Patent Publication Number: US-8538311-B2

Title: Sheet measuring apparatus and image forming apparatus

Description:
CROSS-REFERENCE TO RELATED APPLICATIONS 
     This application is based on and claims priority under 35 USC 119 from Japanese Patent Application No. 2010-272528 filed Dec. 7, 2010. 
     BACKGROUND 
     Technical Field 
     The present invention relates to a sheet measuring apparatus and an image forming apparatus. 
     SUMMARY 
     According to an aspect of the invention, a sheet measuring apparatus includes a first rotating member that includes a first peripheral surface portion that contacts a transported sheet, the first rotating member rotating as the sheet is transported; a second rotating member that includes a second peripheral surface portion, the second peripheral surface portion contacting the first peripheral surface portion and being made of a material different from a material of the first peripheral surface portion, the second rotating member rotating as the first rotating member rotates; a first rotation amount detecting unit that detects a first rotation amount that is a rotation amount of the first rotating member; a second rotation amount detecting unit that detects a second rotation amount that is a rotation amount of the second rotating member; a sheet calculation unit that obtains a first rotating member correction value for correcting an error that is superposed on the second rotation amount due to a radius distribution of the first rotating member in a circumferential direction and that performs calculation related to the transported sheet by using the second rotation amount and the first rotation member correction value; a radius distribution calculating unit that calculates a new radius distribution of the first rotating member in the circumferential direction by using the first rotation amount and the second rotation amount; and an updating unit that updates the first rotating member correction value to a new first rotating member correction value that is obtained on the basis of the new radius distribution. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       Exemplary embodiments of the present invention will be described in detail based on the following figures, wherein: 
         FIG. 1  is a schematic view of an image forming apparatus according to an exemplary embodiment of the present invention; 
         FIG. 2A  is a side view of a length measuring device seen from the front side of the image forming apparatus, and  FIG. 2B  is a top view of the length measuring device seen in the direction IIB of  FIG. 2A ; 
         FIG. 3  is a front view of the length measuring device seen in the direction III of  FIG. 2A  (from the downstream side in the sheet transport direction); 
         FIG. 4  is a block diagram of a controller; 
         FIG. 5  is a flowchart illustrating a process performed by the controller when forming images on both sides of a sheet; 
         FIG. 6A  is a timing chart illustrating the relationship among an upstream edge signal, a first downstream edge signal, a second downstream edge signal, a second A-phase signal, a second Z-phase signal, a first Z-phase signal, a first temperature signal, and a second temperature signal, which are output before and after a sheet passes through the length measuring device; 
         FIG. 6B  is an enlarged view of a region VIB of  FIG. 6A , and  FIG. 6C  is an enlarged view of a region VIC of  FIG. 6A ; 
         FIG. 7  is a flowchart illustrating a process performed by a processor; 
         FIG. 8  is a flowchart illustrating a process for generating a second-roller rotation correction factor table; 
         FIG. 9  illustrates an operation performed in step S 303  of  FIG. 8 ; 
         FIG. 10  illustrates why an error occurs when the length measuring device performs measurement; 
         FIG. 11A  illustrates an example of first-roller radius data, and  FIG. 11B  illustrates an example of second-roller diameter/slit correction data; 
         FIG. 12  is a flowchart illustrating a process for updating the first-roller radius data; 
         FIG. 13  illustrates an operation performed in steps S 401  to S 409  of  FIG. 12 ; 
         FIG. 14  illustrates an operation performed in step S 418  of  FIG. 12 ; and 
         FIG. 15  illustrates an operation performed in steps S 419  to S 423  of  FIG. 12 . 
     
    
    
     DETAILED DESCRIPTION 
     Hereinafter, an exemplary embodiment of the present invention will be described with reference to the drawings. 
       FIG. 1  is a schematic view of an image forming apparatus according to the exemplary embodiment. The image forming apparatus illustrated in  FIG. 1  has a so-called tandem structure, and includes plural image forming units  10  ( 10 Y,  10 M,  10 C,  10 K) that form color toner images by using, for example, an electrophotographic method. The image forming apparatus includes an intermediate transfer belt  20  and a second-transfer device  30 . The color toner images formed by the image forming units  10  are successively transferred (first-transferred) onto the intermediate transfer belt  20 . The second-transfer device  30  simultaneously transfers (second-transfers) the superposed images, which have been transferred to the intermediate transfer belt  20 , onto the sheet S. The image forming apparatus includes a sheet feeder  40 , a fixing device  50 , a cooling device  55 , and a decurler  60 . The sheet feeder  40  feeds the sheet S toward the second-transfer device  30 . The fixing device  50  heats the image, which has been second-transferred to the second-transfer device  30 , and thermally fixes the image onto the sheet S. The cooling device  55  cools the image formed on the sheet S. The decurler  60  corrects a curl of the sheet S that is generated when the sheet S is cooled. In the present exemplary embodiment, the image forming units  10 , the intermediate transfer belt  20 , and the second-transfer device  30  function as an image forming unit. 
     Each of the image forming units  10  includes a photoconductor drum  11 , a charging device  12 , an exposure device  13 , a developing device  14 , a first-transfer device  15 , and a drum cleaner  16 . The photoconductor drum  11  is rotatable. The charging device  12  is disposed near the photoconductor drum  11  and charges the photoconductor drum  11 . The exposure device  13  exposes the photoconductor drum  11  to light and forms an electrostatic latent image. The developing device  14  makes the electrostatic latent image visible by using toner. The first-transfer device  15  transfers color toner images from the photoconductor drum  11  onto the intermediate transfer belt  20 . The drum cleaner  16  removes residual toner from the photoconductor drum  11 . In the following description, the image forming units  10  will be respectively referred to as a yellow image forming unit  10 Y, a magenta image forming unit  10 M, a cyan image forming unit  10 C, and a black image forming unit  10 K. 
     The intermediate transfer belt  20  is looped over three rollers  21  to  23  and rotates. The roller  22  drives the intermediate transfer belt  20 . The roller  23  is disposed opposite a second-transfer roller  31  with the intermediate transfer belt  20  therebetween. The second-transfer roller  31  and the roller  23  constitute the second-transfer device  30 . A belt cleaner  24  is disposed opposite the roller  21  with the intermediate transfer belt  20  therebetween. The belt cleaner  24  removes residual toner from the intermediate transfer belt  20 . 
     The sheet feeder  40  includes a sheet container  41  and a pick-up roller  42 . The sheet container  41  holds the sheet S. The pick-up roller  42  picks up the sheet S from the sheet container  41  and transports the sheet S. Plural transport rollers  43  are disposed in the transport path along which the sheet S is transported from the sheet feeder  40 . The sheet S may be made of any of a paper sheet, a resin sheet that is used for an OHP sheet or the like, and a paper sheet coated with a resin. 
     The fixing device  50  includes a heater for heating the sheet S. In the present exemplary embodiment, the fixing device  50  heats and presses the image, which has been transferred to the sheet S, and thereby fixes the image. 
     The cooling device  55  has a function of cooling the sheet S, which has been heated by the fixing device  50 . The cooling device  55  may be, for example, configured so that the sheet S passes between two metal rollers while being nipped by the metal rollers. 
     The decurler  60  has a function of correcting a curl (warping) that is generated in the sheet S. 
     The image forming apparatus according to the present exemplary embodiment is not only capable of forming an image on one side of the sheet S that is fed from the sheet feeder  40 , but also capable of forming an image on the other side of the sheet S by reversely transporting the one-side recorded sheet S. To perform this function, the image forming apparatus includes a reverse-transport mechanism  70 . After the sheet S has passed through the fixing device  50 , the cooling device  55 , and the decurler  60 , the reverse-transport mechanism  70  flips the sheet S over and reverses the transport direction of the sheet S and returns the sheet S to the second-transfer device  30 . The reverse-transport mechanism  70  is disposed downstream of the decurler  60  in the transport direction of the sheet S. The reversing mechanism includes a switching device  71  that switches the transport path of the sheet S between a path for outputting the sheet S to the outside of the image forming apparatus and a path for reversely transporting the sheet S. The reverse-transport mechanism  70  further includes a reversing device  72  that is disposed in the transport path for reversely transporting the sheet S. The reversing device  72  reverses the transport direction of the sheet S and flips the sheet S over before the sheet S is transported to the second-transfer device  30  again. Plural transport rollers  43  are disposed in the transport path for reversely transporting the sheet S. 
     The image forming apparatus according to the present exemplary embodiment further includes a length measuring device  100  that is disposed downstream of the decurler  60  in the transport direction of the sheet S and upstream of the switching device  71  in the transport direction of the sheet S. The length measuring device  100  measures the length of the sheet S that is transported thereto. The length measuring device  100  need not be disposed at the above-described position, and may be disposed in the transport path for reversely transporting the sheet S. 
     The image forming apparatus further includes a controller  80  and a user interface (UI)  90 . The controller  80  controls the devices and units of the image forming apparatus. The user interface (UI)  90  outputs an instruction received from a user to the controller  80  and provides the user with an instruction received from the controller  80  by using a screen (not shown) or the like. 
       FIGS. 2 and 3  illustrate the length measuring device  100  included in the image forming apparatus illustrated in  FIG. 1 .  FIG. 2A  is a side view of the length measuring device  100  seen from the front side (see  FIG. 1 ) of the image forming apparatus, and  FIG. 2B  is a top view of the length measuring device  100  seen in the direction IIB of  FIG. 2A .  FIG. 3  is a front view of the length measuring device seen in the direction III of  FIG. 2A  (from the downstream side in the transport direction of the sheet S). 
