Abstract:
A motor control device includes a velocity detection unit, a motor control unit, a phase difference detection unit, and a correction value calculation unit. The velocity detection unit detects each velocity of a plurality of driven bodies or of a plurality of motors which independently drives a corresponding one of the driven bodies. The motor control unit independently controls each of the motors based on the velocity and a predetermined velocity directive value. The phase difference detection unit detects a phase difference among each of the driven bodies. The correction value calculation unit calculates a correction value for the velocity or the velocity directive value based on the phase difference.

Description:
CROSS-REFERENCE TO RELATED APPLICATIONS 
   This application claims the benefit of Japanese Patent Application No. 2005-352015 filed Dec. 6, 2005 in the Japan Patent Office, the disclosure of which is incorporated herein by reference. 
   BACKGROUND 
   This invention relates to a motor control device that controls each of a plurality of motors which independently drives a corresponding one of a plurality of driven bodies. Particularly, the present invention concerns a motor control device and a motor control method of controlling a velocity of and a phase difference among each of the plurality of driven bodies to desired values, and an image forming apparatus including the motor control device. 
   In a conventional image forming apparatus that drives and rotates a plurality of photoreceptors to form a color image, it is necessary to control not only a rotational velocity of each of the photoreceptors to a predetermined value but also a rotational phase thereof to be consistent with each other. This is because eccentric rotation of each of the photoreceptors may cause a different surface velocity, even though the rotational velocity of each of the photoreceptors is the same. Therefore, it has been proposed to temporarily change each target velocity value of some of the photoreceptors according to phase difference among each of the photoreceptors so as to correct the phase difference. 
   SUMMARY 
   However, in the conventional image forming apparatus, correction of the phase difference is performed after the rotational velocity of each of the photoreceptors is controlled to a desired velocity. Thus, it takes time to control the rotational velocity of and the phase difference among each of the photoreceptors to the desired values. It has also been proposed in the conventional image forming apparatus to store, in a ROM, a data table composed of control variables for correction of phase differences. However, a huge data table is necessary in order to correct a phase difference while controlling a rotational velocity of each of the photoreceptors to a desired velocity. Accordingly, it is difficult in the conventional image forming apparatus to quickly control the rotational velocity and the phase difference to the desired values. The same problem occurs in various driving systems other than a driving system for an image forming apparatus, as well as in a driving system including reciprocation movement of a piston other than a driving system including rotational movement. 
   The present invention was made to solve the above problems. It would be desirable to provide a motor control device and a motor control method in which each velocity of the driven bodies or the motors is controlled to a desired velocity while a phase difference among each of the driven bodies is corrected, so that the velocity and the phase difference may be promptly controlled to desired values. It would be further desirable to provide an image forming apparatus including such a motor control device. 
   It is desirable that a motor control device of the present invention includes a velocity detection unit, a motor control unit, a phase difference detection unit, and a correction value calculation unit. The velocity detection unit detects each velocity of a plurality of driven bodies or of a plurality of motors which independently drives a corresponding one of the plurality of driven bodies. The motor control unit independently controls each of the motors based on the velocity detected by the velocity detection unit and a predetermined velocity directive value. The phase difference detection unit detects a phase difference among each of the driven bodies. The correction value calculation unit calculates a correction value for the detected velocity or the velocity directive value based on the phase difference detected by the phase difference detection unit. 
   According to the motor control device of the present invention as above, the correction value corresponding to the phase difference is reflected to the control of the motor control unit. Thereby, the phase difference can be corrected while the velocity of each of the driven bodies or motors is brought near to a velocity which corresponds to the velocity directive value. Therefore, the aforementioned velocity and the phase difference can be quickly controlled to the desired values. 

   
     BRIEF DESCRIPTION OF THE DRAWINGS 
     The invention will now be described below, by way of example, with reference to the accompanying drawings, in which: 
       FIG. 1  is a schematic sectional view showing an internal structure of a color laser printer according to the present invention; 
       FIG. 2  is a block diagram showing structures of driving units for photosensitive drums of the printer; 
       FIGS. 3A and 3B  are front and side views, respectively, showing a structure of an index detector of the driving unit; 
       FIG. 4  is a block diagram showing details of a structure of a transfer function calculator of the driving unit; 
       FIGS. 5A and 5B  are graphs respectively showing a change in rotational velocity of the photosensitive drums in case that a secondary photosensitive drum is in phase advance; 
       FIGS. 6A and 6B  are explanatory views respectively showing an index signal generated during control in  FIGS. 5A and 5B ; 
       FIG. 7  is an enlarged view in which the respective index signals shown in  FIGS. 6A and 6B  are superposed on each other; 
       FIGS. 8A and 8B  are graphs respectively showing a change in rotational velocity of the photosensitive drums in case that the secondary photosensitive drum is in phase delay; 
       FIGS. 9A and 9B  are explanatory views respectively showing an index signal generated during control in  FIGS. 8A and 8B ; 
       FIG. 10  is an enlarged view in which the respective index signals shown in  FIGS. 9A and 9B  are superposed on each other; 
       FIGS. 11A and 11B  are graphs showing a change in rotational velocity of the secondary photosensitive drum in case that a gain is set to a relatively low value; 
       FIGS. 12A and 12B  are graphs showing a change in rotational velocity of the secondary photosensitive drum in case that the gain is set to a relatively high value; 
       FIGS. 13A and 13B  are graphs showing a change in rotational velocity of the secondary photosensitive drum in case that the gain is shifted from high to low; 
       FIGS. 14A to 14C  are graphs respectively showing a convergence state of a phase difference in the cases in  FIGS. 11A and 11B ,  12 A and  12 B, and  13 A and  13 B; 
       FIG. 15  is a block diagram showing a structure of a driving unit using software; 
       FIG. 16  is a flowchart illustrating a main routine executed by a control unit of the driving unit; 
       FIG. 17  is a flowchart illustrating a process of calculating a value Cnt_Y2M corresponding to an amount of phase delay in the main routine; 
       FIG. 18  is a flowchart illustrating a process of calculating a value Cnt_M2Y corresponding to an amount of phase advance in the main routine; 
       FIG. 19  is a flowchart illustrating a process of calculating a correction value in the main routine; 
       FIG. 20  is a flowchart illustrating control in consideration of a temperature of a fixing unit; 
       FIG. 21  is a block diagram showing structures of driving units for the photosensitive drums in a variation; 
       FIGS. 22A and 22B  are graphs respectively showing a change in rotational velocity of the photosensitive drums in the variation; and 
       FIGS. 23A and 23B  are graphs respectively showing a change in rotational velocity of a secondary photosensitive drum and a convergence state of a phase difference in the variation. 
   

   DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENT 
   [Structure of Color Laser Printer] 
   Referring to  FIG. 1 , a color laser printer (hereinafter, simply referred to as a printer)  1  has a recoding engine  7  provided with a toner image forming unit  4  and a paper conveying belt  6 , a fixing unit  8 , a paper feeding unit  9 , a stacker  12 , and a control unit  10 . The printer  1  forms an image of four colors on recording paper P in accordance with externally inputted image data. Here, the image data may be text data or code data. 
   The toner image forming unit  4  is provided with four developing units  51 Y,  51 M,  51 C and  51 B. Each of the developing units  51 Y,  51 M,  51 C and  51 B contains a toner of different colors, i.e., yellow, magenta, cyan and black. Each of the development units  51 Y,  51 M,  51 C, and  51 B is provided with a photosensitive drum  3 , a charger  31 , and a scanner unit  41 . The charger  31  uniformly charges the photosensitive drum  3 . The scanner unit  41  exposes a surface of the charged photosensitive drum  3  with laser light to form an electrostatic image in accordance with the image data. Almost all of components of the scanner unit  41  are omitted in  FIG. 1 . Only a component part from which the laser light is emitted is shown in  FIG. 1 . 
   Hereinafter, structures of the components of the printer  1  will be described in detail In the following description, an alphabet of any one of Y for yellow, M for magenta, C for cyan, or B for black is added to a reference number when it is necessary to indicate the color. Otherwise, such alphabet is omitted. 
   Each of the four photosensitive drums  3  ( 3 Y,  3 M,  3 C and  3 B) in the toner image forming unit  4  is formed of a member having a substantially cylindrical shape. The photosensitive drums  3  are rotatably aligned at substantially constant intervals along a horizontal direction. The substantially cylindrical member of the photosensitive drum  3  is constituted of, for example, a substrate made from aluminum and a positively charged photosensitive layer formed on the substrate. The aluminum substrate is grounded on a ground line of the printer  1 . 
   The charger  31  is a so-called scorotoron type charger. The charger  31  is provided with a charging wire  32  that extends in a width direction of the photosensitive drum  3  so as to face the photosensitive drum  3 , and a shield case  33  that houses the charging wire  32  and has an opening on a side facing the photosensitive drum  3 . The charger  31  charges the surface of the photosensitive drum  3  (e.g. to +700V) by applying a high voltage to the charging wire  32 . The shield case  33  has a grid provided at the opening facing the photosensitive drum  3 . The surface of the photosensitive drum  3  is charged to a potential substantially the same as a grid voltage by applying a predetermined voltage to the grid. 
   The scanner unit  41  ( 41 Y,  41 M,  41 C,  41 B) is provided on each of the photosensitive drums  3 . The scanner unit  41  is disposed downstream of the charger  31  in a rotation direction of the photosensitive drum  3 . The scanner unit  41  emits the laser light from a light source for one color of the externally inputted image data, and performs laser light scanning with a mirror surface of a polygon mirror, which is rotationally driven by a polygon motor, to irradiate the surface of the photosensitive drum  3  with the laser light. 
   When the scanner unit  41  irradiates the surface of the photosensitive drum  3  with the laser light according to the image data, a surface potential of the irradiated part is reduced (to +150 to +200 V) to form an electrostatic image on the surface of the photosensitive drum  3 . 
   Each of the development units  51  ( 51 Y,  51 M,  51 C and  51 B) is provided with a development case  56  housing a corresponding color of toner, and a development roller  52 . The development roller  52  is disposed downstream of the scanner unit  41  with respect to the rotation direction of the photosensitive drum  3  in such a manner as to contact the photosensitive drum  3 . Each of the development units  51  positively charges the toner to supply the toner as a uniform thin layer to the photosensitive drum  3 . The positively charged toner is carried to the positively charged electrostatic image formed on the photosensitive drum  3  at the contact part between the development roller  52  and the photosensitive drum  3  by a reverse development method. Thereby, the electrostatic image is caused to be developed. 
   The development roller  52  is made from a base material such as electroconductive silicone rubber. The development roller  52  has a cylindrical shape. A coating layer made from a resin containing fluorine or a rubber material is formed on a surface of the development roller  52 . The toner housed in the development case  55  is a positively charged nonmagnetic one component toner. A yellow toner, a magenta toner, a cyan toner, and a black toner are respectively stored in the development units  51 Y,  51 M,  51 C and  51 B. 
   The paper feeding unit  9  is disposed at a lowermost part of the printer  1 . The paper feeding unit  9  is provided with a housing tray  91  that stores recording paper P and a pickup roller  92  that feeds the recording paper P. The recording paper P stored in the housing tray  91  is fed from the paper feeding unit  9  sheet by sheet by the pickup roller  92  to be sent to the paper conveying belt  6  via conveying rollers  98  and registration rollers  99 . 
   The paper conveying belt  6  has a width which is narrower than that of the photosensitive drum  3 . The paper conveying belt  6  is in the form of an endless belt and runs together with the recording paper P with the recording paper P mounted thereon. The paper conveying belt  6  is held between a driving roller  62  which is driven by a not shown motor and a driven roller  63 . Transfer rollers  61  are also provided on the opposite side of the respective photosensitive drums  3  via the paper conveying belt  6 . As the driving roller  62  is driven and rotated by the motor, the paper conveying belt  6  moves in a counterclockwise direction as indicated by arrows in  FIG. 1 . The recording paper P sent from the registration rollers  99  is sequentially conveyed to between the photosensitive drums  3  and the paper conveying belt  6  so as to be sent to the fixing unit  8 . 
   A toner removal unit  100  including a cleaning roller  105  is provided close to the driven roller  63 , on the side of the paper conveying belt  6  not facing the photosensitive drums  3 . Furthermore, a density detection sensor  111  is provided to face the paper conveying belt  6  on the driving roller  62 . The density detection sensor  111  includes a light source that emits light in the infrared region, a lens that irradiates light from the light source on the paper conveying belt  6 , and a photo transistor that receives reflection of the light. The density detection sensor  111  measures the density of a toner image on the paper conveying belt  6 . 
   The transfer roller  61  transfers a toner image formed on the photosensitive drum  3  on the recording paper P conveyed by the paper conveying belt  6  when a transfer bias (e.g. −10 to −15 μA) which has a polarity reverse to that of the toner is applied between the transfer roller  61  and the photosensitive drum  3  by a current source  112  of a negative voltage. 
   The fixing unit  8  is provided with a thermal roller  81  and a pressure roller  82 . The recording paper P on which the toner image has been transferred is heated and pressurized while being held and conveyed between the thermal roller  81  and the pressure roller  82 . As a result, the toner image is fixed on the recording paper P. The fixing unit  8  also includes a sensor  83  that measures a temperature in the vicinity of the heating roller  81 . 
   The stacker  12  is formed on a top surface of the printer  1 . The stacker  12  is disposed at a discharge side of the fixing unit  8  to retain the recording paper P discharged from the fixing unit  8 . The control unit  10  is provided with a controller with a known CPU and controls an overall operation of the printer  1 . 
   The photosensitive drums  3  are held in such a manner as to be moved upward so that the photosensitive drums  3  can be detached from the paper conveying belt  6 . The photosensitive drums  3  are positioned by a moving member  72  provided to extend over the photosensitive drums  3 . The moving member  72  is formed of a plate-like member having a length sufficient to cover across all of the photosensitive drums  3 . The moving member  72  is held so as to be moved in a horizontal direction in  FIG. 1 . The moving member  72  is provided with four guide holes  72   a  (only two of them are shown in  FIG. 1 ; the other two are omitted) extending in the horizontal direction and having a substantially crank shape. Shafts  3   a  provided on a longitudinal side of the photosensitive drums  3  are fitted into the guide holes  72   a.    
