Patent Publication Number: US-11656074-B2

Title: Calibration method

Description:
CROSS-REFERENCE TO RELATED APPLICATIONS 
     The present application claims priority to Japanese Patent Applications number 2021-065439, filed on Apr. 7, 2021. The contents of this application are incorporated herein by reference in their entirety. 
     BACKGROUND OF THE INVENTION 
     The present disclosure relates to a calibration method for an inspection gauge for CMM. 
     For an inspection of measurement accuracy of a coordinate measuring machine (CMM), an inspection gauge in which a ball is placed at each vertex of a polyhedron or the like is used (for example, see U.S. Pat. No. 6,023,850). A true value of each ball-to-ball distance in the inspection gauge is measured in advance with a reference instrument such as an optical interferometer. As a method of calibrating gauges that have balls as measurement elements such as regular tetrahedron gauges and cubic gauges, a method of calibrating by comparing each ball-to-ball distance of a gauge with a ball bar having a calibration value is known. 
     In a method described in U.S. Pat. No. 6,023,850, it is necessary to measure true values of all of the ball-to-ball distances of an inspection gauge using a reference instrument when calibrating the inspection gauge. Therefore, there is a problem that calibration work of the reference instrument is complicated. 
     SUMMARY 
     The present disclosure focuses on this point and its object is to provide a calibration method capable of reducing the time required for calibration work of an inspection gauge. 
     A calibration method according to the first aspect of the present disclosure is a calibration method for calibrating an inspection gauge in which a ball is arranged at each vertex of a regular tetrahedron, the method includes the steps of: installing the inspection gauge in a first posture on a measurement table of a CMM; measuring a ball-to-ball distance of a plurality of edges of the inspection gauge installed in the first posture using the CMM in which a measurement error at a measurement position, where a first edge among the plurality of edges of the inspection gauge installed in the first posture is located, is identified; installing the inspection gauge on the measurement table in a second posture in which a second edge different from the first edge among the plurality of edges matches the measurement position where the first edge was in the first posture by rotating the inspection gauge about a first rotational symmetry axis after measuring the ball-to-ball distance of the plurality of edges; measuring a ball-to-ball distance of a plurality of edges of the inspection gauge installed in the second posture using the CMM; installing the inspection gauge on the measurement table in a third posture in which a third edge different from the first edge and the second edge among the plurality of edges matches the measurement positon by rotating the inspection gauge about a second rotational symmetry axis different from the first rotational symmetry axis after measuring the ball-to-ball distance of the plurality of edges of the inspection gauge installed in the second posture; measuring a ball-to-ball distance of a plurality of edges of the inspection gauge installed in the third posture using the CMM; and calculating a calibration value calibrating a ball-to-ball distance between two balls at respective ends of a plurality of edges of the inspection gauge by solving simultaneous equations including the ball-to-ball distance of the plurality of edges of the inspection gauge installed in the first posture, the ball-to-ball distance of the plurality of edges of the inspection gauge installed in the second posture, the ball-to-ball distance of the plurality of edges of the inspection gauge installed in the third posture, and the measurement error of the CMM at the measurement position. 
     A calibration method according to the second aspect of the present disclosure is a calibration method for calibrating an inspection gauge in which a ball is arranged at each vertex of a regular tetrahedron, the method comprising the steps of: installing the inspection gauge in a first posture on a measurement table of a CMM; measuring a ball-to-ball distance of a plurality of edges of the inspection gauge installed in the first posture by the CMM in which a measurement error at a first measurement position, where a first edge included in a base of the regular tetrahedron among the plurality of edges of the inspection gauge installed in the first posture is located, and the measurement error at a second measurement position, where a second edge not included in the base is located, are identified; installing, on the measurement table, the inspection gauge in a second posture in which a third edge different from the first edge and the second edge among the plurality of edges matches the first measurement position where the first edge was in the first posture and a fourth edge different from the first edge, the second edge, and the third edge matches the second measurement position where the second edge was in the second posture by rotating the inspection gauge about a rotational symmetry axis orthogonal to the base after measuring the ball-to-ball distance of the plurality of edges of the inspection gauge installed in the first posture; measuring a ball-to-ball distance of a plurality of edges of the inspection gauge installed in the second posture by the CMM; and calculating a calibration value calibrating a ball-to-ball distance between two balls at respective ends of a plurality of edges of the inspection gauge by solving simultaneous equations including the ball-to-ball distance of the plurality of edges of the inspection gauge installed in the first posture, the ball-to-ball distance of the plurality of edges of the inspection gauge installed in the second posture, and the measurement error of the CMM at the first measurement position and the second measurement position. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIGS.  1 A to  1 C  each show an outline of a calibration method according to a first embodiment. 
         FIG.  2    shows a configuration of a CMM. 
         FIG.  3    shows how a measurement error is identified by an identification part. 
         FIG.  4    shows an example of the j-th position of the inspection gauge. 
         FIGS.  5 A to  5 C  each show an example of a position of an edge of the inspection gauge corresponding to a subscript. 
         FIG.  6    is a flowchart showing a procedure for calibrating a ball-to-ball distance of the inspection gauge in the first embodiment. 
         FIGS.  7 A and  7 B  each show an outline of a calibration method according to a second embodiment. 
         FIG.  8    is a flowchart showing a procedure for calibrating the ball-to-ball distance of the inspection gauge in the second embodiment. 
     
