Patent Publication Number: US-2021162584-A1

Title: Three-dimensional measuring device and robotic arm calibration method thereof

Description:
FIELD OF THE INVENTION 
     The present disclosure relates to a measuring device, and more particularly to a three-dimensional measuring device and a robotic arm calibration method thereof. 
     BACKGROUND OF THE INVENTION 
     With the advancement of industrial technology, a wide variety of automatic devices have been extensively developed for use in lives and industries. Generally, a robotic arm is an important component of an automation device. Although the processing stability of the robotic arm is much higher than the manual processing stability, there are still some drawbacks. For example, the robotic arm has many joints. Because of the accumulation and transmission of multiple errors, the precision of the robotic arm is not high. For increasing the precision of the robotic arm, a measuring device is used to calibrate the robotic arm before the robotic arm performs the processing task. At present, the commonly measuring devices include laser interferometers and laser trackers. Generally, the laser interferometer can measure one axis of errors at a time. For measuring the errors of different items, it is necessary to change the lens groups. Consequently, the operation of the laser interferometer is time-consuming. The use of the laser tracker can quickly acquire the detection result at high precision. However, the laser tracker is not cost-effective. 
     Therefore, there is a need of providing a three-dimensional measuring device and a robotic arm calibration method for simultaneously measuring the moving distances of an object in three dimensions and providing the robotic arm calibration parameters so as to address the issues encountered by the prior arts. 
     SUMMARY OF THE INVENTION 
     An object of the present disclosure provides a three-dimensional measuring device. The three-dimensional measuring device includes a ball-shaped structure, an X-axis measuring module, a Y-axis measuring module and a Z-axis measuring module. The ball-shaped structure is contacted with the X-axis measuring module, the Y-axis measuring module and the Z-axis measuring module and assembled with a movable object. Consequently, the three-dimensional measuring device can measure the three-dimensional coordinate of the ball-shaped structure to acquire the working point of the movable object. The use of the three-dimensional measuring device can save the measuring time period and reduce the fabricating cost. 
     Another object of the present disclosure provides a robotic arm calibration method. The robotic arm calibration method is implemented by the three-dimensional measuring device of the present disclosure and has the time-saving and cost-effective efficacy. 
     In accordance with an aspect of the present disclosure, a three-dimensional measuring device is provided. The three-dimensional measuring device is connected with a movable object of an automation device. The three-dimensional measuring device includes a ball-shaped structure, a base, an X-axis measuring module, a Y-axis measuring module and a Z-axis measuring module. The ball-shaped structure is assembled with the movable object. The ball-shaped structure is moved and/or rotated in response to a movement of the movable object. The X-axis measuring module is disposed on the base, and includes a first measuring structure and a first position sensor. The first measuring structure is movable along an X-axis direction and contacted with the ball-shaped structure. When the first measuring structure is pushed by the ball-shaped structure, the first position sensor measures a displacement amount of the first measuring structure. The Y-axis measuring module is disposed on the base, and includes a second measuring structure and a second position sensor. The second measuring structure is movable along a Y-axis direction and contacted with the ball-shaped structure. When the second measuring structure is pushed by the ball-shaped structure, the second position sensor measures a displacement amount of the second measuring structure. The Z-axis measuring module is disposed on the base, and includes a third measuring structure and a third position sensor. The third measuring structure is movable along a Z-axis direction and contacted with the ball-shaped structure. When the third measuring structure is pushed by the ball-shaped structure, the third position sensor measures a displacement amount of the third measuring structure. A measuring space is defined by a movable distance range of the first measuring structure along the X-axis direction, a movable distance range of the second measuring structure along the Y-axis direction and a movable distance range of the third measuring structure along the Z-axis direction. When the ball-shaped structure is moved in the measuring space, a three-dimensional coordinate of the ball-shaped structure is obtained according to sensed results of the first position sensor, the second position sensor and the third position sensor. 