     The length measuring device  100  includes a first roller  110 , a second roller  120 , a support mechanism  130 , and a third roller  140 . The first roller  110  is disposed above a transport path  44  and rotates around a first rotation shaft  110   a . The second roller  120  is disposed above the first roller  110 , is in contact with the first roller  110 , and rotates around a second rotation shaft  120   a . The support mechanism  130  supports the first roller  110  and the second roller  120 . The third roller  140  is disposed opposite the first roller  110  with the transport path  44  therebetween, and rotates around a third rotation axis  140   a . The length measuring device  100  includes a first rotation amount sensor  170  and a second rotation amount sensor  180 . The first rotation amount sensor  170  detects the rotation count and the rotation amount of the first roller  110 . The second rotation amount sensor  180  detects the rotation count and the rotation amount of the second roller  120 . 
     The first roller  110 , which is an example of a first rotating member, includes a first-roller body  111  and a surface layer  112 . The first-roller body  111  is disposed so as to surround the first rotation shaft  110   a . The surface layer  112  is formed on an outer peripheral surface of the first-roller body  111 . The outer peripheral surface of the first roller  110  is a first peripheral surface portion  113  that is a part of the surface layer  112 . In the present exemplary embodiment, the first-roller body  111  and the surface layer  112  are both made of an elastic material such as a rubber or the like. The hardness of the surface layer  112  is larger than that of the first-roller body  111 . In this example, the first roller  110  has two layers. However, the first roller  110  may have only one layer or three or more layers. The first-roller body  111  and the surface layer  112  may be made of a material other than a rubber, such as a plastic, or may be made of different materials. The first-roller body  111  may be made of, for example, a metal such as aluminum. 
     The second roller  120 , which is an example of a second rotating member, includes a second-roller body  121 . The second-roller body  121  is disposed so as to surround the second rotation shaft  120   a , and the entirety of the second-roller body  121 , including the outer peripheral surface, is made of a metal such as aluminum. The outer peripheral surface of the second roller  120  is a second peripheral surface  122  that is a part of the second-roller body  121 . 
     Thus, in the present exemplary embodiment, the first peripheral surface portion  113  of the first roller  110 , which contacts the transported sheet S, is made of a rubber that has a friction coefficient higher than that of a metal. The second peripheral surface  122  of the second roller  120 , which contacts the first peripheral surface portion  113  of the first roller  110 , is made of a metal that has a thermal expansion coefficient smaller than that of a rubber. 
     The support mechanism  130  includes a support shaft  130   a , a first arm  131   a , and a second arm  132   a . The support shaft  130   a  is disposed upstream of the first roller  110  in the transport direction of the sheet S and above the transport path  44 , and extends parallelly to the first rotation shaft  110   a  and the second rotation shaft  120   a . The first arm  131   a  and the second arm  132   a  are rotatable around the support shaft  130   a . The support shaft  130   a  is fixed to and supported by the housing (not shown) of the length measuring device  100 . 
     The first arm  131   a  extends in the transport direction of the sheet S. The support shaft  130   a  is attached to a midstream part of the first arm  131   a  in the transport direction of the sheet S. The first rotation shaft  110   a  of the first roller  110  is rotatably attached to the downstream end of the first arm  131   a  in the transport direction of the sheet S. A through-hole is formed in an end portion of the first arm  131   a  that is located upstream of the support shaft  130   a  in the transport direction of the sheet S. One end of a first spring  131   b  is attached to the through-hole. The first spring  131   b  is a tension spring that extends upward. The other end of the first spring  131   b  is attached to the housing of the length measuring device  100 . Thus, the first spring  131   b  applies to the first arm  131   a  a force that is directed clockwise around the support shaft  130   a  in  FIG. 2A . As a result, the first roller  110  is pressed against the third roller  140  (toward the transport path  44 ). Both the first arm  131   a  and the first spring  131   b  are disposed at each end of the first roller  110  in the axial direction. In the present exemplary embodiment, the first arm  131   a  and the first spring  131   b  constitute a first support portion  131  that supports the first roller  110 . 
     The second arm  132   a  has an L-shape that extends upward from a first end that is in a lower part thereof and then extends downstream in the transport direction of the sheet S. The support shaft  130   a  is attached to the first end of the second arm  132   a . The second rotation shaft  120   a  of the second roller  120  is rotatably attached to a second end of the second arm  132   a , which is located above the first end and downstream of the first end in the transport direction of the sheet S. One end of a second spring  132   b  is attached to an upper end of the second arm  132   a . The second spring  132   b  is a compression spring that extends upward. The other end of the second spring  132   b  is attached to the housing of the length measuring device  100 . Thus, the second spring  132   b  applies to the second arm  132   a  a force that is directed clockwise around the support shaft  130   a  in  FIG. 2A . As a result, the second roller  120  is pressed against the first roller  110 . Both the second arm  132   a  and the second spring  132   b  are disposed at each end of the second roller  120  in the axial direction. In the present exemplary embodiment, the second arm  132   a  and second spring  132   b  constitute a second support portion  132  that supports the second roller  120 . 
     The third roller  140 , including the outer peripheral surface thereof, is made of a metal such as aluminum. When the sheet S is present between the third roller  140  and the first roller  110 , the third roller  140  contacts the sheet S. If not, the third roller  140  contacts the first roller  110 . In the present exemplary embodiment, the third roller  140  is disposed opposite the first roller  110  with the transport path  44  therebetween. Instead of the third roller  140 , a fixed member, such as a metal plate, may be used. 
     The length measuring device  100  includes an upstream sensor  150 , a first downstream sensor  151 , and a second downstream sensor  152 . The upstream sensor  150  is disposed upstream, in the transport direction of the sheet S, of a position at which the first roller  110  contacts the sheet S (or the third roller  140 ), and detects passing of the leading end and the trailing end of the sheet S. The first downstream sensor  151  and the second downstream sensor  152  are disposed downstream, in the transport direction of the sheet S, of a position at which the first roller  110  contacts the sheet S (or the third roller  140 ), and detects passing of the leading end and the trailing end of the sheet S. In the present exemplary embodiment, each of the upstream sensor  150 , the first downstream sensor  151 , and the second downstream sensor  152  is a photoelectric sensor including a light emitting diode (LED) and a photosensor, and optically detects the transported sheet S that is passing an opposite position. The upstream sensor  150 , the first downstream sensor  151 , and the second downstream sensor  152  are attached to the housing (not shown) of the length measuring device  100 . 
     In particular, the upstream sensor  150  and the first downstream sensor  151  are attached to an attachment member  190  that extends in the transport direction of the sheet S. As a result, the upstream sensor  150  and the first downstream sensor  151  are disposed on a straight line extending in the transport direction of the sheet S. The first downstream sensor  151  and the second downstream sensor  152  are disposed opposite each other in a direction perpendicular to the transport direction of the sheet S with a position at which the sheet S contacts the first roller  110  therebetween. In the following description, the term “reference gap length Lg 0 ” refers to the distance between the detection position of the upstream sensor  150  and the detection position of the first downstream sensor  151  at a reference temperature. In the present exemplary embodiment, the upstream sensor  150 , the first downstream sensor  151 , and the second downstream sensor  152  function as an end detecting unit. 
     The length measuring device  100  includes a first temperature sensor  161  and a second temperature sensor  162 . The first temperature sensor  161  measures the ambient temperature around the attachment member  190 . The second temperature sensor  162 , which is an example of a temperature detecting unit, detects the ambient temperature around the second roller  120 . The first temperature sensor  161  is attached to the housing (not shown) of the length measuring device  100 . The second temperature sensor  162  is attached to the second arm  132   a  of the support mechanism  130 . The first temperature sensor  161  and the second temperature sensor  162  may measure, in addition to the ambient temperature, the surface temperatures of the attachment member  190  and the second roller  120 , or may measure the internal temperatures of the attachment member  190  and the second roller  120 . In the present exemplary embodiment, the sheet S that has been heated by the fixing device  50  passes through the length measuring device  100 . Therefore, as an increasing number of sheets S pass through the length measuring device  100 , the internal temperature of the length measuring device  100  may increase. In this example, after the sheet S has passed through the fixing device  50  and the cooling device  55 , the sheet S reaches the length measuring device  100 . If the sheet S has not been sufficiently cooled, the sheet S that retains heat may enter the length measuring device  100 . 
     The first rotation amount sensor  170 , which is an example of a first rotation amount detecting unit, includes a first encoder wheel  171  and a first optical detector  172 . The first encoder wheel  171  has a disk-like shape, is attached to the first rotation shaft  110   a  of the first roller  110 , and rotates together with the first roller  110 . The first optical detector  172  is attached to the first arm  131   a  of the support mechanism  130  so as to face a side surface of the first encoder wheel  171 . Plural first A-phase slits  171   a  and a first Z-phase slit  171   z  extend through the sides (front and back sides) of the first encoder wheel  171 . The first A-phase slits  171   a  are disposed at regular intervals in the circumferential direction. The first Z-phase slit  171   z  is formed at a position that is outside the first A-phase slits  171   a  in the radial direction. The first optical detector  172  optically detects passing of the first A-phase slits  171   a  and passing of the first Z-phase slit  171   z  when the first encoder wheel  171  rotates together with the first roller  110 . In this example, n first A-phase slits  171   a  are formed in the first encoder wheel  171 . 