    
    
     DETAILED DESCRIPTION OF THE DISCLOSURE 
     Hereinafter, the present disclosure will be described through exemplary embodiments, but the following exemplary embodiments do not limit the invention according to the claims, and not all of the combinations of features described in the exemplary embodiments are necessarily essential to the solution means of the invention. 
     First Embodiment 
     [Outline of a Calibration Method] 
     A calibration method according to a first embodiment is a method for calibrating an inspection gauge used for an inspection of measurement accuracy of a coordinate measuring machine (CMM).  FIGS.  1 A to  1 C  each show an outline of the calibration method of the first embodiment. In this calibration method, the inspection gauge is mounted on the CMM such that some edges of the inspection gauge match positions on the CMM identified by a reference instrument such as a laser interferometer. Then, using the CMM, a distance between balls at the ends of each edge of the inspection gauge is measured in a plurality of postures obtained by rotating the inspection gauge about a predetermined rotational symmetry axis for a predetermined number of times. In a case where the CMM is used, since the distance can be automatically measured just by setting a position to be measured in advance, the time required for calibration can be reduced as compared with a case where reference instruments are to be set up according to the number of edges to be calibrated by such a calibration method. 
     In the calibration method according to the first embodiment, for each of three postures, the length of one edge (a distance between balls at respective ends of the edge) is measured at the position of the CMM calibrated by the reference instrument and a distance between balls at respective ends of at least some other edges is measured at an uncalibrated position in the CMM, thereby calibrating the inspection gauge. In this calibration method, the ball-to-ball distance of each edge is measured using the CMM for each of (i) a first posture, (ii) a second posture after rotating the inspection gauge only once about a first rotational symmetry axis from the first posture, and (iii) a third posture after rotating the inspection gauge only once about a second rotational symmetry axis from the second posture.  FIG.  1 A  shows how an inspection gauge  100  is installed in the first posture.  FIG.  1 B  shows how the inspection gauge  100  is installed in the second posture.  FIG.  1 C  shows how the inspection gauge  100  is installed in the third posture. 
     As shown in  FIG.  1 A , the inspection gauge  100  has a regular tetrahedron shape, and measurement elements, balls A to D, are arranged at each vertex of the regular tetrahedron. In an example of  FIG.  1 A , it is assumed that the inspection gauge  100  is installed in the first posture on a measurement table of the CMM. At each vertex of the inspection gauge  100 , a measurement element having a shape other than a sphere may be arranged. The measurement elements are assumed to be capable of measuring coordinates of a feature point such as a center, an edge, or the like of the ball using the CMM. 
     The CMM measures the ball-to-ball distance between two balls. The CMM measures, among the plurality of edges of the regular tetrahedron of the inspection gauge  100 , a first ball-to-ball distance between the ball B and the ball C at respective ends of an edge (hereinafter, also referred to as a first edge) corresponding to a measurement position (a position A indicated by a broken line arrow) indicated by a thick line in  FIG.  1 A . It is assumed that a measurement error when the CMM measures the ball-to-ball distance at this measurement position is identified in advance by a method which will be described later. The CMM also measures ball-to-ball distances corresponding to five edges other than the first edge of the inspection gauge  100  installed in the first posture. 
     After the CMM has measured the plurality of ball-to-ball distances of the inspection gauge  100  installed in the first posture, the inspection gauge  100  is installed on the measurement table in the second posture, which is obtained by rotating about the first rotational symmetry axis, as shown by arrows in  FIG.  1 A . The inspection gauge  100  is installed at the same position as in the first posture in a coordinate system of the CMM. The first rotational symmetry axis is an axis passing through the ball A and orthogonal to a plane including the ball B, the ball C, and the ball D. 
       FIG.  1 B  shows how the inspection gauge  100  is installed in the second posture. In the second posture, an edge (edge sandwiched by the balls B and D, hereinafter also referred to as a second edge) different from the first edge among the plurality of edges of the regular tetrahedron of the inspection gauge  100  moves so as to match with the measurement position where the first edge was in the first posture. The CMM measures a second ball-to-ball distance between the balls B and D at respective ends of the second edge among the plurality of edges of the inspection gauge  100  installed in the second posture. The CMM measures ball-to-ball distances corresponding to five edges other than the second edge of the inspection gauge  100  installed in the second posture. 
     After the CMM has measured the ball-to-ball distances of the plurality of edges of the inspection gauge  100  installed in the second posture, the inspection gauge  100  is installed on the measurement table in the third posture that is obtained by rotating about the second rotational symmetry axis. The inspection gauge  100  is installed at the same position as in the second posture in the coordinate system of the CMM. The second rotational symmetry axis is different from the first rotational symmetry axis. The second rotational symmetry axis is indicated by a broken line in  FIG.  1 B . In an example of  FIG.  1 B , the second rotational symmetry axis passes through the ball D and is orthogonal to a plane including the ball A, the ball B, and the ball C. 