     In accordance with another aspect of present disclosure, a robotic arm calibration method is provided. The robotic arm calibration method includes the following steps. In a step (S 1 ), a three-dimensional measuring device and a robotic arm are provided. The three-dimensional measuring device includes a ball-shaped structure, a base, an X-axis measuring module, a Y-axis measuring module and a Z-axis measuring module. The ball-shaped structure is assembled with the robotic arm, the ball-shaped structure is moved and/or rotated in response to a movement of the robotic arm. The X-axis measuring module is disposed on the base and includes a first measuring structure and a first position sensor. The first measuring structure is movable along an X-axis direction and contacted with the ball-shaped structure. The first position sensor measures a displacement amount of the first measuring structure when the first measuring structure is pushed by the ball-shaped structure. The Y-axis measuring module is disposed on the base and includes a second measuring structure and a second position sensor. The second measuring structure is movable along a Y-axis direction and contacted with the ball-shaped structure. The second position sensor measures a displacement amount of the second measuring structure when the second measuring structure is pushed by the ball-shaped structure. The Z-axis measuring module is disposed on the base and includes a third measuring structure and a third position sensor. The third measuring structure is movable along a Z-axis direction and contacted with the ball-shaped structure. The third position sensor measures a displacement amount of the third measuring structure when the third measuring structure is pushed by the ball-shaped structure. A measuring space is defined by a movable distance range of the first measuring structure along the X-axis direction. A movable distance range of the second measuring structure along the Y-axis direction and a movable distance range of the third measuring structure along the Z-axis direction. When the ball-shaped structure is moved in the measuring space, a three-dimensional coordinate of the ball-shaped structure is obtained according to sensed results of the first position sensor, the second position sensor and the third position sensor. In a step (S 2 ), at least one preset positioning point in the measuring space is measured. In a step (S 3 ), the robotic arm is controlled to be moved from an initial point toward the same preset positioning point with different operation actions for more than two times, and a three-dimensional coordinate of each actual positioning point of the robotic arm at each time is acquired according to the three-dimensional coordinate of the ball-shaped structure measured by the three-dimensional measuring device. In a step (S 4 ), a function equation about each actual positioning point in each operation action of the robotic arm in the step (S 3 ) is calculated according to the forward kinematics, and a predicted positioning point of the robotic arm in each operation action is acquired according to the function equation. Then, a step (S 5 ) is performed to judge whether a difference between the predicted positioning points of the robotic arm in every two different operation actions minus a difference between the actual positioning points of the robotic arm in every two different operation actions is within an acceptable threshold range; if the difference between the two predicted positioning points minus the difference between the two actual positioning points is within the acceptable threshold range, the robotic arm calibration method is ended. In a step (S 6 ), if a judging result of the step (S 5 ) is not satisfied, a Jacobian matrix is generated according to the reached actual positioning point in each operation action of the robotic arm, and a position formula about the predicted positioning point and the actual positioning point in each operation action of the robotic arm is acquired, wherein the Jacobian matrix is a partial derivative of the function equation corresponding to a deviation amount Δα of a shaft size α of each axis of the robotic arm and a deviation amount Δθ of a rotation angle θ of each axis of the robotic arm in the corresponding operation action under the forward kinematics. In a step (S 7 ), a subtraction is performed on the position formulae corresponding every two operation actions of the robotic arm, so that a difference between the deviation amounts Δα and a difference between the deviation amounts Δθ in every two operation actions are calculated. In a step (S 8 ), the shaft size α of each axis and the rotation angle θ of each axis are updated according to the difference between the deviation amounts Δα and the difference between the deviation amounts Δθ, and performing the step (S 4 ) again. 
