatus is usually applied on printers, facsimiles or copying machines, and the recording images include characters or symbols in general. 
     During the recording of images, however, the landing of ink droplets from respective ink-eject heads tends to deviate from predetermined positions. This will be below called the “landing misalignment”. The landing misalignment is divided into two components: the DC component that shifts with at a constant amount; and the AC component that fluctuates periodically. 
     The AC component of a landing misalignment is caused by unevenness in the thickness of a belt or unevenness in the radius of a roller. In order to reduce a landing misalignment with the circumferential length of a belt as one cycle due to the former unevenness or a landing misalignment with the circumferential length of a roller as one cycle due to the latter one, the following methods have been conventionally proposed. Note here that the former cycle will be below called the “belt cycle” and the latter called the “roller cycle”. 
     [Landing Misalignment with Belt Cycle] 
     (1-1) There is proposed a method to enlarge the diameter of a roller or reduce the thickness of a belt so as to make unevenness in the thickness of the belt relatively small. The former, however, makes an image forming device or its components large and heavy and the latter makes its endurance weak. This method can reduce but cannot eliminate a landing misalignment. 
     (1-2) There is proposed a method to measure the rotation speed of a belt by bringing a roller into contact with the surface of the belt without any influence of unevenness in the thickness of the belt. 
     Japanese Patent Publication Laid-open No. 2004-188921 discloses a method to measure the thickness of a belt using a laser Doppler measurement device. 
     The measurement of the rotation speed of a belt at a contact line of the roller and belt, however, frequently gets false detection due to slide or vibrations. Increasing the pressure of the roller to prevent the slide causes the deformation of the belt. Accordingly, the measurement receives some influence of unevenness in the thickness or elasticity of the belt. Also, the laser Doppler measurement device is quite expensive. Moreover, these methods get detection error due to paper dust or ink-mist. 
     (1-3) Japanese Patent Publication Laid-open No. 2000-356875 discloses a method to measure the thickness of a belt in advance and to adjust a recording timing or transfer speed based on obtained data. This method, however, requires a thickness measurement device with high-accuracy so as to accumulate minute unevenness in thickness. Moreover, when a multilayer belt is used, this method cannot deal with landing misalignments other than that due to unevenness in thickness. 
     (1-4) Japanese Patent Publication Laid-open No. H10-186787 discloses a method to measure and correct unevenness in the thickness of a belt by reading a registered pattern on the belt surface at a position different from recording positions. This, however, requires reading means with high-accuracy and high-speed. In addition, the measuring ability of the reading means decreases due to paper dust or ink-mist. 
     [Landing Misalignment with Roller Cycle] 
     (2-1) Japanese Patent Publication Laid-open No. H03-2067 discloses a method to eliminate a landing misalignment with a roller cycle by adjusting intervals between image-recording units to the roller cycle. This method, when combined with the method (1-1) or (1-2), enables to reduce the landing misalignment due to the unevenness of a belt and roller. 
     When the method (2-1) is combined with the method (1-1), an image-forming device is made large because it also requires widening an interval between ink heads. This causes to degrade the accuracy of the manufacturing dimension of head mounting units and to increase the deformation of these units. Also, when the method (2-1) is combined with the method (1-2), it cannot solve the problem of the method (1-2) itself. 
     Japanese Patent Publication Laid-open No. H03-2068 discloses a method to resolve unevenness with a belt cycle by using the method (1-2) and unevenness with a roller cycle by storing the roller cycle and performing correction. However, this method cannot solve the problem of the method (1-2), either. 
     SUMMARY OF THE INVENTION 
     An object of the present invention is to eliminate the AC component of a landing misalignment due to unevenness in the thickness of a belt or unevenness in the radius of a roller. 
     To achieve the above described object, the first aspect of the present invention provides an image-forming apparatus comprising: a transfer unit (for example, a transfer belt according to the first embodiment) that transfers an image-formed medium by rotation thereof; an image-forming unit that includes a first ink head and a second ink head (e.g. ink heads K, C according to the first embodiment) arranged in a transfer direction of the image-formed medium at a predetermined interval and controls the first ink head and the second ink head to eject first and second inks on the image-formed medium; a transfer amount detecting unit (e.g. an encoder according to the first embodiment) that outputs a pulse signal in response to a rotational motion of the transfer unit; a transfer reference position detecting unit (e.g. a belt reference sensor according to the first embodiment) that detects a reference position provided on the transfer unit; an ink landing position calculating unit (e.g. a first accumulator according to the first embodiment) that calculates an ink landing position corresponding to the reference position on the image-formed medium by counting a pulse signal after detection of the reference position; a storage unit that stores correction values for printing timing for one rotational cycle of the transfer unit as a correction table; and a printing timing signal generating unit (e.g. a second accumulator according to the first embodiment) that reads out a correction value for printing timing, which corresponds to the ink landing position on the image-formed medium, from the correction table and generates a printing timing signal by shifting a phase of the pulse signal by the correction value for printing timing; wherein the correction table includes a set of values, each value being calculated based on a measurement value of an interval between a first ink landing position by the first ink head and a second ink landing position by the second ink head in a predetermined image pattern formed on the predetermined image-formed media which are corresponding to a length longer than the one rotational cycle of the transfer unit, and canceling an error of a transfer amount of the predetermined image-formed media by the transfer unit, and wherein the image-forming unit forms an image on the image-formed medium in transfer by controlling the first and second ink heads to eject respective inks in synchronization with the printing timing signal. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIG. 1  is a schematic view of an image-forming apparatus according to the first embodiment of the present invention. 
         FIG. 2  is a view of steps for correcting a landing misalignment. 
         FIG. 3  is a schematic view of the internal structure of a correction circuit according to the first embodiment. 
         FIG. 4  is a view of operations of the correction circuit according to the first embodiment. 
         FIG. 5  is a view of a landing misalignment amount “g” at a position “y′” of a transfer belt when the driving radius “r” [mm] of a driven roller fluctuates in a sine wave form with the circumferential length of the transfer belt as one cycle. 
         FIG. 6  is a view of the landing misalignment amounts of the color inks K, C, M and Y from a value in design at a printing position. 
         FIG. 7  is a view of the landing misalignment amount of K relative to these of C, M and Y at a printing position. 
         FIG. 8  is a view of the measurement values of a relative landing misalignment amount KC(y), and landing misalignment amounts K(y) and C(y) when K(0)=0. 
         FIG. 9  is a view of K(y) when K(y)=0 where 0≦y&lt;110, a wave form S(y) with one cycle (110 [mm]) of K(y), and a landing misalignment amount Δ(y) which is a difference between S(y) and K(y). 
         FIG. 10  is a schematic view of an image-forming apparatus according to the second embodiment of the present invention. 
         FIG. 11  is a schematic view of the internal structure of a correction circuit according to the second embodiment. 
         FIG. 12  is a view of landing misalignment amounts KY(y), KM(y), KC(y) and K(y) obtained from KC(y), K(y) obtained from KM(y), and K(y) obtained from KY(y). 
         FIG. 13  is a view of K(y) obtained from KY(y), KM(y) and KC(y), a wave form S(y) with one cycle (110 [mm]) of K(y), and a landing misalignment amount Δ(y) which is a difference between K(y) and S(y). 
         FIG. 14A  is a view of the landing misalignment amounts of K, C, M and Y from a precise dot pitch. 
         FIG. 14B  is a view of the landing misalignment amounts of C, M and Y relative to that of K. 
         FIG. 14C  is a view of an arrangement of ink heads. 
         FIG. 14D  is a view of a landing position in design and landing positions of K, C, M and Y. 
         FIG. 15A  is a view of the total landing misalignment amounts of K, C, M and Y, from a landing position in design, each of which is a landing misalignment amount with a belt cycle plus a landing misalignment amount with a roller cycle. 
         FIG. 15B  is a view of the total landing misalignment amounts of C, M and Y relative to that of K. 
         FIG. 15C  is a view of the landing misalignment amounts with the belt cycle of C, M and Y relative to that of K, each of which is the total landing misalignment amount minus the landing misalignment amount with the roller cycle. 
     