       FIG.  1 C  shows how the inspection gauge  100  is installed in the third posture. In the third posture, among the plurality of edges of the regular tetrahedron of the inspection gauge  100 , an edge (edge sandwiched by the balls C and D, hereinafter also referred to as a third edge) different from the first edge and the second edge moves to the position that matches the measurement position. The CMM measures a third ball-to-ball distance between the balls C and D at respective ends of the third edge among the plurality of edges of the inspection gauge  100  installed in the third posture. The CMM also measures ball-to-ball distances corresponding to five edges other than the third edge among the plurality of edges of the inspection gauge  100  installed in the third posture. 
     The CMM  200  calculates a calibration value obtained by calibrating a ball-to-ball distance between two balls at respective ends of a plurality of edges of the inspection gauge  100 . The CMM  200  calculates a calibration value by calibrating the ball-to-ball distance between two balls at respective ends of the plurality of edges by solving simultaneous equations including (i) ball-to-ball distances of the plurality of edges of the inspection gauge  100  when installed in the first posture, (ii) ball-to-ball distances of the plurality of edges of the inspection gauge  100  when installed in the second posture, (iii) ball-to-ball distances of the plurality of edges of the inspection gauge  100  when installed in the third posture, (iv) a calibration value (unknown number) obtained by calibrating the ball-to-ball distances between two balls at respective ends of the plurality of edges, and (v) a measurement error of the CMM at the measurement position identified in advance. Details will be described later. By adopting such a calibration method, a user does not need to measure all of the ball-to-ball distances of the inspection gauge  100  using the laser interferometer or the like, so that the time required for calibration work for calibrating the inspection gauge  100  can be reduced. 
     [Configuration of a CMM  200 ] 
       FIG.  2    shows a configuration of a CMM  200 . The CMM  200  includes a communication part  21 , a probe unit  22 , a display part  23 , a storage part  24 , and a control part  25 . The control part  25  includes a measurement part  251 , an identification part  252 , a calculation part  253 , and a display control part  254 . 
     The communication part  21  is a communication interface for communicating with other devices such as the laser interferometer. For example, the communication part  21  communicates with other devices by wireless communication such as a Wi-Fi (registered trademark). The communication part  21  may communicate with other devices via a cable such as a local area network (LAN). 
     The probe unit  22  includes a probe for contacting an object to be measured, a sensor for detecting that the probe has contacted the object to be measured, and a motor for moving the probe or changing an orientation of the probe. The probe unit  22  measures three-dimensional coordinates of a tip of the probe at the time of the probe contacting the object to be measured. The probe unit  22  inputs the measured three-dimensional coordinates to the measurement part  251 . The display part  23  displays characters, an image, or the like. 
     The storage part  24  includes a storage medium such as a read only memory (ROM), a random access memory (RAM), a hard disk, or the like. The storage part  24  stores a program executed by the control part  25 . The control part  25  functions as the measurement part  251 , the identification part  252 , the calculation part  253 , and the display control part  254  by executing the program stored in the storage part  24 . 
     The measurement part  251  measures the ball-to-ball distances of the inspection gauge  100  with the probe unit  22 . More specifically, the measurement part  251  measures coordinates of the center of a ball by measuring three-dimensional coordinates of a plurality of contact positions where the probe contacts the ball of the inspection gauge  100 . 
     The measurement part  251  measures coordinates of the center of each of the four balls of the inspection gauge  100  installed in the first posture. The measurement part  251  calculates distances between the centers of the four balls as the ball-to-ball distances. Among these distances, the measurement part  251  calculates, as the first ball-to-ball distance, a ball-to-ball distance between two balls at respective ends of the first edge corresponding to the measurement position. The measurement part  251  also calculates ball-to-ball distances corresponding to five edges other than the first edge of the inspection gauge  100  installed in the first posture. 
     Similarly, the measurement part  251  respectively calculates ball-to-ball distances between the centers of the four balls arranged at respective vertices of the inspection gauge  100  installed in the second posture. Among these distances, the measurement part  251  calculates, as the second ball-to-ball distance, a ball-to-ball distance between two balls at respective ends of the second edge corresponding to the measurement position. The measurement part  251  also calculates ball-to-ball distances corresponding to five edges other than the second edge of the inspection gauge  100  installed in the second posture. 
     Similarly, the measurement part  251  respectively calculates ball-to-ball distances between the centers of the four balls arranged at respective vertices of the inspection gauge  100  installed in the third posture. Among these distances, the measurement part  251  calculates, as the third ball-to-ball distance, a ball-to-ball distance between two balls at respective ends of the third edge corresponding to the measurement position. The measurement part  251  also calculates ball-to-ball distances corresponding to five edges other than the third edge of the inspection gauge  100  installed in the third posture. The measurement part  251  outputs information indicating the measured ball-to-ball distances to the calculation part  253 . 
     [Identification of the Measurement Error] 