     The above contents of the present disclosure will become more readily apparent to those ordinarily skilled in the art after reviewing the following detailed description and accompanying drawings, in which: 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIG. 1  is a schematic perspective view illustrating a three-dimensional measuring device according to an embodiment of the present disclosure and taken along a first viewpoint; 
         FIG. 2  is a schematic perspective view illustrating the three-dimensional measuring device as shown in  FIG. 1  and taken along a second viewpoint; 
         FIG. 3  is a schematic perspective view illustrating the three-dimensional measuring device as shown in  FIG. 1  and taken along a third viewpoint; 
         FIG. 4  schematically illustrates the application of the three-dimensional measuring device of  FIG. 1  on a robotic arm; 
         FIG. 5  schematically illustrates a measuring space of the three-dimensional measuring device according to the embodiment of the present disclosure, wherein the measuring space is defined by the movable distance range of the first measuring structure along the X-axis direction, the movable distance range of the second measuring structure along the Y-axis direction and the movable distance range of the third measuring structure along the Z-axis direction; and 
         FIG. 6  is a flowchart illustrating a robotic arm calibration method for the three-dimensional measuring device of the present disclosure. 
     
    
    
     DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENT 
     The present disclosure will now be described more specifically with reference to the following embodiments. It is to be noted that the following descriptions of preferred embodiments of this disclosure are presented herein for purpose of illustration and description only. It is not intended to be exhaustive or to be limited to the precise form disclosed. 
     Please refer to  FIGS. 1, 2, 3 and 4 .  FIG. 1  is a schematic perspective view illustrating a three-dimensional measuring device according to an embodiment of the present disclosure and taken along a first viewpoint.  FIG. 2  is a schematic perspective view illustrating the three-dimensional measuring device as shown in  FIG. 1  and taken along a second viewpoint.  FIG. 3  is a schematic perspective view illustrating the three-dimensional measuring device as shown in  FIG. 1  and taken along a third viewpoint.  FIG. 4  schematically illustrates the application of the three-dimensional measuring device of  FIG. 1  on a robotic arm. 
     The three-dimensional measuring device  1  is used for measuring and acquiring the three-dimensional distance or the moving trajectory of a movable object of an automation device. For example, the three-dimensional measuring device  1  is used for measuring a robotic arm  9  of the automation device. The robotic arm  9  is a multi-axis robotic arm with more than three axes. Preferably but not exclusively, the robotic arm  9  is a selective compliance assembly robotic arm (SCARA) or a 6-axis robotic arm. The actions of the robotic arm  9  are controlled by a controller  10 . For example, the controller  10  may control the robotic arm  9  to be moved in an X-axis direction, a Y-axis direction and/or a Z-axis direction. In case that the robotic arm  9  is a multi-axis robotic arm with more than four axes, the controller  10  can adjust the angle of the robotic arm  9 . Moreover, the controller  10  can record the moved position point of the robotic arm  9 . The controller  10  is in communication with the three-dimensional measuring device  1  in a wired transmission manner or a wireless transmission manner. Consequently, the controller  10  acquires the three-dimensional distance and/or the moving trajectory of the robotic arm  9 . 
     In an embodiment, the three-dimensional measuring device  1  includes a base  2 , a ball-shaped structure  3 , an X-axis measuring module  4 , a Y-axis measuring module  5  and a Z-axis measuring module  6 . The base  2  includes a first fixing post  20  and a second fixing post  21 . The first fixing post  20  and the second fixing post  21  are perpendicularly disposed on a top surface  22  of the base  2  and arranged beside each other. A first end of the first fixing post  20  is away from the top surface  22  of the base  2 . A second end of the first fixing post  20  is close to the top surface  22  of the base  2 . The first end of the first fixing post  20  includes a first connection part  200  and a second connection part  201 . The second connection part  201  is connected with the first connection part  200 . The first connection part  200  and the second connection part  201  are arranged in an L-shaped structure. In some embodiments, the first connection part  200  includes a first accommodation hole  202 . The second connection part  201  includes a second accommodation hole  203 . The second fixing post  21  includes a third accommodation hole  210 . 
