Abstract:
A vision correction method for establishing the position of a tool center point (TCP) for a robot manipulator includes the steps of: defining a preset position of the TCP; defining a preset coordinate system T G  with the preset position of the TCP as its origin; capturing a two-dimensional picture of the preset coordinate system T G  to establish a visual coordinate system T V ; calculating a scaling ratio λ of the vision coordinate system T V  relative to the preset coordinate system T G ; rotating the TCP relative to axes of the preset coordinate system T G ; capturing pictures of the TCP prior to and after rotation; calculating the deviation ΔP between the preset position and actual position of the TCP; correcting the preset position and corresponding coordinate system T G  using ΔP, and repeating the rotation through correction steps until ΔP is less than or equal to a maximum allowable deviation of the robot manipulator.

Description:
BACKGROUND 
     1. Technical Field 
     The present disclosure generally relates to correction methods for robot manipulators, and particularly to a vision correction method for tool center point of a robot manipulator. 
     2. Description of Related Art 
     Robot manipulators are used in various industrial fields to fulfill the task requirements for product assembling, welding and other tasks automatically. A robot manipulator of related art may be equipped with a tool, such as a gripper, a cutting device, a glue dispenser or a fixture mounted to a distal end thereof. The tool has a specific defined point, called the Tool Center Point (TCP), that is commanded to move to various positions in the workspace when a robot manipulator performs a task. In use, the positional accuracy of the robot manipulator directly depends on knowing the precise position of the TCP relative to the robot manipulator&#39;s distal end. Since the tool may have been mounted to the distal end of the robot manipulator manually, an alignment error may arise between the tool and the robot manipulator, causing the TCP to deviate from its expected position. Thus, to ensure tasks are performed with high positional accuracy, the position of the TCP relative to the robot manipulator&#39;s distal end must be calibrated to correct for tool misalignment after mounting or collision. Presently, the standard method for calibrating the TCP position requires an operator to visually guide the TCP to a specific point in the workspace with the tool at a minium of four different orientations. This is a time-consuming and inconvenient manual process, and the correction precision is not guaranteed. Automated calibration systems that use light beams to determine the TCP position are known in the art. However, these automated systems are not widely used due to their high cost, laborious setup process, or long calibration process time. 
     Therefore, there is room for improvement within the art. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       The components in the drawings are not necessarily drawn to scale, the emphasis instead being placed upon clearly illustrating the principles of the present disclosure. Moreover, in the drawings, like reference numerals designate corresponding parts throughout several views, and all the views are schematic. 
         FIG. 1  shows a flow chart of an embodiment of a vision correction method for a tool center point of a robot manipulator. 
         FIG. 2  shows an isometric view of the robot manipulator. 
         FIG. 3  shows a partially enlarged view of the robot manipulator of  FIG. 2 . 
         FIGS. 4 through 8  show five different coordinate views of the robot manipulator for illustrating the vision correction method for tool center point of the robot manipulator of the embodiment. 
     
    
    