   The moving member  72  is connected to a lifting motor  74  via a link  73  for converting a rotational force into a horizontal force. The moving member  72  is moved to right or left as the lifting motor  74  rotates in response to an instruction signal from the control unit  10 . When the moving member  72  is moved to the left, the guide holes  72   a are also moved to the left and the shafts  3   a  of the respective photosensitive drums  3  move upward along the substantially crank shape of the guide holes  72   a . As a result, the photosensitive drums  3  are detached from the paper conveying belt  6 . In contrast, when the moving member  72  is moved to the right, the photosensitive drums  3  are brought into contact with the paper conveying belt  6 . Normally, image forming is performed in a state that the photosensitive drums  3  are in contact with the paper conveying belt  6 . 
   An operation of forming an image on recording paper P in the above printer  1  of the present embodiment is as follows. Firstly, a sheet of the recording paper P is supplied from the paper feeding unit  9  by the pickup roller  92  to be sent to the paper conveying belt  6  via the conveying rollers  98  and the registration rollers  99 . Next, the surface of the photosensitive drum  3 Y disposed at the rightmost position in  FIG. 1  is uniformly charged by the charger  31  and then exposed to light by the scanner unit  41 Y based on externally inputted image data for yellow, so that an electrostatic image is formed on the surface of the photosensitive drum  3 Y. Then, a yellow toner which has been positively charged in the development unit  51 Y is supplied to the surface of the photosensitive drum  3 Y for development. The toner image formed in this manner is transferred onto the recording paper P, which is conveyed by the paper conveying belt  6 , by the transfer roller  61  to which the transfer bias has been applied. 
   Subsequently, the recording paper P is conveyed to positions at which the recording paper P faces the respective photosensitive drums  3  for magenta, cyan, and black in turn. Toner images are formed on the surfaces of the photosensitive drums  3  in the same manner as for the yellow toner, and transferred onto the recording paper P by the transfer roller  61  in a superposing manner. Lastly, the toner images of the four colors formed on the recording paper P are fixed on the recording paper P in the fixing unit  8 . The recording paper P is then discharged onto the stacker  12 . 
   In the printer  1 , when execution of calibration is instructed by the control unit  10 , a known measuring patch is formed on the paper conveying belt  6 . Density of the respective colors composing the measuring patch is measured by the density detection sensor  111  of the recording engine  7  at the time of forming the measuring patch. The measuring patch after the density measurements is removed by the cleaning roller  105  of the toner removal unit  100 . 
   [Structure of Driving Unit of Photosensitive Drum] 
     FIG. 2  is a block diagram showing structures of driving units  120  ( 120 Y,  120 M,  120 C and  120 B) of the photosensitive drums  3 . 
   In the present embodiment, the driving units  120 M,  120 C and  120 B have the same structure. The driving unit  120 Y has a different structure than the other driving units  120 M,  120 C and  120 B. Hereinafter, the driving unit  120 Y is referred to as the primary driving unit  120 Y, and the other driving units  120 M,  120 C and  120 B are referred to as the secondary driving units  120 M,  120 C and  120 B. The details of the secondary driving units  120 C and  120 B are omitted in  FIG. 2 . 
   As shown in  FIG. 2 , each of the photosensitive drums  3 Y,  3 M,  3 C and  3 B is connected with each of the motors  121  ( 121 Y,  121 M,  121 C and  121 B; only the motors  121 Y and  121 M are shown in  FIG. 2 ) via a not shown gear. Also, a driving power is inputted to each of the motors  121  via a velocity controller  122  ( 122 Y,  122 M,  122 C,  122 B; only the velocity controllers  122 Y and  122 M are shown in  FIG. 2 ) and a power amplifier  123  ( 123 Y,  123 M,  128 C,  123 B; only the power amplifiers  123 Y and  123 M are shown in  FIG. 2 ). A rotational velocity of each of the motors  121  is detected by a velocity detector  124  ( 124 Y,  124 M,  124 C,  124 B; only the velocity detectors  124 Y and  124 M are shown in  FIG. 2 ). 
   In the primary driving unit  120 Y of the photosensitive drum  3 Y, the velocity detected by the velocity detector  124 Y is subtracted from a velocity directive value as a directive value for control of each of the motors by a subtracter  125 Y. The velocity controller  122 Y performs feedback control of the velocity of the motor  121 Y based on the value obtained by the above subtraction. In contrast, in the secondary driving units  120 M,  120 C and  120 B of the photosensitive drums  3 M,  3 C and  3 B, a feedback control which reflects a phase difference between the photosensitive drum  3 Y and the photosensitive drum  3 M,  3 C or  3 B is performed as follows. 
   That is, each of the photosensitive drums  3  is provided with an index detector  130  ( 130 Y,  130 M,  130 C,  130 D) that generates one index signal per one rotation of the photosensitive drum  3 . Now, a structure of the index detector  130  is explained in detail by way of  FIGS. 3A and 3B . 
   As shown in  FIG. 3A , each of the photosensitive drum  3  is provided with a disk  127  which rotates on the shaft  3   a  together with the photosensitive drum  3 . A slit  128  is bored at a position near the outer periphery of the disk  127 . As shown in  FIG. 3B , the index detector  130  is formed into a U-shape so that the outer peripheral side of the disk  127  can be interposed therethrough. The index detector  130  includes a light emitter  131  that irradiates light toward the disk  127  and a light receiver  132  that detects light passed through the slit  128  when the light emitter  131  faces the slit  28 . When the photosensitive drum  3  is rotated to a predetermined phase where the slit  128  faces the light emitter  131 , the light receiver  132  detects the light and generates an index signal. 
   Referring back to  FIG. 2 , for example, the secondary driving unit  120 M of the photosensitive drum  3 M is provided with a phase difference detector  141 M that compares an index signal generated by the index detector  130 M and an index signal generated by the index detector  130 Y to detect a phase difference between the photosensitive drums  3 Y and  3 M. The phase difference detected by the phase difference detector  141 M is inputted to the subtracter  145 M after calculation by a transfer function calculator (transmission function C(s))  143 M. The subtracter  145 M subtracts the output (hereinafter, referred to as a correction value) of the transfer function calculator  143 M from the velocity detected by the velocity detector  124 M, and then inputs the velocity after the subtraction to the subtracter  125 M. 
   The subtracter  125 M then subtracts the velocity obtained by the subtracter  145 M from the velocity directive value. The velocity controller  122 M performs a feedback control based on the value obtained from the subtracter  125 M. Accordingly, the phase difference between the photosensitive drums  3 Y and  3 M can be corrected while the velocity of the photosensitive drum  3 M is brought near to the velocity which corresponds to the velocity directive value. 
     FIG. 4  is a block diagram showing details of a structure of the transfer function calculator  143  ( 143 M,  143 C,  143 B). As shown in  FIG. 4 , the transfer function calculator  143  is provided with a filter  151  ( 151 M,  151 C,  151 B), a gain correction function unit  153  ( 153 M,  153 C,  153 B), and a multiplier  155  ( 155 M,  155 C,  155 B). The filter  151  removes high frequency component from the phase difference detected by the phase difference detector  141  ( 141 M,  141 C,  141 B). The gain correction function unit  153  outputs a gain Gcomp. The multiplier  155  multiplies the phase difference which has passed the filter  151  by the gain Gcomp. Accordingly, by varying the gain Gcomp, a convergence state of the velocity and the phase difference of each of the photosensitive drums  3 Y,  3 M,  3 C and  3 B change as below. 
   In the following description, various transfer functions of the above control system are assumed as below. Firstly, various constants (electric and mechanical constants) to the motor  121 , that is, wire inductance, wire resistance, input voltage, motor current, torque constant, back electromotive force constant, inertia of a motor shaft, viscosity resistance of a motor shaft, and rotation angular velocity are respectively defined as L, R, V c , i, K t , K e , J 1 , D 1 , and ω 1 . Also, various constants to the photosensitive drums  3 , that is, inertia of the shaft  3   a,  viscosity resistance of the shaft  3   a , and rotation angular velocity are respectively defined as J 2 , D 2 , and ω 2 . A torsion torque constant by gear connection between the motors  121  and the photosensitive drum  3  is defined as K s . Then, the following differential equations become true. 
   