     The X-axis measuring module  4  is fixed on the first connection part  200  of the first fixing post  20 . In an embodiment, the X-axis measuring module  4  includes a first measuring structure  40 , a first position sensor  41 , a first linear track  42  and a first elastic element  43 . The first linear track  42  is disposed on the first connection part  200  along the X-axis direction. The first measuring structure  40  is movable along the X-axis direction. The first measuring structure  40  includes a first contacting part  400  and a sliding part (not shown). The sliding part of the first measuring structure  40  matches the first linear track  42 . Consequently, the sliding part of the first measuring structure  40  can be slid relative to the first linear track  42 . That is, the first measuring structure  40  can be slid on the first linear track  42  along the X-axis direction. The first position sensor  41  is aligned with the first measuring structure  40 . The first position sensor  41  is in communication with the controller  10  in the wired transmission manner or the wireless transmission manner. The first position sensor  41  can measure and acquire the displacement amount of the first measuring structure  40  and transmit the measurement result to the controller  10 . An example of the first position sensor  41  includes but is not limited to an optical rule. A first end of the first elastic element  43  is contacted with the first contacting part  400  of the first measuring structure  40 . A second end of the first elastic element  43  is received within the first accommodation hole  202  of the first connection part  200  and contacted with an inner wall of the first accommodation hole  202 . As the first measuring structure  40  is moved to exert a force on the first elastic element  43 , the first elastic element  43  is compressed to generate an elastic restoring force. When the force is no longer exerted on the first elastic element  43 , the first measuring structure  40  is returned to its original position in response to the elastic restoring force of the first elastic element  43 . 
     The Y-axis measuring module  5  is fixed on the second connection part  201  of the first fixing post  20 . In an embodiment, the Y-axis measuring module  5  includes a second measuring structure  50 , a second position sensor  51 , a second linear track  52  and a second elastic element  53 . The second linear track  52  is disposed on the second connection part  201  along the Y-axis direction. The second measuring structure  50  is movable along the Y-axis direction. The second measuring structure  50  includes a second contacting part  500  and a sliding part (not shown). The sliding part of the second measuring structure  50  matches the second linear track  52 . Consequently, the sliding part of the second measuring structure  50  can be slid relative to the second linear track  52 . That is, the second measuring structure  50  can be slid on the second linear track  52  along the Y-axis direction. The second position sensor  51  is aligned with the second measuring structure  50 . The second position sensor  51  is in communication with the controller  10  in the wired transmission manner or the wireless transmission manner. The second position sensor  51  can measure and acquire the displacement amount of the second measuring structure  50  and transmit the measurement result to the controller  10 . An example of the second position sensor  51  includes but is not limited to an optical rule. A first end of the second elastic element  53  is contacted with the second contacting part  500  of the second measuring structure  50 . A second end of the second elastic element  53  is received within the second accommodation hole  203  of the second connection part  201  and contacted with an inner wall of the second accommodation hole  203 . As the second measuring structure  50  is moved to exert a force on the second elastic element  53 , the second elastic element  53  is compressed to generate an elastic restoring force. When the force is no longer exerted on the second elastic element  53 , the second measuring structure  50  is returned to its original position in response to the elastic restoring force of the second elastic element  53 . 
     The Z-axis measuring module  6  is fixed on the second fixing post  21 . In an embodiment, the Z-axis measuring module  6  includes a third measuring structure  60 , a third position sensor  61 , a third linear track  62  and a third elastic element  63 . The third linear track  62  is disposed on the second fixing post  21  along the Z-axis direction. The third measuring structure  60  is movable along the Z-axis direction. The third measuring structure  60  includes a third contacting part  600  and a sliding part (not shown). The sliding part of the third measuring structure  60  matches the third linear track  62 . Consequently, the sliding part of the third measuring structure  60  can be slid relative to the third linear track  62 . That is, the third measuring structure  60  can be slid on the third linear track  62  along the Z-axis direction. The third position sensor  61  is aligned with the third measuring structure  60 . The third position sensor  61  is in communication with the controller  10  in the wired transmission manner or the wireless transmission manner. The third position sensor  61  can measure and acquire the displacement amount of the third measuring structure  60  and transmit the measurement result to the controller  10 . An example of the third position sensor  61  includes but is not limited to an optical rule. A first end of the third elastic element  63  is contacted with the third contacting part  600  of the third measuring structure  60 . A second end of the third elastic element  63  is received within the third accommodation hole  210  of the second fixing post  21  and contacted with an inner wall of the third accommodation hole  210 . As the third measuring structure  60  is moved to exert a force on the third elastic element  63 , the third elastic element  63  is compressed to generate an elastic restoring force. When the force is no longer exerted on the third elastic element  63 , the third measuring structure  60  is returned to its original position in response to the elastic restoring force of the third elastic element  63 . 