    
    
     DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS 
     Referring to Figures, several embodiments of the invention will be below explained. 
     First Embodiment 
       FIG. 1  is a schematic view of an image-forming apparatus according to the first embodiment of the present invention. The transfer of sheets is performed in the following steps: 
     (1) Picking up a sheet from a paper feeding tray  101  to transfer the sheet to a register roller  103  at rest; 
     (2) Hitting a tip end line of the sheet to the register roller  103  at rest for correcting a landing misalignment in a sub scanning direction; 
     (3) Rotating the register roller  103  and transferring the sheet to a transfer belt  105  at a speed faster than that of the transfer belt  105 ; 
     (4) Transferring the sheet to right of a sub scanning direction in a stuck state over the transfer belt  105  by a negative pressure from an absorbing fan (not shown) just below the transfer belt  105 ; 
     (5) Sequentially printing an image on the sheet with a plurality of line ink-jet heads (ink heads), K (black)  107 , C (cyan)  109 , M (mazenda)  111  and Y (yellow)  113 , aligned above the transfer belt  105 ; and 
     (6) Discharging the sheet to a paper receiving tray (not shown) on the right of the transfer belt  105 . 
     [Specification of Units] 
     
         
         
           
             The circumferential length of the transfer belt  105  in the sub-scanning direction is 1275 mm, the thickness is 0.45 mm, and the width in the main-scanning direction is 355 mm; 
             The diameter of a driven roller  115  is φ40 mm; and 
             The pulse number of a rotary encoder (encoder)  117  is 1500 ppr (pulse/round), the pulse array cycle is 84.67 μm (≅300 dpi), and the interval between neighboring ink heads is 110 mm. 
           
         
       