     
       
         
           
             
               L 
               ⁢ 
               
                 
                   ⅆ 
                   i 
                 
                 
                   ⅆ 
                   t 
                 
               
             
             + 
             
               R 
               · 
               i 
             
           
           = 
           
             
               V 
               c 
             
             - 
             
               
                 K 
                 e 
               
               ⁢ 
               
                 ω 
                 1 
               
             
           
         
       
     
     
       
         
           
             
               K 
               t 
             
             · 
             i 
           
           = 
           
             
               
                 J 
                 1 
               
               ⁢ 
               
                 
                   ⅆ 
                   
                     ω 
                     1 
                   
                 
                 
                   ⅆ 
                   t 
                 
               
             
             + 
             
               
                 D 
                 1 
               
               ⁢ 
               
                 ω 
                 1 
               
             
             + 
             
               
                 K 
                 s 
               
               ⁢ 
               
                 ∫ 
                 
                   
                     ( 
                     
                       
                         ω 
                         1 
                       
                       - 
                       
                         ω 
                         2 
                       
                     
                     ) 
                   
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     ⅆ 
                     t 
                   
                 
               
             
           
         
       
     
     
       
         
           
             
               
                 J 
                 2 
               
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                     2 
                   
                 
                 
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                   t 
                 
               
             
             + 
             
               
                 D 
                 2 
               
               ⁢ 
               
                 ω 
                 2 
               
             
             - 
             
               
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               ⁢ 
               
                 ∫ 
                 
                   
                     ( 
                     
                       
                         ω 
                         1 
                       
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                         ω 
                         2 
                       
                     
                     ) 
                   
                   ⁢ 
                   
                       
                   
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           = 
           0 
         
       
     
   
   From the above, a transfer function of the motor  121  having an input of the input voltage V c  and an output of the location of the photosensitive drum  3  can be expressed as below. In the following function, coefficients a 0  to a 3  and b 0  are determined by the aforementioned electrical and mechanical constants. 
   