     DETAILED DESCRIPTION 
     Referring to  FIGS. 2 and 3 , an embodiment of a robot manipulator  100  includes a main body  10  and a controller  30  connected to the main body  10 . The main body  10  consists of multiple mechanical linkages and includes a base  11  and a drive mechanism  13  that moves the mechanical linkages. The drive mechanism  13  is mounted in the main body  10  and connected with the controller  30 . A tool  15  is assembled to a distal end of the main body  10 . A vision lens  17  is positioned to one side of the tool  15 . The controller  30  includes a preset control software (not shown and not illustrated) and a data storage facility (also not shown and illustrated) installed therein for driving and controlling the drive mechanism  13  to move the tool  15 . In the illustrated embodiment, the drive mechanism  13  is a multi-axis drive mechanism controlled by the controller  30 . The tool  15  has a defined tool center point (TCP)  151  formed at a distal end of the tool  15 . The vision lens  17  is configured for taking a plurality of different images of the tool  15 . The robot manipulator  100  has a predefined basic coordinate system T 0  at the distal end of the main body  10 . The basic coordinate system T 0  includes three coordinate axes (X-axis, Y-axis and Z-axis) intersecting with each other perpendicularly. A point of origin of the basic coordinate system T 0  is defined as O (0, 0, 0). This basic coordinate system is typically centered on a mounting flange at the distal end of the robot manipulator. After mounting the tool  15  onto the mounting flange, the TCP  151  of the tool  15  is positioned within the basic coordinate system T 0  of the robot manipulator  100 . The position of the TCP  151  within the basic coordinate system T 0  is only known approximately due to tool misalignment after mounting; variances in tool fabrication also contributes to uncertainty in the position of the TCP  151 . The object of the invention is to correct the position of the TCP  151  from an approximate value to a highly accurate value. 
     Also referring to  FIG. 1 , a vision correction method of an embodiment for correcting the position of the TCP  151  of the robot manipulator  100  is illustrated as follows. 
     In step S 101 , a preset position of the TCP  151  of the tool  15  is defined as P 0 ; the preset position P 0  is an estimate of the actual position of the TCP  151  of the tool  15 . The corresponding coordinate value of the preset position P 0  of the TCP  151  within the basic coordinate system T 0  is defined as (X g , Y g , Z g ). A preset coordinate system T G  is thus established based on the preset position P 0  of the TCP  151  of the tool  15 . Specifically, the origin of the preset coordinate system T G  is the preset position P 0  of the TCP  151  of the tool  15 . The preset coordinate system T G  includes three coordinate axes X G -axis, Y G -axis and Z G -axis, each coordinate axis of the preset coordinate system T G  is positioned parallel to the corresponding three coordinate axes X-axis, Y-axis and Z-axis of the basic coordinate system T 0 . The coordinate value of the origin/preset position P 0  (X g , Y g , Z g ) of the preset coordinate system T G  within the basic coordinate system T 0  is stored into the preset control software of the controller  30 . During usage, the position of the preset position P 0  of the TCP  151  of the tool  15  can be adjusted and moved by adjusting the position parameters of the preset position P 0  of the TCP  151  of the tool  15  that are stored within the preset control software of the controller  30 . The actual position of the TCP  151  of the tool  15  is defined as P 1 , and thus, the corresponding coordinate value of the actual TCP  151  formed within the preset coordinate system T G  is defined as P 1  (X 1 , Y 1 , Z 1 ). Any deviation between the preset position P 0  of the TCP  151  and the actual position of the TCP  151  (P 1 ) is defined as ΔP (Δx, Δy, Δz), namely, the deviation between the origin of the preset coordinate system T G  (X g , Y g , Z g ) and the actual position of the TCP  151  (P 1 ) is defined as ΔP (Δx, Δy, Δz). 
     In step S 102 , the vision lens  17  is adjusted to aim at and be parallel to an X G Z G  plane of the preset coordinate system T G . The vision lens  17  captures a two-dimensional picture of the X G Z G  plane of the preset coordinate system T G . A two-dimensional visual coordinate system T V  is established based on the two-dimensional picture of the X G Z G  plane of the preset coordinate system T G . The origin of the two-dimensional visual coordinate system T V  is defined as O′ (0, 0). The two-dimensional visual coordinate system T V  includes an x-axis and a z-axis intersecting with the x-axis perpendicularly. The x-axis and the z-axis are parallel to the corresponding X G -axis and Y G -axis of the preset coordinate system T G  respectively. The corresponding coordinate value of the actual TCP  151  captured by the vision lens within the two-dimensional visual coordinate system T V  is defined as P 1 ′ (x 1 , z 1 ). 