     The second rotation amount sensor  180 , which is an example of a second rotation amount detecting unit, includes a second encoder wheel  181  and a second optical detector  182 . The second encoder wheel  181  has a disk-like shape, is attached to the second rotation shaft  120   a  of the second roller  120 , and rotates together with the second roller  120 . The second optical detector  182  is attached to the second arm  132   a  of the support mechanism  130  so as to face a side surface of the second encoder wheel  181 . Plural second A-phase slits  181   a  and a second Z-phase slit  181   z  extend through the sides (front and back sides) of the second encoder wheel  181 . The second A-phase slits  181   a  are disposed at regular intervals in the circumferential direction. The second Z-phase slit  181   z  is formed at a position that is outside the second A-phase slits  181   a  in the radial direction. The second optical detector  182  optically detects passing of the second A-phase slits  181   a  and passing of the second Z-phase slit  181   z  when the second encoder wheel  181  rotates together with the second roller  120 . In this example, m second A-phase slits  181   a  are formed in the second encoder wheel  181 . 
     In the present exemplary embodiment, each of the first rotation amount sensor  170  and the second rotation amount sensor  180  is an incremental rotary encoder. However, any type of sensor may be used, as long as the sensor is capable of measuring the rotation amount of a roller smaller than one rotation (2π(rad)). In the present exemplary embodiment, the first rotation amount sensor  170  and the second rotation amount sensor  180  are sensors that utilize variation in the amount of light. However, the sensors may be sensors that utilize, for example, magnetic variation. 
       FIG. 4  is a block diagram of the controller  80  illustrated in  FIG. 1 . 
     The controller  80  includes a receiving unit  81  and an image signal generator  82 . The receiving unit  81  receives instruction sent from the UI  90  or an external apparatus (not shown) that is connected to the image forming apparatus. When a print instruction is received by the receiving unit  81 , the image signal generator  82  generates color image signals for yellow, magenta, cyan, and black on the basis of image data that has been sent together with the print instruction. The controller  80  includes an image signal output adjustment unit  83  that adjusts timing for outputting the color image signals, which have been generated by the image signal generator  82 , to the image forming units  10  (to be specific, the exposure devices  13  of the image forming units  10 ). Moreover, the image signal output adjustment unit  83  adjusts the magnifications of the color image signals, which have been generated by the image signal generator  82 , in the sub-scanning direction (corresponding to the transport direction of the sheet S). The controller  80  includes an operation controller  84  that controls operations of the units and devices of the image forming apparatus, including the image forming units  10  ( 10 Y,  10 M,  100 ,  10 K), the second-transfer device  30 , the sheet feeder  40 , the fixing device  50 , the cooling device  55 , the decurler  60 , and the reverse-transport mechanism  70 . 
     The controller  80  according to the present exemplary embodiment includes a processor  85  that performs various calculations on the basis of various signals that are input from the length measuring device  100 . The processor  85  includes a length calculator  851 , a velocity calculator  852 , a first-roller radius calculator  853 , a storage unit  854 , a determination unit  855 , and an updating unit  856 . The length calculator  851  calculates a sheet length L that is the length of the sheet S in the transport direction, the sheet S passing through the length measuring device  100 . The velocity calculator  852  calculates a sheet velocity V that is the transport velocity of the sheet S. The first-roller radius calculator  853  calculates the radius of the first roller  110  when the sheet S passes. The storage unit  854  stores various data that is used in the calculations performed by the length calculator  851 , the velocity calculator  852 , and the first-roller radius calculator  853 . The determination unit  855  determines whether or not the first roller  110  has reached the end of its lifetime on the basis of a calculation result obtained by the first-roller radius calculator  853 . The updating unit  856  updates a part of data stored in the storage unit  854  on the basis of the calculation result obtained by the first-roller radius calculator  853 . In the present exemplary embodiment, the length calculator  851  and the velocity calculator  852  are an example of a sheet calculation unit, the first-roller radius calculator  853  is an example of a radius distribution calculating unit, the determination unit  855  is an example of a fault detecting unit, and the updating unit  856  is an example of an updating unit. 
     An upstream edge signal Su that is output from the upstream sensor  150 , a first downstream edge signal Sd 1  that is output from the first downstream sensor  151 , and a second downstream edge signal Sd 2  that is output from the second downstream sensor  152  are input to the processor  85 . A first A-phase signal Sa 1  and a first Z-phase signal Sz 1  that are output from the first optical detector  172  of the first rotation amount sensor  170  are input to the processor  85 . The first A-phase signal Sa 1  is a signal indicating detection of the first A-phase slits  171   a . The first Z-phase signal Sz 1  is a signal indicating detection of the first Z-phase slit  171   z . A second A-phase signal Sa 2  and a second Z-phase signal Sz 2  that are output from the second optical detector  182  of the second rotation amount sensor  180  are input to the processor  85 . The second A-phase signal Sa 2  is a signal indicating detection of the second A-phase slits  181   a . The second Z-phase signal Sz 2  is a signal indicating detection of the second Z-phase slit  181   z . A first temperature signal St 1  that is output from the first temperature sensor  161  and a second temperature signal St 2  that is output from the second temperature sensor  162  are input to the processor  85 . 
     The sheet length L, which has been calculated by the length calculator  851 , is output to the image signal output adjustment unit  83 , and is used to adjust the output of an image signal. The sheet length L is also output to the operation controller  84 , and is used to control the operations of the units and devices included the image forming apparatus. The sheet velocity V (velocity information), which has been calculated by the velocity calculator  852 , is output to the outside and used for performing various operations. 
     The controller  80  includes a central processing unit (CPU), a read only memory (ROM), and a random access memory (RAM). The CPU performs processing on the basis of a program stored in the ROM while exchanging data with the RAM. 
       FIG. 5  is a flowchart illustrating a process performed by the controller  80  when the image forming apparatus illustrated in  FIG. 1  forms images on both sides of the sheet S. Referring to  FIGS. 1 to 5 , the process will be described. 
     When the receiving unit  81  receives a print command from the UI  90  or an external apparatus (step S 101 ), the operation controller  84  activates the units and devices included in the image forming apparatus and causes the units and devices to perform warm-up operations, and the image signal generator  82  generates image signals for color images to be formed on a first side of the sheet S on the basis of input image data. Next, the operation controller  84  causes the sheet feeder  40  to feed the sheet S, and the image signal output adjustment unit  83  outputs the image signals for color images, which have been generated by the image signal generator  82 , to the image forming units  10  (to be specific, the exposure devices  13  of the image forming units  10 ) in sync with feeding of the sheet S (step S 102 ). 
     Thus, the image forming units  10  form images (in this example, toner images) in accordance with the image signals for the first side. To be specific, the operation controller  84  causes the photoconductor drums  11  of the image forming units  10  to rotate, causes the charging devices  12  to charge the rotating photoconductor drums  11 , causes the exposure devices  13  to expose the photoconductor drums  11  with light beams that are emitted in accordance with the color image signals for the first side, thereby forming electrostatic latent images on the surfaces of the photoconductor drums  11 . Next, the operation controller  84  causes the developing devices  14  to develop the electrostatic latent images formed on the photoconductor drums  11  for the corresponding colors, thereby forming color images for the first side. The operation controller  84  causes the first-transfer device  15  to successively first-transfer the images for the first side from the photoconductor drums  11  to the rotating intermediate transfer belt  20  (step S 103 ). Thus, the images for the first side are first-transferred to the intermediate transfer belt  20  in an overlapping manner, and when the intermediate transfer belt  20  rotates further, the images are moved to the second-transfer position in the second-transfer device  30  at which the second-transfer roller  31  and the roller  23  are disposed opposite each other. 
     The sheet S, which has been fed by the sheet feeder  40 , is transported by the transport rollers  43  and reaches the second-transfer position. Then, the operation controller  84  causes the second-transfer device  30  to second-transfer the images for the first side from the intermediate transfer belt  20  to the first side of the sheet S (step S 104 ). 
     Next, the operation controller  84  causes the fixing device  50  to fix the images, which have been transferred to the first side of the sheet S, by, for example, heating and pressing the sheet S. The operation controller causes the cooling device  55  to cool the sheet S, which has been heated by the fixing device  50  (step S 105 ). The sheet S passes through the cooling device  55 , is decurled by the decurler  60 , and is further transported. 
     After the sheet S, on which the image have been fixed on the first side thereof, passes through the cooling device  55  and the decurler  60 , the one-side recorded sheet S is further transported to the length measuring device  100 . In the length measuring device  100 , the first roller  110  and the second roller  120  rotate as the one-side recorded sheet S is transported. The first optical detector  172  of the first rotation amount sensor  170  outputs the first A-phase signal Sa 1  and the first Z-phase signal Sz 1  in accordance with the rotation amount of the first roller  110 . The second optical detector  182  of the second rotation amount sensor  180  outputs the second A-phase signal Sa 2  and the second Z-phase signal Sz 2  in accordance with the rotation amount of the second roller  120 . The upstream sensor  150  outputs the upstream edge signal Su, the first downstream sensor  151  outputs the first downstream edge signal Sd 1 , and the second downstream sensor  152  outputs the second downstream edge signal Sd 2 . 
     The signals output from the length measuring device  100  are input to the processor  85 . The length calculator  851  of the processor  85  calculates the sheet length L of the one-side recorded sheet S, which has passed through the length measuring device  100 , by using the signals input from the length measuring device  100  and data for calculation stored in the storage unit  854  (step S 106 ). Subsequently, the length calculator  851  outputs the calculated sheet length L to the image signal output adjustment unit  83  and the operation controller  84 . Specific calculations performed by the length calculator  851  will be described below. 