    
     During the steps (3)-(5), the step (1) is started for another sheet. Since this shortens a sheet transfer interval, the number of sheets printed per time is increased. Printing an image on a sheet with each ink head is synchronized to a pulse series of the encoder  117 , which is provided along the axis of the driven roller  115 . Accordingly, no influence is given on the image from unevenness in the rotation of a driving roller  121  driven by a driving motor  119 . However, the landing misalignment of each color ink occurs with the circumferential length of the transfer belt  105  as a belt cycle. This is so called the “AC component of the landing misalignment”. The transfer belt  105  has a belt reference mark  123  on a surface thereof. The belt reference mark  123  is detected by a belt reference sensor  125 . Signals from the encoder  117  and the belt reference sensor  125  are sent to a correction circuit  127 . The correction circuit  127  outputs a driving signal to a head driving circuit  129 . A sheet tip sensor  131  measures a tip end line of the sheet. Ink heads  107 ,  109 ,  111 , and  113  respectively have six ink heads of 2-inch each arranged in a houndstooth check shape. 
       FIG. 2  shows the following steps for correcting the landing misalignment of each ink color:
         (1) Printing an image on a sheet in an initial state, and in particular, sequentially printing a predetermined image pattern on a plurality of sheets with their total length longer than a belt rotational cycle (in this case, three A3 sheets with the length of 420 mm and the interval between neighboring sheets of 50 mm);   (2) Measuring the landing misalignment amount of each color ink, and in particular, reading the landing misalignment amount of each color ink relative to that of a reference color ink on the sheets by an external or device-equipped scanner;   (3) Creating a table of correction values (correction table)  37  based on the measurement results and storing this table within the correction circuit  127 ; and   (4) Printing the predetermined image pattern on the sheets with correcting the AC component of the landing misalignment by generating a timing signal that is an encoder pulse signal with a phase shifted based on the correction table  37  stored in the correction circuit  127 .       
     Here, the predetermined image pattern is a pattern for analyzing an amount of color misalignment on an image-forming device such as an ink-jet. The international publication WO2003/082587 discloses a test chart as such a pattern in  FIG. 8 . 
     [Outline of Correction Circuit] 
       FIG. 3  shows the internal structure of the correction circuit  127  and  FIG. 4  shows operations of the correction circuit  127 . According to  FIGS. 3 and 4 , there is explained an example of operations of the correction circuit  127 . 
     [Specification of Phase Shifting of Encoder Pulse at Correction Circuit] 
     
         
         
           
             The phase shifting ability is as follows: the dividing ability is 1/256 pulse (0.33 μm) with data of 10 bits per point; the shift range is 0˜3.996 pulses (0˜338.34 μm). 
             The total volume of the correction table  37  is 640 bits according to 1387 mm, that is, every 256 lines (about 21.7 mm)×64 sections. 
           
         
       
    
     There is explained the internal structure of the correction circuit  127 . 
     The correction circuit  127  comprises an Lck counter (not shown), an Lck×256 frequency multiplier  31 , a belt Lck counter  32 , a first accumulator  33 , a second accumulator  34 , a comparator  35 , a pulse counter  36 , and the correction table  37 . 
     The Lck×256 frequency multiplier  31  generates 256 signals in one cycle of a signal “Lck” and counts them. Here, “Lck” is a pulse signal with a same phase as an encoder phase A generated per one printing cycle (300 dpi). 
     The belt Lck counter  32  is a counter for measuring a current position (of the belt reference mark  123 ) of the transfer belt  105  in a rotating state. A correction value corresponding to the measured current position of the transfer belt  105  is selected from the correction table  37 . 
     The first accumulator  33  adds a counted value of the Lck counter and a counted value of the Lck×256 frequency multiplier  31 . 
     The second accumulator  34  adds a correction value of the correction table  37  to a value of a counted value of the pulse counter  36  plus 1. 
     The comparator  35  compares the added value of the first accumulator  33  and that of the second accumulator  34 . When the former value is greater than or equal to the latter value, the comparator  35  outputs a corrected pulse to the pulse counter  36 . 
     The pulse counter  36  outputs the corrected pulse and counts up its own value. 
     Referring to  FIG. 4 , the above described “addition” and “comparison” operations are explained in detail. 
     The first accumulator  33  adds “a shifted value  41  of the counted value (N+1 in this example) of the Lck counter by 8 bits to the left” and “the counted value  43  of the Lck×256 frequency multiplier  31 ”. The second accumulator  34  adds “a shifted value  45 , (N+1), of the counted value (N) of the pulse counter  36  plus 1 by 8 bits to the left” and “the correction value (the counted value of the Lck×256 frequency multiplier  31 )  47 , 01100100, (corresponding to H N+1 =100)”. The comparator  35  compares the added value of the first accumulator  33  and that of the second accumulator  34 . If the former value is more than or equal to the latter value, the comparator  35  outputs the corrected pulse. 
     The above described structure enables to shift the phase of an encoder pulse based on the position of the transfer belt  105  and the correction values of the correction table  37 . 
     [Method of Creating Correction Table] 
     There is explained a method to create the correction table  37  from a landing misalignment due to unevenness in the thickness of the transfer belt  105 . 
     Now define “y′”[mm] as a distance (position) from a predetermined fixed (static) reference position “O” to the belt reference mark  123  of the transfer belt  105  in a sheet transfer direction (see  FIG. 1 ), and set the circumferential length “L” of the transfer belt  105  as 1275 mm. This definition is equivalent to define “y′” as a distance from the belt reference sensor  125  to the belt reference mark  123  of the belt transfer belt  105  in the transfer direction, but this definition will be below used for explanation as a matter of convenience.  FIG. 5  shows a landing misalignment amount “g” at a position “y′” of the transfer belt  105  when the driving radius “r” [mm] of the driven roller  115  fluctuates in a sine wave form with the belt cycle. 
     Supposing the average radius of the driven roller  115  as 20.225 mm and the amplitude as ±3 μm, we can represent the driving radius r(z′)[mm] at a position “z′”[mm] of the transfer roller  105  as 
     
       
         
           
             
               
                 
                   
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     Thus, when the transfer belt  105  moves from the static reference position “O” to a position “y′”[mm], the transfer amount G(y′)[mm] of a sheet is obtained by 
     
       
         
           
             
               
                 
                   
                     
                       
                         
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     Accordingly, the landing misalignment amount “g”[mm] at the position “y′”[mm] of the transfer belt  105  is given by 
     
       
         