     
       
         
           P 
           = 
           
             
               b 
               0 
             
             
               
                 s 
                 4 
               
               + 
               
                 
                   a 
                   3 
                 
                 · 
                 
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               + 
               
                 
                   a 
                   1 
                 
                 · 
                 s 
               
               + 
               
                 a 
                 0 
               
             
           
         
       
     
   
   A transfer function of phase advance-delay compensation as below can be applied to the velocity controller  122 . In the following function, T 1  and T 2  are designing constants which determine a corner frequency, and α 1  and α 2  are designing constants which determine a low frequency (high frequency) increasing gain. 
   
     
       
         
           
             G 
             ⁡ 
             
               ( 
               s 
               ) 
             
           
           = 
           
             
               K 
               ⁡ 
               
                 ( 
                 
                   
                     
                       
                         T 
                         1 
                       
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                         α 
                         1 
                       
                       · 
                       
                         T 
                         1 
                       
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                 ) 
               
             
             ⁢ 
             
               ( 
               
                 
                   
                     α 
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                     ( 
                     
                       
                         
                           T 
                           2 
                         
                         · 
                         s 
                       
                       + 
                       1 
                     
                     ) 
                   
                 
                 
                   
                     
                       α 
                       2 
                     
                     · 
                     
                       T 
                       2 
                     
                     · 
                     s 
                   
                   + 
                   1 
                 
               
               ) 
             
           
         
       
     
     
       
         
           
             
               where 
               ⁢ 
               
                   
               
               ⁢ 
               
                 α 
                 1 
               
             
             &lt; 
             1 
           
           , 
           
             
               α 
               2 
             
             &gt; 
             1 
           
         
       
     
   
   A well known transfer function of PID (Proportional-Integral-Derivative) control is obtained if put α 1 =0 and α 2 =∞ in the above function. In the present embodiment, a transfer function G(s) is set as below. 
   
     
       
         
           
             G 
             ⁡ 
             
               ( 
               s 
               ) 
             
           
           = 
           
             
               
                 58 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 
                   s 
                   2 
                 
               
               + 
               
                 807 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 s 
               
               + 
               700 
             
             
               
                 s 
                 2 
               
               + 
               
                 100 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 s 
               
             
           
         
       
     
   
   Furthermore, in the present embodiment, the following function having a low pass characteristic is applied to the filter  151  in view of stability. In the following function, a 0 , a 1  and b 0  are designing constants. In the present embodiment, a 0 =b 0 =1 and a 1 =0.02. 
   
     
       
         
           
             
               F 
               l 
             
             ⁡ 
             
               ( 
               s 
               ) 
             
           
           = 
           
             
               b 
               0 
             
             
               
                 
                   a 
                   1 
                 
                 · 
                 s 
               
               + 
               
                 a 
                 0 
               
             
           
         
       
     
   