     Next, the image signal output adjustment unit  83  calculates, on the basis of the sheet length L received from the processor  85  (the length calculator  851 ), the timing at which the color image signals for the second side generated by the image signal generator  82  are output to the exposure devices  13  of the image forming units  10  (positions of the photoconductor drums  11  at which the exposure devices  13  write the images) and the magnifications (or reductions), in the sub-scanning direction, of the color image signals for the second side generated by the image signal generator  82  (step S 107 ). 
     The operation controller  84  causes the switching device  71  to switch the path of the one-side recorded sheet S to a reverse-transport path before the leading end of the sheet S reaches the switching device  71 , and causes the reversing device  72  to reverse the transport direction of the sheet S and to flip the sheet S over. As a result, the reverse-transport mechanism  70  reversely transports the one-side recorded sheet S to a transport path that is upstream of the second-transfer device  30  in the transport direction (step S 108 ). 
     Next, the image signal generator  82  generates color image signals for forming color images on the second side of the sheet S on the basis of input image data. The operation controller  84  causes the one-side recorded sheet S to be reversely transported further. The image signal output adjustment unit  83  adjusts the color image signals for the second side, which have been generated by the image signal generator  82 , in accordance with the writing positions and the magnifications calculated in step S 107 . Then, the image signal output adjustment unit  83  outputs the color image signals to the image forming units  10  (to be specific, the exposure devices  13  of the image forming units  10 ) in sync with feeding of the one-side recorded sheet S (step S 109 ), which is reversely transported. 
     Thus, the image forming units  10  form color images in accordance with the color images signals. To be specific, the operation controller  84  causes the photoconductor drums  11  of the image forming units  10  to rotate, causes the charging devices  12  to charge the rotating photoconductor drums  11 , causes the exposure devices  13  to expose the photoconductor drums  11  with light beams in accordance with the color image signals for the second side, thereby forming electrostatic latent images on the surfaces of the photoconductor drums  11 . Next, the operation controller  84  causes the developing devices  14  for the corresponding colors to develop the electrostatic latent images formed on the photoconductor drums  11 , thereby forming color images for the second side. The operation controller  84  causes the first-transfer devices  15  to successively first-transfer the color images for the second side from the photoconductor drums  11  to the intermediate transfer belt  20 , which rotates together with the photoconductor drums  11  (step S 110 ). The images for the second side, which have been first-transferred to the intermediate transfer belt  20  in an overlapping manner, are moved toward the second-transfer position as the intermediate transfer belt  20  rotates. 
     The one-side recorded sheet S is reversely transported by the transport rollers  43  and reaches the second-transfer position again. The operation controller  84  causes the second-transfer device  30  to second-transfer the images for the second side from the intermediate transfer belt  20  to the second side of the sheet S (step S 111 ). 
     Next, the operation controller  84  causes the fixing device  50  to fix the images onto the sheet S, by, for example, heating and pressing the sheet S, and causes the cooling device  55  to cool the sheet S, which has been heated by the fixing device  50  (step S 112 ). The sheet S passes through the cooling device  55 , is decurled by the decurler  60 , and is transported further. 
     The operation controller  84  causes the switching device  71  to switch the path of the double-side printed sheet S, on both sides of which images have been fixed, to the transport path for outputting the sheet S to the outside of the image forming apparatus before the leading end of the sheet reaches the switching device  71 . Therefore, the double-side recorded sheet S is transported and output to the outside of the image forming apparatus (step S 113 ), and the process is finished. 
     After the above-described double-side image formation process has been performed on each of plural sheets S, a booklet is made by binding the double-side recorded sheets S. At this time, even if the sheet length L differs among the sheets S, image forming conditions such as the writing positions and the magnifications in the sub-scanning direction are adjusted on the basis of the sheet length L measured by the length measuring device  100 . Therefore, displacement amounts among the recorded positions of the sheets S when forming a horizontally double-spread or a vertically double-spread booklet are reduced, whereby a high-quality booklet is bound as compared with the case where the adjustment based on the sheet length L is not performed. 
     In this example, displacement of images formed on the first and second sides of the sheet S is reduced by adjusting the image signals for the second side, which are output to the exposure devices  13 , by using the image signal output adjustment unit  83 . However, a method for reducing displacement of images is not limited thereto. For example, magnifications in the sub-scanning direction may be adjusted by adjusting the ratios of the rotation speeds of the photoconductor drums  11  to the movement speed of the intermediate transfer belt  20 . 
       FIG. 6A  is a timing chart illustrating the relationship among the upstream edge signal Su, the first downstream edge signal Sd 1 , the second downstream edge signal Sd 2 , the second A-phase signal Sa 2 , the second Z-phase signal Sz 2 , the first Z-phase signal Sz 1 , the first temperature signal St 1 , and the second temperature signal St 2 , which are output before and after the sheet S passes through the length measuring device  100 .  FIG. 6B  is an enlarged view of a region VIB of  FIG. 6A , and  FIG. 6C  is an enlarged view of a region VIC of  FIG. 6A . In  FIG. 6A , the first A-phase signal Sa 1  is not illustrated. 
     In the initial state before the sheet S enters the length measuring device  100 , the upstream edge signal Su, the first downstream edge signal Sd 1 , and the second downstream edge signal Sd 2  are each at a high level (H), because the sheet S is not present. In the initial state, the second A-phase signal Sa 2 , the second Z-phase signal Sz 2 , and the first Z-phase signal Sz 1  are each at a certain level (in this example, a low level (L)), because the first roller  110  and the second roller  120  are not rotating. 
     When the leading end of the sheet S in the transport direction (hereinafter, simply referred to as “the leading end”) reaches the detection position of the upstream sensor  150  as the sheet S is transported, the upstream edge signal Su changes from the high level to the low level. 
     Next, when the leading end of the transported sheet S reaches a position at which the sheet S contacts the first roller  110 , the first roller  110  starts rotating due to a force applied by the sheet S. Then, the second roller  120 , which is in contact with the first roller  110 , and the third roller  140 , which faces the first roller  110  with the sheet S therebetween, start rotating. Thus, the first encoder wheel  171  starts rotating together with the first roller  110 , and the second encoder wheel  181  starts rotating together with the second roller  120 . As a result, the second A-phase signal Sa 2  (and the first A-phase signal Sa 1  (not shown)) alternates between the high level and the low level. The first roller  110  does not instantly follow the speed of the sheet S after the first roller  110  starts rotating, but the speed of the first roller  110  gradually increases. Therefore, the speed of the second roller  120 , which is rotated by the first roller  110 , gradually increases. As a result, the intervals between the high level and the low level of the second A-phase signal Sa 2  (and the first A-phase signal Sa 1  (not shown)) gradually decrease. In the following description, a period from the time at which the second A-phase signal Sa 2  changes (hereinafter referred to as “rises”) from the low level to the high level to the next time at which the second A-phase signal Sa 2  rises will be referred to as “one pulse”. 
     Subsequently, at a first time te 1  at which the leading end of the transported sheet S reaches the detection position of the first downstream sensor  151 , the first downstream edge signal Sd 1  changes from the high level to the low level. In this example, at a second time te 2  at which the leading end of the transported sheet S reaches the detection position of the second downstream sensor  152 , the second downstream edge signal Sd 2  changes from the high level to the low level. 
     Which of the first downstream sensor  151  and the second downstream sensor  152  first detects the leading end of the sheet S depends on the orientation (inclination) of the transported sheet S.  FIG. 6A  illustrates an example in which the first downstream sensor  151  detects the leading end of the sheet S before the second downstream sensor  152  does. However, this temporal relationship may be the opposite. In the present exemplary embodiment, irrespective of the temporal relationship, the first time te 1  refers to the time at which the first downstream sensor  151 , which is disposed downstream of the upstream sensor  150 , detects the leading end of the sheet S, and the second time te 2  refers to the time at which the second downstream sensor  152  detects the leading end of the sheet S. The first downstream sensor  151  outputs an analog signal as the first downstream edge signal Sd 1 , and the second downstream sensor  152  outputs an analog signal as the second downstream edge signal Sd 2 . In the present exemplary embodiment, the first time te 1  and the second time te 2  are each determined on the basis of a threshold that is the mean value of the high level and the low level. 
     At the first time te 1  and at the second time te 2 , the upstream edge signal Su maintains the low level. By the second time te 2 , the first roller  110  rotates at a speed corresponding to that of the sheet S, and the second roller  120 , which is rotated by the first roller  110 , rotates at a speed corresponding to that of the sheet S. 
     After the second time te 2 , at a third time te 3  at which the trailing end of the sheet S in the transport direction (hereinafter, simply referred to as “the trailing end”) reaches the detection position of the upstream sensor  150 , the upstream edge signal Su changes from the low level to the high level. In the present exemplary embodiment, for the above-described reason, the third time te 3  is determined by using a threshold that is the mean value of the low level and the high level. 
     At the third time te 3 , the sheet S is passing a position at which the first roller  110  and the third roller  140  are disposed opposite each other, whereby the first roller  110  and the second roller  120  continue rotating. At the third time te 3 , the first downstream edge signal Sd 1  and the second downstream edge signal Sd 2  each maintain the low level. 
     After the third time te 3 , when the trailing end of the transported sheet S has passed the position at which the sheet faces the first roller  110 , the first roller  110  does not receive a force from the sheet S and the second roller  120  does not receive a force from the first roller  110 . However, the first roller  110  does not immediately stop rotating, but gradually decelerates and then stops rotating. As a result, the intervals between the high level and the low level of the second A-phase signal Sa 2  (and the first A-phase signal Sa 1 ) gradually increase, and finally the level becomes constant (in this example, at the low level). 