     Also referring to  FIG. 4 , in step S 103 , the drive mechanism  13 , controlled by the controller  30 , causes the TCP  151  of the tool  15  to move along the X G -axis direction and to finally stop at a position P 2  with a X G -axis directional distance of about X 2 . The corresponding coordinate value of the actual TCP  151  positioned at the position P 2  relative to the preset coordinate system T G  can be figured as P 2  (X 1 −X 2 , 0, 0). The vision lens  17  then captures a picture of the TCP  151  of the tool  15  positioned at the position P 2 . The corresponding coordinate value of the TCP  151  formed within the two-dimensional visual coordinate system T V  can be calculated as P 2 ′ (x 1 −x 2 , z 1 ). And thus, the scaling ratio λ of the vision coordinate system T V  relative to the preset coordinate system T G  can be easily calculated as λ=(X 1 −X 2 )/(x 1 −x 2 ). 
     Also referring to  FIG. 5 , in step S 104 , the TCP  151  of the tool  15  is firstly moved back to its original position P 1 , and then the TCP  151  of the tool  15  is driven to rotate about the Z G -axis of the preset coordinate system T G  by about 180 degrees, and finally stop at a position P 3 . The corresponding coordinate value of the actual TCP  151  positioned at the position P 3  relative to the preset coordinate system T G  can be figured as P 3  (X 3 , Y 3 , Z 3 ). The vision lens  17  captures a picture of the TCP  151  of the tool  15  positioned at the position P 3 . The corresponding coordinate value of the TCP  151  formed within the two-dimensional visual coordinate system T V  can be defined as P 3 ′ (x 3 , z 3 ). The distance between the x 3  and x 1  as shown in visual coordinate system T V  can be easily calculated. Since the TCP  151  of the tool  15  is rotated relative to the Z G -axis direction of the preset coordinate system T G  to about 180 degrees, the following mathematical functional relationships can be easily established: X 1 −X 3 =2Δx; Z 3 =Z 1 ; 2Δx==λ(x 1 −x 3 ). Thus the value of Δx can be easily calculated by means of the aforementioned mathematical functional relationships. 
     Also referring to  FIG. 6 , in step S 105 , the TCP  151  of the tool  15  is firstly moved back to its original position P 1 , and then the TCP  151  of the tool  15  is driven to rotate by about 90 degrees relative to the Z G -axis direction of the preset coordinate system T G , and finally stop at a position P 4 . The corresponding coordinate value of the actual TCP  151  positioned at the position P 4  relative to the preset coordinate system T G  can be marked or designated as P 4  (X 4 , Y 4 , Z 4 ). The vision lens  17  captures a picture of the TCP  151  of the tool  15  positioned at the position P 4 . The corresponding coordinate value of the TCP  151  formed within the two-dimensional visual coordinate system T V  can be defined as P 4 ′ (x 4 , z 4 ). The distance between the x 4  and x 1  as shown in visual coordinate system T V  can be easily calculated. Since the TCP  151  of the tool  15  is rotated relative to the Z G -axis direction of the preset coordinate system T G  by about 90 degrees, the following mathematical function relationships can thus be easily defined: X 1 −X 4 =Δx+Δy; Y 1 −Y 4 =Δy−Δx; Z 4 =Z 1 ; Δx+Δy=X 1 −X 4 =λ(x 1 −x 4 ). Thereby, since the value of Δx has been calculated in the previous step S 104 , the value of Δy can be calculated by means of the aforementioned functional relationships. 
     Also referring to  FIG. 7 , in step S 106 , the TCP  151  of the tool  15  is firstly moved back to its original position P 1 , and then the TCP  151  of the tool  15  is driven to rotate about 90 degrees about the Y G -axis direction of the preset coordinate system T G , and finally stop at a position P 5 . The corresponding coordinate value of the actual TCP  151  at the position P 5  relative to the preset coordinate system T G  can be designated or marked as P 5  (X 5 , Y 5 , Z 5 ). The vision lens  17  captures a picture of the TCP  151  of the tool  15  when at the position P 5 . The corresponding coordinate value of the TCP  151  formed within the two-dimensional visual coordinate system T V  can be defined as P 5 ′ (x 5 , z 5 ). The distance between the z 5  and z 1  as shown in visual coordinate system T V  can be easily calculated. Since the TCP  151  of the tool  15  is rotated relative to the Y G -axis direction of the preset coordinate system T G  by about 90 degrees, thus the following functional relationships can be easily established: Z 1 −Z 5 =Δx+Δz; X 1 −X 5 =Δx−Δz; Δx+Δz=Z 1 −Z 5 =λ(z 1 −z 5 ). Thus value of Δz can be easily calculated by means of the aforementioned relationships. 