   Under the conditions as above, the effect of the gain Gcomp in the transfer function calculator  143  is investigated. Firstly, the gain Gcomp is fixed to an intermediate value so as to learn the change in rotational velocity of the photosensitive drum  3 Y and another photosensitive drum  3  (e.g.,  3 M; this can be  3 C or  3 B).  FIGS. 5A and 5B  are graphs respectively showing the change in rotational velocity of the photosensitive drum  3 Y and the another photosensitive drum  3  (e.g.,  3 M) in case that the another photosensitive drum  3  (e.g.,  3 M) is in phase advance. The velocity of the photosensitive drum  3 Y is feedback controlled regardless of a phase difference from the another photosensitive drum  3  (e.g.,  3 M). Accordingly, as shown in  FIG. 5A , the velocity of the photosensitive drum  3 Y smoothly converges to a target directive velocity. In contrast, the velocity of the another photosensitive drum  3  (e.g.,  3 M) is feedback controlled in reflection of the phase difference from the photosensitive drum  3 Y. Accordingly, as shown in  FIG. 5B , the velocity of the another photosensitive drum  3  (e.g.,  3 M) converges to the target directive velocity in an oscillating manner. 
     FIGS. 6A and 6B  are explanatory views showing index signals respectively generated by the index detector  130 Y and another index detector  130  (e.g.,  130 M; this can be  130 C or  130 B) during the above control.  FIG. 7  is an enlarged view in which the respective index signals in  FIGS. 6A and 6B  are shown in a superposed manner. In  FIG. 7 , a “primary rotation body” shown in a dotted line represents the index signal from the index detector  130 Y, and a “secondary rotation body” shown in a solid line represents the index signal from the another index detector  130  (e.g.,  130 M). As can be seen from  FIGS. 5A ,  5 B and  7 , the phase difference between the photosensitive drum  3 Y and the another photosensitive drum  3  (e.g.,  3 M) is corrected while each of the photosensitive drum  3 Y and the another photosensitive drum  3  (e.g.,  3 M) is accelerated to the target directive velocity. 
   When the another photosensitive drum  3  (e.g.,  3 M) is in phase delay, the same result was obtained as well.  FIGS. 8A and 8B  are graphs respectively showing the change in rotational velocity of the photosensitive drum  3 Y and the another photosensitive drum  3  (e.g.,  3 M) under the above control.  FIGS. 9A and 9B  are explanatory views respectively showing an index signal generated by the index detector  130 Y and the another index detector  130  (e.g.,  130 M) under the same control.  FIG. 10  is an enlarged view in which the index signals in  FIGS. 9A and 9B  are shown in a superposed manner. Also in  FIG. 10 , the “primary rotation body” and the “secondary rotation body” respectively represent the index signals of the index detector  130 Y and the another index detector  130  (e.g.,  130 M). As can be seen in  FIGS. 8A ,  8 B and  10 , even in the case that the another photosensitive drum  3  (e.g.,  3 M) is in phase delay, the phase difference between the photosensitive drum  3 Y and the another photosensitive drum  3  (e.g.,  3 M) is corrected while each of the photosensitive drum  3 Y and the another photosensitive drum  3  (e.g.,  3 M) is accelerated to the target directive velocity. 
     FIG. 11A  is a graph showing the change in rotational velocity of the another photosensitive drum  3  (e.g.,  3 M) in case that the gain Gcomp is set to a relatively low value.  FIG. 11B  is a partially enlarged view of  FIG. 11A . As seen from  FIGS. 11A and 11B , when the gain Gcomp is set to be low, convergence of the phase difference is late but oscillation (amplitude) of the rotational velocity is small as compared to the case in which the gain Gcomp is set to a relatively high value. Also, after the phase difference is converged, no large oscillation occurs even by fluctuation due to disturbance. It was found that stability of the rotational phase (velocity) of the another photosensitive drum  3  (e.g.,  3 M) is favorable. 
     FIG. 12A  is a graph showing the change in rotational velocity of the another photosensitive drum  3  (e.g.,  3 M) in case that the gain Gcomp is set to a relatively high value.  FIG. 12B  is a partially enlarged view of  FIG. 12A . As seen from  FIGS. 12A and 12B , when the gain Gcomp is set to be high, it was found that convergence of the phase difference is quick but the rotational phase (velocity) is easy to deviate even after the convergence of the phase difference. 
   Accordingly in the present embodiment, the gain correction function unit  153  is designed to output the variable gain Gcomp which is large at the startup of the control and small at the convergence of the phase difference.  FIGS. 13A and 13B  show the change in rotational velocity of the another photosensitive drum  3  (e.g.,  3 M) when the gain Gcomp set high at the startup is linearly decreased after the startup, and maintained at a constant value by stopping the change of the gain Gcomp after 0.8 seconds. As can be seen by comparison between the case of  FIGS. 13A and 13B , and the cases of  FIGS. 11A ,  11 B,  12 A and  12 B, the time taken for the phase convergence is clearly shorter than the case at low gain. The velocity fluctuation at the phase convergence is smaller than the case at high gain on both oscillation amplitude and vestigial amplitude. As noted above, by setting the gain Gcomp to be high at the startup and low at the phase convergence, the phase difference is quickly converged and the rotational velocity and the phase difference can be reliably controlled to desired values. 
     FIGS. 14A to 14C  are graphs showing a converging state of the phase difference (error) in each of the above cases.  FIG. 14A  shows the case at low gain,  FIG. 14B  shows the case at high gain, and  FIG. 14C  is the case in which the gain Gcomp is changed from high to low as explained above. As shown in  FIG. 14A , convergence of the phase difference is slow in the case at low gain. As seen from  FIG. 14B , the phase difference is reliably maintained at zero even after the convergence. Also, vestigial wave occurs, and oscillation occurs due to slight disturbance. To the contrary, as shown in  FIG. 14C , the time taken for convergence and the amount of oscillation are well balanced in case that the gain Gcomp is changed from high to low as explained above. 
   The gain Gcomp may be changed in various manners. As shown below, for example, the gain Gcomp may be switched by two steps, depending on whether the time T elapsed after the startup has exceeded a threshold δ.
 
G comp =g 1  when T&lt;δ
 
G comp =g 2  when T≧δwhere g 1 &gt;g 2 &gt;0
 
   In case that the gain correction function unit  153  is defined by the above equations, the gain Gcomp is set at a high gain g 1  so as to quickly converge the phase difference until time δ elapses after the startup. Then, after the time δ has elapsed, the gain Gcomp is set at a low gain g 2  to reliably converge the rotational velocity and the phase difference to desired values. 
   Also as shown below, the gain Gcomp may be set at the high gain g 1  until time δ 1  elapses after the startup, and then may be linearly decreased until time δ 2  to be maintained at the low gain g 2  after time δ 2  has elapsed. 
   
     
       
         
           
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   Moreover, as shown below, the gain Gcomp may be decreased along an asymptote of an exponential function between δ 1  and δ 2 .
 