     When the trailing end of the transported sheet S passes the detection position of the first downstream sensor  151 , the first downstream edge signal Sd 1  changes from the low level to the high level. When the trailing end of the transported sheet S passes the detection position of the second downstream sensor  152 , the second downstream edge signal Sd 2  changes from the low level to the high level. Thus, when one sheet S has passed through the length measuring device  100 , the signals (excluding the first temperature signal St 1  and the second temperature signal St 2 ) that are output from the length measuring device  100  return to the initial state, and stand by until transportation of the next sheet S starts. 
     The first time te 1 , at which the first downstream sensor  151  detects the leading end of the sheet S, is not necessarily the same as the timing at which the second A-phase signal Sa 2  rises (see  FIG. 6B ). In the following description, the period between the first time te 1  and the timing at which the second A-phase signal Sa 2  rises right after the first time te 1  will be referred to as a leading-end fractional pulse period T 1 , and one pulse period of the second A-phase signal Sa 2  that includes the leading-end fractional pulse period T 1  will be referred to as a leading-end one pulse period T 2 . 
     The third time te 3 , at which the upstream sensor  150  detects the trailing end of the sheet S, is not necessarily the same as the timing at which the second A-phase signal Sa 2  rises (see  FIG. 6C ). In the following description, the period between the third time te 3  and the timing at which the second A-phase signal Sa 2  has risen right before the third time te 3  will be referred to as a trailing-end fractional pulse period T 3 , and one pulse period of the second A-phase signal Sa 2  that includes the trailing-end fractional pulse period T 3  will be referred to as a trailing-end one pulse period T 4 . 
     In the following description, a period between the first time te 1  and the second time te 2  will be referred to as an inclination detection period T 5 . The inclination detection period T 5  is calculated with respect to the first time te 1 . Therefore, the inclination detection period T 5  may have a positive value (when the second time te 2  is after the first time te 1 ) and may have a negative value (when the second time te 2  is before the first time te 1 ). 
     Although not described above, every time the first encoder wheel  171  rotates once together with the first roller  110 , the first Z-phase signal Sz 1  changes between the low level and the high level. Every time the second encoder wheel  181  rotates once together with the second roller  120 , the second Z-phase signal Sz 2  changes between the low level and the high level. In this example, as is clear from  FIG. 2  and other figures, the diameter of the second roller  120  is smaller than that of the first roller  110 , so that one period of the second Z-phase signal Sz 2  is shorter than one period of the first Z-phase signal Sz 1 . 
       FIG. 7  is a flowchart illustrating a process performed by the processor  85 . 
     The processor  85  determines whether or not a calibration mode has been set through the UI  90  (step S 201 ). If the calibration mode has been set, the image forming apparatus according to the present exemplary embodiment transports the sheet S through the length measuring device  100 . An image need not be formed on the transported sheet S. 
     If the determination in step S 201  is “yes”, as the sheet S passes through the length measuring device  100 , the upstream edge signal Su, the first downstream edge signal Sd 1 , the second downstream edge signal Sd 2 , the first A-phase signal Sa 1 , the first Z-phase signal Sz 1 , the second A-phase signal Sa 2 , the second Z-phase signal Sz 2 , the first temperature signal St 1 , and the second temperature signal St 2 , which are illustrated in  FIG. 6A , are input to the processor  85  (step S 202 ). 
     The first-roller radius calculator  853  of the processor  85  calculates a first-roller radius data r 1 _new on the basis of these signals and various data read from the storage unit  854 . Then, the updating unit  856  stores the calculated first-roller radius data r 1 _new in the storage unit  854 , thereby updating the first-roller radius data r 1 _new (step S 203 ). The details of the first-roller radius data r 1 _new and step S 203  will be described below. 
     Next, the determination unit  855  of the processor  85  detects whether or not an irregularity in the diameter of the first roller  110  exists on the basis of the first-roller radius data r 1 _new, which has been calculated by the first-roller radius calculator  853  (step S 204 ). 
     If the determination in step S 204  is “no”, i.e., if an irregularity in the diameter is not detected, the processor  85  finishes the process in the calibration mode. 
     If the determination in step S 204  is “yes”, i.e., if an irregularity in the diameter is detected, the determination unit  855  outputs a control signal to the operation controller  84  to stop the operation of the image forming apparatus (step S 205 ), outputs a control signal to the UI  90  to cause the UI  90  to perform fault notification (step S 206 ), and subsequently finishes the process. 
     If the determination in step S 201  is “no”, the processor  85  determines whether or not a command for starting an image forming operation (job) has been received through the UI  90  or the like (step S 207 ). 
     If the determination in step S 207  is “yes”, as the sheet S passes through the length measuring device  100  during the image forming operation, the upstream edge signal Su, the first downstream edge signal Sd 1 , the second downstream edge signal Sd 2 , the first A-phase signal Sa 1 , the first Z-phase signal Sz 1 , the second A-phase signal Sa 2 , the second Z-phase signal Sz 2 , the first temperature signal St 1 , and the second temperature signal St 2 , which are illustrated in  FIG. 6A , are input to the processor  85  (step S 208 ). 
     The first-roller radius calculator  853  of the processor  85  calculates the first-roller radius data r 1 _new on the basis of these signals and various data read from the storage unit  854 . Then, the updating unit  856  stores the calculated first-roller radius data r 1 _new in the storage unit  854 , thereby updating the first-roller radius data r 1 _new (step S 209 ). 
     Next, the determination unit  855  of the processor  85  detects whether or not an irregularity in the diameter of the first roller  110  exists on the basis of the first-roller radius data r 1 _new, which has been calculated by the first-roller radius calculator  853  (step S 210 ). The operations performed in step S 209  and step S 210  are the same as those performed in step S 203  and step S 204 , respectively. 
     If the determination in step S 210  is “no”, i.e., if an irregularity in the diameter is not detected, the length calculator  851  of the processor  85  calculates the sheet length L, which is the length of the sheet S in the transport direction, on the basis of various signals input from the outside and various data read from the storage unit  854  (including the first-roller radius data r 1 _new, which has been updated in step S 209 ) (step S 211 ). 
     Then processor  85  determines whether or not the job has been finished (step S 212 ). If the determination in step S 212  is “no”, the process returns to step S 208 , and the sheet length L of the next sheet S is calculated. If the determination in step S 212  is “yes”, the process of the job is finished. If the determination in step S 207  is “no”, the process is finished without calculating the sheet length L. 
     If the determination in step S 210  is “yes”, i.e., if an irregularity in the diameter is detected, the determination unit  855  outputs a control signal to the operation controller  84  to stop the operation of the image forming apparatus (step S 205 ), outputs a control signal to the UI  90 , causes the UI  90  to perform fault notification (step S 206 ), and subsequently finishes the process. 
     Referring  FIGS. 4 ,  6 , and other figures, a process for calculating the sheet length L (step S 211 ), which is performed by the length calculator  851 , will be described in detail. In the present exemplary embodiment, when calculating the sheet length L by using the length measuring device  100 , correction is performed to reduce an error due to an irregularity in the diameter of the first roller  110 , an error due to an irregularity in the diameter of the second roller  120 , and an error due to displacement of the positions of the second A-phase slits  181   a , which are used for measuring the sheet length L. 
     As the sheet S passes through the length measuring device  100 , the upstream edge signal Su, the first downstream edge signal Sd 1 , the second downstream edge signal Sd 2 , the second A-phase signal Sa 2 , the second Z-phase signal Sz 2 , the first Z-phase signal Sz 1 , the first temperature signal St 1 , and the second temperature signal St 2 , which are illustrated in  FIG. 6A , are input to the length calculator  851 . 
     The length calculator  851  obtains the first time te 1  from the first downstream edge signal Sd 1 , the second time te 2  from the second downstream edge signal Sd 2 , and the third time te 3  from the upstream edge signal Su, respectively. 
     Next, the length calculator  851  calculates the inclination detection period T 5  on the basis of the first time te 1  and the second time te 2 ; calculates a first temperature Temp 1  on the basis of the first time te 1 , the third time te 3 , and the first temperature signal St 1 ; and calculates the second temperature Temp 2  on the basis of the first time te 1 , the third time te 3 , and the second temperature signal St 2 . The first temperature Temp 1  is the average of the first temperature signal St 1  during the period from the first time te 1  to the third time te 3 . The second temperature Temp 2  is the average of the second temperature signal St 2  during the period from the first time te 1  to the third time te 3 . 
     Next, the length calculator  851  counts a second-roller rotation count N of the second roller  120  on the basis of the first time te 1 , the third time te 3 , the second A-phase signal Sa 2 , and the second Z-phase signal Sz 2 . The second-roller rotation count N represents the rotation count of the second roller  120  during the period from the first time te 1  to the third time te 3 . In this example, the first rotation is defined as the 0-th rotation.  FIG. 6A  illustrates the 0-th rotation (represented by &lt;0&gt; in  FIG. 6A ) to the 3rd rotation (represented by &lt;3&gt; in  FIG. 6A ) (N=3). Hereinafter, a rotation of the second roller  120  during the period from the first time te 1  to the third time te 3  will be referred to as a “j-th rotation”. Therefore, j is in the range of 0≦j≦N (where j and N are integers). 
     The length calculator  851  counts an initial pulse count n 1  and a terminal pulse count n 2  on the basis of the first time te 1 , the third time te 3 , the second A-phase signal Sa 2 , and the second Z-phase signal Sz 2 . The initial pulse count n 1  is the number of pulses of the second A-phase signal Sa 2  that is counted during the 0-th rotation (j=0) of the second roller  120 . The initial pulse count n 1  is represented by an integer by omitting a fractional pulse right after the first time te 1 . The terminal pulse count n 2  is the number of pulses of the second A-phase signal Sa 2  that is counted during the final rotation (in this example, j=N=3) of the second roller  120 . The terminal pulse count n 2  is represented by an integer by omitting a fractional pulse right before the third time te 3 . 