     The identification part  252  communicates with the laser interferometer via the communication part  21 . The identification part  252  identifies in advance a measurement error of the CMM  200  at the measurement position.  FIG.  3    shows how the measurement error is identified by the identification part  252 . The identification part  252  measures the same distance at the measurement position using two devices, i.e., the probe unit  22  and the laser interferometer  35 . By comparing measurement results measured by the two devices, the identification part  252  identifies a measurement error when the probe unit  22  performs the measurement. 
     In an example of  FIG.  3   , a rail  31  is installed at the same position as the measurement position ( FIGS.  1 A to  1 C ) of the measurement table of the CMM  200 . It is supposed that a slider  32  that is movable on the rail  31  moves along the measurement position. An object to be measured  33  and a reflector  34  are fixed to the slider  32 . The laser interferometer  35  is arranged such that its optical axis matches a direction in which the rail  31  extends. The optical axis of the laser interferometer  35  is defined by the center of a spherical portion of a tip of the object to be measured  33  and the center of a reflector plate of the reflector  34 . 
     First, it is assumed that the slider  32  is at a first reference position on the rail  31 . In a state in which the slider  32  is at the first reference position, a position of the spherical portion of the object to be measured  33  is approximately coincident with a position of the ball B (see  FIG.  1 A ) arranged at one end of the first edge when the inspection gauge  100  is arranged in the first posture. The identification part  252  measures the coordinates of the spherical portion of the object to be measured  33  with the probe unit  22  in the state where the slider  32  is at the first reference position. As indicated by a broken line arrow in  FIG.  3   , a distance from the laser interferometer  35  to the reflector  34  is measured by the laser interferometer  35  in a state where the slider  32  is at the first reference position. The identification part  252  obtains, from the laser interferometer  35 , a measurement result of the distance from the laser interferometer  35  to the reflector  34  when the slider  32  is at the first reference position. 
     Next, the slider  32  is moved to a second reference position on the rail  31 . In a state in which the slider  32  is at the second reference position, the position of the spherical portion of the object to be measured  33  is approximately coincident with a position of the ball C (see  FIG.  1 A ) arranged at one end of the first edge when the inspection gauge  100  is arranged in the first posture. The identification part  252  measures coordinates of the spherical portion of the object to be measured  33  with the probe unit  22  in the state where the slider  32  is at the second reference position. The distance from the laser interferometer  35  to the reflector  34  is measured by the laser interferometer  35  in the state where the slider  32  is at the second reference position. The identification part  252  obtains, from the laser interferometer  35 , a measurement result of the distance from the laser interferometer  35  to the reflector  34  when the slider  32  is at the second reference position. 
     The identification part  252  calculates, as the calibration value, a reference distance from the first reference position to the second reference position on the basis of a difference between (i) the distance (broken line arrow in  FIG.  3   ) from the laser interferometer  35  to the reflector  34  when the slider  32  is at the first reference position and (ii) the distance from the laser interferometer  35  to the reflector  34  when the slider  32  is at the second reference position. 
     The identification part  252  is not limited to an example in which the calibration value of the reference distance is measured by the laser interferometer  35 . For example, the identification part  252  may measure the calibration value of the reference distance using a ball bar (not shown). 
     The identification part  252  calculates a measurement value of the reference distance from the first reference position to the second reference position on the basis of a difference between (i) coordinates of the object to be measured  33  measured by the probe unit  22  when the slider  32  is at the first reference position and (ii) coordinates of the object to be measured  33  measured by the probe unit  22  when the slider  32  is at the second reference position. 
     In the example of this disclosure, measurement accuracy of the laser interferometer  35  is higher than measurement accuracy of the probe unit  22 . Therefore, the identification part  252  assumes that a difference between the calibration value measured by the laser interferometer  35  and the measurement value measured by the probe unit  22  is caused by the measurement error of the probe unit  22 . The identification part  252  identifies a measurement error of the measurement value measured by the probe unit  22  at the measurement position on the basis of the difference between the calibration value of the reference distance measured by the laser interferometer  35  and the measurement value of the reference distance measured by the probe unit  22 . The identification part  252  outputs the identified measurement error to the calculation part  253 . 
     [Calculation of a Calibration Value of the Ball-to-Ball Distance] 
     The calculation part  253  calculates a calibration value by calibrating the ball-to-ball distance between two balls at respective ends of each edge of the inspection gauge  100 . The calculation part  253  calculates calibration values g 1  to g 6  of the ball-to-ball distances of each edge of the regular tetrahedron of the inspection gauge  100  and measurement errors m 1  to m 6  that occur when the measurement part  251  measures the ball-to-ball distances of each edge of the regular tetrahedron of the inspection gauge  100  by solving the following Equation 1. 
     