G comp =g 1  when T&lt;δ 1  
 
 G   comp   =Be   −AT   +C  when δ 1   ≦T&lt;δ   2  
 
G comp =g 2  when T≧δ 2  
 
where g 1 &gt;g 2 &gt;0
 
   Much smoother convergence of the phase difference is achieved in the latter two cases in which the Gcomp is consecutively decreased, as compared to the former case in which the gain Gcomp is decreased stepwisely. 
   [Variation of Driving Unit of Photosensitive Drum] 
   The above control can be also executed by a software process using a microcomputer.  FIG. 15  is a block diagram showing a structure of a driving unit for use in executing the software process.  FIG. 15  only shows the structure relevant to the photosensitive drums  3 Y and  3 M (driving units  120 Y and  120 M). However, the driving units  120 C and  120 B for the photosensitive drums  3 C and  3 B are designed in the same manner. 
   As shown in  FIG. 15 , in this control system, a signal from the control unit  10  (see  FIG. 1 ) is inputted to the power amplifier  123 , signals from the velocity detector  124  and the index detector  130  are inputted to the control unit  10 . The control unit  10  is constituted of a known microcomputer including a CPU  10   a , a ROM  10   b  and a RAM  10   c . The control unit  10  executes the following process based on a program stored in the ROM  10   b.  Other than the components shown in  FIG. 10 , various components like an operation panel of the printer  1  are connected to the control unit  10 . Illustrations of those components are omitted since there is not direct relationship with the process explained hereafter. 
     FIG. 16  is a flowchart illustrating a main routine of a velocity control process, executed by the control unit  10 , to adjust the velocity of the photosensitive drum  3 M. This process is started when a print directive is inputted from an external computer or the like to generate a driving directive for each of the photosensitive drums  3 Y,  3 M,  3 C and  3 B. 
   When the process is started, firstly in S 1 , a velocity directive value of the photosensitive drum  3 M is set. In S 2 , calculation of a correction value is started by another routine. Detailed explanation will be later given on this another routine. Here, the correction value corresponds to the output of the transfer function calculator  143 M. 
   Next in S 3 , the correction value at the time is subtracted from the detection velocity inputted from the velocity detector  124 M. Based on the velocity after the subtraction, a known feedback calculation process is performed in S 4 . That is, in S 4 , a voltage inputted to the motor  121 M is calculated so that the velocity calculated in S 3  is consistent with the velocity corresponding to the above velocity directive value. When the input voltage is calculated in this manner, a signal corresponding to the input voltage is outputted to the power amplifier  123 M by another routine. 
   In S 5 , it is determined whether the image data is processed and driving of the photosensitive drum  3 M is complete. If not (S 5 : N), the process returns to S 3  and the above steps are repeated. Otherwise (S 5 : Y), the process moves to S 7  to set zero to the velocity directive value. In S 8 , a feedback calculation process in accordance with the velocity directive value is performed. In S 9 , it is determined whether the photosensitive drum  3 M is stopped. If not (S 9 : N), the feedback calculation process in S 8  is repeated. Otherwise (S 9 : Y), the process is ended. 
   The calculation of the correction value which is started in S 2  is made up of three processes performed in parallel as shown in  FIGS. 17 to 19 . Firstly,  FIG. 17  is a flowchart illustrating a process of calculating a counter value Cnt_Y2M which corresponds to a phase delay amount of the photosensitive drum  3 M to the photosensitive drum  3 Y. 
   As shown in  FIG. 17 , when the process is started, firstly, it is determined in S 21  whether an index signal is generated by the index detector  130 Y (HP_Y edge detection). If not (S 21 : N), the process stands by at S 21 . Otherwise (S 21 : Y), a counter value P_cnt is cleared to zero in S 22 . 
   In S 23 , the counter value P_cnt is incremented by one. In S 24 , it is determined whether an index signal is generated by the index detector  130 M (HP_M edge detection). If not (S 24 : N), the process returns to S 23  to stand by while the counter P_cnt is incremented one by one. If an index signal is generated by the index detector  130 M (S 24 : Y), the counter value P_cnt at the time is stored as the counter value Cnt_Y2M in S 25 . 
   Subsequently in S 26 , it is determined whether the driving of the photosensitive drums  3 M and  3 Y is completed. If not (S 26 : N), the process returns to S 21  and the above steps are repeated. Otherwise (S 26 : Y), the counter value Cnt_Y2M is cleared to zero in S 27 . The process is ended. 
     FIG. 18  is a flowchart illustrating a process of calculating a counter value Cnt_M2M which corresponds to a phase advance amount of the photosensitive drum  3 M to the photosensitive drum  3 Y. As shown below, this process is designed substantially the same with the process in  FIG. 17 . 
   That is, when this process is started, firstly, it is determined in S 31  whether an index signal is generated by the index detector  130 M. If (S 31 : N), the process stands by in S 31 . If an index signal is generated by the index detector  130 M (S 31 : Y), a counter value N_cnt is cleared to zero in S 32 . 
   Subsequently in S 33 , the counter value N_cnt is incremented by one. It is determined in S 34  whether an index signal is generated by the index detector  130 Y. If not (S 34 : N), the process returns to S 38  to stand by while the counter value N_cnt is incremented one by one. 
   When an index signal is generated by the index detector  130 Y (S 34 : Y), the counter value N_cnt at the time is stored as the counter value Cnt_M2Y in S 35 . Until the driving of the photosensitive drums  3 M and  3 Y is completed (S 36 : N), the above steps are repeated. When the driving is completed (S 36 : Y), the counter value Cnt_M2Y is cleared to zero in S 37 . The process is ended. 
     FIG. 19  is a flowchart illustrating a process of calculating the correction value from the counter values Cnt_Y2M and Cnt_M2Y stored at the time. As shown in  FIG. 19 , when the process is started, firstly in S 41 , variables Cnt and sgn are cleared to zero. Next in S 42 , it is determined which of the counter values Cnt_Y2M and Cnt_M2Y is larger. If Cnt_M2Y&lt;Cnt_Y2M (S 42 : Y), −1 is set to sgn and Cnt_M2Y is set to Cnt in S 43 . If Cnt_M2Y≧Cnt_Y2M (S 42 : N), +1 is set to sgn and Cnt_Y2M is set to Cnt in S 44 . 
   In this manner, when the variables Cnt and sgn are set in S 43  or S 44 , the process moves to S 46  so that the correction value is calculated by C(s)*sgn*Cnt. Here, C(s) corresponds to a transfer function in the transfer function calculator  143 M, which is the result of multiplication of the filter element (e.g., filter element for removing high frequency component) and the gain Gcomp. The correction value calculated in this manner is stored in a predetermined area in the RAM  10   c  to be used in the process in  FIG. 16 . 
   Subsequently in S 47 , it is determined whether the driving of the photosensitive drums  3 M and  3 Y is completed. If not (S 47 : N), the process returns to S 42  to repeat the above steps. Otherwise (S 47 : Y), the correction value is cleared to zero in S 48 . The process is ended. 