           
             
               
                 
                   
                     
                       
                         
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     Since the pulse number of the encoder  117  is 1500 ppr and the average of a driving radius r(y′) is 20.225 mm, the average interval of neighboring dots (average printing interval) is given by 84.718 μm (=20.225×2π/1500). When this value is set as a dot interval in design (target), the landing misalignment amount “g”[mm] from a dot position in design is represented by (3). 
     Thus, it is considered the integration value of the deviations of the driving radius “r” from the average radius in the transfer direction of the transfer belt  105  as a landing misalignment amount (that is, the expansion and contraction of an image) from a value in design. 
     For example, in  FIG. 1 , when an A3 sheet (length 420 mm) is transferred lengthwise on the transfer belt  105  from the left hand and the printing of the predetermined image pattern on the sheet is started with the ink head “K” arranged above the position y′=100 mm of the transfer belt  105 , the printing range of each color ink, which depends on the ink head positions (interval: 110 mm) of K, C, M and Y, is as follows: 
     Printing range of K: 100 mm≦y′≦520 mm (line  501  in  FIG. 5 ); 
     Printing range of C: 210 mm≦y′≦630 mm (line  502  in  FIG. 5 ); 
     Printing range of M: 320 mm≦y′≦740 mm (line  503  in  FIG. 5 ); and 
     Printing range of Y: 430 mm≦y′≦850 mm (line  504  in  FIG. 5 ). 
     The next step is to conform the landing misalignment amounts K(y), C(y), M(y) and Y(y) of K,C,M and Y to each other at a printing position “y” of an A3 sheet, that is, K(y)=C(y)=M(y)=Y(y). As shown in  FIG. 1 , “y” is a position measured backward from a tip end line of the A3 sheet in the transfer direction. Then, as shown in  FIG. 6 , K(y), C(y), M(y) and Y(y) at a dot position “y” in design are represented by
 
 K ( y )= g ( y+ 100),  (4-1)
 
 C ( y )= g ( y+ 210)= K ( y+ 110),  (4-2)
 
 M ( y )= g ( y+ 320)= K ( y+ 220),  (4-3)
 
 Y ( y )= g ( y+ 430)= K ( y+ 330).  (4-4)
 
In  FIG. 6 , the interval from a dotted line  601  to a dotted line  602  corresponds to the length of an A3 sheet. Note here that g(y+100) in (4-1) is obtained by substituting y′=y+100 into (3) (See  FIG. 1 ).
 
     It is possible to specify a position “y” of the transfer belt  105  to which the position “y” of a point of the predetermined image pattern on the A3 sheet corresponds from a positional relationship between the register roller  103  and each ink head. This is achieved by starting the rotation of the register roller  103  to transfer the A3 sheet to the transfer belt  105  in the instance when the belt reference mark  123  passes through the belt reference sensor  125 . This is still achieved by starting the rotation when particular counts of encoder pulse are generated after the belt reference mark passes through the belt reference sensor  125 . 
     Suppose here that KC(y) represents the landing misalignment amount of K relative to that of C at a dot position “y” in design; KM(y) the landing misalignment amount of K relative to that of M; and KY(y) the landing misalignment amount of K relative to that of Y. 
     As shown in  FIG. 7 , they are expressed as
 
 KC ( y )= K ( y )− C ( y )= K ( y )− K ( y+ 110)= g ( y+ 100)− g ( y+ 210),  (5-1)
 
 KM ( y )= K ( y )− M ( y )= K ( y )− K ( y+ 220)= g ( y+ 100)− g ( y+ 320),  (5-2)
 
 KY ( y )= K ( y )− Y ( y )= K ( y )− K ( y+ 330)= g ( y+ 100)− g ( y+ 430)  (5-3)
 
     KY(y) tends to be the largest in them since the ink-head interval between K and Y is the widest. The landing misalignment amount changes depending on a sheet position. 
     Accordingly, if a landing misalignment amount g(y′) with the belt cycle of the transfer belt  105  is obtained from (3), it is possible to make KC(y), KM(y) and KY(y) zero, that is K(y)=C(y)=M(y)=Y(y), by setting the values of the correction table  37  as −g(y′). 
     Next, to get g(y) from K(y) using the measurement value KC(y), solving the recurrence equation (5-1), K(y+110)=K(y)−KC(y), with y=110n, we obtain 
                             K   ⁡     (   110   )       =       ⁢       K   ⁡     (   0   )       -     KC   ⁡     (   0   )           ,                   K   ⁡     (   220   )       =       ⁢         K   ⁡     (   110   )       -     KC   ⁡     (   110   )         =       K   ⁡     (   0   )       -     KC   ⁡     (   0   )       -     KC   ⁡     (   110   )             ,                 K   ⁡     (   330   )       =       ⁢         K   ⁡     (   220   )       -     KC   ⁡     (   220   )         =       K   ⁡     (   0   )       -     KC   ⁡     (   0   )       -                       ⁢         KC   ⁡     (   220   )       -     KC   ⁡     (   330   )         ,                     ⁢   ⋮                 K   ⁡     (     110   ⁢   n     )       =       ⁢       K   ⁡     (   0   )       -       ∑     i   =   0     n     ⁢           ⁢     KC   ⁡     (     110   ⁢   i     )                         (   6   )               
where “n” is an integer that satisfies the relationship d×n≧L and “i” is an integer that satisfies the inequality 0≦i≦n.
 