     In step S 107 , the TCP  151  of the tool  15  is moved back to its original position P 1 . The deviation ΔP (Δx, Δy, Δz) between the preset position P 0  of the TCP  151  and the actual position of the TCP  151  (P 1 ) is calculated by means of the aforementioned functional relationships and compared with the maximum allowable deviation of the robot manipulator  100 . 
     In step S 108 , if the deviation ΔP (Δx, Δy, Δz) is less than or equal to the maximum allowable deviation of the robot manipulator  100 , the preset position P 0  of the TCP  151  then can be considered as the actual position of the TCP  151  (P 1 ) thereby completing the coordinate correction work. By means of this procedure, there is no need to further correct the coordinates of the TCP  151  of the robot manipulator  100 . The corresponding coordinate value (X g , Y g , Z g ) of the preset position P 0  of the TCP  151  within the basic coordinate system T 0  can be directly considered as the actual position of the TCP  151  (P 1 ). 
     Also referring to  FIG. 8 , in step S 109 , if the magnitude of the deviation ΔP (Δx, Δy, Δz) is greater than the maximum allowable deviation of the robot manipulator  100 , the position parameters of the preset position P 0  of the TCP  151  that are stored within the control software of the controller  30  then can be amended by adding a deviation ΔP. Specifically, the preset position P 0  of the TCP  151  relative to the basic coordinate system T 0  is changed into (Xg+Δx, Yg+Δy, Zg+Δz). Thereby, the newly-changed coordinate value (Xg+Δx, Yg+Δy, Zg+Δz) of the preset position P 0  of the TCP  151  can be considered as the new preset position of the TCP  151 , the aforementioned steps S 101 ˜S 109  are repeated until the deviation between the preset position P 0  of the TCP  151  and the actual position of the TCP  151  is less than or equal to the maximum allowable deviation of the robot manipulator  100 , to altogether complete the coordinate correction to the TCP  151  of the robot manipulator  100  and obtain the correct position of the actual TCP  151  relative to the basic coordinate system T 0 . Correction usually cannot be completed in a single iteration through steps S 101 ˜S 109 , because alignment error between the vision coordinate system T V  and the preset coordinate system T G , vision lens distortion, and error in calibrating λ will produce error in the calculation of ΔP (Δx, Δy, Δz). The error is reduced successively through each iteration of steps S 101 ˜S 109 . 
     In another embodiment, the coordinate axes of the preset coordinate system T G  are not parallel to the corresponding coordinate axes of the basic coordinate system T 0 . The basic coordinate system T 0  can also be defined at other positions, such as at the base  11 . 
     In another embodiment, the vision lens  17  can also include a picture analysis unit to facilitate the calculation of the scaling ratio λ of the vision coordinate system T V  relative to the preset coordinate system T G , so that the step  103  is rendered non-fundamental or mandatory. The sequence of order of the steps S 103 , S 104  and S 105  is not fixed. 
     In another embodiment, the deviation ΔP (Δx, Δy, Δz) between the preset position P 0  of the TCP  151  and the actual position of the TCP P 1  can be established by other means. Another method to calculate the deviation ΔP (Δx, Δy, Δz) between the preset position of the TCP  151  and the actual position of the TCP  151  is as follows: 
     Firstly, the actual position of the TCP  151  of the tool  15  is defined as P T , and the coordinate value of the actual TCP  151  formed within the preset coordinate system T G  is defined as P T  (X T , Y T , Z T ). The preset position of the TCP  151  of the tool  15  is defined as P G , and the preset coordinate system T G  is thus established based on the preset position P G  of the TCP  151  of the tool  15 . Since the origin of the preset coordinate system T G  is the preset position P G  of the TCP  151  of the tool  15 , thus, the deviation ΔP (Δx, Δy, Δz) between the preset position of the TCP P G  and the actual position of the TCP P T  is calculable according to the following equations:
 