       
         
           
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                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
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                             0 
                           
                           
                             0 
                           
                           
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                             0 
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                           
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                             0 
                           
                           
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                             0 
                           
                           
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                             1 
                           
                           
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                             1 
                           
                           
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                             1 
                           
                           
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                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             1 
                           
                           
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                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                         
                       
                       ] 
                     
                     ⨯ 
                     
                       [ 
                       
                         
                           
                             
                               m 
                               1 
                             
                           
                         
                         
                           
                             
                               m 
                               2 
                             
                           
                         
                         
                           
                             
                               m 
                               3 
                             
                           
                         
                         
                           
                             
                               m 
                               4 
                             
                           
                         
                         
                           
                             
                               m 
                               5 
                             
                           
                         
                         
                           
                             
                               m 
                               6 
                             
                           
                         
                         
                           
                             
                               g 
                               1 
                             
                           
                         
                         
                           
                             
                               g 
                               2 
                             
                           
                         
                         
                           
                             
                               g 
                               3 
                             
                           
                         
                         
                           
                             
                               g 
                               4 
                             
                           
                         
                         
                           
                             
                               g 
                               5 
                             
                           
                         
                         
                           
                             
                               g 
                               6 
                             
                           
                         
                       
                       ] 
                     
                   
                 
               
               
                 
                   ( 
                   1 
                   ) 
                 
               
             
           
         
       
     
     In Equation 1, m j  (j=1, 2, . . . , 6) represents a measurement error (unknown number) when the probe unit  22  measures a ball-to-ball distance corresponding to the j-th position in a measurement coordinate system of the CMM  200 . 
       FIG.  4    shows examples of the j-th position in the measurement coordinate system of the CMM  200 . The inspection gauge  100  is installed such that vertices of the regular tetrahedron of the inspection gauge  100  are aligned with positions P 1 , P 2 , and P 3 , shown in  FIG.  4   , in the first posture, the second posture, and the third posture.
 
 FIG.  4    shows the respective j-th (j=1, 2, . . . , 6) positions in the measurement coordinate system of the CMM  200 .
 