     In an embodiment, the ball-shaped structure  3  is connected with a distal end of an end shaft  90  of the robotic arm  9  directly or indirectly. As shown in  FIG. 4 , the ball-shaped structure  3  is indirectly connected with the distal end of the end shaft  90  of the robotic arm  9  through a linkage  8 . As the robotic arm  9  is moved, the ball-shaped structure  3  is synchronously moved along the three-dimensional directions and synchronously rotated at the specified angles. The displacement of the ball-shaped structure  3  indicates the displacement of the working point of the robotic arm  9 . The three-dimensional coordinate of the center of the ball-shaped structure  3  indicates the three-dimensional position point of the robotic arm  9 . The ball-shaped structure  3  is arranged among the X-axis measuring module  4 , the Y-axis measuring module  5  and the Z-axis measuring module  6 . Moreover, the ball-shaped structure  3  is contacted with the first contacting part  400  of the first measuring structure  40 , the second contacting part  500  of the second measuring structure  50  and the third contacting part  600  of the third measuring structure  60 . While the ball-shaped structure  3  is moved in at least one of the X-axis direction, the Y-axis direction and the Z-axis direction, the corresponding measuring structure is pushed by the ball-shaped structure  3 . For example, if the ball-shaped structure  3  is moved in the X-axis direction to push the first contacting part  400 , the first measuring structure  40  is moved. If the ball-shaped structure  3  is moved in the Y-axis direction to push the second contacting part  500 , the second measuring structure  50  is moved. If the ball-shaped structure  3  is moved in the Z-axis direction to push the third contacting part  600 , the third measuring structure  60  is moved. After the first position sensor  41  measures the displacement amount of the first measuring structure  40 , the second position sensor  51  measures the displacement amount of the second measuring structure  50  and the third position sensor  61  measures the displacement amount of the third measuring structure  60 , the sensed results are transmitted to the controller  10 . After the controller  10  acquires the three-dimensional coordinate and the displacement of the ball-shaped structure  3 , the controller  10  acquires the three-dimensional coordinate and the displacement of the working point of the end shaft  90  of the robotic arm  9 . Consequently, the operating condition of the robotic arm  9  is calculated, and the corresponding control and/or calibration is performed. 
       FIG. 5  schematically illustrates a measuring space of the three-dimensional measuring device according to the embodiment of the present disclosure, wherein the measuring space is defined by the movable distance range of the first measuring structure along the X-axis direction, the movable distance range of the second measuring structure along the Y-axis direction and the movable distance range of the third measuring structure along the Z-axis direction. As mentioned above, the movement of the ball-shaped structure  3  is driven by the robotic arm  9 , and the ball-shaped structure  3  is contacted with the first contacting part  400  of the first measuring structure  40 , the second contacting part  500  of the second measuring structure  50  and the third contacting part  600  of the third measuring structure  60 . Consequently, a rectangular measuring space  7  is defined by the movable distance range X 1  of the first measuring structure  40  along the X-axis direction, the movable distance range Y 1  of the second measuring structure  50  along the Y-axis direction and the movable distance range Z 1  of the third measuring structure  60  along the Z-axis direction. While the ball-shaped structure  3  is moved in the measuring space  7 , the sensed results of the first position sensor  41 , the second position sensor  51  and the third position sensor  61  reflect the three-dimensional displacement of the ball-shaped structure  3  and the three-dimensional coordinate of the ball-shaped structure  3 . Consequently, the three-dimensional position point of the robotic arm  9  is acquired. In case that the position of the origin is defined in the measuring space  7 , the three-dimensional coordinate of the ball-shaped structure  3  that is sensed by the first position sensor  41 , the second position sensor  51  and the third position sensor  61  is the absolute coordinate. Preferably, each of the movable distance range X 1  of the first measuring structure  40  along the X-axis direction, the movable distance range Y 1  of the second measuring structure  50  along the Y-axis direction and the movable distance range Z 1  of the third measuring structure  60  along the Z-axis direction is equal to the radius of the ball-shaped structure  3 . Consequently, while the ball-shaped structure  3  is moved in the measuring space  7 , the ball-shaped structure  3  is contacted with the first contacting part  400  of the first measuring structure  40 , the second contacting part  500  of the second measuring structure  50  and the third contacting part  600  of the third measuring structure  60 . 