     The length calculator  851  obtains the leading-end fractional pulse period T 1  and the leading-end one pulse period T 2  on the basis of the first time te 1  and the second A-phase signal Sa 2 , and obtains the trailing-end fractional pulse period T 3  and the trailing-end one pulse period T 4  on the basis of the third time te 3  and the second A-phase signal Sa 2 . 
     The length calculator  851  generates a second-roller rotation correction factor table R[j, i] for correcting an error due to an irregularity in the diameter of the first roller  110 , an error due to an irregularity in the diameter of the second roller  120 , and an error due to displacement of the positions of the second A-phase slits  181   a , which are used for measuring the length of the sheet S. The second-roller rotation correction factor table R[j, i] is made on the basis of the phase difference between the first roller  110  and the second roller  120  (see Δθ=x[j] in  FIG. 6A ) during the period from the first time te 1  to the third time te 3 . The process for generating the second-roller rotation correction factor table R[j, i] will be described below. 
     The length calculator  851  calculates the sheet length L by using various numerical values and various data obtained in the above-described process. The following equations are used to calculate the sheet length L.
 
 L=f 4( Lm,T 5)  (1)
 
 Lm=Lg+Lr   (2)
 
 Lg=Lg 0*α*Temp1  (3)
 
 Lr =( Y 1 +Y 2 +Y 3)*λ*β*Temp2  (4)
 
 Y 1 =f 1( N,n 1 ,n 2 ,x[ 0 ]˜x[N ])  (5)
 
 Y 2 =f 2( T 1 /T 2 ,n 1 ,x[ 0])  (6)
 
 y 3 =f 3( T 3 /T 4 ,n 2 ,x[N ])  (7)
 
     As shown in equation (1), the sheet length L is represented by a skew correction function f 4  having a corrected measured length Lm and the inclination detection period T 5  as variables. As shown in equation (2), the corrected measured length Lm is the sum of a corrected gap length Lg and a measured roller length Lr. 
     The corrected gap length Lg, which corresponds to the period during which the sheet S is detected by only one of the upstream sensor  150  and the first downstream sensor  151 , is obtained on the basis of the reference gap length Lg 0  (see  FIG. 2B ), which is the distance between the upstream sensor  150  and the first downstream sensor  151 . The measured roller length Lr, which corresponds to the period during which the sheet S is detected by the upstream sensor  150  and the first downstream sensor  151 , i.e., the period from the first time te 1  to the third time te 3 , is obtained on the basis of the rotation amount of the second roller  120  due to the rotation of the first roller  110 . 
     To be specific, as shown in equation (3), the corrected gap length Lg is the product of the reference gap length Lg 0 , the thermal expansion coefficient α of the attachment member  190 , and the first temperature Temp 1 . The reference gap length Lg 0  and the thermal expansion coefficient α are stored in the storage unit  854  beforehand. 
     As shown in equation (4), the measured roller length Lr is the product of the sum of a roller first pulse count Y 1 , a roller second pulse count Y 2 , and a roller third pulse count Y 3 ; the resolution λ (see  FIG. 6A ) of the second A-phase slits  181   a ; the thermal expansion coefficient β of the second roller  120 ; and the second temperature Temp 2 . The resolution λ and the thermal expansion coefficient β are stored in the storage unit  854  beforehand. 
     The roller first pulse count Y 1  corresponds to the pulse count of the second A-phase signal Sa 2  during the period from the end of the leading-end fractional pulse period T 1  to the start of the trailing-end fractional pulse period T 3 . The roller second pulse count Y 2  corresponds to the pulse count of the second A-phase signal Sa 2  during the leading-end fractional pulse period T 1 . The roller third pulse count Y 3  corresponds to the pulse count of the second A-phase signal Sa 2  during the trailing-end fractional pulse period T 3 . 
     As shown in equation (5), the roller first pulse count Y 1  is represented by a roller-encoder correction function f 1  having the second-roller rotation count N of the second roller  120 , the initial pulse count n 1 , the terminal pulse count n 2 , and the phase difference between rollers x[j] (0≦j&lt;N) as variables. 
     As shown in equation (6), the roller second pulse count Y 2  is represented by a leading-end pulse count function f 2  having the ratio between the leading-end fractional pulse period T 1  and the leading-end one pulse period T 2 , the initial pulse count n 1 , and the 0-th phase difference between rollers x[0] as variables. 
     As shown in equation (7), the roller third pulse count Y 3  is represented by a trailing end pulse count function f 3  having the ratio between the trailing-end fractional pulse period T 3  and the trailing-end one pulse period T 4 , the terminal pulse count n 2 , and the N-th phase difference between rollers x[N] as variables. 
     In the present exemplary embodiment, the pulse count of the second A-phase signal Sa 2 , which is used to calculate the measured roller length Lr when calculating the sheet length L, is corrected by using the second-roller rotation correction factor table R[j, i], which is obtained on the basis of the phase difference between rollers x[j], and thereby the roller first pulse count Y 1 , the roller second pulse count Y 2 , and the roller third pulse count Y 3  are obtained. 
       FIG. 8  is a flowchart illustrating a process for generating the second-roller rotation correction factor table R[j, i].  FIG. 9  illustrates an operation performed in step S 303  of  FIG. 8 .  FIGS. 10 ,  11 A, and  11 B illustrate an operation performed in step S 307  of  FIG. 8 . 
     First, the length calculator  851  calculates the second-roller rotation count N of the second roller  120  on the basis of the first time te 1 , the third time te 3 , the second A-phase signal Sa 2 , and the second Z-phase signal Sz 2  (step S 301 ). Next, the length calculator  851  sets j=0 (step S 302 ), and calculates a second-roller pulse interval p 2 [j, i] (0≦i≦n) of the second A-phase signal Sa 2  for the j-th rotation with respect to the j-th rise of the second Z-phase signal Sz 2  (step S 303 , see also  FIG. 9 ). Next, the length calculator  851  calculates the pulse count of the second A-phase signal Sa 2  during the period from the j-th rise of the second Z-phase signal Sz 2  to the j-th rise of the first Z-phase signal Sz 1  as Δθ=x[j] (step S 304 , see  FIG. 9 ). 
     Next, the length calculator  851  reads the first-roller radius data r 1 _new[i] (0≦i&lt;INT(n*r 1 /r 2 )) from the storage unit  854  (step S 305 , see  FIG. 11A ). The length calculator  851  reads a second-roller diameter/slit correction data r 2 [i] (0≦i≦n) from the storage unit  854  (step S 306 , see  FIG. 11B ). 
     Subsequently, the length calculator  851  generates the second-roller rotation correction factor table R[j, i]=r 2 [i]*r 1 _new[g]/r 1 _new[i] on the basis of two sets of data (r 1 _new[i] (x[j]≦i≦x[j]+n−1 (mod INT(n*r 1 /r 2 ))) and r 1 _new[g] (x[j]+θ10≦g≦x[j]+θ10+n−1 (mod INT(n*r 1 /r 2 ))) (see  FIG. 11A ), which have been obtained from the first-roller radius data r 1 _new[i] in step S 305 , and the second-roller diameter/slit correction data r 2 [i] (0≦i&lt;n), which has been read in step S 306  (see  FIG. 11B ) (step S 307 ). 
     Next, the length calculator  851  corrects the pulse intervals by using the second-roller pulse interval p 2 [j, i] obtained in step S 303  and the second-roller rotation correction factor table R[j, i] obtained in step S 307 , and thereby calculates a corrected second-roller pulse interval p 2 [j, i]″=p 2 [j, i]*R[j, i] (0≦i≦n) that corresponds to the j-th rotation (step S 308 ). The length calculator  851  updates j to j+1 (step S 309 ), and determines whether or not the updated value of j is equal to or smaller than the second-roller rotation count N (step S 310 ). If the determination in step S 310  is “yes”, the process returns to step S 303  and the process continues. 
     If the determination in step S 310  is “no”, the length calculator  851  calculates a rise timing t 2 [i]″ of the corrected second A-phase signal Sa 2  during the period from the first time te 1  to the third time te 3  on the basis of the corrected second-roller pulse interval p 2 [j, i]″ (0≦j≦N, 0≦i≦n) (step S 311 ), which has been obtained in step S 308  for each second-roller rotation count N, and finishes the process. 
       FIGS. 10 to 11B  will be described.  FIG. 10  illustrates why an error occurs when the length measuring device  100  performs measurement.  FIG. 11A  illustrates an example of the first-roller radius data r 1 _new[i], and  FIG. 11B  illustrates an example of the second-roller diameter/slit correction data r 2 [i].  FIG. 10  does not illustrate the first encoder wheel  171  and the second encoder wheel  181 , and schematically illustrates the first A-phase slits  171   a , the first Z-phase slit  171   z , the second A-phase slits  181   a , and the second Z-phase slit  181   z.    
     In the following description, the position at which the sheet S contacts the first roller  110  will be referred to as a sheet nip Ns, and the position at which the first roller  110  contacts the second roller  120  will be referred to as a roller nip Nr. A radius of the first roller  110  extending from the first rotation shaft  110   a  to the sheet nip Ns will be referred to as a first sheet nip radius R 11 , and the radius of the first roller  110  extending from the first rotation shaft  110   a  to the roller nip Nr will be referred to as a first-roller nip radius R 12 . A radius of the second roller  120  extending from the second rotation shaft  120   a  to the roller nip Nr will be referred to as a second-roller nip radius R 20 . 