     Thus, by accumulating the measurement value KC(y), we obtain K(y) and C(y) with the interval of 110 mm. In  FIG. 8 , K(y) is denoted as the symbol “●”, and C(y) is denoted as the symbol “X” when K(0)=0. The equation (6) becomes 
                     K   ⁡     (     110   ⁢   n     )       =     -       ∑     i   =   0     n     ⁢     KC   ⁡     (     110   ⁢   i     )                   (   7   )               
when K(0)=0.
 
     It is possible to generalize (6) for arbitrary y=110n+a where “a” is a real number which satisfies the inequality 0≦a≦110. For example, if K(y)=0 for 0≦y&lt;110, we obtain, in the same way to obtain (7), 
                     K   ⁡     (   y   )       =       K   ⁡     (       110   ⁢   n     +   a     )       =     -       ∑     i   =   0     n     ⁢     KC   ⁡     (       110   ⁢   i     +   a     )                     (   8   )               
where “i” is an integer which satisfies the inequality 0≦i≦n. It is shown as a discontinuous function  901  in  FIG. 9 .
 
     From the discontinuous function  901 , we can construct a continuous function 
                     S   ⁡     (   y   )       =       1   L     ⁢       ∑     i   =   0       L   -   1       ⁢     K   ⁡     (       110   ⁢   i     +   m     )                   (   9   )               
where “m” represents the remainder of y/110 and “L” the frequency of 110 mm within the belt cycle. In this case, L=12 is set by using a KC(y) wave form with 1320 mm, which is longer than the transfer belt cycle of 1275 mm. This continuous function (9) is shown as a waveform  902  in  FIG. 9 .
 
     Subtracting (9) from (8), we obtain
 
Δ( y )= K ( y )− S ( y ).  (10)
 
This is the very landing misalignment amount of K to be obtained. This function is shown as a continuous function  903  in  FIG. 9 .
 
     Here is explained (9) in detail. The equations (5-1)-(5-3) show that a 110 mm periodic component of K(y) is canceled from KC(y), KM(y) and KY(y). It is therefore impossible to restore the 110 mm periodic component of K(y). This means that there is no influence on KC(y), KM(y) and KY(y) even though any kind of 110 mm periodic wave is added to or subtracted from K(y). The continuous function S(y) may be considered as an arbitrary wave with 110 mm cycle which makes K(y) continuous. It is however preferable to use the waveform S(y) of (9) so as to prevent the image expansion and contraction of a single color as much as possible. 
     Since the equations (4-1)-(4-4) show that K(y) only differs from g(y) in phase, it is easy to obtain g(y) from K(y). Storing “−g(y)” as a value in the correction table  37  in the correction circuit  127  and then adjusting the timing of ink-eject from the ink heads  107 ,  109 ,  111 ,  113  enables to eliminate a landing misalignment. 
     It is possible to create the correction table  37  before product shipment. It is preferable to urge users or service persons to recreate the correction table  37  regularly, such as every one-year, through a display on the image-forming apparatus. 
     A method of printing in an initial state before measuring the landing misalignment can be any one of the methods: 
     (a) Printing in a corrected state using a already stored correction table; and 
     (b) Printing in non-corrected state after clearing the already stored correction table. 
     The method (a) is comprised of: 
     (1) At the step (1) of  FIG. 2 , printing the predetermined image pattern in a corrected state with an already used correction table; 
     (2) At the step (2), measuring a landing misalignment based on the printed predetermined image pattern; and 
     (3) At the step (3), calculating a new correction value based on the measured landing misalignment and adding the new correction value to the already used correction table in the correction circuit  127 . 
     For example, supposing that, although printing with the correction value of +100 μm at a certain position of the transfer belt  105  is done, the landing misalignment amount of −30 μm is still remained. Then, it is possible to add 30 μm to the previous correction value of +100 μm to get the new correction value of +130 μm. 
     When the transfer belt  105  is used continuously, the method (a) is better because of the less amount of change in correction values. On the other hand, when the transfer belt  105  is exchanged, the method (b) is better because of clearing the previous correction values. 
     The above initial state makes some gaps (lack of data) corresponding to sheet intervals in measured data of landing misalignment. As an example of initial setting, printing three A3 sheets sequentially with the sheet interval of 50 mm enables to measure the landing misalignment of a length longer than the belt cycle (1275 mm), but makes the two gaps of 50 mm. It is therefore preferable to set appropriate sheet intervals depending on a sheet length and a belt cycle. 
     As an example of proper setting, printing four A3 sheets with the sheet interval of 220 mm between the first and second sheets and the third and forth sheets and the sheet interval of 535 mm between the second and third sheets enables to measure landing misalignment with no gap in data over the belt cycle (1275 mm) with about 100 mm overlapped. 
     With a correction value obtained from the above method, it is also possible to eliminate periodical landing misalignment due to unevenness in units other than the transfer unit composed of the transfer belt and the rotary encoder. Such a unit is a cylindrical sheet transfer unit that rotates with clamping a tip of a sheet, a liner encoder with unevenness in a slit or ruled line on the belt surface thereof, or the like. 
     [Correction Circuit] 
     The correction circuit  127  shown in  FIG. 3  comprises a FPGA (Field Programmable Gate Array), an ASIC (Application Specific Integrated Circuits), and a CPU (Central Processing Unit). Although the correction circuit  127  is separated from the head driving circuit  129  in  FIG. 1 , it is possible to configure them on an identical device or an identical plate. The belt Lck counter  32  in  FIG. 3  selects a collection value from the correction table  37  every 256 lines. It is however preferable to change a correction value more frequently than every 256 lines using a liner interpolation value calculated from anteroposterior correction values. 
     [Creation of Correction Table] 
     As described above, a sine wave as the rotational cycle of the transfer belt  105  can be used to create the correction table  37 . However, it is also possible to use other waveforms. In addition, it is possible to use a color ink other that K as a reference color ink and to use other calculation methods other than the present calculation method, depending on the size of machines. 
     Second Embodiment 
       FIG. 10  shows a schematic picture of an image forming apparatus according to the second embodiment. This image forming apparatus differs from that according to the first embodiment in  FIG. 1  in the following points:
         A roller reference mark  229  and a roller reference sensor  223  are added for the driven roller  115 ;   An encoder reference mark  227  and an encoder reference sensor  225  are added for the encoder  117 ;   An encoder speed-reducing device (not shown), which makes two rotations of the encoder correspond to one rotation of the roller, is added; and   The pulse number of the encoder  117  is 750 ppr instead of 1500 ppr.       
     Compared with the first embodiment, the above structure further enables to correct the unevenness of the driven roller  115  and the encoder  117  and costs less for the encoder  117 . 
     [Outline of Correction Circuit] 
       FIG. 11  shows the internal structure of a correction circuit  221  according to the second embodiment. Referring to  FIG. 11 , there is explained an example of operations of the correction circuit  221 . 
     [Specification of Phase Shifting of Encoder Pulse at Correction Circuit] 
     