     In an embodiment, the X-axis measuring module  4  further includes a first proximity sensor  44  corresponding to the first measuring structure  40 . When the first measuring structure  40  is within the sensing range of the first proximity sensor  44 , the first proximity sensor  44  issues a prompt signal to the controller  10 . Due to the arrangement of the first proximity sensor  44 , the controller  10  takes a corresponding action when the first measuring structure  40  is moved to the limit position. Similarly, the Y-axis measuring module  5  further includes a second proximity sensor  54  corresponding to the second measuring structure  50 . When the second measuring structure  50  is within the sensing range of the second proximity sensor  54 , the second proximity sensor  54  issues a prompt signal to the controller  10 . Due to the arrangement of the second proximity sensor  54 , the controller  10  takes a corresponding action when the second measuring structure  50  is moved to the limit position. Similarly, the Z-axis measuring module  6  further includes a third proximity sensor  64  corresponding to the third measuring structure  60 . When the third measuring structure  60  is within the sensing range of the third proximity sensor  64 , the third proximity sensor  64  issues a prompt signal to the controller  10 . Due to the arrangement of the third proximity sensor  64 , the controller  10  takes a corresponding action when the third measuring structure  60  is moved to the limit position. 
     In an embodiment, the ball-shaped structure  3  further includes a fastening hole  30 . The fastening hole  30  is concavely formed on the surface of the ball-shaped structure  3 . The inner wall of the fastening hole  30  has an inner thread structure (not shown). Consequently, the ball-shaped structure  3  can be assembled with the linkage  8  through the fastening hole  30  or assembled with the distal end of the end shaft  90  of the robotic arm  9  through the fastening hole  30 . 
       FIG. 6  is a flowchart illustrating a robotic arm calibration method for the three-dimensional measuring device of the present disclosure. Since the three-dimensional coordinate of the center of the ball-shaped structure  3  is correlated to the actual point of the working point of the robotic arm  9 , the actual point of the working point of the robotic arm  9  is acquired according to the three-dimensional coordinate of the center of the ball-shaped structure  3 . The robotic arm calibration method is applied to the controller  10  of the robotic arm  9 . The operating principles of the robotic arm calibration method will be described as follows. 
     In the robot kinematics analysis, the forward kinematics of the existing technology can be used to establish a mathematical model based on the arm length of each axis of the robotic arm  9  and the rotation angle of each axis of the robotic arm  9  in order to estimate the position of the working point of the robotic arm  9  (i.e., to predict the coordinate of the positioning point). If the robotic arm  9  is moved toward the same preset positioning point with different operation actions, the predicted positioning point of the robotic arm  9  in the space can be estimated directly by using a mathematical model according to the rotation angle of each axis of the robotic arm  9 . According to the ideal mathematical model of the robotic arm  9 , the function equation FK(θi) about the rotation angle θ of each axis of the robotic arm  9  in each operation action under the forward kinematics can be used to calculate the predicted positioning point {circumflex over (P)} of the robotic arm  9  in the space. The predicted positioning point {circumflex over (P)} can be expressed by the mathematic formula (1): 
         {circumflex over (P)}   i = (θ i )  (1);
 
     In the above mathematic formula, {circumflex over (P)} is the predicted positioning point of the working point of the robotic arm  9  under the forward kinematics, i is the i-th operation action executed by the robotic arm  9 , and  (θ i ) is the function equation FK(θi) about the rotation angle θ of each axis of the robotic arm  9  in each operation action under the forward kinematics. 