     Regarding the first roller  110 , the angle between the position of the first Z-phase slit  171   z  and the detection position of the first optical detector  172  for detecting the first Z-phase slit  171   z  around the first rotation shaft  110   a  will be referred to as a first-roller rotation angle θ 1 . Regarding the first roller  110 , the angle between the detection position of the first optical detector  172  for detecting the first Z-phase slit  171   z  and the sheet nip Ns around the first rotation shaft  110   a  will be referred to as a first-roller first set angle θ 11 . Regarding the first roller  110 , the angle between the sheet nip Ns and the roller nip Nr around the first rotation shaft  110   a  will be referred to as a first-roller second set angle θ 12 . The sum of the first-roller first set angle θ 11  and the first-roller second set angle θ 12 , i.e., the angle between the detection position of the first optical detector  172  for detecting the first Z-phase slit  171   z  and the roller nip Nr around the first rotation shaft  110   a  will be referred to as a first-roller set angle θ 10 . The first-roller rotation angle θ 1 , the first-roller first set angle θ 11 , and the first-roller second set angle θ 12  are defined so that the positive directions thereof are clockwise in  FIG. 10 , which is opposite to the rotation direction of the first roller  110  (counterclockwise in  FIG. 10 ). The magnitude of the first-roller rotation angle θ 1  changes in accordance with the rotation of the first roller  110 . The magnitudes of the first-roller first set angle θ 11  and the first-roller second set angle θ 12  are fixed. 
     Regarding the second roller  120 , the angle between the position of the second Z-phase slit  181   z  and the detection position of the second optical detector  182  for detecting the second Z-phase slit  181   z  around the second rotation shaft  120   a  will be referred to as a second-roller rotation angle θ 2 . Regarding the second roller  120 , the angle between the detection position of the second optical detector  182  for detecting the second Z-phase slit  181   z  and the roller nip Nr around the second rotation shaft  120   a  will be referred to as a second-roller set angle θ 20 . The second-roller rotation angle θ 2  and the second-roller set angle θ 20  are defined so that the positive directions thereof are clockwise in  FIG. 10 , which is opposite to the rotation direction of the second roller  120  (counterclockwise in  FIG. 10 ). The magnitude of the second-roller rotation angle θ 2  changes in accordance with the rotation of the second roller  120 . The magnitude of the second-roller set angle θ 20  is fixed. 
     The first roller  110  and the second roller  120  used in the present exemplary embodiment are made beforehand with an accuracy within a predetermined tolerance. Therefore, the first sheet nip radius R 11  and the first-roller nip radius R 12  of the first roller  110  may differ from each other. Because the first roller  110  rotates when a measuring operation is performed, the relationship between the first sheet nip radius R 11  and the first-roller nip radius R 12  may change from moment to moment in accordance with the rotation of the first roller  110 . Because the second roller  120  rotates when a measuring operation is performed, the second-roller nip radius R 20  may change from moment to moment in accordance with the rotation of the second roller  120 . If the radii of the first roller  110  and the second roller  120  are designed to be different from each other (in this example, the radius of the first roller  110  is larger than that of the second roller  120 ), depending on the states (phases) of the first roller  110  and the second roller  120 , the relationship between the first-roller nip radius R 12  and the second-roller nip radius R 20  may change from moment to moment in accordance with the rotations of the first roller  110  and the second roller  120 . 
     The second encoder wheel  181  used in the present exemplary embodiment is also manufactured with an accuracy within a predetermined tolerance. Therefore, the intervals between the second A-phase slits  181   a , which are supposed to be formed at regular intervals in the circumferential direction of the second encoder wheel  181 , may be deviated from a design value. 
     If, for example, the first roller  110  has eccentricity, the surface velocity of the first peripheral surface portion  113  at the sheet nip Ns (referred to as a sheet nip velocity) may differ from the surface velocity of the first peripheral surface portion  113  at the roller nip Nr (referred to as a roller nip velocity). To be specific, the roller nip velocity is the product of the sheet nip velocity and (first-roller nip radius R 12 /first sheet nip radius R 11 ). 
     If the second roller  120  has eccentricity, the rotation amount of the second encoder wheel  181  at the sheet nip Ns may differ from the rotation amount of the second encoder wheel  181  at a position corresponding to the second optical detector  182 . Moreover, if the second A-phase slits  181   a  are not formed at regular intervals in the second encoder wheel  181 , a difference arising therefrom is superposed on the difference due to the eccentricity. 
     Therefore, in the present exemplary embodiment, before shipping the image forming apparatus, measurement for determining the correspondence between the phase (rotation angle) of the first roller  110  and the radius distribution of the first roller  110  with respect to the position of the first Z-phase slit  171   z  is performed by using the length measuring device  100 . The result of the measurement is stored in the storage unit  854  as initial first-roller radius data r 1 _init[i], which is an example of a reference radius distribution. The initial first-roller radius data r 1 _init[i], which is an example of a reference radius distribution, is used as the initial data for the first-roller radius data r 1 _new[i]. 
     Moreover, in the present exemplary embodiment, before shipping the image forming apparatus, measurement for determining the correspondence among the phase (rotation angle) of the second roller  120 , the radius distribution of the second roller  120 , and the distribution of intervals between adjacent slits of the second A-phase slits  181   a  of the second encoder wheel  181  with respect to the position of the second Z-phase slit  181   z  is performed by using the length measuring device  100 . The second-roller diameter/slit correction data r 2 [i], which is obtained by reversing the sign of the result of the measurement and then normalizing the result, is stored in the storage unit  854 . 
       FIG. 11A  illustrates an example of the first-roller radius data r 1 _new[i], and  FIG. 11B  illustrates an example of the second-roller diameter/slit correction data r 2 [i]. The first-roller radius data r 1 _new[i] and the second-roller correction data r 2 [i] are each stored in the storage unit  854  as numerical data representing the correspondence. For ease of understanding,  FIGS. 11A and 11B  illustrate the graphs of the data. 
     In  FIG. 11A , the horizontal axis represents the first-roller rotation angle θ 1  (rad), and the vertical axis represents the radius of the first roller  110  (mm). Referring to  FIGS. 10 and 11A , when the first-roller rotation angle θ 1  is, for example, π/2 (rad), the sheet nip Ns of the first roller  110  is at a position that is retarded from the first-roller rotation angle θ 1  by the first-roller first set angle θ 11  (π (rad) in the example of  FIG. 11A ), so that the first sheet nip radius R 11  at this time has a value corresponding to θ 1 =3π/2 (rad). The roller nip Nr of the first roller  110  is at a position that is retarded from the first-roller rotation angle θ 1  by the sum of the first-roller first set angle θ 11  (π (rad) in the example of  FIG. 11A ) and the first-roller second set angle θ 12  (3π/4 (rad) in the example of  FIG. 11A ), so that the first-roller nip radius R 12  at this time has a value corresponding to θ 1 =9π/4 (rad), i.e., θ 1 =π/4 (rad). The first-roller rotation angle θ 1  changes in accordance with the rotation of the first roller  110 , while the first-roller first set angle θ 11  and the first-roller second set angle θ 12  (and the first-roller set angle θ 10 ) do not change. Therefore, by obtaining the first-roller rotation angle θ 1  of the first roller  110  by using the first Z-phase slit  171   z , the first sheet nip radius R 11  and the first-roller nip radius R 12  at this time are obtained. 
     In  FIG. 11B , the horizontal axis represents the second-roller rotation angle θ 2  (rad), and the vertical axis represents the correction factor. Referring to  FIGS. 10 and 11B , when the second-roller rotation angle θ 2  is, for example, π/2 (rad), the roller nip Nr of the second roller  120  is at a position that is retarded from the second-roller rotation angle θ 2  by the second-roller set angle θ 20  (5π/4 (rad) in the example of  FIG. 11B ), so that the correction factor at this time has a value corresponding to θ 2 =7π/4 (rad). 
     In the present exemplary embodiment, when the length calculator  851  calculates the sheet length L, the second-roller rotation correction factor table R[j, i], which is generated on the basis of the phase difference between rollers x[j] by determining the correspondence between the first-roller radius data r 1 _new[i] and the second-roller diameter/slit correction data r 2 [i] read from the storage unit  854 , is used to calculate the roller first pulse count Y 1 , the roller second pulse count Y 2 , and the roller third pulse count Y 3 . Thus, occurrence of error in the measured roller length Lr due to insufficient accuracy of the first roller  110 , the second roller  120 , or the second encoder wheel  181  is reduced, so that an error included in the sheet length L calculated by using the measured roller length Lr is reduced. 
     In the present exemplary embodiment, when the velocity calculator  852  calculates the sheet velocity V, the second-roller rotation correction factor table R[j, i], which is generated on the basis of the phase difference between rollers x[j] by determining the correspondence between the first-roller radius data r 1 _new[i] and the second-roller diameter/slit correction data r 2 [i] read from the storage unit  854 , is used. Therefore, occurrence of an error in the sheet velocity V is reduced. 