   The aforementioned processes are also performed to each of the motors  121 C and  121 B. In this manner, the same control as in each of the driving units  120  shown in  FIG. 2  can be performed. Also in the above process, the control is performed such that the smaller of the values Cnt_M2Y and Cnt_Y2M becomes zero. Accordingly, the phase difference can be quickly converged. In the process for the motor  121 Y, the step S 3  in  FIG. 16  and the processes in  FIGS. 17 to 19  may be omitted. The correction value is always equal to zero if the same program is applied. Accordingly, even if the step S 3  in  FIG. 16  and the processes in  FIGS. 17 to 19  are performed to the motor  121 Y, the same control can be performed as in the case in which there is such omission. 
   Also in the case of using a software program as above, the following control may be performed based on the temperature of the fixing unit  8  detected by the sensor  83 . That is, while the temperature of the fixing unit  8  is low, it is necessary to converge the phase difference so quickly since image forming is unable to be performed. Also, when the temperature of the fixing unit  8  is low and the ambient temperature of the photosensitive drums  3  is low, it is preferable to cause oscillation in rotational velocity of the photosensitive drums  3  by setting a high gain, since toner remained on the surface of the photosensitive drums  3  is hard and a load applied when the photosensitive drums  3  are rotated is high due to friction with the paper conveying belt  6 . 
     FIG. 20  is a flowchart illustrating the control in consideration of the temperature of the fixing unit  8 . As shown in  FIG. 20 , when the process is started, firstly in S 51 , the process stands by until a print directive is received (S 51 : N). If a print directive is received (S 51 : Y), the process moves to S 52 . It is determined in S 52  whether the temperature of the fixing unit  8  is equal to or more than a predetermined degree, particularly equal to or more than the softening temperature of toner, based on the signal from the sensor  83 . If the temperature of the fixing unit  8  is less than the predetermined temperature (S 52 : N), low velocity drive is directed in S 53 . The process returns to S 52 . The predetermined temperature may be higher than the softening temperature of toner, e.g., equal to or more than the melting temperature of toner. 
   When low drive is directed in S 53 , the gain Gcomp is set to be relatively low. The velocity directive value is also set to be relatively low. Therefore, large stress is inhibited from being applied to the surface of the photosensitive drums  3 . Life of the photosensitive drums  3  can be prolonged. 
   While the temperature of the fixing unit  8  is less than the predetermined temperature (S 52 : N), low velocity drive is continued. When the temperature of the fixing unit  8  is raised to the predetermined temperature (S 52 : Y), high velocity drive is directed. Then, the velocity directive value is set to the value at normal image forming. The gain Gcomp is set high at first, and low at the convergence of the phase difference as previously noted. The gain Gcomp may be changed in any of the previously described manners. 
   Subsequently, it is determined in S 56  whether the driving of the photosensitive drums  3  is completed. If not (S 56 : N), the process moves to S 55  to continue high velocity drive, Otherwise (S 56 : Y), the velocity directive value is set to zero in S 57 . The process is ended. That is, in the present process, after the temperature of the fixing unit  8  is raised to the predetermined temperature, that is, the temperature at which toner is softened and the load of the photosensitive drums  3  is small enough, a transition to high velocity drive (S 55 ) takes place. Therefore, friction of the photosensitive drums  3  with the paper conveying belt  6  having stiff toner therebetween can be inhibited, and life of the photosensitive drums  3  can be prolonged. Particularly, at the startup of rotation of each of the photosensitive drums  3 , the peripheral velocity of each of the photosensitive drums  3  (until reaching to a constant velocity rotation, there is variation in peripheral velocity among each of the photosensitive drums  3 ) and the moving velocity of the paper conveying belt  6  do not coincide with each other. However, the influence of the difference to the life of the photosensitive drums  3  can be limited to the minimum. 
   Also in the above embodiments, the normal feedback control is performed in the motor  121 Y. However, the correction value in accordance with the phase difference may be reflected to the control of the motor  121 Y. That is, as shown in  FIG. 21 , an adder  145 Y is provided between the velocity detector  124 Y and the subtracter  125 Y, and the output of the transfer function calculator  143  (one of the transfer function calculators  143 M,  143 C and  143 B;  143 M in  FIG. 21 ) is also inputted to the adder  145 Y. In this manner, the correction value can be reflected to both of the controls of the motor  121 Y and another motor  121  (one of the motors  121 M,  121 C and  121 B;  121 M in  FIG. 21 ). Therefore, the phase difference can be converged all the more faster. In  FIG. 21 , the same reference number is given to the same components as in  FIG. 2 , and explanation thereof is repeated. 
   To realize such a control system, little ingenuity may be required for application in the case of three or more motors. 
   In the case of applying the control system to the printer  1  having four photosensitive drums  3 , for example, two motors may be respectively connected to two photosensitive drums  3  via gears so that four photosensitive drums  3  are driven by the two motors. 
   In the case of applying the structures of the driving units  120  shown in  FIG. 2  as well, two motors may be respectively connected to two photosensitive drums  3  via gears so that four photosensitive drums  3  are driven by the two motors. 
   In this case, these two motors may be controlled in the same manner as the two motors  121 Y and  121 M shown in  FIG. 21 . 
     FIGS. 22A and 22B  are graphs respectively showing the rotational velocity of each of the photosensitive drums  3 Y and  3 M in the control system of  FIG. 21 .  FIG. 23A  is a partially enlarged view of  FIG. 22B .  FIG. 23B  is a graph showing convergence of the phase difference. As shown in  FIGS. 22A ,  22 B,  23 A and  23 B, the phase difference can be reliably converged all the more faster by applying the correction value to both of the controls of the motors  121 Y and  121 M. Accordingly, in the control system in  FIG. 21 , the rotational velocity of each of the motors  121 Y and  121 M can be inhibited from exceeding the target rotational velocity. For this purpose, the output of the transfer function calculator  143 M may be inputted to the subtracter  145 M if the symbol of the output is positive, or to the adder  145 Y if the symbol is negative. 
   The present invention is not limited to the above described embodiments. The present invention can be practiced in various manners without departing from the technical scope of the invention. 
   For instance, the present invention can be applied to various driving systems other than a driving system for an image forming apparatus, as well as a driving system including reciprocation movement of a piston other than a driving system including rotation movement. 
   Also in the above embodiments, the output of the velocity detector  124  is corrected in accordance with the output of the transfer function calculator  143 . However, the velocity directive value may be corrected instead. 
   Moreover, the phase difference may not be necessarily controlled to be zero (timing at which each index signal is simultaneously generated). The phase difference may be controlled to be a specific value. For example, if a desired value which reflects eccentricity of each of the photosensitive drums  3  can be obtained by the aforementioned calibration, the phase difference may be controlled to the obtained value.