Δ x=X   T   ; Δy=Y   T   ; Δz=Z   T .  Equation (1)
 
The vision lens  17  then takes a two-dimensional picture of the X G Z G  plane of the preset coordinate system T G . A two-dimensional visual coordinate system T V  is established based on the two-dimensional picture of the X G Z G  plane of the preset coordinate system T G . The corresponding coordinate value of the actual TCP  151  formed within the two-dimensional visual coordinate system T V  is defined as P T ′ (X T ′, X T ′).
 
     Secondly, the TCP  151  of the tool  15  is rotated by the drive mechanism  13  relative to the Z G -axis direction of the preset coordinate system T G  about θ 1  degrees, and finally stop at a position P T1 . The corresponding coordinate value of the actual TCP  151  positioned at the position P T1  relative to the preset coordinate system T G  can be denoted or designated as P T1 (X T1 , Y T1 , Z T1 ). The vision lens  17  captures a picture of the TCP  151  of the tool  15  positioned at the position P T1 . The corresponding coordinate value of the TCP  151  formed within the two-dimensional visual coordinate system T V  can be defined as P T1 ′ (X T1 ′, Z T1 ′). The functional relationships between the coordinate value P T  (X T , Y T , Z T ) of the actual TCP  151  and the coordinate value P T1 (X T1 , Y T1 , Z T1 ) of the position P T1  of the TCP  151  can be presented as follow in equation (2): 
                         X   T     -     X     T   ⁢           ⁢   1         =       X   T     -           X   T   2     +     Y   T   2         ·     cos   (       θ   1     +     arc   ⁢           ⁢   cos   ⁢       X   T           X   T   2     +     Y   T   2               )           ;           Equation   ⁢           ⁢     (   2   )                 
since X T −X T1 =λ*(x T ′−x T1 ′); Δx=X T ; and Δy=Y T ; the value of x T ′−x T1 ′ can be calculated from the two-dimensional visual coordinate system T V ; and thus, a new formula, referred to as Equation (3):
 
                     λ   *     (       X   T   ′     -     X     T   ⁢           ⁢   1     ′       )       =       Δ   ⁢           ⁢   x     -           Δ   ⁢           ⁢     x   2       +     Δ   ⁢           ⁢     y   2           ·     cos   (       θ   1     +     arc   ⁢           ⁢   cos   ⁢       Δ   ⁢           ⁢   x           Δ   ⁢           ⁢     x   2       +     Δ   ⁢           ⁢     y   2                 )                 Equation   ⁢           ⁢     (   3   )                 
can be established based on the aforementioned functional relationships.
 
     Thirdly, the TCP  151  of the tool  15  is then moved back to its original position, and then the TCP  151  of the tool  15  is rotated relative to the Y G -axis direction of the preset coordinate system T G  about θ 2  degrees, and finally stops at a position P T2 . The corresponding coordinate value of the actual TCP  151  positioned at the position P T2  relative to the preset coordinate system T G  can be figured as P T2  (X T2 , Y T2 , Z T2 ). The vision lens  17  captures a picture of the TCP  151  of the tool  15  positioned at the position P T2 . The corresponding coordinate value of the TCP  151  formed within the two-dimensional visual coordinate system T V  can be defined as P T2 ′ (X T2 ′, z T2 ′). The functional relationships between the coordinate value P T  (X T , Y T , Z T ) of the actual TCP  151  and the coordinate value P T2  (X T2 , Y T2 , Z T2 ) of the position P T2  of the TCP  151  can be established as follows, in equations (4) and (5): 
     
       
         
           
             
               
                 
                   
                     
                       
                         X 
                         T 
                       
                       - 
                       
                         X 
                         
                           T 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           2 
                         
                       
                     
                     = 
                     
                       
                         X 
                         T 
                       
                       - 
                       
                         
                           
                             
                               X 
                               T 
                               2 
                             
                             + 
                             
                               Z 
                               T 
                               2 
                             
                           
                         
                         · 
                         
                           cos 
                           ( 
                           
                             
                               θ 
                               2 
                             
                             + 
                             
                               arc 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               cos 
                               ⁢ 
                               
                                 
                                   X 
                                   T 
                                 
                                 
                                   
                                     
                                       X 
                                       T 
                                       2 
                                     
                                     + 
                                     
                                       Z 
                                       T 
                                       2 
                                     
                                   
                                 
                               
                             
                           
                           ) 
                         
                       
                     
                   
                   ; 
                 
               
               