     Ideally, the controller  10  has the following settings about the command of the preset positioning point. If the robotic arm  9  is moved toward the same preset positioning point with more than two different operation actions, the difference between the predicted positioning points {circumflex over (P)} of the robotic arm  9  in every two different operation actions minus the difference between the actual positioning points P of the robotic arm  9  in every two different operation actions is equal to zero. That is, ({circumflex over (P)} i+1 −{circumflex over (P)} i )−(P i+1 −P i ) is equal to 0. However, because of the production errors and the assembling errors, the controller  10  has the following settings about the command of the preset positioning point. For example, if the robotic arm  9  is moved toward the same preset positioning point with more than two different operation actions (or gestures), there is a position deviation between the actual positioning points of the robotic arm  9  in every two different operation actions. That is, the robotic arm  9  is not moved toward the same actual position. If the above situation occurs, ({circumflex over (P)} i+1 −{circumflex over (P)} i )−(P i+1 −P i ) is not equal to 0. In accordance with the robotic arm calibration method of the present disclosure, the controller  10  performs the calibration on the robotic arm  9  when ({circumflex over (P)} i+1 −{circumflex over (P)} i )−(P i+1 −P i ) is not within the acceptable threshold range. 
     For performing the calibration, the existing technology is used to generate a Jacobian matrix according to the actual positioning point of the robotic arm  9  in each operation action. Consequently, the relationship between the predicted positioning point {circumflex over (P)} and the actual positioning point P may be expressed by the position formula (2): 
         P   i   ={circumflex over (P)}   i +[Δ a ,Δθ]= (θ i )+ J   i [Δ a ,Δθ]  (2)
 
     In the above mathematic formula, J is a Jacobian matrix, Ji|Δα,Δθ| is the first partial derivative of the function equation  (θ i ) corresponding to the deviation amount Δα of the shaft size α of each axis of the robotic arm  9  and the deviation amount Δθ of the rotation angle θ of each axis of the robotic arm  9  in the i-th operation action under the forward kinematics. 
     Then, according to the mathematic formula (2), a subtraction is performed on the position formula of the (i+1)-th operation action and the position formula of the i-th operation action. Consequently, the mathematic formula (3) is acquired. 
       ( P   i+1   −P   i )−{ (θ i+1 )− (θ i )}=( J   i+1   −J   i )[Δα,Δθ]  (3);
 
     In the mathematic formula (3), P i+1  and P i  are the three-dimensional coordinates of the actual positioning points that are measured by the three-dimensional measuring device  1 . 
     Consequently, according to the sensed results of the three-dimensional measuring device  1 , the controller  10  acquires the value of (P i+1 −P i ) (i.e., the difference between the actual positioning points P of the robotic arm  9  in every two different operation actions). Since  (θ i+1 ) and  (θi) are known values according to the ideal mathematic model of the robotic arm  9 , the relationship between (P i+1 −P i ) and (J i+1 −J i )[Δα,Δθ] can be realized. 
     If the value of (P i+1 −P i ) is close to 0 through the calibration of the robotic arm  9 , the actual positioning points of the robotic arm  9  are nearly identical when the robotic arm  9  is moved toward the same preset positioning point with more than two different operation actions. According to the mathematic formula (3), the controller  10  calculates the difference between the deviation amounts Δα and the difference between the deviation amounts Δθ in every two different operation actions. According to the difference between the deviation amounts Δα and the difference between the deviation amounts Δθ, the shaft size a of each axis and the rotation angle θ of each axis obtained according to the ideal mathematic model of the robotic arm  9  under the forward kinematics are updated. That is, the shaft size α of each axis and the rotation angle θ of each axis in the corresponding function equation are updated. Consequently, the value of (P i+1 −P i ) is gradually converged to 0. In such way, the calibration of the robotic arm  9  is completed. 