         
         
           
             The phase shifting ability is as follows: the dividing ability is 1/256 pulse (0.33 μm) with data of 10 bits per point; the shift range is 0˜3.996 pulses (0˜338.34 μm). 
             The total volume of a correction table for a belt is 640 bits corresponding to 1387 mm, that is, every 256 lines (about 21.7 mm)×64 sections. That of a correction table for a roller is 320 bits corresponding to 173 mm, that is, every 64 lines (about 5.4 mm)×32 sections. That of a correction table for an encoder is 160 bits corresponding to 86 mm, that is, every 64 lines (about 5.4 mm)×16 sections. 
           
         
       
    
     There is explained the internal structure of the correction circuit  221 . 
     The correction circuit  221  comprises an Lck counter (not shown), an Lck×256 frequency multiplier  201 , a belt Lck counter  202 , a roller Lck counter  207 , an encoder Lck counter  208 , a first accumulator  203 , a second accumulator  204 , a comparator  205 , a pulse counter  206 , a correction table for belt (belt correction table)  211 , a correction table for roller (roller correction table)  212 , and a correction table for encoder (encoder correction table)  213 . 
     The first accumulator  203  adds a counted value of the Lck counter and a counted value of the Lck×256 frequency multiplier  201 . 
     The second accumulator  204  adds respective correction values of the above correction tables  211 ,  212  and  213  to a value of a counted value of the pulse counter  206  plus 1. 
     The roller Lck counter  207  is a counter for measuring a current position (of the roller reference mark  229 ) of the driven roller  115 . A correction value (roller correction value) corresponding to the measured current position of the roller reference mark  229  is selected from the roller correction table  212 . 
     The encoder Lck counter  208  is a counter for measuring a current position (of the encoder reference mark  227 ) of the encoder  117 . A correction value (encoder correction value) corresponding to the current position of the encoder reference mark  227  is selected from the encoder correction table  213 . 
     With the above structure, it is possible to shift the phase of the encoder pulse by the total value of the correction values: a belt correction value corresponding to the current position of the transfer belt reference mark  123  of the transfer belt  105 ; a roller correction value of the driven roller  115 ; and a encoder correction value of the encoder  117 . 
     [Creation of Correction Table] 
     There is the unevenness of the rotations of three kinds of transfer units (the transfer belt  105 , the roller  115 , and the encoder  117 ). The transfer units have the belt cycle of 1275 mm, the roller cycle of 126 mm, the encoder cycle of 64 mm, respectively. 
     In the first embodiment, there is explained how to obtain K(y) from KC(y). In the second embodiment, there is explained how to obtain K(y) from KC(y), KM(y) and KY(y) with more accuracy than that of the first embodiment. 
     Under the condition K(0)=0 as same as the first embodiment, we obtain K(y) for y=330n [mm] (squares  231  with the interval of 330 mm in  FIG. 12 ) using KY(y) as follows: 
                     K   ⁡     (     330   ⁢   n     )       =     -       ∑     i   =   0     n     ⁢     KY   ⁡     (     110   ⁢   i     )                   (   11   )               
where “n” is an integer more than or equal to 0.
 
     Then, using (11), KC(y) and KM(y), we obtain a value shifted by 110 mm from the respective values of (11) (circles  232  in  FIG. 12 ) and a value shifted by 220 mm from the respective values of (11) (triangles  233  in  FIG. 12 ) as follows:
 
 K (330 n+ 110)= K (330 n )− KC (330 n ),  (12-1)
 
 K (330 n+ 220)= K (330 n )− KM (330 n ).  (12-2)
 