     In Equation 1, l #i,k  (i=1, 2, 3, and k=1, 2, . . . , 6) represents a measurement value obtained by the measurement part  251  measuring a ball-to-ball distance at the k-th position of the inspection gauge  100  arranged in the i-th posture.  FIGS.  5 A to  5 C  each show an example of the positions of the edges of the inspection gauge  100  corresponding to the subscript k.  FIG.  5 A  shows the positions of the edges of the inspection gauge  100  corresponding to the subscript k (k=1, 2, . . . , 6) when the inspection gauge  100  is arranged in the first posture (i=1). Arrows in  FIG.  5 A  indicate a direction in which the inspection gauge  100  is rotated when installing the inspection gauge  100  in the second posture. 
       FIG.  5 B  shows the positions of the edges of the inspection gauge  100  corresponding to the subscript k (k=1, 2, . . . , 6) when the inspection gauge  100  is arranged in the second posture (i=2). A broken line in  FIG.  5 B  indicates the second rotational symmetry axis around which the inspection gauge  100  is rotated when installing the inspection gauge  100  in the third posture. Arrows in  FIG.  5 B  indicate a direction in which the inspection gauge  100  is rotated when installing the inspection gauge  100  in the third posture.  FIG.  5 C  shows the positions of the edges of the inspection gauge  100  corresponding to the subscript k (k=1, 2, . . . , 6) when the inspection gauge  100  is arranged in the third posture (i=3). g k  (k=1, 2, . . . , 6) indicates a calibration value (unknown number) obtained by calibrating a ball-to-ball distance between two balls at respective ends of the k-th edge of the inspection gauge  100 . 
     In Equation 1, l 2, sc1  indicates a value identified by the identification part  252  for a measurement error when the measurement part  251  measures the ball-to-ball distance at the measurement position. The calculation part  253  calculates a calibration value g k  and a measurement error m j  of the ball-to-ball distance by solving simultaneous equations corresponding to a matrix shown in Equation 1. In the example of this disclosure, the calculation part  253  calculates the calibration value g k  of the ball-to-ball distance by the method of least squares. Similarly, the calculation part  253  calculates the measurement error m j  of the ball-to-ball distance by the method of least squares. In this manner, the calculation part  253  can calculate the most probable value as the calibration value g k  or the like. The calculation part  253  outputs information indicating the calculated calibration value g k  and the measurement error m j  of the ball-to-ball distance to the display control part  254 . 
     The display control part  254  displays characters, an image, or the like on the display part  23 . The display control part  254  displays the calibration value g k  and the measurement error m j  of the ball-to-ball distance calculated by the calculation part  253 . 
     [Calibration Procedure of the Inspection Gauge] 
       FIG.  6    is a flowchart showing a procedure for calibrating the ball-to-ball distance of the inspection gauge  100  in the first embodiment. First, the inspection gauge  100  is installed in the first posture on the measurement table of the CMM  200  in which the measurement error at the measurement position of the first edge is identified (S 101 ). The measurement part  251  measures each of the ball-to-ball distances between two balls at respective ends of the plurality of edges of the inspection gauge  100  installed in the first posture (S 102 ). The inspection gauge  100  is rotated about the first rotational symmetry axis by 120 degrees and installed in the second posture (S 103 ). The measurement part  251  measures each of the ball-to-ball distances between two balls at respective ends of the plurality of edges of the inspection gauge  100  installed in the second posture (S 104 ). 
     Next, the inspection gauge  100  is rotated about the second rotational symmetry axis by 120 degrees and installed in the third posture (S 105 ). The measurement part  251  measures each of the ball-to-ball distances between two balls at respective ends of the plurality of edges of the inspection gauge  100  installed in the third posture (S 106 ). The calculation part  253  calculates the calibration value of the ball-to-ball distance by solving simultaneous equations including (i) the ball-to-ball distances of the plurality of edges of the inspection gauge  100  which are measured when installed in the first posture, (ii) the ball-to-ball distances of the plurality of edges of the inspection gauge  100  which are measured when installed in the second posture, (iii) the ball-to-ball distances between the plurality of edges of the inspection gauge  100  which are measured when installed in the third posture, (iv) a calibration value obtained by calibrating the ball-to-ball distances corresponding to the plurality of edges of the inspection gauge  100 , and (v) a measurement error at the measurement position identified in advance (S 107 ), and ends the process. 
     Second Embodiment 
     In the first embodiment, an example in which the measurement part  251  measures the ball-to-ball distances of the inspection gauge  100  installed in the first posture, the second posture, and the third posture has been described. On the other hand, in a second embodiment, the inspection gauge  100  is calibrated by measuring lengths of a total of two edges that are (i) one edge of a base and (ii) one edge of a surface other than the base and not included in the base, measured at two positions of the CMM calibrated with the reference instrument, and by measuring a distance between balls at respective ends of at least one other edge at an uncalibrated position in the CMM, in each of two postures. That is, the procedure of installing the inspection gauge  100  in a third posture is omitted, and the measurement part  251  measures the ball-to-ball distances of the inspection gauge  100  installed in a first posture and a second posture. 
       FIGS.  7 A and  7 B  each show an outline of a calibration method according to the second embodiment of the present disclosure.  FIG.  7 A  shows a state in which the inspection gauge  100  is installed in the first posture.  FIG.  7 B  shows a state in which the inspection gauge  100  is installed in the second posture. 
     Similar to the first embodiment, the inspection gauge  100  is installed on the measurement table of the CMM  200  in the first posture. In the second embodiment, it is assumed that a measurement error in a case where the measurement part  251  measures ball-to-ball distances at a first measurement position (a position A indicated by a broken line arrow) and a second measurement position (a position B indicated by a broken line arrow), both indicated by a thick line in  FIG.  7 A , is identified in advance by the identification part  252  in the same manner as in the first embodiment. The measurement part  251  measures a first ball-to-ball distance between the balls B and C at respective ends of a first edge corresponding to the first measurement position among the plurality of edges of the inspection gauge  100  installed in the first posture. Similarly, the measurement part  251  measures a second ball-to-ball distance between the balls A and C at respective ends of a second edge corresponding to the second measurement position. The measurement part  251  also measures ball-to-ball distances corresponding to four edges other than the first edge and the second edge among the plurality of edges of the inspection gauge  100  installed in the first posture. 
     After the measurement part  251  has measured the ball-to-ball distances of the plurality of edges of the inspection gauge  100  installed in the first posture, the inspection gauge  100  is installed on the measurement table in the second posture in which the inspection gauge  100  is rotated about its rotational symmetry axis, as shown by arrows in  FIG.  7 A . The rotational symmetry axis is an axis extending approximately in the depth direction in  FIG.  7 A . The rotational symmetry axis passes through the ball A and is orthogonal to a plane including the ball B, the ball C, and the ball D. In an example of the second embodiment, when the plane (plane including the balls B, C, and D) orthogonal to the rotational symmetry axis in the inspection gauge  100  is a base, the first edge corresponding to the first measurement position is included in the base, but the second edge corresponding to the second measurement position is not included in the base. By arranging the first measurement position, the second measurement position, and the rotational symmetry axis for rotating the inspection gauge  100  in such a manner, the user can restrict an occurrence of rank deficiency in which the solution is not uniquely determined when calculating the ball-to-ball distance using an equation described later. 
       FIG.  7 B  shows how the inspection gauge  100  is installed in the second posture. In the second posture, an edge (edge sandwiched by the balls B and D, hereinafter also referred to as a third edge) different from the first edge and the second edge among the plurality of edges of the regular tetrahedron of the inspection gauge  100  moves such that it matches the first measurement position where the first edge is arranged in the first posture. At this time, an edge (edge sandwiched by the balls A and B, hereinafter also referred to as a fourth edge) different from the first edge, the second edge, and the third edge among the plurality of edges of the regular tetrahedron of the inspection gauge  100  matches the second measurement position where the second edge is arranged in the first posture. 
     The measurement part  251  measures a third ball-to-ball distance between the balls B and D at respective ends of the third edge corresponding to the first measurement position among the plurality of edges of the inspection gauge  100  installed in the second posture. The measurement part  251  measures a fourth ball-to-ball distance between the balls A and B at respective ends of the fourth edge corresponding to the second measurement position among the plurality of edges of the inspection gauge  100  installed in the second posture. The measurement part  251  also measures ball-to-ball distances corresponding to four edges other than the third edge and the fourth edge among the plurality of edges of the inspection gauge  100 . 
     The calculation part  253  calculates a calibration value by calibrating the ball-to-ball distance between two balls at respective ends of the plurality of edges of the inspection gauge  100  by solving simultaneous equations including (i) the ball-to-ball distances of the plurality of edges of the inspection gauge  100  when installed in the first posture, (ii) the ball-to-ball distances between the plurality of edges of the inspection gauge  100  when installed in the second posture, (iii) a calibration value (unknown number) obtained by calibrating the ball-to-ball distances between two balls at respective ends of the plurality of edges of the inspection gauge  100 , and (iv) a measurement error of the CMM  200  at the first measurement position and the second measurement position identified by the identification part  252 . 
     The calculation part  253  calculates calibration values g 1  to g 6  of the ball-to-ball distance between the two balls at respective ends of each edge of the regular tetrahedron of the inspection gauge  100  and measurement errors m 1  to m 6  that occur when the measurement part  251  measures the ball-to-ball distances with the following Equation 2. 
     