     In the present exemplary embodiment, as described above, the surface layer  112  of the first roller  110  is made of an elastic material such as rubber, so that the first roller  110  may easily follow the transported sheet S. On the other hand, if the surface layer  112  of the first roller  110  is made of an elastic material, the surface layer  112  easily wears as compared with the case where the surface layer  112  is made of a metal or the like. In this case, wear that occurs on the surface layer  112  of the first roller  110  may be overall wear in which the entire periphery of the surface layer  112  is worn or may be local wear in which a part of the periphery of the surface layer  112  is worn. When the surface layer  112  of the first roller  110  is worn and the radius distribution of the first roller  110  changes, deviation of the actual radius distribution of the first roller  110  from the first-roller radius data r 1 _new[i] stored in the storage unit  854  increases, and thereby errors in the above-described calculations of the sheet length L and the sheet velocity V may increase. When local wear occurs on the first roller  110 , the first roller  110  and the second roller  120  vibrate as the first roller  110  rotates, and thereby an error in the above-described calculations of the sheet length L and the sheet velocity V may increase. 
     Therefore, in the present exemplary embodiment, as described above with reference to  FIG. 7 , the first-roller radius data r 1 _new[i] is updated, and detection of an irregularity in the diameter of the first roller  110  is performed on the basis of the updated first-roller radius data r 1 _new[i]. 
       FIG. 12  is a flowchart illustrating a process for updating the first-roller radius data r 1 _new[i] for the first roller  110 , which is performed in steps S 203  and S 209  illustrated in  FIG. 7 .  FIG. 13  illustrates an operation performed in steps S 401  to S 409  of  FIG. 12 .  FIG. 14  illustrates an operation performed in step S 418  of  FIG. 12 . 
       FIG. 15  illustrates an operation performed in steps S 419  to S 423  of  FIG. 12 . 
     First, the first-roller radius calculator  853  calculates the second-roller rotation count N of the second roller  120  on the basis of the first time te 1 , the third time tea, the second A-phase signal Sa 2 , and the second Z-phase signal Sz 2  (step S 401 ). Next, the first-roller radius calculator  853  calculates the second-roller pulse interval p 2 [j, i] (0≦j≦N, 0≦i≦n) of the second A-phase signal Sa 2  with respect to the rise of the second Z-phase signal Sz 2  (step S 402 ). 
     Next, the first-roller radius calculator  853  reads the second-roller diameter/slit correction data r 2 [i] (0≦i≦n) from the storage unit  854  (step S 403 , see  FIG. 11B ). 
     The first-roller radius calculator  853  corrects the pulse intervals by using the second-roller pulse interval p 2 [j, i] obtained in step S 402  and the second-roller diameter/slit correction data r 2 [i] read in step S 403 , and thereby calculates the corrected second-roller pulse interval p 2 [j, i]′=p 2 [j, i]*r 2 [i] (0≦j≦N, 0≦i≦n) (step S 404 , see  FIG. 13 ). For this correction, the first-roller radius data r 1 _new[i] is not taken into account because the second-roller rotation correction factor table R[j, i] is not used. 
     Next, the first-roller radius calculator  853  calculates the rise timing t 2 [i]′ of the corrected second A-phase signal Sa 2  during the period from the first time te 1  to the third time te 3  on the basis of the corrected second-roller pulse interval p 2 [j, i]′ (0≦j≦N, 0≦j≦n) obtained in step S 404  (step S 405 , see  FIG. 13 ). 
     The first-roller radius calculator  853  calculates a first-roller rotation count M of the first roller  110  on the basis of the first time te 1 , the third time te 3 , and the first Z-phase signal Sz 1  (step S 406 ). Next, the first-roller radius calculator  853  calculates the first-roller pulse interval p 1 [j, i] (0≦j≦M, 0≦i≦m) of the first A-phase signal Sa 1  with respect to the rise of the first Z-phase signal Sz 1  (step S 407 , see  FIG. 13 ). 
     Next, the first-roller radius calculator  853  reads the second-roller radius data from the storage unit  854  (step S 408 ). The second-roller radius data represents the correspondence between the second-roller rotation angle θ 2  of the second roller  120  and the radius of the second roller  120 . 
     Then, the first-roller radius calculator  853  calculates the first-roller pulse interval d 1 [j, i] (1≦j≦M, 0≦i≦m) by using the first-roller pulse interval p 1 [j, i] obtained in step S 407 , the rise timing t 2 [i]′ of the corrected second A-phase signal Sa 2  obtained in step S 405 , and the average of the second-roller radius data read in step S 408  (step S 409 ). Subsequently, the first-roller radius calculator  853  reads the number k of stored update data items from the storage unit  854  (step S 410 ). 
     Next, the first-roller radius calculator  853  substitutes the first-roller pulse interval d 1 [j, i] (1≦j≦M, 0≦i&lt;m) obtained in step S 409  into the update data e[j, i] (k≦j&lt;k+M, 0≦i&lt;m), and stores the result in the storage unit  854  (step S 411 ). Then, the first-roller radius calculator  853  updates the number k of stored update data items to k+M (step S 412 ), and reads the number K of update data items from the storage unit  854  (step S 413 ). The first-roller radius calculator  853  determines whether or not the number k of stored update data items updated in step S 412  is equal to or larger than the number K of update data items read in step S 413  (step S 414 ). 
     If the determination in step S 414  is “yes”, the first-roller radius calculator  853  sets the number k of stored update data items at 0 (step S 415 ), and reads update data e[j, i] (0≦j&lt;K, 0≦i&lt;m) stored in the storage unit  854  in step S 411  (step S 416 ). Then, the first-roller radius calculator  853  performs averaging of K update data items e[j, i] (0≦j&lt;K, 0≦i&lt;m) read in step S 416 , and calculates the average d_avg[i] (0≦i&lt;m) of the first-roller pulse interval (step S 417 , see the upper part of  FIG. 14 ). Next, the first-roller radius calculator  853  changes the array number of the average value d_avg[i] (0≦i&lt;m) of the first-roller pulse interval calculated in step S 417  from m to INT(n*r 1 /r 2 ), and calculates the average d_avg[i]′ (0≦i&lt;INT(n*r 1 /r 2 )) of the changed first-roller pulse interval (step S 418 , see the lower part of  FIG. 14 ). Next, the first-roller radius calculator  853  reads the initial first-roller pulse interval d_init[i] (0≦i&lt;INT(n*r 1 /r 2 )) from the storage unit  854  (step S 419 , see the upper part of  FIG. 15 ). 
     The first-roller radius calculator  853  calculates first-roller wear amount data Δr 1 [i] (0≦i&lt;INT(n*r 1 /r 2 )) by calculating the difference between the average d_avg[i]′ (0≦i&lt;INT(n*r 1 /r 2 )) of the changed first-roller pulse interval, which has been obtained in step S 418 , and the initial first-roller pulse interval d_init[i] (0≦i&lt;INT(n*r 1 /r 2 )) read in step S 419  (step S 420 , see the left middle part of  FIG. 15 ). 
     Next, the first-roller radius calculator  853  reads the initial first-roller radius data r 1 _init[i] (0≦i&lt;INT(n*r 1 /r 2 )) from the storage unit  854  (step S 421 , see the right middle part of  FIG. 15 ). The first-roller radius calculator  853  calculates new first-roller radius data r 1 _new[i] (0≦i&lt;INT(n*r 1 /r 2 )) by calculating the difference between the initial first-roller radius data r 1 _init[i] (0≦i&lt;INT(n*r 1 /r 2 )) read in step S 421  and the first-roller wear amount data Δr 1 _init[i] (0≦i&lt;INT(n*r 1 /r 2 )) obtained in step S 420  (step S 422 , see the lower part of  FIG. 15 ). Then, the first-roller radius calculator  853  stores the new first-roller radius data r 1 _new[i] (0≦i&lt;INT(n*r 1 /r 2 )) in the storage unit  854  (step S 423 ), and finishes the process. If the determination in step S 414  is “no”, the process is finished without performing the above-described operations. 
     Detection of an irregularity in the diameter of the first roller  110  in step S 204  and step S 210  of  FIG. 7  is performed as follows. 
     First, the determination unit  855  obtains the updated first-roller radius data r 1 _new[i] (see the lower part of  FIG. 15 ) from the first-roller radius calculator  853 . Next, the determination unit  855  determines whether or not the updated first-roller radius data r 1 _new[i] is deviated from a design value (for example, 15.0 mm) of the radius of the first roller  110  beyond a predetermined range (for example, 15.0±0.3 mm). If at least a part of the updated first-roller radius data r 1 _new[i] is deviated from the design value of the radius of the first roller  110  beyond the predetermined range, the determination unit  855  determines that an irregularity in the diameter has occurred in the first roller  110 , and causes the UI  90  to perform fault notification. The determination unit  855  calculates the perimeter of the first peripheral surface portion  113  of the first roller  110  by using the updated first-roller radius data r 1 _new[i] (see the lower part of  FIG. 15 ), determines whether or not the calculated perimeter is smaller than a predetermined lower limit (for example 91.0 mm) of the design value of the perimeter of the first roller  110  (about 92.25 mm if the design value of the radius of the first roller  110  is 15.0 mm). If the calculated perimeter of the first roller  110  is smaller than the lower limit, the determination unit  855  determines that an irregularity in the diameter has occurred in the first roller  110 , and causes the UI  90  to perform fault notification. In this example, a first rotating member correction value is obtained on the basis of the first-roller radius data r 1 _new[i], and a second rotating member correction value is obtained on the basis of the second-roller diameter/slit correction data r 2 [i]. 
     The foregoing description of the exemplary embodiments of the present invention has been provided for the purposes of illustration and description. It is not intended to be exhaustive or to limit the invention to the precise forms disclosed. Obviously, many modifications and variations will be apparent to practitioners skilled in the art. The embodiments were chosen and described in order to best explain the principles of the invention and its practical applications, thereby enabling others skilled in the art to understand the invention for various embodiments and with the various modifications as are suited to the particular use contemplated. It is intended that the scope of the invention be defined by the following claims and their equivalents.