     Further, we obtain K(y) and S(y) to get a difference Δ(y) in the same way to obtain K(y) from KC(y) in the first embodiment (expressions (9), (10)).  FIG. 13  shows K(y) as a discontinuous function  241 , S(y) as a continuous function  242 , and the difference Δ(y) as a line  243 . 
     Like this, we can obtain the unevenness of the longest cycle, that is, the belt cycle. This unevenness includes that of the roller cycle (126 mm) and that of the encoder cycle (64 mm). It is therefore possible to extract each cycle of the roller and the encoder from the unevenness of the belt cycle to create each correction table: the belt correction table  211 ; the roller correction table  212 ; and the encoder correction table  213 . 
     There is no need to provide the encoder reference position sensor  225  and the encoder correction table  213  when the position of the roller reference mark  229  and the position of the encoder reference mark  227  are not to be misaligned. 
       FIG. 14A  shows the landing misalignment amounts of K, C, M and Y from a precise dot pitch.  FIG. 14B  shows the landing misalignment amount of C, M and Y from K.  FIG. 14C  shows an arrangement of ink heads.  FIG. 14D  shows a landing position in design and landing positions of K, C, M and Y. 
     Firstly, the printing position of y=0 mm is explained. 
     As shown in  FIG. 14A , at the printing position y=0 mm, the landing misalignment amount of K is more than 20 μm, that of C is less than 20 μm, that of M is 0 μm, and that of Y is less than −20 μm. 
     As shown in  FIG. 14B , at the printing position y=0 mm, the landing misalignment amount of C from K is less than 20 μm, that of M from K is more than 20 μm, and that of Y from K is more than 40 μm. 
       FIG. 14D  shows the landing position in design and the landing positions of K, C, M and Y (1400) at the printing position of y=0 mm. 
     Next, the printing position of y=110 mm is explained. 
     As shown in  FIG. 14A , at the printing position y=110 mm, the landing misalignment amount of K is less than 20 μm, that of C is 0 μm, that of M is less than −20 μm, and that of Y is more than −20 μm. 
     As shown in  FIG. 14B , at the printing position y=110 mm, the landing misalignment amount of C from K is less than 20 μm, that of M from K is more than 20 μm, and that of Y from K is more than 40 μm. 
       FIG. 14D  shows the landing position in design and the landing positions of K, C, M and Y (1411) at the printing position of y=110 mm. 
     Next, the printing position of y=220 mm is explained. 
     As shown in  FIG. 14A , at the printing position y=220 mm, the landing misalignment amount of K is 0 μm, that of C is less than −20 μm, that of M is more than −20 μm, and that of Y is −30 μm. 
     As shown in  FIG. 14B , at the printing position y=220 mm, the landing misalignment amount of C from K is less than 20 μm, that of M from K is more than 20 μm, and that of Y from K is 30 μm. 
       FIG. 14D  shows the landing position in design and the landing positions of K, C, M and Y (1422) at the printing position of y=220 mm. 
     Next, the printing position of y=330 mm is explained. 
     As shown in  FIG. 14A , at the printing position y=330 mm, the landing misalignment amount of K is less than −20 μm, that of C is more than −20 μm, that of M is −30 μm, and that of Y is more than −20 μm. 
     As shown in  FIG. 14B , at the printing position y=330 mm, the landing misalignment amount of C from K is less than 20 μm, that of M from K is less than 20 μm, and that of Y from K is less than 20 μm. 
       FIG. 14D  shows the landing position in design and the landing positions of K, C, M and Y (1433) at the printing position of y=330 mm. 
     Next, the printing position of y=440 mm is explained. 
     As shown in  FIG. 14A , at the printing position y=440 mm, the landing misalignment amount of K is more than −20 μm, that of C is −30 μm, that of M is more than −20 μm, and that of Y is less than −20 μm. 
     As shown in  FIG. 14B , at the printing position y=440 mm, the landing misalignment amount of C from K is less than 20 μm, that of M from K is less than −20 μm, and that of Y from K is less than −20 μm. 
       FIG. 14D  shows the landing position in design and the landing position of K, C, M and Y (1444) at the printing position of y=440 mm. 
     Next, the printing position of y=550 mm is explained. 
     As shown in  FIG. 14A , at the printing position y=550 mm, the landing misalignment amount of K is −30 μm, that of C is more than −20 μm, that of M is less than −20 μm, and that of Y is less than 20 μm. 
     As shown in  FIG. 14B , at the printing position y=550 mm, the landing misalignment amount of C from K is less than −20 μm, that of M from K is less than −20 μm, and that of Y from K is more than −20 μm. 
       FIG. 14D  shows the landing position in design and the landing positions of K, C, M and Y (1455) at the printing position of y=550 mm. 
     As described above, the landing misalignment amount of Y from the landing position in design at the printing position y=220 mm is −30 μm, the landing misalignment amount of M from the landing position in design at the printing position y=330 mm is −30 μm, the landing misalignment amount of C from the landing position in design at the printing position y=440 mm is −30 μm, and the landing misalignment amount of K from the landing position in design at the printing position y=550 mm is −30 μm. 
     Shifting the timing of ejecting the ink Y earlier by 30 μm at the printing position y=220 mm enables to shift: the timing of ejecting the ink M earlier by 30 μm at the printing position y=330 mm, which is 110 mm apart from the printing position y=220 mm; the timing of ejecting the ink C earlier by 30 μm at the printing position y=440 mm, 220 mm apart; and the timing of ejecting the ink K earlier by 30 μm at the printing position y=550 mm, 330 mm apart. As a result, all the color inks come to land at the position in design. 
       FIG. 15A  shows the total landing misalignment amount of K, C, M and Y from a landing position in design, each of which is a landing misalignment amount with the belt cycle plus a landing misalignment amount with the roller cycle. 
     A thin line  1511  represents the landing misalignment amounts of K with the belt cycle, a thin line  1512  that of C with the belt cycle, a thin line  1513  that of M with the belt cycle, and a thin line  1514  that of Y with the belt cycle. 
     A thick line  1521  represents the total of the landing misalignment amount of K with the belt cycle and that with the roller cycle, a thick line  1522  that of C with the belt cycle and that with the roller cycle, a thick line  1523  that of M with the belt cycle and that with the roller cycle, and a thick line  1524  that of Y with the belt cycle and that with the roller cycle. 
       FIG. 15B  shows the total landing misalignment amounts of C, M and Y relative to that of K, as same as in  FIG. 15A . 
     A thin line  1531  represents the landing misalignment amount of C from K with the belt cycle, a thin line  1532  that of M from K with the belt cycle, and a thin line  1533  that of Y from K with the belt cycle. 
     A thick line  1541  represents the total of the landing misalignment amount of C from K with the belt cycle and that with the roller cycle, a thick line  1542  that of M from K with the belt cycle and that with the roller cycle, and a thick line  1543  that of Y from K with the belt cycle and that with the roller cycle. 
     As described above, a total landing misalignment amount that includes a landing misalignment amount with the roller cycle is shown as the thick lines  1541 ,  1542  and  1543  in  FIG. 15B . Supposing KC(y) as the thick line  1541  representing the landing alignment amount of C relative to that of K, KM(y) as the thick line  1542  representing the landing alignment amount of M relative to that of K, and KY(y) as the thick line  1543  representing the landing alignment amount of Y relative to that of K, we obtain the waveforms Rkc(y), Rkm(y) and Rky(y) with the cycle of 126 mm such that 
                       Rkc   ⁡     (   y   )       =       1   P     ⁢       ∑     i   =   0       P   -   1       ⁢     kc   ⁡     (       126   ⁢   i     +   q     )             ,           (     13   ⁢     -     ⁢   1     )                   Rkm   ⁡     (   y   )       =       1   P     ⁢       ∑     i   =   0       P   -   1       ⁢     km   ⁡     (       126   ⁢   i     +   q     )             ,           (     13   ⁢     -     ⁢   2     )                 Rky   ⁡     (   y   )       =       1   P     ⁢       ∑     i   =   0       P   -   1       ⁢     ky   ⁡     (       126   ⁢   i     +   q     )                   (     13   ⁢     -     ⁢   3     )               
where “q” represents the remainder of y/126 and “P” the frequency of 126 mm in the cycle (1275 mm) of the transfer belt cycle. In this case, P=10≈1275/126.
 