                 
                   Equation 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     ( 
                     4 
                     ) 
                   
                 
               
             
             
               
                 
                   
                     
                       
                         Z 
                         T 
                       
                       - 
                       
                         Z 
                         
                           T 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           2 
                         
                       
                     
                     = 
                     
                       
                         Z 
                         T 
                       
                       - 
                       
                         
                           
                             
                               X 
                               T 
                               2 
                             
                             + 
                             
                               Z 
                               T 
                               2 
                             
                           
                         
                         · 
                         
                           cos 
                           ( 
                           
                             
                               θ 
                               2 
                             
                             + 
                             
                               arc 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               cos 
                               ⁢ 
                               
                                 
                                   Z 
                                   T 
                                 
                                 
                                   
                                     
                                       X 
                                       T 
                                       2 
                                     
                                     + 
                                     
                                       Z 
                                       T 
                                       2 
                                     
                                   
                                 
                               
                             
                           
                           ) 
                         
                       
                     
                   
                   ; 
                 
               
               
                 
                   Equation 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     ( 
                     5 
                     ) 
                   
                 
               
             
           
         
       
     
     Since X T −X T1 =λ*(x T ′−x T1 ′); Z T −Z T1 =λ*(z T ′−z T1 ′); Δx=X T ; Δy=Y T ; and Δz=Z T ; the value of x T ′−x T2 ′ and z T ′−z T2 ′ can be calculated from the two-dimensional visual coordinate system T V ; and thus, the following equations (6) and (7) can be formulated based on the aforementioned functional relationships: 
     
       
         
           
             
               
                 
                   
                     λ 
                     * 
                     
                       ( 
                       
                         
                           X 
                           T 
                           ′ 
                         
                         - 
                         
                           X 
                           
                             T 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             2 
                           
                           ′ 
                         
                       
                       ) 
                     
                   
                   = 
                   
                     
                       Δ 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       x 
                     
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                             Δ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               x 
                               2 
                             
                           
                           + 
                           
                             Δ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               z 
                               2 
                             
                           
                         
                       
                       · 
                       
                         cos 
                         ( 
                         
                           
                             θ 
                             2 
                           
                           + 
                           
                             arc 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             cos 
                             ⁢ 
                             
                               
                                 Δ 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 x 
                               
                               
                                 
                                   
                                     Δ 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     
                                       x 
                                       2 
                                     
                                   
                                   + 
                                   
                                     Δ 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     
                                       z 
                                       2 
                                     
                                   
                                 
                               
                             
                           
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   Equation 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     ( 
                     6 
                     ) 
                   
                 
               
             
             
               
                 
                   
                     λ 
                     * 
                     
                       ( 
                       
                         
                           Z 
                           T 
                           ′ 
                         
                         - 
                         
                           Z 
                           
                             T 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             2 
                           
                           ′ 
                         
                       
                       ) 
                     
                   
                   = 
                   
                     
                       Δ 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       z 
                     
                     - 
                     
                       
                         
                           
                             Δ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               x 
                               2 
                             
                           
                           + 
                           
                             Δ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               z 
                               2 
                             
                           
                         
                       
                       · 
                       
                         cos 
                         ( 
                         
                           
                             θ 
                             2 
                           
                           + 
                           
                             arc 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             cos 
                             ⁢ 
                             
                               
                                 Δ 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 z 
                               
                               
                                 
                                   
                                     Δ 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     
                                       x 
                                       2 
                                     
                                   
                                   + 
                                   
                                     Δ 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     
                                       z 
                                       2 
                                     
                                   
                                 
                               
                             
                           
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   Equation 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     ( 
                     7 
                     ) 
                   
                 
               
             
           
         
       
     
     The values of Δx, Δy, Δz can be determined by solving the aforementioned formulas in equations (3), (6) and (7). And thus, the deviation ΔP (Δx, Δy, Δz) between the preset position of the TCP  151  and the actual position of the TCP  151  is expressly defined. 
     While various embodiments have been described and illustrated, the disclosure is not to be construed as being limited thereto. Various modifications can be made to the embodiments by those skilled in the art without departing from the true spirit and scope of the disclosure as defined by the appended claims.