     Please refer to the flowchart of  FIG. 6 . The robotic arm calibration method includes the following steps. 
     Firstly, in a step S 1 , a robotic arm (e.g., the robotic arm  9  as shown in  FIG. 4 ) and a three-dimensional measuring device  1  as shown in  FIG. 1  are provided. The ball-shaped structure  3  is connected with an end shaft  90  of the robotic arm  9 , and located at an initial point. 
     In a step S 2 , at least one preset positioning point in a measuring space  7  of the three-dimensional measuring device  1  is calculated. The measuring space  7  is defined by the movable distance range X 1  of the first measuring structure  40  along the X-axis direction, the movable distance range Y 1  of the second measuring structure  50  along the Y-axis direction and the movable distance range Z 1  of the third measuring structure  60  along the Z-axis direction. 
     In a step S 3 , the robotic arm  9  is moved from the initial point toward the same preset positioning point with different operation actions for more than two times, the three-dimensional coordinate of the actual positioning point of the robotic arm  9  at each time is acquired according to the three-dimensional coordinate of the ball-shaped structure  3  measured by the three-dimensional measuring device  1 . 
     In a step S 4 , a function equation about each actual positioning point in each operation action of the robotic arm  9  in the step S 3  (i.e., the mathematic formula (1)) is calculated according to the forward kinematics, and the predicted positioning point of the robotic arm  9  in each operation action is acquired according to the function equation. 
     Then a step S 5  is performed to judge whether the difference between the predicted positioning points {circumflex over (P)} of the robotic arm  9  in every two different operation actions minus the difference between the actual positioning points P of the robotic arm  9  in every two different operation actions (i.e., ({circumflex over (P)} i+1 −{circumflex over (P)} i )−(P i+1 −P i )) is within an acceptable threshold range. 
     If the judging result of the step S 5  is not satisfied, a step S 6  is performed. In the step S 6 , a Jacobian matrix is generated according to the reached actual positioning point in each operation action of the robotic arm  9 , and a position formula about the predicted positioning point and the actual positioning point in each operation action of the robotic arm  9  (i.e., the mathematic formula (2)) is acquired. The Jacobian matrix is a partial derivative of the function equation corresponding to the deviation amount Δα of the shaft size α of each axis of the robotic arm  9  and the deviation amount Δθ of the rotation angle θ of each axis of the robotic arm  9  in the corresponding operation action under the forward kinematics. 
     After the step S 6 , a step S 7  is performed. In the step S 7 , a subtraction is performed on the position formulae corresponding to every two operation actions of the robotic arm  9 . Consequently, the difference between the deviation amounts Δα and the difference between the deviation amounts Δθ in every two different operation actions are calculated. 
     After the step S 7 , a step S 8  is performed. In the step S 8 , the shaft size α of each axis and the rotation angle θ of each axis obtained according to the ideal mathematic model of the robotic arm  9  under the forward kinematics are updated according to the difference between the deviation amounts Δα and the difference between the deviation amounts Δθ. 
     After the step S 8 , the step S 4  is repeatedly done. If the judging result of the step S 5  is satisfied, a step S 9  is performed. Meanwhile, the robotic arm calibration method is ended. 
     From the above descriptions, the present disclosure provides a three-dimensional measuring device and a robotic arm calibration method thereof. The three-dimensional measuring device includes a ball-shaped structure, an X-axis measuring module, a Y-axis measuring module and a Z-axis measuring module. The ball-shaped structure is contacted with the X-axis measuring module, the Y-axis measuring module and the Z-axis measuring module and assembled with a movable object. Consequently, the three-dimensional measuring device can measure the three-dimensional coordinate of the ball-shaped structure to acquire the working point of the movable object. The use of the three-dimensional measuring device can save the measuring period and reduce the fabricating cost. Moreover, the robotic arm calibration method is implemented by the three-dimensional measuring device of the present disclosure and has the time-saving and cost-effective efficacy.