       FIG. 15C  shows the landing misalignment amounts with the roller cycle of C, M and Y relative to that of K, each of which is the total landing misalignment amount minus the landing misalignment amount with the roller cycle. A dotted line  1551  represents Rkc(y) obtained from (13-1), a dotted line  1552  Rkm(y) obtained from (13-2), and a dotted line  1553  Rky(y) obtained from (13-3). As the above, when the cycle of a landing misalignment is known, it is possible to extract the landing misalignment amount with the cycle. 
     The above (7) to (12-1), (12-2) are for a method to obtain the table of correction values K(y) from arbitrary relative landing misalignment. By applying this method to the dotted lines  1551 - 1553  in  FIG. 15C , it is possible to obtain correction values with the roller cycle. The method is explained below. 
     [Method to Obtain K from KC] 
     Supposing Rkc(y) as KC(y) in (7), we obtain K(y) using (7)˜(10). K(y) for y=0˜126 mm is the correction values of the roller correction table  212 . 
     [Method to Obtain K from KC, KM, KY] 
     Supposing Rky(y) as KY(y) in (1), Rkc(y) as KC(y) in (2), and Rkm(y) as KM(y) in (2), we obtain K(y) using (9), (10). K(y) for y=0˜126 mm is the correction values of the roller correction table  212 . 
     As is clear from the above explanation, according to the present invention, it is possible to eliminate a landing misalignment by correcting a timing of ink-ejection from an ink head depending on a rotational position of a rotational component. 
     It should be noted that the above explanation is just done with several examples, so that the technical scope of the invention is not limited by them. 
     For example, K(y) is obtained using KC(y) in the first embodiment. However with Fourier transformation, it is possible to obtain respective graphs of K(y) of the three kinds of transfer units in any of the following methods:
         (1) Obtaining K(y) by using each graph of the cycles (64 mm, 1.26 mm, and 1275 mm) which is extracted from KC(y).   (2) Extracting each cycle (64 mm, 1.26 mm, and 1.275 mm) from K(y) to be obtained. In particular, the methods are as indicated below.       

     In the method (1), there are obtained three KC(y)s from one KC(y): the cycle of 64 mm extracted from KC(y), the cycle of 126 mm extracted from KC(y), and the cycle of 1275 mm extracted from KC(y). Then, from the three KC(y)s, there are created three tables of correction values respectively: the correction table K(y) for encoder; the correction table K(y) for roller, and the correction table K(y) for belt. 
     In the method (2), there is obtained one K(y) from one KC(y). From the one K(y), there are obtained three K(y)s: “the cycle of 64 mm extracted from the one K(y)”; “the cycle of 126 mm extracted from the one K(y)”; and “the cycle of 1275 mm extracted from the one K(y)”. 
     This application is based on the Japanese Patent Applications No. 2006-106252, filed on Apr. 7, 2006, the entire content of which is incorporated by reference herein.