       
         
           
             [ 
             
               Equation 
               ⁢ 
                    
               2 
             
             ] 
           
         
       
       
         
           
             
               
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   l 
                                   
                                     
                                       # 
                                       ⁢ 
                                       1 
                                     
                                     , 
                                     1 
                                   
                                 
                               
                             
                             
                               
                                 
                                   l 
                                   
                                     #1 
                                     , 
                                     2 
                                   
                                 
                               
                             
                             
                               
                                 
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                                       # 
                                       ⁢ 
                                       1 
                                     
                                     , 
                                     3 
                                   
                                 
                               
                             
                             
                               
                                 
                                   l 
                                   
                                     #1 
                                     , 
                                     4 
                                   
                                 
                               
                             
                             
                               
                                 
                                   l 
                                   
                                     #1 
                                     , 
                                     5 
                                   
                                 
                               
                             
                             
                               
                                 
                                   l 
                                   
                                     #1 
                                     , 
                                     6 
                                   
                                 
                               
                             
                             
                               
                                 
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                                     #2 
                                     , 
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                                     , 
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                                   l 
                                   
                                     2 
                                     , 
                                     scl 
                                   
                                 
                               
                             
                             
                               
                                 
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                                     5 
                                     , 
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                           ] 
                         
                         = 
                         
                           
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                                   1 
                                 
                                 
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                             ] 
                           
                           ⨯ 
                           
                             [ 
                             
                               
                                 
                                   
                                     m 
                                     1 
                                   
                                 
                               
                               
                                 
                                   
                                     m 
                                     2 
                                   
                                 
                               
                               
                                 
                                   
                                     m 
                                     3 
                                   
                                 
                               
                               
                                 
                                   
                                     m 
                                     4 
                                   
                                 
                               
                               
                                 
                                   
                                     m 
                                     5 
                                   
                                 
                               
                               
                                 
                                   
                                     m 
                                     6 
                                   
                                 
                               
                               
                                 
                                   
                                     g 
                                     1 
                                   
                                 
                               
                               
                                 
                                   
                                     g 
                                     2 
                                   
                                 
                               
                               
                                 
                                   
                                     g 
                                     3 
                                   
                                 
                               
                               
                                 
                                   
                                     g 
                                     4 
                                   
                                 
                               
                               
                                 
                                   
                                     g 
                                     5 
                                   
                                 
                               
                               
                                 
                                   
                                     g 
                                     6 
                                   
                                 
                               
                             
                             ] 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   2 
                   ) 
                 
               
             
           
         
       
     
     In Equation 2, m j  (j=1, 2, . . . , 6) represents a measurement error (unknown number) when measuring a ball-to-ball distance corresponding to the j-th position by the probe unit  22  in the measurement coordinate system of the CMM  200 . In Equation 2, l #i,k  (i=1, 2, 3, and k=1, 2, . . . , 6) represents a measurement value obtained by the measurement part  251  measuring a ball-to-ball distance at the k-th position of the inspection gauge  100  arranged in the i-th posture. g k  (k=1, 2, . . . , 6) represents a calibration value (unknown number) obtained by calibrating a ball-to-ball distance between two balls at respective ends of the k-th edge of the inspection gauge  100 . In Equation 2, l 2, scl  represents a value identified by the identification part  252  for a measurement error when the measurement part  251  measures the ball-to-ball distance at the first measurement position. In Equation 2, l 5, scl  represents a value identified by the identification part  252  for a measurement error when the measurement part  251  measures the ball-to-ball distance at the second measurement position. 
     [Calibration Procedure of the Inspection Gauge] 
       FIG.  8    is a flowchart showing a procedure for calibrating the ball-to-ball distance of the inspection gauge  100  in the second embodiment. First, the inspection gauge  100  is installed in the first posture on the measurement table of the CMM  200  in which measurement errors are identified at the first measurement position and the second measurement position (S 201 ). The measurement part  251  measures each of the ball-to-ball distances between two balls at respective ends of the plurality of edges of the inspection gauge  100  installed in the first posture (S 202 ). 
     Next, the inspection gauge  100  is rotated about its rotational symmetry axis by 120 degrees and installed in the second posture (S 203 ). The measurement part  251  measures each of the ball-to-ball distances between two balls at respective ends of the plurality of edges of the inspection gauge  100  installed in the second posture (S 204 ). 
     The calculation part  253  calculates a calibration value by calibrating the ball-to-ball distances between two balls at respective ends of the plurality of edges of the inspection gauge  100  by solving simultaneous equations including (i) the ball-to-ball distances of the plurality of edges of the inspection gauge  100  installed in the first posture, (ii) the ball-to-ball distances of the plurality of edges of the inspection gauge  100  installed in the second posture, (iii) calibration values obtained by calibrating the ball-to-ball distances corresponding to the plurality of edges of the inspection gauge  100 , and (iv) measurement errors at the measurement positions identified in advance (S 205 ), and ends the process. 
     In the first and second embodiments, examples where the calculation part  253  of the CMM  200  calculates the calibration value obtained by calibrating the ball-to-ball distances corresponding to the plurality of edges of the inspection gauge  100  have been described. However, the present disclosure is not limited to the examples in which the CMM  200  includes the calculation part  253 . For example, a calibration apparatus (not shown) different from the CMM  200  may be provided with a calculation part, and the calculation part may calculate a calibration value obtained by calibrating the ball-to-ball distances corresponding to the plurality of edges of the inspection gauge  100 . 
     In the first embodiment, an example has been described in which the measurement part  251  measures the ball-to-ball distances of the inspection gauge  100  installed in the first posture, the second posture, and the third posture. However, the present disclosure is not limited to this, and the user may increase the number of times the posture of the inspection gauge  100  is changed. For example, the measurement part  251  may measure the ball-to-ball distance of the inspection gauge  100  installed in the first posture, the second posture, the third posture, and the fourth posture. In this manner, the calculation part  253  can increase redundancy of the calibration value to be calculated for the ball-to-ball distance of the inspection gauge  100 , and reduce influences of measurement noise in the calculation of the calibration value. 
     [Effect of the Calibration Method of the Present Disclosure] 
     According to the calibration method of the present disclosure, the calculation part  253  calculates the calibration value of the ball-to-ball distances by solving simultaneous equations. By adopting such a calibration method, the user does not need to measure all the ball-to-ball distances of the inspection gauge  100  using the laser interferometer  35  or the like, so that the time required for calibration work for calibrating the inspection gauge  100  can be reduced. 
     The present disclosure is explained on the basis of the exemplary embodiments. The technical scope of the present disclosure is not limited to the scope explained in the above embodiments and it is possible to make various changes and modifications within the scope of the disclosure. For example, all or part of the apparatus can be configured to be functionally or physically distributed and integrated in arbitrary units. Further, new exemplary embodiments generated by arbitrary combinations of them are included in the exemplary embodiments of the present disclosure. The effect of the new embodiment caused by the combination has the effect of the original embodiment together.